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The Pennsylvania State University The Graduate School Eberly College of Science

COVARIANT FORMULATIONS OF THE LIGHT-ATOM

PROBLEM

A Dissertation in Physics by Craig Chandler Price

© 2018 Craig Chandler Price

Submitted in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

August 2018 The dissertation of Craig Chandler Price was reviewed and approved∗ by the following:

Nathan D. Gemelke Assistant Professor of Physics Dissertation Advisor, Chair of Committee

David S. Weiss Professor of Physics Associate Head for Research

Marcos Rigol Professor of Physics

Zhiwen Liu Professor of Electrical Engineering

Nitin Samarth Professor of Physics Department Head

∗Signatures are on file in the Graduate School.

ii Abstract

We explore the ramifications of the light-matter interaction of ultracold neutral atoms in a covariant, or coordinate free, approach in the quantum limit. To do so we describe the construction and initial characterization of a novel atom-optical system in which there is no obvious preferred coordinate system but is fed back upon itself to amplify coherent quantum excitations of any emergent spontaneous order of the system. We use weakly dissipative, spatially incoherent light that is modulated by amplified fluctuations of the phase profile of a Sagnac interferometer whose symmetry is broken by the action of a generalized Raman sideband cooling process. The cooling process is done in a limit of low probe intensity which effectively produces slow light and forms a platform to explore general relativistic analogues. The disordered potential landscape is populated with k-vector defects where the atomic density contracts with an inward velocity potentially faster than the speed of the slow light at that region. We show preliminary results of characterizing the system that show the effect of atomic participation in the feedback loop that cement the analogy to Kuramoto locked oscillators. In addition, we examine novel formulations of quantum field theory with the aim to capture a simultaneous quantization of covariant light - atom interaction. Instead of using field-theoretic building blocks composed of harmonic excitations, we use nonlinear, complex excitations similar to those used in the Kuramoto model. Generalizing this approach into a continuous, coordinate free version, we consider the effect of electromagnetic induced transparency (EIT) with probe fields that have a vanishingly small amplitude. The associated slow light phenomenon can be exploited as it passes through a dielectric to create a general relativistic analogue whose dynamics may capture limits of dynamical gravity. Further, the topology of such light-atom physics can be used as a new, conceptually simplifying vehicle to understand the propagation of light through more general, and familiar phenomena such as imaging systems and cavities.

iii Table of Contents

List of Figures viii

Acknowledgments x

Chapter 1 Introduction 1

Chapter 2 Generalized quantum field theory 4 2.1 QFT as a bath of coupled, classical springs ...... 4 2.1.1 Complex Lagrangians ...... 6 2.2 System of oscillators ...... 7 2.2.1 Clifford algebra with matrices ...... 9 2.3 Field theory ...... 10 2.3.1 Adiabatic invariants ...... 12 2.4 Modified least action principle ...... 14 2.5 Nonlinear oscillators ...... 16 2.5.1 Self-injection locked oscillators ...... 17 2.5.2 Conserved microscopic Hamiltonians ...... 22 2.5.3 Microscopic relativistic Lagrangian ...... 24 2.5.4 L as a metric ...... 25 2.6 Observables behave like response functions ...... 27 2.6.1 Complex valued observables ...... 28

Chapter 3 Dielectrics and curved spacetime 32 3.1 Introduction ...... 32 3.2 Nonlinear media as a covariant tensor ...... 33 3.2.1 Maxwell’s equations in media ...... 33 3.2.2 Covariant formulation of Maxwell’s equations ...... 35

iv 3.2.3 Coordinate transformations of covariant media ...... 37 3.2.4 χ can include nonlinear dielectrics ...... 38 3.3 When the linear susceptibility tensor is equivalent to a spacetime metric ...... 40 3.3.1 Relating the susceptibility to the Riemannian tensor . . . . 40 3.3.2 Topological changes and the susceptibility tensor ...... 41 3.3.3 Transformations of both the metric and the susceptibility . . 42 3.4 Atom-light interaction through EIT ...... 47 3.4.1 Magnetic couplings and the Zeeman shift ...... 49 3.4.2 The electric dipole interaction ...... 50 3.4.2.1 Rotating wave approximation ...... 56 3.4.3 Scalar and vector light shifts from far-detuned light . . . . . 58 3.4.4 Density matrix elements ...... 59 3.5 Light propagation and topology ...... 63 3.5.1 At boundaries, rays are ramified ...... 64 3.5.1.1 Defining cohomology classes ...... 66 3.5.1.2 Images define Cut Loci ...... 67 3.5.1.3 Cohomology defines discrete modes in physical optics 68 3.5.1.4 Nonlinear interactions permit ramification and gluing 69 3.5.2 Time delay correlation, oscillation, and lasing ...... 70 3.5.2.1 Topological classification by laser and parametric oscillation ...... 70 3.5.2.2 Lasing thresholds tell us when to ramify and glue . 73 3.5.3 Geometric optics and quantum mechanics ...... 74 3.5.3.1 Geometric pictures of classical optics ...... 74 3.5.3.2 Polarization of optical variables ...... 75 3.5.3.3 Geometric quantization ...... 76 3.5.3.4 EBK quantization ...... 77 3.5.3.5 Geometric interpretation of the principle quantum number ...... 78 3.5.3.6 The one-dimensional laser cavity ...... 79 3.5.3.7 The role of nonlinearities ...... 80 3.5.3.8 Reconnecting with laser optics ...... 80 3.5.3.9 Self-consistent lasing and topology ...... 81

Chapter 4 Feedback to probe excitations of an atom light system 84 4.1 Apparatus to manipulate cold atoms ...... 85 4.1.1 Vacuum chamber ...... 85 4.1.2 Atom sources ...... 89

v 4.1.3 Laser systems ...... 89 4.1.3.1 Diode lasers ...... 90 4.1.3.2 Laser frequency locking ...... 90 4.1.3.2.1 Absorption lock ...... 91 4.1.3.2.2 Beat note locks ...... 91 4.1.3.3 Double pass tapered amplifier ...... 92 4.1.3.4 Optical switchyard ...... 93 4.1.3.5 High power, 1064 nm light ...... 94 4.1.4 Water cooling ...... 95 4.1.5 Magnetic field controllers ...... 97 4.1.6 Timing system ...... 98 4.1.7 Trapping and cooling atoms ...... 98 4.1.8 Imaging and probing the atoms in the UHV cell ...... 99 4.2 Generalized Raman sideband cooling ...... 102 4.2.1 Characterization of the optical fiber modes ...... 103 4.2.2 Expansion of modes in Cartesian coordinates ...... 108 4.2.3 Effective number of degrees of freedom ...... 112 4.3 Synthetic thermal body ...... 113 4.3.1 Preliminary experiments ...... 116 4.4 Constructing a light-atom oscillator ...... 119 4.4.1 Feedback loop - 1064 nm section ...... 121 4.4.2 Feedback loop - 780 nm section ...... 123 4.4.3 Feedback loop - electronic backend ...... 123 4.5 Light-atom oscillator characterization ...... 128 4.5.1 Injection Locking ...... 130 4.6 Atomic participation in the feedback path ...... 138 4.6.1 Modeling the propagator ...... 139 4.6.2 Adding atoms to the closed loop ...... 143

Appendix A Experimental Apparatus 2.0 148 A.1 Experimental apparatus - Collisional Microscope ...... 149 A.1.1 Transport and Imaging ...... 149

Appendix B Novel, UHV compatible Glass Bonding Techniques 153 B.1 Introduction ...... 153 B.2 Method ...... 155 B.3 Conclusion ...... 159

vi Bibliography 161

vii List of Figures

2.1 Mass on a spring QFT ...... 8

3.1 Folding, or manifold ramification in optics ...... 65

4.1 Vacuum chamber ...... 86 4.2 Lock table optics ...... 92 4.3 Laser switchyard ...... 94 4.4 1064 nm optical table schematic ...... 96 4.5 Fiber launches into the UHV cell ...... 100 4.6 High resolution images of the atom cloud ...... 101 4.7 The projection of the light that generates the disordered potential . 103 4.8 Roots of the characteristic equation ...... 107 4.9 Synthetic Thermal Body ...... 114 4.10 Synthetic bath of photons to maximize entropy and lose energy . . 116 4.11 Landau Zener crossings in a disordered potential ...... 117 4.12 Stretch the optical fiber to make an engineered optical bath . . . . 117 4.13 Disordered magnetic field ...... 118 4.14 Overview schematic of the experiment ...... 120 4.15 780 nm optical pumping optics ...... 124 4.16 Overview of electronics ...... 125 4.17 Electronic schematic for the closed loop response ...... 127 4.18 Power spectrum of closed and open loop response ...... 129 4.19 Power spectrum of closed loop response at increasing gain . . . . . 131 4.20 Power spectrum of a clipped beat ...... 132 4.21 4 ms of the closed loop response ...... 134 4.22 Relative phase of unlocked oscillators ...... 135 4.23 Relative phase of locked oscillators ...... 137 4.24 Unwrapped phase difference at high gain ...... 138 4.25 Adler wedges ...... 139 4.26 Frequency of closed loop response ...... 140

viii 4.27 Phase of the closed loop response during RSC ...... 144 4.28 Unwrapped phase of multiple shots with and without atoms . . . . 145 4.29 Relative number of phase slips with atoms ...... 146

A.1 Experiment chamber for collision microscopy ...... 151 A.2 Experiment chamber with added pumps ...... 152

B.1 Glass substrates after bonding ...... 157 B.2 Glass substrate with pocket drilled into center ...... 157 B.3 Pressure versus time after shutting off the ion pump for the nanopar- ticle hydroxide-catalyzed bonded glass ...... 158

ix Acknowledgments

Graduate school has been an intense, very educational experience and it would be remiss not to single out the key figures who helped me in this story. My fellow travelers, Qi, Jianshi, Rene, and George were great companions and peers. Without their hard work, cold atoms, much less these experiments, would never have happened. George’s cheerfulness and competent precision, Rene’s thoroughness and quiet excellence, Jianshi’s thoughtfulness and sudden stabs of genius, and Qi’s level-headed hard work and tenacity - were instrumental for the work in this thesis. Nate, my advisor, marshaled us all and taught me everything I know about how to create an experiment. From teaching technical competence to making abstract concepts “just normal physics” his perpetually creative energy has been and will be indispensable to my scientific career. My other scientific mentors were Mrs. Overland who I still remember fondly, Paolo, and of course Natasha. Several people along the way have proved to be invaluable in my life as well. Justin, Mike, and Ashley I am particularly grateful for their companionship over the years. I am grateful for my brothers who have shaped who I am today, my father for his curiosity and my mother for her support. Lastly I am grateful for Natasha for keeping me grounded, staying positive, and always having my back - and for little Fiona in providing some squees of delight as her father struggled along, finishing this manuscript. And finally of course I am grateful for the support from NSF for the first few years of research, and the Kaufman Foundation for support in the later years of the research.

x Chapter 1 | Introduction

The free evolution of an atom interacting with a light field is a well-studied problem that is the basis of a number of fields of study in atomic physics. The conventional treatment of the problem is to quantize the light and the atom separately as simple harmonic oscillator modes. A mode of the light is defined by a wavevector k and a polarization  fixed by some boundary such as cavity mirrors that define a box. Fixing the boundary conditions and normalizing with respect to the total energy in the box leads to modes of the light that are analogous to the modes of a simple harmonic oscillator. Even in very general situations, harmonic modes can be defined because the problem reduces to solving the Helmholtz equation on various geometries [1,2]. The Helmholtz equation in any finite cavity has an orthonormal, complete set of eigenfunctions which can be labeled by a discrete set of multi-index labels, κ = (κ1, κ2, κ3, κ4) that label the normal mode functions Eκ and Bκ. The normal mode functions are real, transverse vector fields that satisfy the boundary conditions,

n(r) × E (r) = 0 (1.1) n(r) · B(r) = 0 (1.2) for each r on an interior wall of a perfectly conducting cavity. The wavefunction of the atom can be analogously treated leading to quantized transitions between fixed states in the atom. However, one regime in which this prescription may lead to ambiguity is one where neither the atom nor the light has a regular, static, topologically simple confining envelope which can support specific well-defined modes. We approach this

1 limit with our experiment where we have a closed feedback loop that connects a strongly disordered, dynamic 3D conservative potential with slow light phenomena. The atoms that provide the connection between the slow light and the conservative potential laser fields are accelerated by the exchange of photons. At the sites where atoms reside within the potential, they make random walks in spin- and real- space in order to support nonlinear oscillations of collective light-atom modes. While in all well-behaved confining geometries, an appropriate basis can be chosen from which electromagnetic modes can be defined, we find that new approaches to the quantization of atom-light systems may be useful to simplify the mathematical machinery and provide useful physical insight. Consider an atom in free space addressed by a coherent beam of light. Upon absorption or emission of a photon of light, the atom must react to conserve momentum and accelerate. To physically describe the system, we must be able to articulate how an accelerated atom interacts with modes of light. This thought is not novel [3, 4]; Unruh and others showed through simple arguments that the vacuum must appear to a moving atom to be full of low-energy radiation. This immediately necessitates a covariant formulation of atom-light quantization. At base, a covariant formulation cannot have a preferred reference frame; there cannot be fixed light modes which are quantized before considering the atom and vice versa. Canonical quantization would have us use the correspondence principle, however since there is no preferred frame from either the light modes or from the atom, the free atom-light quantization yields considerable ambiguity. In the second chapter of this thesis we seek to formulate a novel method of quantizing an atom light system that is applicable to accelerating reference frames and topologically non-trivial boundaries. Our problem is that of laser cooling a dynamic, disordered atomic gas which is, by standard quantum field theory conception, a box of springs that are all driven by an oscillating source. We begin the analysis by assuming that the oscillators are all uncoupled. We write down the standard formulations of the symplectic form and integrable trajectories on a Poisson manifold. To capture more complicated dynamics, we consider those interactions perturbatively as deforming the algebra and breaking the integrability. Ordinarily in QFT we consider that observables are hermitian operators with real valued eigenvalues. However, it is more natural to describe light propagating on a dielectric as a complex response function like we commonly do with dispersion

2 and the index of refraction. But more generally, in order to measure any observable such as energy, one must perturb the system by attaching a classical probe which must also have densely packed energy levels which dissipates some energy. Complex coordinates are the natural vehicle to capture this phenomenon. A measurement of the energy must necessarily measure the imaginary part, the lifetime, as well. From Lorentz invariance, a complex coordinate merely means that there is a correlation, or localization of events in spacetime. We explore extensions to quantum field theory that might explain the light-atom scattering problem in a conceptually simpler manner. The Kuramoto model in combination with a multi-form Lagrangian may provide a natural vehicle that captures aspects of a covariant quantization of field theory. We also explore a novel experimental approach where we build up quantum- covariant dynamics by feeding the output of a slow-light cooling process of neutral atoms back to a modulation of their confining potential. We profile the behavior of the oscillator by introducing a controlled auxiliary feedback path to position the system on the verge of oscillation. The inclusion of atoms in the feedback network then encourages a coherent light-atom state that can build up to be representative of the modes that play an analogue role to gravitational dynamics. We injection lock the system with a stable oscillator and sweep the injection frequency to analyze the closed loop response. Finally we show that the addition of atoms to the feedback path couples more gain into the oscillator paving the way for future refinements.

3 Chapter 2 | Generalized quantum field the- ory

Standard treatments of quantum field theory [5] often begin conceptually with the idea that fields are simple harmonic excitations. For our experiment that coherently addresses atomic transitions, we model our problem as a bath of coupled oscillators. This has conceptual explanatory power because the standard picture of a driving electromagnetic wave forcing an oscillatory response in the electron that is bound to a rigid atom is quite clear as written into introductory textbooks. While realistic models make use of an open system to permit the loss of energy and entropy to the environment, we will begin with the simplest model, a closed bath of coupled oscillators, yet damped non specifically to permit the loss of energy. Even though this sort of model does have plenty of explanatory power, it must suffer at least from the limitation that the system is integrable which limits the phenomenon that we can describe. To capture chaotic phenomena, we introduce the Kuramoto Hamiltonian which has nonlinear couplings to clamp any one oscillator from absorbing unlimited energy, and to support soliton modes where discrete, non-interacting modes can persist indefinitely.

2.1 QFT as a bath of coupled, classical springs

We first begin by describing the atom coupled to the light field as the motion of a 1D damped harmonic oscillator consisting of a ball attached to a fixed point by a spring. The ball has mass, m, and its spring constant k. Its motion away from

4 equilibrium is denoted by X = X(t). Newton’s second law, plus a damping term proportional to the velocity, implies,

d2X dX m = − kX − c (2.1) dt2 dt 0 =X¨ + ΛX˙ + Ω (2.2)

We can equivalently examine the motion of the oscillators with complex coordinates, q, q∗ which helps us view the dynamics of the oscillator as a combination of an in-phase and an out-of-phase part. The transformation from coordinates (x, x˙) to (q, q∗) can be done with the transformation, q = −x˙ − iωx + λx [6] 1. Or,

      q λ − iω −1 x   =     (2.3) q∗ λ + iω −1 x˙

Also, we must satisfy Eq. 2.2 which requires,

      x 0 1 x ∂t   =     (2.4) x˙ −Ω −Λ x˙

To determine what differential equation q, q∗ should satisfy, we calculate what combination of x, x˙ must be used.

      q λ − iω −1 x ∂t   = ∂t     (2.5) q∗ λ + iω −1 x˙       λ − iω −1 0 1 x =       (2.6) λ + iω −1 −Ω −Λ x˙      −1   λ − iω −1 0 1 λ − iω −1 q =         (2.7) λ + iω −1 −Ω −Λ λ + iω −1 q∗     1 (−λ − iω)(λ + Λ − iω) − Ω(λ − iω)(λ + Λ − iω) + Ω q =     2iω (−λ − iω)(λ + Λ + iω) − Ω(λ − iω)(λ + Λ + iω) + Ω q∗ (2.8)

1The reference, [6] has a sign error.

5 and if we want to demand that the resulting equations for q, q∗ are independent from each other which permits direct integration of the first order system of equations, then the last matrix must be diagonal. This is essentially choosing the free parameters of ω and λ to be a specific choice which fixes the "rotation" matrices,   λ − iω −1  , such that we have rotated our x, x˙ coordinates into a basis that λ + iω −1 describes both positive and negative frequencies. Then if ω 6= 0, we have,

Λ = −2λ (2.9) Ω = λ2 + ω2 (2.10) which yields,

0 =q ˙ + iωq + λq (2.11) 0 =q ˙∗ − iωq∗ + λq∗ (2.12) and,

q˙ = q(−iω − λ) (2.13) q˙∗ = q∗(iω − λ) (2.14) which we can solve to obtain,

(−iω−λ)t q =q0e (2.15) ∗ ∗ (iω−λ)t q =q0e (2.16) which are analogous to both components of a damped, rotating wave. We use complex coordinates in this way to be able to capture the dissipation of energy of the system of oscillators without needing to introduce microscopic interaction mechanisms with a bath.

2.1.1 Complex Lagrangians

Now to go further and identify canonical coordinates and momenta we need to form a Lagrangian. Forming a complex Lagrangian that describes these equations

6 of motion is not without precedent [6,7]. It can be shown that if such a Lagrangian describes the equations of motion of both a variable and its complex conjugate, then the Lagrangian can be real valued [6]. We include dissipation by adding an imaginary part to the Lagrangian. We use two separate Lagrangians, L¯ and L ∗, which govern the equations of motion for q, or, q∗ respectively.

i L¯ = (q∗q˙ − qq˙∗) − ωq∗q + iλq∗q (2.17) 2 −i L ∗ = (qq˙∗ − q∗q˙) − ωqq∗ − iλqq∗ (2.18) 2 where q∗ is the complex conjugate of q. For a system with no dissipation (Λ = λ = 0), the Lagrangians reduce to the same real-valued quantity. Eq. 2.11 can be reproduced through variation of q∗ in Eq. 2.17 and Eq. 2.12 is derived through variation of q in Eq. 2.18.

2.2 System of oscillators

We can extend this description of one ball and a spring to describe a 1D system of coupled, damped harmonic oscillators consisting of a chain of identical balls and springs. Each ball has a mass, m, and is connected to its nearest neighbor balls with a spring constant of k, and are located at an equilibrium separation of ζ, and a displacement from equilibrium of δ as shown in Fig. 2.1. Its motion away from equilibrium is denoted by Xi = Xi(t). Newton’s second law, plus a general damping term proportional to the velocity of any of the three masses, implies for the ith mass,

d2X m i = − k(X − X ) dt2 i i−1

− k(Xi − Xi+1) dX dX dX + c i−1 + c i + c i+1 (2.19) 1i dt 2i dt 3i dt

We can write the equations of motion for the whole system2 as a second order,

2with open boundary conditions

7

Figure 2.1: Mass on a spring QFT. k is the spring constant, ζ is the distance between masses, which are index by i. matrix, differential equation,

X¨ + ΛX˙ + Ω(2)X = 0 (2.20)

where X is a vector of n degrees of freedom, (X1,X2, ··· ,Xn) where Xi is the position from equilibrium of the ith mass within its oscillation, and n is the number of balls. Λ and Ω(2) are respectively matrices of the coefficients for the oscillator’s damping and frequency. Ω(2) is named in this suggestive way to remind the reader that these coefficients for a harmonic oscillator are proportional to the square of the oscillation frequency. We can equivalently examine the motion of the oscillators with complex coordi- nates (using Einstein’s repeated index summation convention),

˙ qi = Xi − iωjiXi + λjiXi (2.21)

and if we choose λji and ωji to commute with each other along with the specific choice of Λ = 2λ, and Ω(2) = ω2 + λ2, then we can split the second order differential equation into two first order differential equations,

q˙i + iωjiq + λjiqi = 0 (2.22) ∗ ∗ ∗ ∗ ∗ q˙i − iωjiq + λjiqi = 0 (2.23)

Using complex coordinates helps us view the dynamics of the oscillator as a combination of an in-phase and an out-of-phase part.

8 2.2.1 Clifford algebra with matrices

To solve this differential equation in more general cases, we wish to use a similar trick to what Dirac used for splitting a second order differential equation into a pair of first order ones,

0 = X¨ + ΛX˙ + Ω(2)X (2.24) 2 (2) = (∂t + Λ∂t + Ω )X (2.25) ˜ ˜ 2 = hA0∂t + A5i X (2.26)

2 ∗ ˜∗ ˜∗ ˜ ˜ where we define h·i ≡ hαβDαDβ and, Dα ≡ (D ,D) ≡ (A0∂t + A5, A0∂t + A5), and ∗ is the conjugate (not transpose), and hαβ is a 2 × 2 matrix. Then, to solve the differential equation, we have,

˜∗ ˜ ˜∗ ˜ ˜∗ ˜ ˜∗ ˜ hαβDαDβ = A0∂tA0∂t + A0∂tA5 + A5A0∂t + A5A5 ˜ ˜∗ ˜ ˜∗ ˜ ˜∗ ˜ ˜∗ + A0∂tA0∂t + A5A0∂t + A0∂tA5 + A5A5 (2.27) 2 (2) = ∂t + Λ∂t + Ω (2.28)

˜ ˜ (2) This requires that A0 and A5 be independent of time, Ω to be Hermitian and positive semi-definite (to permit a Cholesky decomposition), and,

˜ ˜ [A0, A0]+ = 1 (2.29) ˜ ˜ (2) [A5, A5]+ = Ω (2.30) ˜ ˜ 2 ∗ [A0, A5]+ = Λ = 0 (2.31)

˜ ˜ where [F,G]+ ≡ FG + GF is the anti-commutator. We can rescale A0, A5 by defining Ω from the Cholesky decomposition, Ω(2) = ΩΩ∗ with ∗ the conjugate ∗ ˜ transpose, and Ω, Ω , invertible, and letting, Aµ = AµΩ, so that,

[Aµ,Aν]+ = ηµν1 (2.32)

  1 0 with ηµν =  . 0 1   1 0 The Aµ’s form an associative algebra over C with the identity, I =   0 1

9 since all n × n matrices form an associative algebra with the multiplication being

the ordinary matrix multiplication. However, if Cn is an associative algebra with unit, I, generated by an n-dimensional vector subspace V n, if h, i is any quadratic n n form on V , and if V has a basis, e1, . . . , en satisfying ejek + ekej = 2gjkI where n n gjk ≡ hej, eki, then Cn = C(V ) is called the Clifford algebra generated by V with

the quadratic form h, i. [8] Thus A0,A5 form a Clifford algebra, analogous to the Dirac matrices. In the limit when Λ 6= 0, it is much less convenient to decompose, but as Λ →  from 0, it deforms the Clifford algebra. One can connect this to deformation quantization through the deformation of the Poisson algebra.

2.3 Field theory

To see how springs and masses could look after second quantization, we move to a field theoretic approach. Before, q and q∗ represented the in-phase and out-of- phase motion of a damped mass connected by springs. Now let q = q(x, y, z, t) be the magnitude of the in-phase/out-of-phase extension of the spring at position (x, y, z, t). Here we have assumed a continuum model of the individual springs, and declared that our interest is only in the amount of excitement at a particular point. This could represent a dipole moment of an atom interacting with an electric field, or many other specific local interaction processes. Let’s consider a field ψ that  ∂q ∂q ∂q ∂q  describes excitations of waves over the field of q’s. We take, ψ = ψ q, ∂x , ∂y , ∂z , ∂t , ∗ ∗  ∗ ∂q∗ ∂q∗ ∂q∗ ∂q∗  ∗ and ψ = ψ q , ∂x , ∂y , ∂z , ∂t , but we require that the gradients of q and q are not completely independent. In analogue with relativistic metrics that are slave 2 µ ν  s to the invariant interval s with, s = ηµνx x , we demand that ψ = ψ q, q where qs is notational convenience for our specific covariant derivative using Aµ.

s ∂q X µ ∂q µ q ≡ ≡ A µ ≡ A ∂µ (2.33) ∂s µ ∂x

This says that the quantities, Aµ, serve as the underlying metric which in the case of flat space time reproduce the Minkowski metric, but can also accommodate curved space in a covariant manner. In fact, we can reproduce the Schrödinger equation in this formalism by beginning

10 with a judicious choice of L¯ and L ∗, the two Lagrangian densities. Let,

ˆ µ 1 s s HµA L¯ = (ψψ∗ − ψ∗ψ) + ψ∗ ψ (2.34) 2 ~ ˆ µ 1 s s HµA L ∗ = (ψ∗ψ − ψψ∗) + ψ ψ∗ (2.35) 2 ~ where we have also defined a second, complimentary Lagrangian similar to how we argued earlier with the equation for a harmonic oscillator. Now let us derive what the Euler-Lagrange equations look like using s derivatives [9]. The action is the integral over the Lagrangian density,

Z s S[ψ] = L (ψ, ψ) d4x (2.36)

Now we set the variation of the action to zero,

Z " # δL δL 4 δS = δψ + δ(∂µψ) d x = 0 (2.37) δψ δ(∂µψ)

µ µ Then integrating by parts and assuming δ(A ∂µψ) = A ∂µδψ,

Z ! Z ! 4 δL δL 4 δL δS = d x − ∂µ δψ + d x∂µ δψ (2.38) δψ δ(∂µψ) δ(∂µψ) where the second term is a total derivative and is zero on the boundary. Thus we have, δL δL ∂µ − = 0 (2.39) δ(∂µψ) δψ Using these Euler-Lagrange equations, we can reproduce Schrodinger’s equation s ∗ ˆ µ ∗ by variation of ψ assuming that HµA is independent of ψ ,

  ∂ ∂L¯ ∂L¯  ∗  = ∗ (2.40) ∂µ ∂(∂µψ ) ∂ψ   ∂ ∂L¯  −1 ∗  = (2.41) ∂µ ∂(Aµ ∂sψ )  ¯ ∂ µ ∂L A s  = (2.42) ∂µ ∂ψ∗

11 µ ∂ −1 1 s Hˆ A Aµψ = ψ + µ ψ (2.43) ∂µ 2 2 ~ s ˆ µ −~ψ = HµA ψ (2.44) where in Eq. 2.43 we have used the derivative with respect to the Lagrangian, µ Eq. 2.34, and passed ∂µ through A on the left hand side. Similarly we have,

! ∂ ∂L ∗ ∂ = L ∗ (2.45) ∂µ ∂(∂µψ) ∂ψ s ∗ ˆ µ ∗ −~ψ = HµA ψ (2.46)

The final equation has the derivative with respect to the interval which serves as a generalized time on the left hand side, and the operator on the field is on the right hand side.

2.3.1 Adiabatic invariants

If the field, ψ, depends on an external parameter, λ, that changes slowly and periodically in time, then one of the more natural frameworks to describe the dynamical motion of the system is by making a canonical transformation to action angle variables and identifying the adiabatic invariants [10, 11]. In the following we follow Landau’s logic. λ could be the envelope parameter of a near resonant driving force, or gauge field. Let H = H (ψ, π, ψ∗, π¯; λ), where π and π¯ are the canonical momenta to ψ and ψ∗. If we use the assumption that λ varies adiabatically, and use ˙ ∂H Hamilton’s equations on shell, ψ = ∂π , we can show that for any of our canonical field variables ψ, ψ∗, π, π¯,

dI h i i = 0 (2.47) dt 1 I I ≡ π dψ (2.48) 0 2π 1 I I ≡ π¯ dψ∗ (2.49) 1 2π where π is the mathematical constant, and the integral goes over the path for the system at constant energy and λ. Thus Ii remains constant when λ varies, i.e. I is

12 an adiabatic invariant. It can also be written as an area integral,

1 ZZ I = dπ dψ (2.50) i 2π

We take the action to be,

Z t2 S ≡ L dt (2.51) t1 Z = (π dψ +π ¯ dψ∗ − H dt) (2.52) and we can derive Hamilton’s equations from the principle of least action in this form. Any canonical transformation with new variables, Πi, Ψi must also satisfy the same equation,   Z X 0 0 = δ  Πi dΨi − H dt (2.53) i And since we can add a total differential of a function, F , to one side without chang- ing the equivalence under the variation, we can specify any canonical transformation as, π dψ +π ¯ dψ∗ − H dt = Π dΨ + Π¯ dΨ∗ − H 0 dt + dF (2.54)

F is called the generating function of the transformation. First we will consider the motion of the oscillator in phase space such that it completes a revolution over some period. In this case, there is a particularly useful canonical transformation to action-angle variables that simplifies the motion of the canonical momentum to be a constant, and for the conjugate variable to grow linearly in time. We take the momentum variables to be Ii, and the generating function is S0, the abbreviated action. Further, the generating function is set to the abbreviated action, Z Z S0 = π dψ + π¯ dψ (2.55)

From these choices, we can use the following formulas from the generating functions,

∂F ∂F 0 ∂F π = , Ψi = ,H = H + (2.56) ∂ψ ∂Πi ∂t

13 to calculate the canonical momentum and conjugate variable.

∂S (ψ, I; λ) π = 0 (2.57) ∂ψ ∂S (ψ, I; λ) w = 0 (2.58) ∂I

Since the generating function is now, like the parameter λ, an explicit function of time, the new Hamiltonian H0 is different from the old one. We calculate H 0 as,

∂S0 H 0 = E(I; λ) + (2.59) ∂t = E(I; λ) + Λλ˙ (2.60)

  where, Λ = ∂S0 . Hamilton’s equations are thus, ∂λ q,I

! ∂H 0 ∂Λ I˙ = − = − λ˙ (2.61) ∂w ∂w I,λ ! ∂H 0 ∂Λ w˙ = = ω(I; λ) + λ˙ (2.62) ∂I ∂I w,λ

  where, ω = ∂E . ∂I λ

2.4 Modified least action principle

From our work so far in describing a dissipative box of springs, we see that we can use two different Lagrangians. To make that compatible with the principle of least action, we form a composite Lagrangian from the two microscopic ones by combining L¯ and, L ∗ with a metric,

q α β L ≡ `αβL L (2.63) which enable us to write the action in the usual way,

Z S = L dt (2.64)

14 where L α is a vector with two components which we will take to be, (L¯, L ∗), and ∗ `αβ is a 2 × 2 metric independent of q, q that will dictate how to combine the two Lagrangians. The Euler-Lagrange equations of motion take the form,

d ∂L ∂L = (2.65) dt ∂q˙γ ∂qγ where qγ ∈ (q, q∗). The right hand side can be written in terms of components as,

β ∂L α α ∂L β 1 (L γ + L γ )`αβ = ∂q ∂q (2.66) 2 L  α β  `αβ β ∂L ∂L α = L + L  (2.67) 2L ∂qγ ∂qγ and the left hand side is,

   α β  d `αβ β ∂L ∂L α  L + L  = (2.68) dt 2L ∂q˙γ ∂q˙γ  renaming dummy indices and putting the two sides together we get,

α α !  α α ! 1 β ∂L β ∂L d 1 β ∂L β ∂L `αβL + `βαL =  `αβL + `βαL  2L ∂qγ ∂qγ dt 2L ∂q˙γ ∂q˙γ (2.69) α " α # 1 β ∂L d 1 β ∂L L (`αβ + `βα) = L (`αβ + `βα) (2.70) 2L ∂qγ dt 2L ∂q˙γ

s we can add `αβ + `βα = `αβ to obtain a symmetrized version of `. Differentiating,

 s β  α  s β  α  s β  α `αβL ∂L d `αβL ∂L `αβL d ∂L   =   +   (2.71) 2L ∂qγ dt 2L ∂q˙γ 2L dt ∂q˙γ

Collecting terms and rearranging,

α α !  s β  α  s β  ∂L d ∂L `αβL ∂L d `αβL −   =   (2.72) ∂qγ dt ∂q˙γ 2L ∂q˙γ dt 2L

15 where we now define,

α  ∗  α ∂L i q −q P = =   (2.73) γ ∂q˙γ 2 q∗ −q ∂L α d ∂L α Eα = − (2.74) γ ∂qγ dt ∂q˙γ

α α where Pγ is a matrix of the conjugate momenta, and Eγ are the Euler-Lagrange equations for each component of L . We obtain,

!  s β  α α d `αβL E − P   = 0 (2.75) γ γ dt 2L

The conditions on the above equation that reproduce our equations of motion are α that the off-diagonal components of Eγ must be zero, and the diagonal components must be nonzero. If we consider the quantity,

d F α ≡ Eα − P α (2.76) γ γ γ dt

α which we name suggestively as a "fluctuation operator" that for Eα = 0 represents the classical Euler-Lagrange equations of motion. Further, we can think of Eq. 2.75 as a vector equation with, s β ˆ `αβL Lα ≡ (2.77) 2L α ˆ as a unit vector. We find that Fγ Lα = 0. α Note that Eγ is a tensored version of F = ma, which classically in an inertial reference frame should be equal to zero. Under a frame transformation, we should

dvf experience an inertial force of mAf with Af the frame acceleration, or, mAf = m dt .

2.5 Nonlinear oscillators

A simple harmonic oscillator is the building block for much of modern quantum field theory. In the first few sections we explored how a bath of damped harmonic oscillators and correspondingly complex Lagrangians can reproduce aspects of standard QFT. We found that even though we could construct a field theory for the damped oscillators that simply lost energy, we could not capture an equilibration of

16 energy and a maximization of entropy. If our damped harmonic oscillators are not connected to a bath, then the coupling matrices can always be rotated into a basis that diagonalizes the couplings. Our model was missing essential elements of real systems that can support saturation and oscillation. Let us now turn to a nonlinear set of degrees of freedom that clearly exhibit saturation and phase locking.

2.5.1 Self-injection locked oscillators

Synchronization is not commonly used in the context of many-body quantum theory. But for laser physics or electronic oscillators, where one has explicitly the problem of a non-linear oscillator driven by a periodic force, synchronization is often studied quite deeply. We wish to apply these ideas to the field of many-body quantum systems, where instead of strictly appealing to thermodynamic arguments, we also introduce the non-equilibrium concepts such as injection locking. In fact, the general phenomenon of coherence in many-body states like a laser or a Bose-Einstein condensate raise the idea of many classical oscillators driving one another. Coupled oscillators are widely known to synchronize to each other from labora- tory systems like the 1064 fiber laser used in this thesis, to more common classical events like phase locking of two pendulum that hang on the same beam [12, 13], chirping crickets [14], flashing of fireflies [15], flame dynamics [16], clapping hands at a concert [17], and arrays of Josephson junctions [18]. For two coupled oscillators, a wide variety of behavior can arise with different amplitudes of drive or strength of coupling. For our purposes, we want to explore how the coherence of many body systems can be related to the idea of oscillator synchronization. In particular, we intend on treating the particles of a many-body system as individual classical oscillators. A simple, well-studied model of injection locked oscillators originates from Adler [19, 20]. He showed how an external oscillation superimposed on another freely oscillating system can pull the frequency and phase lock the free oscillator to an external drive. He found that the instantaneous phase difference between the two oscillators, α, obeyed,

dα E ω = ∆ω − 1 0 sin α (2.78) dt 0 E 2Q which is a nonlinear differential equation and describes the oscillator’s phase

17 difference as a function of time. ∆ω0 = ω0 − ω1 is the natural beat frequency with

ω0 the frequency of the oscillator and ω1 and E1 the frequency and voltage of the driving signal. Q and E are the quality factor of the free running resonator and the amplitude of the electric field. When,

E ∆ω 1 0 > 2Q (2.79) E ω0

then the free running oscillator phase locks to the injected signal. This relation is closely related to the Arnold tongue [21] which roughly gives the width of the lock-in region for an assortment of recursive functions similar to Eq. 2.78. In an open bath of coupled oscillators, one would expect that the system should exhibit features like locked oscillation or synchronization. The reason is that from the perspective of any one oscillator, it looks as if it is driven by a constant power input from the rest of the bath of oscillators. That is to say that the single oscillator feels a fluctuating force that is either in phase or out of phase with its own motion. For small amplitudes, it would be most energetically favorable to lock in phase. Motion out of phase would be resisted by the inertia of the dominant modes, and its energy would be damped. This is a similar picture to that of clock pendulums that synchronize or even the human circadian rhythm [22,23]. Such a bath of coupled oscillators that spontaneously undergoes synchronization is naturally related to the Kuramoto model [24] which was originally conceived to understand the coordination of biological processes [25], and its mean field description would follow individual decoupled driven oscillators, like the Adler model. York connected the two by considering an equation of motion [20],

Z Z ˙ ˙ V + β V dt + γV = αVinj + δ1Vinj + δ2 Vinj dt (2.80)

(originally for δi = 0), that describes a forced RLC circuit with a stage of power gain. This is recast into phase and amplitude variables by assuming V = <(Aeiθ)

iθinj and Vinj = <(Ainje ), which yields the Adler equation in traditional form,

dθ = ω + K sin(θ − θ) (2.81) dt inj

18 In this case, ω A K ≡ 0 inj (2.82) 2Q A so that K is the forcing amplitude, dependent on the dissipation through Q. This

ω0 2 ω0 happens when α ≡ Q , β = ω0, and γ = Q . Usually the oscillator amplitude is discarded since saturation usually is present, but it can be modeled by the Van der Pol equation [26], d2x dx − µ(1 − x2) + x = 0 (2.83) dt2 dt The restoring force is chosen as,

ω R (|V |) γ = 0 (1 − d ) (2.84) Q RL where −Rd is the negative resistance (or the gain stage) and depends nonlinearly on the amplitude of the system. If,

Rd 2 2 1 − ≡ µ(α0 − |V | ) (2.85) RL then York’s model gives a damping term that flips sign at |V | = α. This shows how the system clamps an oscillator’s response. For our purposes, passive oscillators are more sensible, so it would make more sense to describe saturation via the δ, and α terms which intuitively should describe the equilibration of oscillators. While the Adler equation describes injection locking of oscillators, the Kuramoto model captures the self-synchronization of passive nonlinear oscillators [24,27]. The Kuramoto model is simply,

N dθi X = ωi + Kij sin(θj − θi) i = 1,...,N (2.86) dt j=1 where θi is the phase of the ith oscillator, ωi is its natural frequency, and Kij represents the strength of the nonlinearity between two oscillators. The model is just simple enough to be mathematically tractable, yet can capture the transition between oscillators running incoherently and a spontaneously synchronized, or condensed state when K is made sufficiently large. The Kuramoto angular variable can represent the angular orientation of a pendulum, the phase of the voltage in a nonlinear RLC circuit, the phase of

19 the order parameter in a U(1)-broken symmetry like the circle map recursion relationship that is the paradigm for the Arnold tongue [28], or a Bose-Einstein condensate. Eq. 2.86 has a simple form if the second term is neglected. It is simply one of several identical equations that describe decoupled oscillators. For K 6= 0, it describes the motion of the phase of the ith oscillator in an all-to-all coupling network where the coupling depends on a sinusoidal function of the relative phase difference between the two oscillators. In analyzing the model, there are an assorted number of approximations that are commonly made. Often one assumes that the ωi are all of the same frequency or very close in value, that K  ω is weak, or that a mean-field K = Ki can capture the coarse behavior. Further, it is exactly solvable under certain transformations [24]. In the mean field analysis, there are two states as a function of K. At small K, the oscillators are incoherent and the oscillators drift randomly with independent phases and frequencies. At large K, the system crosses a phase transition into a fully synchronized state where the oscillators pull each other sufficiently hard to keep them all at a common frequency and phase. This is reminiscent of the laser oscillation problem. The laser gain medium consists of many different particles all with a different phase and natural frequency from thermal broadening or other effects to form inhomogeneous gain profiles. Macroscopic coherent states, or lasing, only occurs when a large portion of the medium are phase and frequency locked together. Qualitatively the Kuramoto model solves many of the deficiencies of our simple oscillators-in-a-box model that we began with. It depicts driving each oscillator by the sum of all of the others. Depending on the coupling strength, it captures a phase transition between a thermal and a condensed state. Further, it provides a source of nonlinearity which can clamp, or restrict the movement of energy into a single oscillator. Other nonlinear systems may also be good candidates to describe quantum field theories as well. In particular they generally support solitons such as with the Korteweg-de Vries equation [29], and contain specific parameter spaces in which the system does not thermalize completely. We can gain more insight into the utility of the Kuramoto model for our light- atom problem by examining the Hamiltonian of the system that yields the equations

20 of motion of the Kuramoto model [30].

N X ω` 2 2 L 2 2 2 H = (q` + p` ) + (q` + p` ) `=1 2 4 N 1 X 2 2 2 2 + K`,m(q`pm − qmp`)(qm + pm − q` − p` ) (2.87) 4 `,m=1 where qi is the coordinate of an oscillator, and pi is the corresponding momentum. This can be cast into action angle variables by the substitution of,

1   I = q2 + p2 (2.88) ` 2 ` ` ! q` φ` = arctan (2.89) p` which yields,

N N X 2 X q H = ω`I` + L`I` − K`,mK`,m ImI`(Im − I`) sin(φm − φ`) (2.90) `=1 `,m=1 with the equations of motion of,

˙ ∂H Ij = − (2.91) ∂φj N X q = −2 Km,j ImIj(Im − Ij) cos(φm − φj) (2.92) m=1 ˙ ∂H φj = (2.93) ∂Ij

= ωj + LIj N  s  X q Im + Km,j 2 IjIm sin(φm − φj) − (Im − Ij) sin(φm − φj) (2.94) m=1 Ij

The dynamics for a particular Ij is,

N ˙ X φj = ωj + LI + 2IK`,j sin(φ` − φj) (2.95) `=1

21 2.5.2 Conserved microscopic Hamiltonians

We will take L = 0 for the moment and examine the dynamics of the Lagrangian from the perspective of a single oscillator looking out at a bath of driving oscillators. The Kuramoto Hamiltonian (Eq. 2.87) can be re-written as,

X X H (pi, qi) = ωihi + `ij(hi − hj)Kij (2.96) i ij

where hi describes the free Hamiltonian (the diagonal terms), and `ij = qipj − qjpi describes the free Lagrangian of linear oscillators described in Sec. 2.1.1 such that i Lj = −m 2 `ij. If the Hamiltonian (Eq. 2.96) came from a microscopic L , it would be,

X L = piq˙i − H (2.97) ∂L i pi= ∂q˙i

X X X = − ωihi + piq˙i − ∆hij`ijKij (2.98) ∂L i i ij pi= ∂q˙i

X X = − ωihi − 2pi(qj∆hijKij − δijq˙i) (2.99) ∂L i ij pi= ∂q˙i

where in the last term we assume that the product of ∆hij`ijKij is symmetric under i, j, permitting us to extend the sum by a factor of 2 and gathering the terms. We insert a Kronecker delta in order to condense the index notation. To ∂L actually make the substitution, pi → , we would need to substitute it into ∂q˙i 2 2 hi = qi + pi , obtaining a largely unsolvable nonlinear partial differential equation 0 for L . Instead, we fix hi ≡ hi to a value in the integrable limit. This is reasonable because q2 + p2 = h is the action variable when cast into action angle variables (Eq. 2.88). Now we have, with implied summations,

0 0 L = −ωihi − 2pi[(∆h K)ij − δij∂t]qj (2.100)

and we define, 0 Tij ≡ (∆h K)ij − δij∂t (2.101)

22 which is a matrix valued differential operator. Rearranging,

1 0 ∂L − (L + ωihi ) = Tijqj (2.102) 2 ∂q˙i where the right hand side looks like a term of pq˙ where the T operator is switching the indices around and providing the time derivative on q. Now we make this compatible with the development from Sec. 2.4 and also q α β demand that L be of the form, hαβL L so that,

q α β q ∂ hαβL L 1 α β 0 − ( hαβL L + ωihi ) = Tijqj (2.103) 2 ∂q˙i α 1 s β ∂L = hαβL Tijqj (2.104) L ∂q˙i where the L ’s are dissipative. A difference here is that we are taking the derivative with respect to the microscopic variables. We will need to utilize the chain rule in order to take the derivatives with respect to a field mode, Ψ and Π. The right hand side of Eq. 2.104 looks like Eq. 2.77, where the unit vector is dotted with a matrix. We are only taking the derivative with respect to the real part of q, so there isn’t a lower index on the corresponding E (Eq. 2.74), or equivalently, we can set it to zero. Then we have, 1 0 1 ˆ α − (L + ωih ) = LαP Tijqj (2.105) 2 i 2 i where ∂L α looks similar to the P matrix (Eq. 2.73) multiplying and we used, ∂q˙i L

s β ˆ hαβL Lα ≡ (2.106) 2L We also find, ˆ ˆα αβ ˆ ˆ LαL = h LαLα = 1/4 (2.107) where we have raised the index by using the metric, hαβ and,

q α β L = hαβL L (2.108) q ˆ α = L LαL (2.109)

23 so that, ˆ α L = LαL (2.110)

α ˆ where we see that L is a dot product between L and the unit vector, Lα. We let, ˙ α α S0 ≡ Pi Tijqj (2.111) so that, 0 ˆ ˙ α −(L + ωihi ) = LαS0 /4 (2.112) ˙ where, we have chosen the name, S0, because of the similarity to the time derivative of an abbreviated action. Multiply through by L , to eliminate the denominator of Lˆ, and it becomes a quadratic equation.

2 0 ˙ α L + ωihi L + LαS0 = 0 (2.113)

2.5.3 Microscopic relativistic Lagrangian

Let us call, 0 Σ ≡ ωihi (2.114) the energy. Then you have this quadratic equation,

2 ˙ α L + ΣL + LαS0 = 0 (2.115) whose roots are,

s Σ Σ2 ˙ α L± = − ± − LαS (2.116) 2 4 0  s  ˙ α Σ 4LαS = −1 ± 1 − 0  (2.117) 2  Σ2 

˙ α 2 The root become imaginary when 4LαS0 > Σ . The two roots that are possible choices for the Lagrangian look like the special relativity metric with an additive factor. The additive factor, −1, to the Lagrangian can be ignored because it can be captured by the choice of extremizing the action as a maximum or a minimum which is not physically significant.

24 We can interpret this as a general relativity Lagrangian with Σ = 2mc2, and,

s ˙ α 1 4LαS ≡ −1 ± 1 − 0 (2.118) γ Σ2 which is analogous to the Lagrangian for a free particle in special relativity. The numerator represents mv2 and the denominator, m2c2. In special relativity, there are two ways to think of the action, either as the interval, or to realize that the interval is the proper time. Since

v2 ! dτ = 1 − dt (2.119) c2 and that the action is the integral of the Lagrangian against time,

Z Z Z ds ∝ dτ = L dt = S (2.120)

This is an invariant quantity, it represents the proper time of a particle moving with a velocity v. Thus not only has our bath of oscillators given a picture of oscillators driving each other and locking up, but also that the inherent dynamics capture time dilation. ˙ α 2 If LαS0 = mv , then time dilates as a function of that speed, v. From earlier,

˙ α α LαS0 = LαPi Tijqj (2.121) α ∂L 0 = Lα (Kij∆hijqj − q˙i) (2.122) ∂q˙i where we see that the only place that K enters is as an offset to the velocity, akin to a gauge potential.

2.5.4 L as a metric

Alternatively, instead of solving for the roots of the quadratic L equation (Eq. 2.113), we can substitute in the actual form of the Lagrangian again to investigate the nature of hαβ.

q α β α β β ˙ α hαβL L + Σ hαβL L + L hαβS0 = 0 (2.123)

25 Now rearranging and combining,

q β α ˙ α α β hαβL (L + S0 ) = −Σ hαβL L (2.124) squaring both sides,

β β¯ α ˙ α α¯ ˙ α¯ 2 α β hαβhα¯β¯L L (L + S0 )(L + S0 ) = Σ hαβL L (2.125) a sufficient condition for a solution is,

β 2 α β¯ α ˙ α α¯ ˙ α¯ hαβL [Σ L − hα¯β¯L (L + S0 )(L + S0 )] = 0 (2.126) which means, 2 α β¯ α ˙ α α¯ ˙ α¯ Σ L = hα¯β¯L (L + S0 )(L + S0 ) (2.127) renaming subscripts, and pulling them down with the metric,

α 1 β ˙ β α ˙ α L = Lβ(L + S )(L + S ) (2.128) Σ2 0 0 now we write this as a dot product with the metric hαβ. It takes the overlap of L ˙ with L + S0. ˙ α hL , L + S0i ˙ α L = (L + S0) (2.129) Σ2 where this last equation is a vector equation. This is a vector which is equal to a scalar times a vector. From this we conclude that L should lie along the direction ˙ ˙ of L + S0. Thus all the unit vectors for L and S0 must be the same, that is,

ˆ ˆ˙ L = S0 (2.130)

The magnitude of L is,

α hL , L i = LαL (2.131) ˙ hL , L + S0i = (2.132) Σ2 ˙ hL , S0i = (2.133) 1 − Σ2

26 and also, ˙ hL , S0i 1 = 1 − (2.134) hL , L i Σ2 or, ˆ ˙ ˙ hL , S0i 1 S0 = 1 − = (2.135) L Σ2 L This means that the equations of motion can be written in terms of Sˆ˙

d EαSˆ˙ β = P α Sˆ˙ β (2.136) β 0 β dt 0

α ˙ α with the magnitudes of L and S0 given by the above equation. Note that the q’s and p’s should be the position coordinates, X, that is, real-valued spring extensions.

2.6 Observables behave like response functions

Observables are dynamic variables that can be measured. Conventionally these are associated with a Hermitian operator that acts on the state of the system. However, it also must be true that in measuring quantities like the index of refraction, or polarizability of an atom, we interact with it by probing and touching it. What we observe, however, is the response of the observable. In fact, if we are modeling quantum field excitations as simple oscillators, then observables such as mass, charge, energy, and momentum must also be the response that the observable would have to other entities pushing on it. Therefore, it makes sense to consider modeling an oscillator as using a (nonlinear) response function that naturally carries the concepts such as susceptibility and impedance. This perspective requires a shift from considering quantum mechanical observables as representing an intrinsic amount of quanta, to representing the method in which external measuring devices interact with the quanta. This concept of observables lends itself naturally to the analytic continuation of observables. The propagator for a free scalar field is the function φ(x) that satisfies

2 (x + m )φ(x) = δ(x) (2.137)

27 the Fourier transform yields,

(−p2 + m2)φ(p) = 1 (2.138)

and we can invert and solve for φ(x), but to do so we need to analytically continue by adding i in the denominator.  is taken to zero after the integration, but the important point remains that to make sense of the propagator, we had to treat the particle excitation as if it had some non-infinite lifetime. This extension to a non- purely unitary world comes naturally by viewing the observables as an analytic (possibly nonlinear) response function.

2.6.1 Complex valued observables

If one takes a mechanics perspective, a unitary perspective, that the energy an oscillator carries is a conserved degree of freedom, then no matter what happens to the oscillator it will have the same time-average amount of energy. We normally phrase this as the operator that describes the time evolution of the system must be,

Uˆ = e−iHt/ˆ ~ (2.139)

a time-independent unitary operator. Since it is related to time symmetry, this then means that there is no time dependent external force applied to the system. However, to measure a mechanical oscillator in a laboratory, the only way to determine the energy is to couple it to another system. For example, to determine the energy you must perturb the system by drawing a little bit of energy out of it. When you measure a oscillator, an observer must couple power out of the system to form a representation of the oscillation frequency, but this reduces the energy of the oscillator and necessarily also indirectly measures the oscillator’s decay time. Measuring something with a lot of inertia will not affect the oscillation much, but this point becomes more apparent as one probes the oscillation of only a few electrons. Thus when you are measuring the energy, you are actually measuring the complex energy, which is both the oscillation and the damping rate. Revisiting our choice of complex Lagrangian in Sec. 2.4, we mentioned before that the q and q∗ could be thought of as the in-phase and the out-of-phase position of a harmonic oscillator. The in-phase component transfers power in and accelerates

28 the particle while the out-of-phase part dissipates energy away. Now if complex coordinates in time and energy are useful descriptions of an oscillator, then Lorentz symmetry demands that it is equally useful for spatial coordinates as well. Complex spatial coordinates would mean that a spatial oscillation decays away when examined from a boosted reference frame. For our atoms being driven by an oscillating field, there is always a finite width to the linewidth. It is necessary to scatter entropy out, and to bend light. The resonance from a linewidth of the atoms is necessary to write down an index of refraction. Conventionally, we would describe the light atom scattering process unitarily and use that as input into the open, thermodynamics problem. But by treating all observables as complex response functions, we might be able to introduce dissipation in the problem and keep it there during the procedure of quantization. We want to generalize the Kuramoto Hamiltonian so as to describe all other symmetries, Hµ. Let’s begin again with the Kuramoto Hamiltonian, with p and q, now understood to represent a generic field excitation, such as the polarizability of an atom. X X Hµ = ωµihµi + Kµ,ij(qipj − qjpi)(hµi − hµj) (2.140) i ij where, ˙ hµi = {hµi, Hµ=0} (2.141) where {·, ·} are the Poisson brackets, and Hµ=0 is the Hamiltonian for the time symmetry. We insist on this form for the action angle variables with hµi being the action variable so we can associate each µ with another canonical change of variables to another action angle formalism under which the Hamiltonian would take the form, Hµ.

We can complexify the Kuramoto coordinates by taking Hµ to be,

X X h i Hµ = ωµihµi + = (qi + ipi)(qj − ipj) (hµi − hµj)Kµ,ij (2.142) i ij where = is the imaginary part. Let zi ≡ qi + ipi to give,

  X X X ∗ Hµ = <  ωµihµi − i zi zj (∆hKµ)ij (2.143) i i j

29   X d = <  ωµihµi − izivµ,i (2.144) i where we have suggestively defined,

d X ∗ vµi ≡ zj (∆hKµ)ij (2.145) j as a dissipative velocity. Let, c X d Hµ ≡ ωµihµi − izivµi (2.146) i be the analytically continued version of Hµ and we use a Legendre transformation to obtain the Lagrangian. We appeal to an established way of concatenating integrable systems developed independently by Suris [31–34] and Nijhoff [35–39], and in the Legendre transformation take the derivatives with respect to a multi-time coordinate and let, µ ∂qi qi ≡ (2.147) ∂tµ be a derivative of q with respect to a multi-time variable. The Legendre transfor- mation becomes,

c X µ c Lµ ≡ piqi − Hµ (2.148) i X µ d = piqi − ωiµhiµ + izivµi (2.149) i X µ d d = pi(qi − vµi) − ωiµhiµ + iqivµi (2.150) i

X ∂L µ d = µ (qi − vµi) − Σµ (2.151) i ∂qi where, X d Σµ = ωiµhiµ − iqivµi (2.152) i The Lagrangian must be that of a composite form, formed out of all the α q µν c c individual contributions of Hµ and L . We combine L = η Lµ Lν with our

30 q α β previous notion that L = hαβL L , and expand L as,

q α β µν L = hαβLµ Lν η (2.153) where ηµν is diagonal. In this way, the action functional is computed over a p- dimensional surface with the independent variables of tµ. This will only be true for simultaneous solutions of the Euler-Lagrange equations δL/δu = 0 [40]. Substituting in L ,

α  µ¯ α¯  α X Lα¯ ∂Lµ¯ µ d α Lµ =  µ (qi − vµi) − Σµ (2.154) i L ∂qi

d ∗ Note that vµi is a function of z , and not of z. This means that though we have d continued the Kuramoto hamiltonian into complex coordinates, quantities like vµi are holomorphic functions. Derivatives of v would be the response to perturbing v with respect to some complex coordinates. These being analytic functions could coorespond to being an observable that is a response function.

31 Chapter 3 | Dielectrics and curved spacetime

3.1 Introduction

In the previous chapter we showed that a nonlinear, complexified model of springs and masses is necessary to support a model that exhibits phenomena like self-locked synchronization. It also provides a natural extension to relativity. It should be straightforward to apply this to our experiment because many light-atom phenomena require nonlinear effects like saturation in pumping and dissipation in forming a linewidth. Now we want to discuss our model of springs and masses, but in a coordinate free manner. In the following we will show that the effect of a medium in which light propagates can make spacetime look like a gravitational metric for the light. To an observer in media in flat spacetime, the light can appear to be propagating through vacuum but, in a curved spacetime. This concept of media-spacetime equivalence has been reinvented a number of times [41–44], starting as early as 1923 by Gordon [45]. Others have gone further [46–48], and connected the dynamics of the excitations of ultracold gasses1 to the dynamical interplay between mass and spacetime. This is particularly enticing because it is those dynamics which make Einstein’s general relativity so difficult to process: the field equations are eighth order nonlinear partial differential equations. Maxwell’s equations provide a complete description of a macroscopic description of light propagation and interaction with macroscopic matter. Analogues to gravity are not new [49], waves in water is the most familiar to our everyday knowledge,

1specifically the nonlinear GPE model of a BEC

32 but our point here is that the tools of standard-model physics and gravity can be useful for understanding atom-light interactions.

3.2 Nonlinear media as a covariant tensor

3.2.1 Maxwell’s equations in media

From Maxwell’s equations [50] for electric and magnetic fields we have established,

ρ ∂B ∇ · E = ∇ × E = − ε0 ∂t ∂E ∇ · B = 0 ∇ × B = ε µ + µ J (3.1) 0 0 ∂t 0 where E, and B are three-component vectors that indicate the x, y, and z spatial 4π h H i 1 h F i components of the electric and magnetic fields, µ0 = 7 , ε0 = 2 , 10 m µ0(299792458) m h C i h A i ρ is the charge density m3 and J is the current density m2 . We introduce the scalar, V , and vector, A, potentials of the electric and magnetic fields as,

∂A E = −∇V − (3.2) ∂t B = ∇ × A (3.3)

In free space (ρ = 0, J = 0),

∂B ∇ · E = 0 ∇ × E = − ∂t ∂E ∇ · B = 0 ∇ × B = ε µ (3.4) 0 0 ∂t where we can take the curl of the curl equations, and use the curl of the curl identity2, to give us the familiar wave equations,

∂2E ε µ − ∇2E = 0 (3.5) 0 0 ∂t2 ∂2B ε µ − ∇2B = 0 (3.6) 0 0 ∂t2 2∇ × (∇ × A) = ∇(∇ · A) − ∇2A

33 In matter, we must account for the complication of the media interacting with the field to amplify, attenuate, or redirect the fields (which in turn modifies the distribution of the media and so on). We combine the electromagnetic interactions in media into D, the electric displacement field, and H, the magnetic field strength,

D ≡ ε0E + P (3.7) 1 H ≡ B − M (3.8) µ0 with P, the dipole moment per unit volume of the medium or polarization, and M, the magnetic dipole moment per unit volume or magnetization. Inside linear, isotropic media we empirically have,

P = ε0χeEM = χmH (3.9) where, introducing even more notational quantities, we can subsume P and M into two scalar quantities, 1 D = εEH = B (3.10) µ where χe is the electric susceptibility, χm is the magnetic susceptibility, and χe, χm are dimensionless and material dependent. Then Maxwell’s equations in general media are,

∂B ∇ · D = ρ ∇ × E = − (3.11) ∂t ∂D ∇ · B = 0 ∇ × H = + J (3.12) ∂t where we have redefined ρ and J to be the free charge density and the free current density, and incorporated ε and µ3 into the displacement field and the field strength. In Cartesian coordinates we can equivalently write this as,

i i X ∂D X ijk ∂Ek ∂B i = ρ  j = − (3.13) i ∂x jk ∂x ∂t i i X ∂B X ijk ∂Hk ∂D i i = 0  j = + j (3.14) i ∂x jk ∂x ∂t

3which are possibly tensor valued

34 where the sums using Latin indices e.g. i, j, k run over 1, 2, 3, and ijk is the Levi-Civita symbol: ijk = 1 if (i, j, k) is an even permutation of (1, 2, 3), −1 if it is an odd permutation, and 0 if any index is repeated. But this has been treating space and time as independent quantities where interactions with the fields propagate independent of the location, velocity, and acceleration of the interacting entities. This is not correct, and we must introduce the spacetime metric and associated curvature constraints to accurately depict the interaction and propagation of electromagnetic fields [51].

3.2.2 Covariant formulation of Maxwell’s equations

The metric, g, is a function that defines the distance between two vectors. It defines the infinitesimal stretch between two basis vectors to maintain the constant interval, 2 µ ν ds = gµνdx x (3.15)

For concreteness, let g be the spacetime metric which we will take for the moment to be Minkowski,   1 0 0 0     µν 0 −1 0 0  g =   (3.16)   0 0 −1 0    0 0 0 −1 With g, we can begin to combine space and time quantities into a mathematical formalism that details their interplay through matter and fields. Vectors with indices on top, Aα, denote contravariant vectors - vectors that scale inversely by a change in basis, and vectors with with indices on bottom, Aα, indicate covariant vectors - those that co-vary with a change in basis. We can recast the fields into a field tensor to succinctly capture how all four components of the fields transform in various bases,

 √ √ √  0 −Ex ε0µ0 −Ey ε0µ0 −Ez ε0µ0    √  µν Ex ε0µ0 0 −Bz By  F =   (3.17)  √  Ey ε0µ0 Bz 0 −Bx   √  Ez ε0µ0 −By Bx 0

35 Importantly we also can write down the electromagnetic dual to the field tensor,

  0 −Bx −By −Bz    √ √  µν Bx 0 Ez ε0µ0 −Ey ε0µ0 F˜ =   (3.18)  √ √  By −Ez ε0µ0 0 Ex ε0µ0   √ √  Bz Ey ε0µ0 −Ex ε0µ0 0 √ √ through the explicit transformation, E ε0µ0 → B and, B → −E ε0µ0. The duality transform in electromagnetic theory refers to the symmetry highlighted by relativity where expressions in terms of electric fields have a directly analogous expression in terms of magnetic fields and vice versa. This is simply due to the fact that with relativistic transformations, the electric field is often transformed into the magnetic field and vice versa. This casting of the fields into covariant notation preserves Maxwell’s equations,

αβ α ∂βF = µ0J (3.19) ˜αβ ∂βF = 0 (3.20) where the sums using Greek indices e.g. α, β, γ run over 1, 2, 3, 4,

α ρ J = (√ ,Jx,Jy,Jz) (3.21) ε0µ0

∂ is the 4-current and ∂α ≡ ∂xα . The 4-vector potential is,

α √ A = ( ε0µ0V,Ax,Ay,Az) (3.22)

We can in addition rewrite the field tensor in terms of the vector potential as,

∂Aβ ∂Aα F αβ = − (3.23) ∂xα ∂xβ

36 We can similarly write down the excitation field tensor for the fields in media,

  0 −Dxc −Dyc −Dzc     µν Dxc 0 −Hz Hy  G = µ   (3.24) 0   Dyc Hz 0 −Hx    Dzc −Hy Hx 0 and define a polarization tensor, P µν, and a susceptibility tensor, χ, that relates the vector valued electric and magnetic field strengths to the vector valued polarization and magnetization,

  0 −Pxc −Pyc −Pzc     µν Pxc 0 Mz −My  P = µ   (3.25) 0   Pyc −Mz 0 Mx    Pzc My −Mx 0 with the relationship, Gµν = F µν + P µν (3.26)

.

3.2.3 Coordinate transformations of covariant media

We can introduce coordinate transformations by [52,53],

xα → xα0 (xα) (3.27)

0 α t α0 ∂xα where x = ( √ , r) is a coordinate vector of spacetime. Let us define Λ ≡ α , ε0µ0 α ∂x α0 and |Λ| ≡ det(Λα ), the Jacobian matrix of the transformation and its determinant. For convenience we will write products of the transformations as,

0 0 0 0 0 0 Λα1 Λα2 ... Λαn = Λα1 α2 ...αn (3.28) α1 α2 αn α1 α2 ...αn

The fields transform as,

µ0ν0 Fµ0ν0 = Λµ ν Fµν (3.29)

37 0 0 1 0 0 Gµ ν = Λµ ν Gµν (3.30) |Λ| µ ν

and finally the coefficients in the constitutive relationship must transform like,

0 0 0 1 0 0 0 χα1 α2 ...αn = Λα1 α2 ...αn χα1α2...αn (3.31) |Λ| α1 α2 ...αn where this encapsulates the general transformation law for linear and nonlinear materials in spacetime coordinates. We apply the idea of coordinate transformations to specifically that of media which is moving. For example, if the medium is moving in the x-direction with a α0 α0 α constant velocity u, the coordinate transformation x = Λα x takes the form of a Lorentz boost in the minus-x direction.  √  γ γu ε0µ0 0 0    √  α0 γu ε0µ0 γ 0 0 Λ =   , |Λ| = 1 (3.32) α    0 0 1 0   0 0 0 1 with γ = √ 1 . 2 1−u ε0µ0

3.2.4 χ can include nonlinear dielectrics

Often, the equation for the polarization in nonlinear optics is defined in terms of higher powers of the electric field [54].

P (ω) = χ(1)E(ω) X (2) + D2 χijk(ωn + ωm, ωn, ωm)Ej(ωn)Ek(ωm) jk X (3) + D3 χijkl(ωo + ωn + ωm, ωo, ωn, ωm)Ej(ωn)Ek(ωm)Eo(ωm) jkl + ... (3.33)

where Da is the degeneracy factor and is equal to the number of distinct permuta- (1) (2) (3) tions of the applied field frequencies, ωn,m,o,.... χ , χijk, χijkl are the linear, second-, and third-order susceptibility tensors. Like the nonlinear (electric) polarization defined in Eq. 3.33, the polarization tensor can be expanded in terms of field tensor

38 as [52,53],

µν µνσκ µνσκαβ µνσκαβγδ P = χ Fσκ + χ FσκFαβ + χ FσκFαβFγδ + ... (3.34) ∞ X µνα1β1...αnβn = χ Fα1β1 Fαnβn (3.35) n=1 where χ is a second, fourth, sixth, etc. rank tensor that couples each component of E or B to each other in successive powers of the field. We can relate the fields in Gµν which appear directly in Maxwell’s equations to P µν by incorporating F µν through,

Gµν = F µν + P µν (3.36) 1 ∞ µα1 νβ1 να1 µβ1 X µνα1β1...αnβn = (g g − g g )Fα1β1 + χ Fα1β1 ...Fαnβn (3.37) 2 n=1 ∞ X µνα1β1...αnβn = χ Fα1β1 ...Fαnβn (3.38) n=1 where we have made a redefinition of χ. We can break these terms down into familiar nonlinear effects [52,53],

i 0iσκαβ P (2) = χ FσκFαβ (3.39) 0i0k0m 0i0kmn 0iklmn = 4χ F0kF0m + 4χ F0kFmn + χ FklFmn (3.40) ijk ijk ijk = a EjEk + b EjBk +c BjBk (3.41) | {z } | {z } Pockels effect Faraday effect and,

i 0iσκαβµν P (3) = χ FσκFαβFµν (3.42) ijkl ijkl ijkl ijkl = a EjEkEl +b EjEkBl + c EjBkBl +d BjBkBl (3.43) | {z } | {z } Kerr effect Cotton-Mouton effect

For general boosts of nonlinear material, the permittivities and permeabilities can become mixed so that a Pockels medium at rest can display a Faraday effect if it is moved relative to the observer - or vice-versa.

39 3.3 When the linear susceptibility tensor is equivalent to a spacetime metric

3.3.1 Relating the susceptibility to the Riemannian tensor

The relationship between Maxwell’s equations and the (linear) permittivity tensor can be written as [49],  1/2ij µ1/2εµ1/2  ij   g =   (3.44)  det(µε)  in the case where ε ∝ µ. If we combine the electric and magnetic permittivities into one object, Z, such that [49],

µανβ ∂α(Z Fνβ) = 0 (3.45) expresses Maxwell’s equations then also, Z can be written in terms of the spacetime metric as [52] √ Zµναβ = K −g(gµαgνβ − gµβgνα) (3.46) where g is the determinant of the metric, gµν. We can try and identify that four index object, Z with R, the scalar part of the Riemann curvature tensor, which is the object that measures how much the components of a vector change when it is parallel transported along a small closed curve on the manifold. The Riemann curvature tensor associates a tensor to each point of a Riemannian manifold (i.e. a tensor field) that measures the extent to which the metric tensor is not locally isometric to Euclidean space. For example, the electromagnetic tensor field, Fµν is a tensor field. We can decompose the Riemann tensor via the Ricci decomposition into three parts,

Rabcd = Sabcd + Eabcd + Cabcd (3.47) Rm S = m (g g − g g ) (3.48) abcd n(n − 1) ac db ad cb

m where gab is the metric tensor, and Rm is the Ricci curvature, n is the dimensionality. From Post [52], the susceptibility tensor is the quantity that relates the field

40 tensor to the excitation tensor as,

1 Gλν = χλνσκF (3.49) 2 σκ

Then, we can relate χ to a metric through,

λνσκ √ λσ νκ λκ νσ χ = Y0 g(g g − g g ) (3.50) which we can relate to the Riemann curvature tensor for manifolds that have a λσ νκ Riemann scalar R = Rλνσκg g which is a constant. The Riemann tensor is then given by, √ 1 √ Rλνσκ g = R g(gλσgνκ − gλκgνσ) (3.51) 12

3.3.2 Topological changes and the susceptibility tensor

Pick a basis {σi} of 1-forms, not necessarily orthonormal. Choose any set of 1-forms, i i ωj to be the connection 1-forms. The curvature 2-form, Ωj, is defined by the Cartan structure equation, i i i m Ωj = dωj + ωm ∧ ωj (3.52) and we can expand with respect to our basis to write,

1 Ωi ≡ Ri σk ∧ σl (3.53) j 2 jkl 1 g Ωi = R σk ∧ σl (3.54) µi j 2 µjkl

i where R is the Riemann curvature tensor. Since Ωj is a 2-form, a matrix represen- tation of it would be anti-symmetric, yielding,

2 3 gµiΩµj = i23σ ∧ σ (3.55)

A representative of each Chern class, ck(V ) are given as the coefficients of the characteristic polynomial of the curvature form, Ω of V ,

! itΩ X k det + I = ck(V )t (3.56) 2π k

41 which we can use the matrix identity of,

tr(ln X)) = ln(det(X)) (3.57) and obtain,

" 2 2 X k tr(Ω) tr(Ω ) − tr(Ω) 2 ck(V )t = I + i t + 2 t k 2π 8π −2tr(Ω3) + 3tr(Ω2)tr(Ω) − tr(Ω)3 # + i t3 + ··· (3.58) 48π3 " tr(Ω2) −2tr(Ω3) # = I + t2 + i t3 + ··· (3.59) 8π2 48π3

Thus, some gross changes of the Riemann tensor should be apparent as a change of the Chern class.

3.3.3 Transformations of both the metric and the susceptibility

Focusing our attention on χ, let us conceptually separate the effects of changing dielectrics from changing metrics. As we have seen from the previous sections, the susceptibility tensor in flat spacetime, which describes the refraction of light, relates to the gravitational metric of vacuum, which also describes the refraction of light. The field of Transformation Optics most recently has used this relation to construct materials that bend light into exotic configurations like cloaking devices [55,56]. By making use of the geometric nature of Maxwell’s equations and differential geometry, Thompson et al., have comprehensively shown how metrics and susceptibilities transform together [57–60], and we shall follow their lead. In vacuum, we will define the Hodge dual, ?, such that,

1q (?F) = |g| gαγgβδF (3.60) µν 2 µναβ γδ G = ?F (3.61) where F and G are both two-forms. In this way, ? contains the information of the mapping to the dual space of the underlying manifold and incorporates the metric structure. It also shows that ? is a linear map and provides the constitutive relations.

42 Together, these yield Maxwell’s equations as,

dF = 0 (3.62) dG = J (3.63) where J is the current 3-form. In component form, J is,

q µ Jαβγ = |g|αβγµj (3.64)

To extend to Maxwell’s equations in linear media, we take the minimal approach and define a tensor that takes ?F to G. That is,

G = χ(?F) (3.65)

Thus the tensor χ contains all the information of the dielectric’s material properties. It is averaging over all the material contributions to the action that would contribute in a more microscopic quantum field theory. Here we emphasize that we use χ and ? as the two separate entities that encode both the material effects and the space-time effects. We can compare the above constitutive equation with the vector calculus equation to identify χ as [58],

  A ∗ ∗ ∗     σρ 1 BA ∗ ∗  χ =   (3.66) γδ   2 CDA ∗    EFGA where,   0 0 0 0     0 0 0 0 A =   (3.67)   0 0 0 0   0 0 0 0

43  −1 −1 −1  0 −µxx −µxy −µxz    −1  µxx 0 −γ1xz γ1xy  B =   (3.68)  −1  µxy γ1xz 0 −γ1xx  −1  µxz −γ1xy γ1xx 0

 −1 −1 −1  0 −µyx −µyy −µyz    −1  µyx 0 −γ1yz γ1yy  C =   (3.69)  −1  µyy γ1yz 0 −γ1yx  −1  µyz −γ1yy γ1yx 0   0 −γ2zx −γ2zy −γ2zz     γ2zx 0 −zz zy  D =   (3.70)   γ2zy zz 0 −zx    γ2zz −zy zx 0

 −1 −1 −1  0 −µzx −µzy −µzz    −1  µzx 0 −γ1zz γ1zy  E =   (3.71)  −1  µzy γ1zz 0 −γ1zx  −1  µzz −γ1zy γ1zx 0   0 γ2yx γ2yy γ2yz     −γ2yx 0 yz −yy F =   (3.72)   −γ2yy −yz 0 yx    −γ2yz yy −yx 0   0 −γ2xx −γ2xy −γ2xz     γ2xx 0 −xz xy  G =   (3.73)   γ2xy xz 0 −xx    γ2xz −xy xx 0 where the ∗ indicate entries that are antisymmetric on either the first or second σρ indices of χγδ . Eq. 3.66 illustrates χ as a matrix of matrices. We can match com- ponents of the constitutive equation, G = χ(?F), with the constitutive equations in flat space to yield,

−1 ∗ ∗ ∗ H =µ ˇ B +γ ˇ1 ED =ε ˇ E +γ ˇ2 B (3.74)

44 where aˇ is used to denote a 3 × 3 matrix. This can be rearranged into a more familiar form as,

B =µ ˇH +γ ˇ1ED =ε ˇE +γ ˇ2H (3.75) and we can switch between the two representations using the following relations,

−1 −1 ∗ ∗ ∗ µˇ = (ˇµ ) εˇ =ε ˇ − γˇ2 µˇγˇ1 (3.76) ∗ ∗ γˇ1 = −µˇγˇ1 γˇ2 =γ ˇ2 µˇ (3.77)

In writing down a coordinate free representation of a quantum field theory for atoms and light, we want to know what does the (vacuum) spacetime metric look like that is formed by a moving, nonlinear dielectric. One way to conceptualize this is as a set of general transformations from one manifold to another. We can understand specifically how the effects of curved spacetime and that of dielectric media affect the propagation of light in the same way. By making clear how the insertion of a medium transforms the electromagnetic fields on one manifold with a particular metric into another set of fields on a different manifold that can be described by another metric and a different material susceptibility. We introduce a mapping T from the space of an initial space-time manifold, M to an image manifold, M˜ , T : M → M˜ ⊆ M (3.78)

We have to be careful here because the mapping may not be simple. Although the effect of some dielectric may be to open a hole in where the electromagnetic fields propagate, imagining puncturing a hole in space-time is unphysical. We must define a specific mapping that acts on the electromagnetic fields,

T : M˜ → M (3.79) such that it takes the electromagnetic fields in M˜ and places them back in the original manifold M. For the other contributions, we make another map that takes the new metric into the original one while remaining in the same manifold.

T : M → M (3.80)

Although spacetime may curve or deform, we are not considering changes in the

45 underlying topology of spacetime itself. Those possible changes are pushed into the mapping of the electromagnetic fields, T . While it is sometimes possible to set T = T −1 when T is simple, it is correct in general to refer to the map that takes the fields F to F˜ as the pullback of T , T ∗. Thus,

T ∗(F) = F˜ (3.81) T ∗(g) = g˜ (3.82)

The mappings of the pullback of T and T take the metric and the field configurations of (g, ?, χ, F, G, J) where dF = 0, dG = J, and G = χ(?F) to a new configuration of (gˆ, ?ˆ, χ˜, F˜, G˜ , J˜) where dF˜ = 0, dG˜ = J˜, and G˜ = χ˜(?ˆF˜). To physically create this mapping we have the choice of two knobs: curve spacetime, or add dielectric media. We focus on the effect of the media here. At a point x ∈ M˜ , we have,

˜ ∗ Gx = T (GT (x)) (3.83) ∗ = T (χT (x) ◦ ?T (x) ◦ FT (x)) (3.84) where we have used Eq. 3.65 in the second line. We also have,

˜ ∗ Gx = χ˜ x ◦ ?ˆx ◦ T (FT (x)) (3.85) from the image constitutive relation, G˜ = χ˜(?ˆF˜). The right hand side of Eq. 3.84 and Eq. 3.85 must be equal to fulfill the requirements of our mappings. 2 ˜ Consider the action of the mapping on a bivector, Vx ∈ Tx (M),

∗ χT (x) ◦ ?T (x) ◦ FT (x) ◦ dT (Vx) = [χ˜ x ◦ ?ˆx ◦ T (FT (x))](Vx) (3.86)

µ Let Λν be the Jacobian matrix of T , then we can re-write the above equation in component form as,

µν σρ α β λκ ξζ γδ σ ρ λκ χαβ|T (x) ?µν |T (x)Fσρ|T (x)(ΛλΛκ)|xVx = (˜χλκ?ˆξζ )|xFσρ|T (x)(Λγ Λδ )|xVx (3.87)

46 λκ Cancelling Fσρ|T (x) and V |x,

µν σρ α β ξζ γδ σ ρ χαβ|T (x) ?µν |T (x)(ΛλΛκ)|x = (˜χλκ?ˆξζ )|x(Λγ Λδ )|x (3.88)

Moving the Lorentz transformations to the right hand side,

µν σρ ξζ γδ σ ρ −1 π −1 θ χαβ|T (x) ?µν |T (x) = (˜χλκ?ˆξζ )|x(Λγ Λδ )|x((Λ )σ(Λ )ρ)|x (3.89)

σρ Multiplying both sides by ?µν|T (x),

µν σρ τη ξζ γδ σ ρ −1 π −1 θ τη χαβ|T (x) ?µν |T (x) ?σρ |T (x) = (˜χλκ?ˆξζ )|x(Λγ Λδ )|x((Λ )σ(Λ )ρ)|x ?πθ |T (x) (3.90)

Using the fact that, ?? F = −F, we have,

µν ξζ γδ σ ρ −1 π −1 θ τη χαβ|T (x) = (−1)(˜χλκ?ˆξζ )|x(Λγ Λδ )|x((Λ )σ(Λ )ρ)|x ?πθ |T (x) (3.91)

The most general χ has 36 independent components but the metric only has 10 components. Therefore not every dielectric has a analogue as a spacetime metric. It T must have  = µ and γ2 = γ1 [42,43,59]. This now only leaves nine free parameters so it is only possible to determine the metric up to an overall scale factor.

3.4 Atom-light interaction through EIT

We have shown in the previous section how the susceptibility of a medium influences the null geodesics like a curved spacetime metric. Our experiment will explore the domain of atom-light interactions where the atom couples to the vacuum modes in a way that creates electromagnetically induced transparency (EIT) conditions [61]. EIT produces very slow group velocities of light in partnership with atoms that are dynamically moving through their trapping potential. Environments like these provide opportunities to explore gravitational analogues. [49,62–65] Our system consists of a single Rb87 atom interacting with electromagnetic fields. Rubidium is hydrogen-like, having a single valence electron which oscillates like a mass on a spring around the heavy atomic nucleus in the presence of electromagnetic fields. The orbital angular momentum of the electron couples its intrinsic spin to the field to form the discrete levels known as the Fine structure, and it couples to

47 the spin of the nucleus to form the hyperfine levels. Since the electron is classified 2 ˆ 4 ˆ as 5 S1/2, it has spin S = 1/2 , the nucleus has spin I = 3/2, and orbital angular momentum Lˆ = 0, then Fˆ = Jˆ + ˆI = 2 or 1 in the ground state. Of which, we work with the smaller of the two. We couple this manifold to the electronically excited, 2 ˆ ˆ 5 P3/2 F = 0, L = 1, state. With a small (compared to the hyperfine energies) applied magnetic field, the degenerate ground state breaks into three, denoted by

differing mF numbers, 1, 0, −1 and the excited state remains unchanged. We label the excited state as |00i, and the mF= 1, 0, −1 states as |1i, |0i, |−1i. We apply optical pumping light along the zˆ axis (which we also pick as the quantization axis) 2 2 close to the 5 S1/2 → 5 P3/2 transition. The magnetic field is oriented such that it

remains in the x, z plane and is parameterized by an angle, θB, such that,

B = B(sin θBex + cos θBez) (3.92)

and B ∼ 60 mG. We can formulate the Hamiltonian operator that describes the time evolution of the light-field-atom system as,

ˆ ˆ ˆ ˆ ˆ H = H0 + H1064 + HB + Hint (3.93)

ˆ where H0 is the energy of the unperturbed, degenerate, atomic levels in vacuum,

  ~ω0 0 0 0      0 ~ω2 0 0  Hˆ =   (3.94) 0    0 0 ~ω2 0    0 0 0 ~ω2

ˆ where we have ordered the basis for the bare atomic states in the order of, H0 = ˆ 0 (H0)ij where i and j index over |0 i, |1i, |0i, |−1i. ~ω0 is energy of the excited, 2 2 5 P3/2 F = 0 state, and ~ω2 is energy of the ground states in the,5 S1/2 F = 1 manifold. 4I am using boldface, E to denote a 3-vector, a hat, Hˆ to denote an operator, and both, Fˆ to denote an operator that possesses components.

48 3.4.1 Magnetic couplings and the Zeeman shift

Adding in the effect of the external magnetic field we write down the energy due to the total atomic magnetic moment.

ˆ HB = µˆ B · B (3.95) µB = (gsSˆ + gLLˆ + gIˆI) · B (3.96) ~ where, µ = e~ is the Bohr magneton, g , g , g are respectively the electron B 2me s L F spin, electron orbital, and nuclear g-factors that capture any modifications to the respective magnetic dipole moments due to relativistic effects and the slightly bumpy and misshapen structure of the atom. If the energy shift from the external magnetic field is small compared to the hyperfine splittings5, then we are in the linear, weak Zeeman regime and can model the energy as,

ˆ µB HB ∼ gF Fˆ · B (3.97) ~ µB ˆ ˆ = gF B(Fx sin θB + Fz cos θB) (3.98) ~   0 0 0 0  √    0 0 2 cos θB sin θB 0  = B   (3.99) ~   0 sin θB 0 sin θB   √  0 0 sin θB − 2 cos θB where, B0 = µB√gF B , and, ~ 2     0 0 0 0 0 0 0 0         ~ 0 0 1 0 0 1 0 0  Fˆ = √   Fˆ =   (3.100) x   z ~   2 0 1 0 1 0 0 0 0      0 0 1 0 0 0 0 −1

  0 0 0 0     ~ 0 0 1 0 Fˆ = √   (3.101) y   2i 0 −1 0 1   0 0 −1 0

51/105 in the ground state manifold, 1/104 in the excited state

49 from embedding the angular momentum matrices in our higher dimensional Hilbert space.

3.4.2 The electric dipole interaction

We apply an optical pumping electric field which is a co-linear running wave forwards and backwards across the atoms plus a field from the vacuum modes of light with an electric field of,

p X v X E = E cos(ωpt − kp · r) + E cos(ωvt + kv · r) (3.102) kp kv where p indexes the optical pumping, v the vacuum. Since the wavelength of light, λ, is much longer than the size of the atom, cos(k · r) ∼ 1, sin(k · r) ∼ 0. This simplifies the field to,

p X h i E = E cos(ωpt) cos(kp · r) − sin(ωpt) sin(kp · r) kp v X   + E cos(ωvt) cos(kv · r) − sin(ωvt) sin(kv · r) (3.103) kv p v ∼ E cos(ωpt) + E cos(ωvt) (3.104)

This electromagnetic wave shakes the electron by adding the term to the Hamiltonian that is the energy associated with the alignment of the electric field and the atomic dipole moment with a separation of r,

ˆ ˆ Hint = −d · E (3.105) where dˆ = −|e|ˆr is the atom’s dipole moment operator, and e is the unit of electric charge. Due to the general rotational symmetry of the potential formed by the nucleus, we will profit by expanding r in terms of the spherical basis. The spherical basis unit vectors are defined in terms of the Cartesian basis vectors (ex, ey, ez) as [66],

1 e1 = −√ (ex + iey) (3.106) 2 ∗ = −(e−1) (3.107)

50 1 e−1 = √ (ex − iey) (3.108) 2 ∗ = −(e1) (3.109)

e0 = ez (3.110) ∗ = (e0) (3.111)

Similarly for the components of a Cartesian vector A such that, A = Axex + Ayey +

Azez, then the components of A in the spherical basis are,

1 A1 = −√ (Ax + iAy) (3.112) 2 1 A−1 = √ (Ax − iAy) (3.113) 2

A0 = Az (3.114)

with Aq ≡ eq · A, and,

X q A = (−1) Aqe−q (3.115) q X ∗ = Aq(eq) (3.116) q with q ∈ {1, 0, −1}. The above transformations were defined in part because we must be careful in defining complex vectors so that their dot product evaluates to a real number. Now explicitly, the dot product is,

X q A · B = (−1) Aqe−q · B (3.117) q X q = (−1) AqB−q (3.118) q X ∗ = Aq(Bq) (3.119) q

We can also re-write the components of the position operator as,

s 4π r = r Y q(θ, φ) (3.120) q 3 1 foreshadowing the connection between the dipole matrix elements and the spherical

51 harmonics. Having defined the spherical basis, the respective amplitude of the p P p polarization components of the electric field, E = q Eq eq, are written in the spherical basis and depend on the orientation of the magnetic field, θB. They are given by,

p p E−1 = E cos θB(1 + sgn(cos θB))/2 (3.121) p p E1 = E cos θB(1 − sgn(cos θB))/2 (3.122) p p E0 = E | sin θB| (3.123) v v E−1 = E cos θB(1 + sgn(cos θB))/2 (3.124) v v E1 = E cos θB(1 − sgn(cos θB))/2 (3.125) v v E0 = E | sin θB| (3.126)

p p + − where E−1(E1 ) is otherwise known as the component of the field that is σ (σ ) polarized. ˆ ˆ 0 0 The matrix elements of Hint are given by, hF mF |Hint|F mF i which describe the 0 0 overlap integral between the wave functions of the initial state, |F mF i, and the

final state, |F mF i, mediated by a spin-1 photon. We define the Rabi frequency by,

ˆ p 0 0 p hF mF |d · E |F mF i Ωm m0 ≡ − (3.127) F F ~ −1 X p 0 0 = Eq hF mF |dq|F mF i (3.128) ~ q and,

ˆ v 0 0 v hF mF |d · E |F mF i Ωm m0 ≡ − (3.129) F F ~ −1 X v 0 0 = Eq hF mF |dq|F mF i (3.130) ~ q where p refers to the component from the optical pumping fields, and v the component from the vacuum fields. To further simplify this matrix element, we make use of the Wigner-Eckart theorem which concretizes the concept that angular momentum is conserved in a transition [66]. Suppose two angular momentum eigenstates, |αjmi and |α0j0m0i are coupled by an irreducible tensor operator, T(k),

52 k with components, Tq , and α are any number of additional quantum numbers that do not depend on the angular momentum that are needed to define the state. We can split this transition into a part that contains the specifics of the transition, and another purely geometric part that arises from the spherical symmetry of the atom [67–69],

(k) 0 0 0 2k (k) 0 0 0 0 hαjm|Tq |α j m i = (−1) hαj||T ||α j ihjm|j m ; kqi (3.131) where the coefficients, hαj||T(k)||α0j0i do not depend on m, m0, or, q, and are called the reduced matrix elements. The coefficient on the right side is the Clebsch-Gordan coefficient which is defined through,

X 0 0 0 0 |j1j2; jmi = |j1m1; j2m2ihj1m1; j2m2|j1j2; jmi (3.132) 0 0 j1j2m1m2 where, |j1m1; j2m2i ≡ |j1m1i|j2m2i. We can also express the Clebsch-Gordan coefficient more conveniently by,

√   2j3 + 1 j1 j2 j3 hj1m1; j2m2|j3−m3i =   (3.133) j1−j2−m3 (−1) m1 m2 m3

  j1 j2 j3 where   is the Wigner 3 − j symbol. m1 m2 m3 Even though the matrix elements we are interested in are in terms of the total atomic angular momentum, F , the Wigner-Eckart theorem still applies and we can write,

0 0 0 0 0 hF mF |dq|F mF i = hF ||d||F ihF mF |F mF ; 1qi (3.134) s 0 F 0−F +m0 −m 2F + 1 0 0 = hF ||d||F i(−1) F F hF m |F m ; 1−qi 2F 0 + 1 F F (3.135) where q = m − m0. With the electric dipole approximation, the laser only interacts with the orbital angular momentum, Lˆ, of the total electron angular momentum,

Jˆ = Lˆ + Sˆ, not the nuclear state |ImI i. We expand out F to include explicitly the

53 dependence on quantum number I by using the decomposition,

 0  0 q j j k  (k) 0 j +j1+k+j2 0 1 1 (k) 0 hj||T ||j i = δ 0 (−1) (2j + 1)(2j + 1) hj ||T ||j i j2j2 1 0 1 1 j j j2 (3.136) to simplify,

hF ||d||F 0i ≡ hJIF ||d||J 0I0F 0i (3.137)

 0  0 q  JJ 1 = hJ||d||J 0i(−1)F +J+1+I (2F 0 + 1)(2J + 1) (3.138) F 0 FI

   JJ 0 1 where is the Wigner 6 − j symbol defined by, F 0 FI

  j1+j2+j3+j j1 j2 j12 (−1) ≡ q hj1j23; jm|j12j3; jmi (3.139) j3 j j23 (2j12 + 1)(2j23 + 1)

Wigner 6 − j symbols are used as a notationally convenient way to keep track of coupling three angular momenta together [66,67,70]. Using the empirically measured Einstein’s A coefficient, or the linewidth, Γ, we can solve for the matrix element as [66,71],

Γ = A (3.140) 3 ω0 2 = 3 |hg|d|ei| (3.141) 3π0~c 3 ω0 2Jg + 1 2 = 3 |hJg||d||Jei| (3.142) 3π0~c 2Je + 1 rearranging, 3 ! 3π0~c 2Je + 1 2 Γ 3 = |hJg||d||Jei| (3.143) ω0 2Jg + 1 Now we can fully evaluate the matrix elements of the electric dipole interaction Hamiltonian in terms of well-defined Rabi frequencies which in turn only depends on the excited state lifetime, Γ = Ag←e = 2π∆f = 1/τ, and the laser field and

54 polarization.

p 1 X p 0 0 Ω 0 = E hF m |d |F m i (3.144) mF m q F q F F ~ q s 1 0 0 2F + 1 X 0 F −F +mF −mF p 0 0 = hF ||d||F i(−1) 0 Eq hF mF |F mF ; 1−qi ~ 2F + 1 q (3.145)

 0  1 0 q  JJ 1 = hJ||d||J 0i(−1)F +J+1+I (2F 0 + 1)(2J + 1) ~ F 0 FI s 0 0 2F + 1 X F −F +mF −mF p 0 0 × (−1) 0 Eq hF mF |F mF ; 1−qi (3.146) 2F + 1 q v   u 3 0 ! 0 1 u 3π0~c 2J + 1 F 0+J+1+I q  JJ 1 = tΓ (−1) (2F 0 + 1)(2J + 1) 3 0 ~ ω0 2J + 1 F FI s 0 0 2F + 1 X F −F +mF −mF p 0 0 × (−1) 0 Eq hF mF |F mF ; 1−qi (3.147) 2F + 1 q v   u 3 0 ! 0 1 u 3π0~c 2J + 1 F 0+J+1+I q  JJ 1 = tΓ (−1) (2F 0 + 1)(2J + 1) 3 0 ~ ω0 2J + 1 F FI √   s 0 0 0 0 2F + 1 2F + 1 X F 1 F F −F +mF −mF p × (−1) 0 Eq   0 F −1+mF 0 2F + 1 (−1) q mF −q −mF (3.148) and similarly for Ωv. 0 0 0 For our problem, the excited state has F = 0,F = 1, mF = 0,I = I = 3/2, j = 1/2, j0 = 3/2 which yields,

v u 3 ! p 1 u 3π0~c 2(3/2) + 1 0+1/2+1+3/2q Ω 0 = tΓ (−1) (2(0) + 1)(2(1/2) + 1) mF m 3 F ~ ω0 2(1/2) + 1   v q u   1/2 3/2 1  u2(1) + 1 2(0) + 1 X 1 1 0 0−1+0−mF t p × (−1) 1−1+0 Eq    0 1 3/2 2(0) + 1 (−1) q mF −q 0 (3.149) s     −1 36π c3 1/2 3/2 1  1 1 0 0~ −1−mF X p = Γ 3 (−1) Eq   ~ ω0  0 1 3/2 q mF −q 0 (3.150)

55   −1s 3π c3 1 1 0 0~ −mF X p = Γ 3 (−1) Eq   (3.151) ~ ω0 q mF −q 0 1 s 3π c3 1 = Γ 0~ E p √ (3.152) 3 mF ~ ω0 3 and finally,

p p Ω10 = E1 K (3.153) p p Ω00 = E0 K (3.154) p p Ω−10 = E−1K (3.155) where, s 3 1 3Γπ0~c K = √ 3 (3.156) ~ 3 ω0 With the Rabi frequency defined, the electric dipole Hamiltonian is,

  0 c.c. c.c. c.c.    p v   E1 cos(ωpt) + E1 cos(ωvt) 0 0 0  Hˆ = K   (3.157) int ~  p v   E0 cos(ωpt) + E0 cos(ωvt) 0 0 0   p v  E−1 cos(ωpt) + E−1 cos(ωvt) 0 0 0

3.4.2.1 Rotating wave approximation

We now apply the rotating wave approximation which we will do by only considering ˆ the effect from H0. We expand the cosines as exponentials, and transform to the ˆ rotating wave basis by multiplying Hint by,

ˆ ˆ ˆ ˆ † HRW = UHintU (3.158) where,

d i |ψ(t)i = − H|ψ(t)i (3.159) dt ~ |ψ(t)i = e−iHt/~|ψ(0)i (3.160)

56 so,

ˆ Uˆ = e−iH0(t)t/~ (3.161)   e−iω0t 0 0 0    −iω2t   0 e 0 0  =   (3.162)  −iω2t   0 0 e 0    0 0 0 e−iω2t which gives non-zero matrix elements of,

1   it(ω0−ω2) p  2itωp  −itωp v  2itωv  −itωv (Hˆ ) = e Ω 0 1 + e e + Ω 0 1 + e e (3.163) RW 10 2 10 10 1   it(ω0−ω2) p  2itωp  −itωp v  2itωv  −itωv (Hˆ ) = e Ω 0 1 + e e + Ω 0 1 + e e (3.164) RW 20 2 00 00 1   it(ω0−ω2) p  2itωp  −itωp v  2itωv  −itωv (Hˆ ) = e Ω 0 1 + e e + Ω 0 1 + e e RW 30 2 −10 −10 (3.165) + c.c. we apply the rotating wave approximation and drop the fast moving terms ∼ e2itω which average away to zero.

1 it(ω0−ω2)  p −itωp v −itωv  (Hˆ ) = e Ω 0 e + Ω 0 e (3.166) RW 10 2 10 10 1 it(ω0−ω2)  p −itωp v −itωv  (Hˆ ) = e Ω 0 e + Ω 0 e (3.167) RW 20 2 00 00 1 it(ω0−ω2)  p −itωp v −itωv  (Hˆ ) = e Ω 0 e + Ω 0 e (3.168) RW 30 2 −10 −10 + c.c.

We transform back to obtain,

1  p −itωp v −itωv  (Hˆ ) = Ω 0 e + Ω 0 e (3.169) RW 10 2 10 10

1  p −itωp v −itωv  (Hˆ ) = Ω 0 e + Ω 0 e (3.170) RW 20 2 00 00

1  p −itωp v −itωv  (Hˆ ) = Ω 0 e + Ω 0 e (3.171) RW 30 2 −10 −10 + c.c.

57 Excluding a small effect from the tensor light shift, the EIT Hamiltonian is then [72],

  ω c.c. c.c. c.c. 0 √  1 0 0   Ω 0 (ω + B 2 cos θ ) B sin θ 0  Hˆ =  2 10 2 B B  , (3.172) ~  1 0 0   Ω 0 B sin θ ω B sin θ   2 00 B 2 √ B   1 0 0  0 2 Ω−10 0 B sin θB (ω2 − B 2 cos θB) where,

 p −itωp v −itωv  Ω100 = K E1 e + E1 e (3.173)

 p −itωp v −itωv  Ω000 = K E0 e + E0 e (3.174)

 p −itωp v −itωv  Ω−100 = K E−1e + E−1e (3.175)

3.4.3 Scalar and vector light shifts from far-detuned light

We can explicitly include the effect from the tensor light shift from the far off- resonant light, though the exact couplings are not known on a site-by-site basis. The atomic energy-level shift from second order perturbation theory with the rotating wave approximation is,

ˆ 2 1 X |hi|d · E(x)|ji| ∆Ei = (3.176) 4 j ~∆j where ∆j is the detuning from the excited state. Taking into account different initial and final states ,we have the light-shift operator,

ˆ† ∗ ˆ ˆ 1 X |jihj|d · E (x)|lihl|d · E(x)|iihi| Uij = (3.177) 4 l ~∆l which has a simple form in the limit of large detuning from the excited state hyperfine splittings [73,74].

ˆ U = U(x)1ˆ + gF DFSBeff(x) · Fˆ (3.178) where 1 is the identity operator and,

2 U(x) = U0|(x)| (3.179)

58 i ∗ Beff(x) = U0 [ (x) × (x)] (3.180) 2~ ! ~Γ2 I U0 = (3.181) 12∆avg IS  −1 1 1 ∆avg =  +  (3.182) 2∆1/2 ∆3/2   ∆3/2 − ∆1/2 DFS =   (3.183)  ∆3/2  2 + ∆1/2 where IS is the saturation intensity, and  is a dimensionless local polarization vector. For light at 1064 nm, we calculate the relative size of the effective magnetic

field to the scalar potential, DFS, to be ∼ 1/20. The 1064 nm confining potential is formed by four beams of strongly disordered, multimode light. Though the axis of the beams are coplanar, the light is tightly focused by 0.4 NA microscope objectives and thus has significant components in all 3 spatial directions. Further, the mode distribution is random, and hence the orientation of  is random as well. The Hamiltonian contribution due to the light in matrix form looks like,

 √  U(x) 2 0 0 0  gF DFS~ √   U(x) 2  gF DFS  0 + Beffz(x) Beffx(x) − iBeffy(x) 0  √ ~  gF DFS~ √   U(x) 2  2  0 Beffx(x) + iBeffy(x) Beffx(x) − iBeffy(x)  gF DFS~ √   U(x) 2  0 0 Beffx(x) + iBeffy(x) − Beffz(x) gF DFS~ (3.184) ˆ and we will write as, (H1064(x))ij.

3.4.4 Density matrix elements

To determine the evolution of the system, we solve for the density matrix. If this was a problem of a unitary processes, we would examine the time propagator of the Hamiltonian, or the quantum Liouville equation,

d ρˆ(t) = −i[Hˆ (t), ρˆ(t)] (3.185) dt

However, since this problem fundamentally involves dissipation of energy from the atoms into the vacuum through spontaneous emission, we must begin by using an

59 equation that takes into consideration the coupling to environmental degrees of freedom [75]. Formally we would consider solving for the equations of motion for the system under study by tracing over the degrees of freedom of the bath as the following, d ρˆ (t) = −itr [Hˆ (t), ρˆ(t)] (3.186) dt S B where S and B denotes "system", and "bath" respectively. If there exists a dynamical map, V (t), that takes,

V (t)ˆρS(0) =ρ ˆS(0) → ρˆS(t) (3.187) then, n ˆ ˆ † o V (t)ˆρS(0) ≡ trB U(t, 0)[ˆρS(0) ⊗ ρˆB]U (t, 0) (3.188)

For many distinct t, we have a one parameter family of {V (t)|t ≥ 0} of dynamical maps with V (0) the identity map. This family describes the entire future time evolution of the open system. However, if we consider that the stochastic fluctuations of the Bath quickly decohere from the influence of the System, like,

V (t1)V (t2) = V (t1 + t2), t1, t2 ≥ 0 (3.189) then we can model this system and bath interaction as a Markov process which in general is a process that rapidly forgets its past history. The Lindblad equation describes a first order differential equation for the reduced density matrix of the open system [75],

dρˆ (t) S = L[ˆρ (t)] (3.190) dt S which is often split into two parts,

i ˆ ˆ L[ˆρS(t)] = − [H, ρˆS(t)] + D(ˆρS) (3.191) ~ ˆ where D(ˆρS) is the dissipator.

N 2−1 ! ˆ X ˆ ˆ† 1 ˆ† ˆ 1 ˆ† ˆ D(ˆρS ) = γk AkρˆS(t)Ak − AkAkρˆS(t) − ρˆS(t)AkAk (3.192) k=1 2 2 and {|ki} is an orthonormal basis for the bath, and N is the dimension of the

60 Hilbert space. For a System that is coupled to a thermal Bath with continuous modes of exchange of energy with the System [75],

3 ! ˆ X 4ω ˆ ˆ† 1 n ˆ† ˆ o D(ˆρS ) = 3 (1 + N(ω)) A(ω)ˆρSA (ω) − A (ω)A(ω), ρˆS ω>0 3~c 2 3 ! X 4ω ˆ ˆ† 1 n ˆ† ˆ o + 3 (N(ω)) A(ω)ˆρSA (ω) − A (ω)A(ω), ρˆS (3.193) ω>0 3~c 2

ˆ ˆ† where the Lindblad operators, Ai(ω)(Ai (ω)) lower (raise) the atomic energy by ˆ the amount ~ω, and Ai(ω) describe spontaneous and thermally induced emission 3 3 ˆ† processes which occur with the rate 4ω (1 + N(ω))/3~c while Ai (ω) describe thermally induced absorption processes taking place with the rate 4ω3N(ω)/3~c3. For a two-level system,

1 1 ! Dˆ(ρ ) = γ (N + 1) σˆ ρ(t)ˆσ − σˆ σˆ ρˆ(t) − ρˆ(t)ˆσ σˆ S 0 − + 2 + − 2 + − 1 1 ! + γ N σˆ ρˆ(t)ˆσ − σˆ σˆ ρˆ(t) − ρˆ(t)ˆσ σˆ (3.194) 0 + − 2 − + 2 − +

3 ˆ 2 4ω0 |d| with, γ0 = 3 . Where, σˆ+ = |eihg| and σˆ− = |gihe| . In our four-level system of 3~c EIT we neglect the thermal excitation process, and linearly add three such Lindblad operators that account for each of the transitions between the excited state and the ground states. This becomes,

γ001 D(ρ ) = [2ˆσ 0 ρσˆ 0 − σˆ 0 0 ρ − ρσˆ 0 0 ] S 2 10 0 1 0 0 0 0 γ000 + [2ˆσ 0 ρσˆ 0 − σˆ 0 0 ρ − ρσˆ 0 0 ] 2 00 0 0 0 0 0 0 γ00−1 + [2ˆσ 0 ρσˆ 0 − σˆ 0 0 ρ − ρσˆ 0 0 ] (3.195) 2 −10 0 −1 0 0 0 0 where σij = |iihj|, the atomic projection operator. Or,

 γ001 h 0 0 0 0 i Dˆ(ρ ) = 2|1iρ 0 0 h1| − |0 ih0 |ρ − ρ|0 ih0 | S 2 0 0 γ000 h 0 0 0 0 i + 2|0iρ 0 0 h0| − |0 ih0 |ρ − ρ|0 ih0 | 2 0 0  γ00−1 h 0 0 0 0 i + 2|−1iρ 0 0 h−1| − |0 ih0 |ρ − ρ|0 ih0 | (3.196) 2 0 0

61 rearranging,

 = γ001|1iρ0000 h1| + γ000|0iρ0000 h0| + γ00−1|−1iρ0000 h−1|

1 0 0 − (γ 0 + γ 0 + γ 0 ) ρ|0 ih0 | 2 0 1 0 0 0 −1 # 1 0 0 − (γ 0 + γ 0 + γ 0 ) |0 ih0 |ρ (3.197) 2 0 1 0 0 0 −1 writing this explicitly in matrix form by taking matrix elements formed from the ˆ basis elements as hi|D(ρS )|ji, we have,

 1 1 1  0 0 0 0 0 −γρ0 0 − 2 γρ0 1 − 2 γρ0 0 − 2 γρ0 −1  1   − γρ 0 γ 0 ρ 0 0 0 0  (Dˆ(ρ )) =  2 10 0 1 0 0  (3.198) S ij  1   0 0 0 0   − 2 γρ00 0 γ0 0ρ0 0 0   1  0 0 0 0 − 2 γρ−10 0 0 γ0 −1ρ0 0 where, γ = γ001 + γ000 + γ00−1. We can then solve this system of equations by looking for the steady state equation and setting ρ˙ = 0. Let ρij be the solutions to this set of equations. ˆ 0 0 ˆ The polarization for an atom is given by P = hdi = hF mF |d|F mF i which is the simple dipole moment operator. We calculate the electric susceptibility

∂Pi by differentiating, χij = v , which is simply related to the permittivity tensor, ∂Ej  = (1 + χ)0 Therefore we must calculate the polarization of our EIT system by

P = tr[ˆρijPˆ jk] .

ˆ 0 0 ˆ Pq=−1 =ρ ˆ001h1|d−1|0 i +ρ ˆ100 h0 |d−1|1i (3.199) ˆ 0 0 ˆ Pq=0 =ρ ˆ000h0|d0|0 i +ρ ˆ000 h0 |d0|0i (3.200) ˆ 0 0 ˆ Pq=1 =ρ ˆ00−1h−1|d1|0 i +ρ ˆ−100 h0 |d1|−1i (3.201) which simplifies by using K defined above in Eq. 3.156 to,

Pq=1 = K(ˆρ001 +ρ ˆ100 ) (3.202)

Pq=0 = K(ˆρ000 +ρ ˆ000 ) (3.203)

Pq=−1 = K(ˆρ00−1 +ρ ˆ−100 ) (3.204)

62 3.5 Light propagation and topology

We can further illustrate the utility and physicality of complex coordinates began in Sec. 2.6.1 by describing light propagation that reflects off of material boundaries. Let us consider a ray encountering a dielectric boundary at which it reflects [76]. To proceed with ray tracing, the optician unfolds the ray upon a curved surface. This can be captured by geometric flow used by topologists, combined with the mathematics of topology or manifold classification. When we extend these ideas to optically non-reciprocal systems, we naturally introduce features that look like gauge fields or manifolds with both metrics and auxiliary connections. In optics, one has two traditional means to attack any problem - geometric optics, tracing the dominant rays of light representing the normal to optical phase- fronts, and physical optics, in which we solve the wave equation of light in the medium of interest. The former is usually thought of as approximate, and is in many ways similar to the approximation classical mechanics makes of quantum mechanics for particles. The easiest connection between these two is the Eikonal approximation made in either, in which geometric ray paths, or particle trajectories, are quantified by a path-dependent quantity - the transit time of a ray in optics, or the action of a particle in non-relativistic quantum mechanics. In optics, the Eikonal approximation consists of taking the solutions to the wave equation as,

ψ ≈ eA(r)+ik(S(r)−ct) (3.205) where we assume that the wavevector is along the z direction, and A and S are real functions denoting the amplitude and the optical path length of the wave. Substituting it into the wave equation,

1 ∂2ψ n2 ∂2ψ ∇2ψ − = ∇2ψ − = 0 (3.206) u2 ∂t2 c2 ∂t2 we get,

ik[2∇A · ∇S + ∇2S]ψ + [∇2A + (∇A)2 − k2(∇S)2 + n2k2]ψ = 0 (3.207)

63 Since A and S are real, the quantities in the brackets can be set to zero separately,

∇2A + (∇A)2 + k(n2 − (∇S)2) = 0 (3.208) ∇2 + 2∇A · ∇ · S = 0 (3.209)

From the assumption of geometric optics, we assume that A changes slowly compared to a wavelength. Therefore we have,

n2 = (∇S)2 (3.210)

This implies that waves are locally planar, ie. has a well-defined wavefront. The Eikonal approximation in quantum field theory captures the large momentum, small angle scattering limit by keeping the action equal to the scalar, (∇S)2 = 2m(E −V ).

3.5.1 At boundaries, rays are ramified

Both of the above topics become more subtle when either light or particles can reflect or diffract at boundaries where the optical index or potential energy suddenly changes. This sudden change leads to multiple paths that satisfy the equations of motion and force the Eikonal to be no longer single-valued. One resolution of this problem is to ramify space at these boundaries, splitting it into two or more manifolds to keep track of multiple trajectories [77–79]. The trick keeps the Eikonal single-valued, and prevents ray-paths from developing caustics where they cross and create a multi-valued Eikonal. Many optical design software programs use this to make practical calculations, and it is what many optical engineers choose to do when they unfold a lens or mirror system on paper. In the early years of quantum mechanics, it is also what Einstein, Brillouin and Keller (EBK) did to modify Bohr-Sommerfeld quantization to rectify it with non- relativistic higher-dimensional harmonic oscillators, square-well reflection, and the relativistic fine structure of hydrogen [80]. Consider a light wave incident on a transmissive diffraction grating. Treating the grating as thin, we would often draw an incident ray as split into two or more parts past the grating: a zeroth order transmitted beam, and one or more higher-order diffracted beams with altered direction or k-vector. In the ramified picture, we would draw each order on a separate leaf of paper. The same is done at a partially

64 reflective dielectric boundary, introducing another leaf for each split of the ray into parts as shown in Fig. 3.1. However, the Eikonal can still cross and be multi-valued in an arbitrary system; perfect reflection from a curved surface is one of many examples that show this is true. In fact, an image point in an optical system is a zero-dimensional caustic surface.

Figure 3.1: Folding, or manifold ramification in optics. For optical rays incident on a totally internally reflecting dielectric boundary (left), or refracting (middle), or diffracting from a grating (like the acoustic grating formed by a sound wave in an acousto-optic medium), one can form a bifurcated, or ramified, manifold on which the ray trace is simpler. In a physical optics viewpoint, this eliminates the caustics (places parallel rays would cross one another), and preserves the single-valuedness of the the transit time taken between two points, or the Eikonal.

The effect of driving a medium’s susceptibility for optical ray paths transiting through the driven medium can be understood with an Eikonal approximation. The optical index-of-refraction, the transit time for light, and the tendency for light to split at diffraction zones depends on the local value of the drive intensity. Since the optical wave speed is determined by the index, the diffraction effect can be incorporated by a smoothly varying effective local metric on the ramified manifold without media. The light will propagate along lines of minimal path length according to this metric. For any dielectric material that has boundaries, the unfolding procedure must result in a curved manifold even without any smooth variations in the index of refraction [76]. With a flat index profile, rays propagate as straight lines on each curved, immersed surface6. With the addition of a smooth index modulation with spectral components much larger than a wavelength, perhaps from a long- wavelength sound field, the rays curve to follow paths of least time by Fermat’s

6An immersion is a differentiable function between differentiable manifolds whose derivative is one-to-one. A Klein bottle can be immersed in R3. A figure 8 loop can be immersed in R2. The map t → (t3, t2) from R to R2 has a cusp at t = 0 and so is not an immersion.

65 principle on the spacetime metric. With a grating, diffraction occurs and the ray splits, but the caustics can again be eliminated by attachment of new manifolds. To a geometric optician, this constant unfolding or ramification is annoying, and even a ray trace becomes progressively more difficult to represent on paper as immersed manifolds in three-or-fewer dimensions. But to a topologist, the unfolded ray trace is still useful - it lies on a cover7 of the original optical medium and the attachments build a ramification of the original space. It is natural to ask how and whether such a drawing, or immersion of this space, could or must work. Certainly the paper (a Riemannian8 space) can be higher-dimensional, and Whitney’s embedding theorem [81] says that for any immersion to be a more useful embedding, the “paper” need not be more than twice the dimension of the optical medium. If the “drawing” can be made into an embedding9, we can use elements of differential calculus in order to solve differential wave equations on the ramified manifold.

3.5.1.1 Defining cohomology classes

One can also ask how many different types of such manifolds there are, and which type is defined by Bragg gratings and smooth index variations. For example, a Mach-Zender interferometer requires manifold attachments at both gratings, and ray paths departing one should recombine on the other to describe how an interferometer behaves. Our attachment then requires a reconnection or crossing with the manifold of unrefracted waves from the first grating at the position of the second. We can characterize the Mach-Zender by the number holes it possesses, which is characterized by its cohomology10. After many such diffractions, many such attachments of ray bundles are necessary to describe a network of Mach-Zender interferometers. By extension, if we want to understand the physical optics of that system, the optical modes we are interested

7A covering, C, of a manifold, M, means that there is a continuous map, f, that takes: f : C → M. 8A Riemannian manifold is a manifold equipped with an inner product on its tangent space that is smooth. That is, it has a metric tensor. 9An embedding is a smooth immersion that has a continuous inverse function. Neither a Klein bottle in R3 nor a figure 8 in R2 is an embedding since the inverse at the self-crossings is not single-valued. 10Homology and cohomology are roughly the process of identifying a shape of a topological space by identifying and characterizing its holes and boundary conditions.

66 in should also be defined by cohomology classes of that more complex manifold. After all, they should be the solutions of the physical optics wave-equation, which are harmonic functions on that space. This is precisely the topic of de-Rahm cohomology. If one includes the additional information of the optical k-vector as a position along extra dimensions in the immersion of this manifold, the complete manifold of ray spaces must be formed by many self-intersections corresponding to the closures of these interferometer paths. On each lives a metric describing smooth variations of the refractive index introduced by media. The space we immerse into by Whitney’s theorem must now be four times the number of spatial dimensions our ray trace began in. We can also think of the metric’s smooth variations as inherited from its immersion, and attribute slower light propagation to more curvature in its immersion. Thus the media introduces a bumpy, locally six-dimensional, self-intersecting manifold immersed in some higher-dimensional space, but on which rays are straight and take the shortest transit time. The manifold now represents the classical phase- space of a ray, with both a local position and momentum defined; therefore this is the symplectic manifold11 of classical Hamiltonian mechanics applied to light. The media can be time-dependent so we can also extrude it along a fourth space-time dimension. We want to classify this manifold’s coarse structure to understand if and how different cohomology classes affect a system’s gross dynamics.

3.5.1.2 Images define Cut Loci

Geometric ray paths away from caustics correspond to extrema in the Eikonal (paths of least time), and we can view the process of optical path unfolding as a way to preserve this fact in a system with caustics. In many optics problems, we are primarily interested in cases in which images are formed, where a continuous distribution of rays intersect at something very close to a common point. Above, we defined this point as the caustic of some sharp optical boundary surface, but in geometry, there is another useful way to refer to this image point. For any smooth

11A symplectic manifold is a smooth manifold with a 2-form. The phase space of classical mechanics is an example with the 2-form being the relationship between the conjugate variables, pi, and qi.

67 metric, it is the cut locus12 of the source point. In fact, an imperfect imaging system, or one far from an image condition has a cut locus which is not a point, but a line. If the optical medium is linear at this cut locus point or line, there is no need to ramify the medium to avoid this point of intersection. Consider the introduction of a material boundary surface introduced at this image point. Flipping the direction of one of the rays, to represent the action of a specular reflection at this boundary, and thereby coupling one of these rays back into the other at the image point, gives a closed cycle on the manifold. If that cycle can be smoothly shrunk to a point, the flipped ray belongs to the same sequence of ramifications off of optical boundaries13 as the original ray. If it cannot be shrunk to a point, it represents a different cohomology class of one-dimensional curves in the ramified manifold, and is non-contractible because the sequence of ramification surfaces it followed differs from the first ray. In this latter interpretation, it is important to note that the boundary introduced at our new image caustic does not split the manifold further, but actually glues two parts together. We will return to this when we discuss optical nonlinearity and lasing systems.

3.5.1.3 Cohomology defines discrete modes in physical optics

In a confined optical system, we know we should obtain discrete modes from closed ray paths. But in the ramified picture above, for there to be any chance that a ray path closes, such a gluing is necessary. If it sounds like this violates a tenant of thermodynamics by decreasing the phase-volume along an optical path, one should remember we increased the phase volume, or classical etendue, at ramifications to begin with, so it is not the first apparent violation. A natural question to ask is what happens when the Eikonal approximation is lifted, and how can our cohomology classes can be understood when we demand a full solution to the wave-equation in this medium? In classical photonics, this is the natural way one defines discrete modes as solutions of the optical wave equation subject to boundary conditions precisely at our ramification boundaries. Allowing these discontinuities is a weak, or viscosity solution to the wave equation [82].

12The cut locus of a point on a manifold is the set of all other points for which there are multiple minimizing geodesics. For example, the cut locus on a sphere is the point directly opposite. The cut locus of a point on an infinite cylinder is the line on the opposite side of the point. 13like gratings or TIR events

68 Since solving the wave equation requires differential calculus to be well-defined, we must first answer the question of how to do calculus on topologically non trivial manifolds. This is the essence of de Rahm cohomology.

3.5.1.4 Nonlinear interactions permit ramification and gluing

An under-appreciated fact in classical optics is that each of these ramifications we started with involves a nonlinear interaction of some type. This is apparent by considering the conservation of momentum when rays are diffracted or bent. In nonlinear optical media, the diffraction occurs in three- or four-wave mixing; one light-field forms a grating for the other. At dielectric boundaries for refraction, there is a transfer of momentum to the medium, so the material providing the index must interact with the light through a nonlinearity. For a detector (or a source) of light, the absorption (emission) of energy by the detector (source) cannot be completely linear either, but rather mediated by interaction with some other field, for example the excitation of electrons in a photoelectric medium. Thus the opportunity to split a manifold, or re-glue it where it crosses itself, is related to the nonlinearities present in the medium. In all this discussion, we treated the medium as fixed and therefore the metric, its discontinuities, and the ramification points were considered fixed. However if you consider the acoustic pressure in an acousto-optic modulator as the field that defines the analogue spacetime metric, then we know that this field evolves by its own equation of motion, and its equation could not be independent of the light. The transfer of momentum from the light at a point of diffraction must imply this. Or even more pertinent, let us imagine that we have an atomic gas that is able to move around in a potential as it interacts with a running laser beam. There must be a nonlinearity in the system because of atom-light coupling even if the atoms and the light by themselves behave linearly. Thus treating the metric as a proxy for the optical index as an auxiliary field is a convenience to decouple those problems, similar to ramifying the manifold where it is discontinuous. When we ask if ramifying or gluing should occur at these points, we are really asking something about the dynamics of the other field, and their influence from the field at hand. It is a self-consistency problem between the fields to ask if the manifold should ramify. Further, for the phase-volume to be maintained along any eventually closed path, the contributions from splitting and

69 gluing must balance.

3.5.2 Time delay correlation, oscillation, and lasing

We can drive the dielectric actively to parametrically control its gain which drives it toward some extremum. Often this involves getting the source to lase. This is reminiscent of a geometric flow, and since the dielectric is driven with feedback from its own sensed profile, it can be thought of intrinsically, like Ricci flow14 [83]. To geometric optical engineers the fact that closed ray-paths correspond to cohomology classes can be physically useful. To each such closed curve belongs an optical round-trip transit time that can be measured by its contribution to the weight of a time-delay-correlation (TDC) distribution for light sent out and returned, for example, when profiling a surface with a laser diode. To receive any back-reflected signal in the laser-diode from its own emission, this surface must lie in the cut locus of the diode, as defined by the driven optical index and corresponding metric. If the surface lies at a projective focus, paraxial rays will exist with equal transit times to the surface. For any two paraxial rays in a single image bundle used to form a loop, the transit time is the same and a cluster of weight is obtained at twice that common transit time. A different pair of rays chosen both from a single different bundle contributes a separate transit time cluster. For one ray chosen from each of two bundles and connected together to form a closed path, another cluster of transit times occurs at the sum of transit times along each. Thus the number of peaks in the TDC signal appearing at the sum of two others is a measure of the number of cohomology classes of this manifold.

3.5.2.1 Topological classification by laser and parametric oscillation

The count of cohomology classes must be based on a non-arbitrary decision point about how strong a cluster must be represented in this measurement for it to be counted. Equivalently, we need a binary decision about when the manifold is actually glued. This is about the strength of a nonlinearity that couples two optical rays together at a manifold self-intersection. It is natural to connect this to something like a lasing threshold condition [84,85], and answer in the affirmative

14Ricci flow is the smoothing flow of a metric analogous to heat flow.

70 when there is lasing, and in the negative below threshold. The appearance of sums and differences for transit times also is important for four-wave mixing inside nonlinear media. In either lasing or parametric oscillation, our manifold’s coarse structure, as represented by its cohomology classes, is altered by the gluing necessary to define a lasing mode. In passive optics like the modes of a waveguide, we have no trouble seeing that modes should be discrete, but we typically associate it with boundary conditions at a surface of a structure or cavity. But this too is actually a nonlinearity that couples the medium to the light. A confined optical mode in a loss-less material will propagate inside forever, similar to a laser where a mode’s gain must dominate its losses. So the binary decision on how to discretely count modes again rely on the strength of a nonlinearity. An experimental apparatus probing this topological flow might use a acoustic drive over a crystal towards a narrowing of individual TDC clusters. As the TDC signals become better-defined, it is more likely that any given imaging path represents an extended cavity laser system which can oscillate. Put another way, we use feedback to modify or flow the metric, driving it toward a condition that puts the source laser in its own cut locus in the glued manifold. Sensing the acoustic field and feeding back the time-delayed correction signal implies in a continuum limit that, ∂tg = f(g), where g is the metric and f is some function of the metric defining a filter. Introducing a control, c, of this

filter using the laser-diode feedback can be represented by a parameter in f = fc.

If the filter produced the Ricci-tensor fc = −2Ric(g), for example, this would exactly define Ricci flow [86]. However, the freedom available in the acoustic field is certainly too limited to allow fc to be chosen precisely to implement Ricci flow. However, in Sec. 4.4, we create a very general nonlinear system that might support such topology There is reason to believe a generic algorithm driving TDC signals toward tighter clustering would be related to Ricci flow. Consider two rays joining the same point q of the cut-locus (lying on an external surface) of p (at the emission point of the source laser). One is created by a smooth geodesic ray, pq, joining them, and a second, qrp, created by a sharp refraction or diffraction in the acoustic medium at a well-defined point r. Toponogov’s theorem [87] tells us that the difference in lengths between these is shorter in a medium of more negative curvature. Hence

71 driving TDC clusters narrower by any choice of filter will favor making the Ricci curvature near these paths more negative, or equivalently expand regions of negative curvature, as in Ricci flow [86]. Many forms of intrinsic geometric flow, like the Ricci flow, can be versed as gradient flow problems. Often one defines a target measurement to extremize and drive control parameters toward its minimum. One could define the acoustic waveform as multi-channel wavelet, and dither its component amplitudes at different rates, demodulating their effect on the second moment of the TDC distribution to derive error signals, and thereby driving toward a minimum the TDC width using them. In equilibrium, the error signals should converge to zero. Associating the TDC width (second moment) with an energy functional, one can relate this to doing calculus of variations on it.

Z b E(u) = L(u, ux) dx (3.211) a

The convergence to zero error is similar to the approach toward a fixed point obeying the Euler-Lagrange equation minimizing that functional.

∂L d ∂L − = 0 (3.212) ∂u dt ∂ux

Ricci flow minimizes a functional formed by a volume integral of the scalar curvature of a manifold, plus some gauge-like quantities [88]. The experimental system above minimizes the width of the distribution of geodesics starting and ending at the source laser itself, constrained by the freedom available in the acoustic drive parameters. So they are not the same. But for a positive-curvature region near a surface- imaging focus, the system drives toward a state in which the laser and a point on the surface are like the antipodes of a sphere, and would approach that same state regardless of which of many initial metrics one started from, so it is playing some role of uniformization. With that in mind, perhaps we are in a better position to understand ramification and gluing by nonlinear optical and acousto-optical processes. Geometric flows in general often result in singularities after a finite time. If we look at the feedback process from this perspective, what are the roles of these singularities?

72 3.5.2.2 Lasing thresholds tell us when to ramify and glue

In the experimental system, it is clear that the measure of TDC signals is simple for sufficiently small back-coupled signals, as the optical field returned to the laser from the reflective surface interferes in a simple way with the laser’s field. However, as the strength of the back-coupled field increases by better focus, and a TDC cluster becomes sufficiently narrow, eventually the laser will achieve an injection-lock and lase on the extended mode to the surface and back. This condition is binary, and there exists a threshold condition above which, it will, and below which, it will not. Since the lasing mode changes discretely, so does our TDC optimization signal and the character of our flow problem. In Ricci flow problems, there is also prescription for how to flow past the point where singularities occur based on surgery [89]. One cuts the manifold and continues to flow with the pieces. There is no reason to believe the physical system continues its geometric flow past a singularity in precisely the way surgery has been used with Ricci flow, just as there is no reason to choose the physical system to flow precisely as Ricci flow does in the first place. But the process of geometric flow, sudden change, and continuation is similar, and the continuous flow intervals are punctuated by dynamic lasing transitions. Lasing occurs for well-defined optical modes, and those modes correspond to geometric ray paths. In our unfolded space, those ray paths cannot cross or close without representing a member of its cohomology class. With no stable extended lasing modes, the prescription above gives us no cause to glue, and there can be no nontrivial cohomology classes on the manifold. A single unstable mode caused by driving toward better feedback (geometric flow) gives cause to glue at one self-intersection, and one or more non-trivial cohomologies can result. The prescription to glue when one obtains a lasing threshold is useful because it relates lasing modes to closed cohomology representatives. It would be best from this standpoint if the physical system better separated the flowing from the gluing. This can be arranged if the acoustic waveform parameters are separated into parameters for temporal and spatial frequencies, and amplitudes of drive. After converging to optimum in the first of these, we can consider a flow to have completed; subsequently increasing amplitudes increases the strength of acoustically diffracted fields, and can take the system through a discrete transition. As multiple modes begin to meet the small signal condition for oscillation,

73 multiple paths in an extended laser system may begin to lase simultaneously at different frequencies. These are notoriously difficult laser physics problems, relying on the detailed nonlinear behavior of a lasing medium near saturation. We have a choice in this situation, either to represent the manifold above as glued in cases where the small-signal condition for lasing is met, or to represent the ultimate configuration of the lasing modes. The latter is clearly the choice more directly tied to the TDC as a measurable, since we must accept its behavior which is determined by the lasing mode. It is also possible to synchronize oscillation at a common frequency along many paths. This is a discrete transition to a lasing state consistent with multiple active feedback paths, and is easily detected by the spectral content (temporal coherence), or a sudden jump in intensity or hysteresis with current modulation. A somewhat esoteric arena of laser physics has tried to explain this behavior for a range of systems such as high-power multimode emitters, like broad-stripe lasers [90, 91]. Most of these models resemble the classical Kuramoto model of synchronization in a network, which is a simple extension of the Adler problem of phase-locking (or laser injection-locking) in a pair of real-world oscillators to many. Naturally, the extended cavity mode chooses a well-defined oscillator phase, breaking a symmetry by the nonlinearity in the gain.

3.5.3 Geometric optics and quantum mechanics

3.5.3.1 Geometric pictures of classical optics

At both refractive boundaries and diffractive attachments, it is important to remember that the k-vector changes discretely as the ray is bent, and that the momentum is absorbed from the acoustic field. In the corresponding physical optics picture, where we would solve the optical wave equation, the viscosity solution admits weak discontinuities at diffraction or refraction boundaries. If we kept track of the k-vectors by doubling the dimensionality of manifold and representing diffraction as a continuous redistribution of amplitude at different k-vectors, we could appeal to classical thermodynamics. We could represent patches of the phase-volume’s propagation like a thermodynamic flow, which would help predict when a well-defined power density of optical noise might amplify enough to take over as a lasing state during Kuramoto transitions. However, in this symplectic

74 space when you include both the optical ray position and the angle15, the new manifold ramification is not directly attached to the original. Thus the optical system would be represented by a manifold which is not connected, and the flows of which are harder to define. Which is more useful? The ramified spatial manifold described above, with an added k-space and a correspondence to the symplectic space of classical mechanics? Or the spatial manifold where an Eikonal approximation leads to discontinuous jumps that require discontinuous viscosity solutions to the wave equation? All of these frameworks get periodic use in even the most rigorous analysis of lasing systems, and often multiple pictures are used in the same analysis without checking consistency. The reason is simple; usually the medium the laser operates in is taken as fixed, and the optical nonlinearities are treated as secondary topics, or at best perturbatively. That is, the field propagates on a fixed manifold and metric, which is not solved self-consistently with the lasing behavior. The physical optics treatment, solving the optical wave equation on the original spatial coordinates, would account for the smooth transfer of momentum to and from acoustic modes during diffraction. We could capture the same physics in the ramified space by introducing complex k-vectors to capture the growth or decay of new component waves with distance. These could describe evanescent fields at TIR boundaries found in the viscosity solutions to the physical optics problem or with the propagation through dissipative or gain media where a complex k-vector is natural. So the complexification of k-vectors is seen repeatedly as useful, and it is helpful to consider the ramified spatial manifold above as carrying a single-valued complexified line bundle16 representing the optical ray direction and its growth rate. So, a ramified space with a complex bundle seems of comparable use as the symplectic space if we are not to entirely forgo the Eikonal approach.

3.5.3.2 Polarization of optical variables

Keeping the ramified spatial manifold requires that we maintain a distribution of complex k-vectors and keep track of the discrete phases and k-vector jumps at ramification points since they might affect the overall lasing profiles. We could also work the other way, stretching out the boundaries where diffraction occurs

15k-vector 16A line bundle is a choice of a continuous one-dimensional vector for each point on the manifold.

75 and shrinking the spatial parts where we are sure to keep track of the total phase accumulated between the diffraction and refraction zones. Either of these, real space or k-space, is a specific choice of polarization17, and keeps track of precisely half of the classically conjugate mechanical degrees of freedom. Going between the two requires a symplectic or canonical transformation of geometric ray variables. Importantly, at these dielectric boundaries and diffraction zones, there is a tacit change in language to our optical ray traces. Each ray bundle undergoes a sudden and discrete change in k-vector and optical phase. To calculate the coarse change, we need to have good definition of the k-vector itself. This too is a symplectic or canonical transform of variables that is associated with the ramification boundary. This introduces a natural local symplectic form for the mechanics of the system.

3.5.3.3 Geometric quantization

In geometric quantization of the Hilbert space, the polarization18 comes from projecting a ramified version of the symplectic manifold onto the ramified spatial one described above in a way that no information is lost. Consider, for example, the phases imprinted by diffractive acoustic components defining a series of ramifications on the manifold that form a closed loop. For the optical field, a complex bundle, L, on the ramified manifold, to be single-valued, its sections19 must agree at the self-intersection point. For this to be generally assured, the local symplectic form, ω, can be chosen as the curvature of a connection,∇,20 [92]

ω curv ∇ = (3.213) ~ This imposes an integrality limitation21 on the phase accrued between two self- intersections, like in a Ramsey interferometer, and leads to topological numbers and geometric phases. From geometric quantization, it has another implication; the natural flow of any physical quantity, f, on this ramified manifold is altered by this connection.

17Polarization in geometric quantization refers to the choice of n out of 2n variables of the classical phase space for representation as operators in the Hilbert space. 18in geometric quantization sense 19A section is a continuous inverse of the projection from the bundle to the base space. Here, the base space is the ramified manifold, and the bundle is the field for the ray. 20A connection defines how one may parallel transport. 21called the prequantization condition

76 One associates an operator o(f) with each observable f, offset from its classical form by,

o(f) = −i~∇Xf + f (3.214) where this connection, ∇, is projected along the classical flow, Xf . Because the commutator of two observables, [o(f), o(g)], now includes this effect, the operator algebra forms a deformed version of the classical mechanics Poisson algebra, in that, [o(f), o(g)] = i~o({f, g}) (3.215) This should be familiar from all forms of canonical quantization, geometric quanti- zation in particular. Note that outside of the action of the classical Kuramoto (lasing) transition for a system of oscillators, there is geometric quantum mechanics in the optical system. Even though quantum measurement or state collapse are not evident in this analogy, we can form a meaningful Hilbert space and operator algebra from classical light propagation.

3.5.3.4 EBK quantization

There is another way to see aspects of quantization schemes. By the reasoning above, we are forced to admit our classical ray traces have a tacit local canonical change of variables very near the ramification boundaries. If we smooth our ray-traces by physical optics effects in small regions near the boundaries, we are forced to consider all symplectic manifolds, or the Lagrangian Grassmanian22 (LG), smoothly connected to our ramified spatial one. Since an optical mode corresponds to some closed path in a ray trace, we obtain a path of local symplectomorphisms23 for any path on the manifold, which is a path somewhere in the LG. We can then ascribe a Maslov index24 [93, 94] to it. This Maslov invariant then is a number that keeps track of our ray trace’s discrete caustic phase jumps and k-vector rotations at ramification boundaries.

22The Lagrangian Grassmanian is the set of all polarization subspaces of the symplectic manifold. The Lagrangian is one particular subspace. The name comes from “Lagrange brackets” in classical mechanics. 23An isomorphism in the sympletic manifold. It is a volume preserving transformation of phase-space, or a canonical transformation. 24The Maslov indexes a path of symplectomorphisms, or the orbit of the Hamiltonian vector flow.

77 Therefore we can work with projected orbits on the simpler ramified spatial manifold, provided we keep track of these Maslov indices. This resembles the data necessary to quantize the action in EBK quantization [95], in which the action is a weighted-sum of Maslov indices, added to the principle quantum number, and multiplying Planck’s constant. In this sense, the Maslov indices can be considered an outgrowth of geometric considerations, but in quantization, they offset the quantum number. So it leads one to ask if the quantum number itself is geometric.

3.5.3.5 Geometric interpretation of the principle quantum number

Perhaps we can understand the appearance of the quantum number as an extra index by incorporating an index for navigating the branches of the ramified space similar to the caustic phase shifts the Maslov indices account for. Accordingly, they both change rapidly near the folds and bifurcations of the space. As we noted before, for the optical modes to be defined by nontrivial cohomology, a ray must traverse multiple sheets before closing, and the space cannot continuously bifurcate at boundaries and diffraction zones without eventually regluing. This is important for the underlying optics ideas leading to the conditions for laser oscillation because the nonlinearities in the medium dictate when to glue the manifold. If a lasing mode is to be represented by a closed path on this ramified and glued manifold, the branch-index must return to its original value. Thus we can choose the representation of the branch-index change near a splitting- or gluing-boundary as a rapid local modification of this new index, but it must also be single-valued once the manifold is glued to represent lasing modes. The gain integrated along this closed path (including a factor for each traversal through a ramification boundary) must be unity. Traditionally, lasing is understood to occur when the small signal gain along a closed optical path is equal to one; loosely one can think of the contributions to out-coupled modes of a laser as forming a geometric series of waves amplified after multiple passes, and unit single-pass gain as the condition for the series to diverge. Physically, gain eventually saturates due to depletion from stimulated emission, and is therefore dependent on the strength of the optical field itself. Without incorporating this fact, a lasing state as defined above would be modeled by an optical field with divergent amplitude.

78 3.5.3.6 The one-dimensional laser cavity

Consider briefly a one-dimensional laser cavity consisting of two mirrors. Traditional cavity analysis examines the round trip gain and loss and requires the unity round trip gain to be, [96],

 ωp  g (ω) ≡ r exp α (ω)p − α p − i − i∆β (ω)p (3.216) rt m m 0 c m m = e−iq2π (3.217)

where αm is the gain coefficient from any gain medium with a corresponding phase shift, −i∆βm(ω)pm, and pm is the round trip length of the gain medium. r is the total contribution to the loss from the mirror reflections, and p is the round trip path length. Then we simply require the condition for lasing to be when both the magnitude of the gain to be 1, and the phase to be a integer multiple, q, of 2π. However, if the single pass gain is 1, then while we know that the system can lase, it is not evident which mode it will lase on, or perhaps on which several modes. Which modes it picks depend on the nonlinear aspect of the gain medium and cavity. Choosing which particular mode to lase on can also be viewed as picking which manifold to glue back to the original. All n modes for a 1D laser to pick to lase then map onto which specific way we reglue the manifold. In this case, the laser cavity would be a ramified manifold of infinitely many folds along one direction. If we re-glue the manifold to itself after a single ramification, and again along the time direction, we regain equivalently the phase-space picture of a simple harmonic oscillator or the mode structure of an optical cavity. However, one can consider the condition for lasing to be when a ramified manifold such as if we reglued the pth copy of the medium back to the first, after n twists in the manifold along the time direction before regluing. Demanding the gain must be unit along some path in its cohomology25, one finds the gain must be the `th root of unity where ` is the least common multiple of (p, n). So, in this new gauge, obtained by new choices of ` gluing, the cohomological structure is not fully sufficient to identify the modes, because one also requires knowledge of the gain to fully define the mode. This is evident since the nonlinearity26 effects both

25that of a torus with curves of two integral winding numbers 26the field-strength dependence

79 the gain, and its effect on mode structure.

3.5.3.7 The role of nonlinearities

In any real laboratory system incorporating dispersive, or laser gain media, optical nonlinearities always play an important role. It is noteworthy then that they have never been incorporated in a quantum description of laser physics in an entirely consistent way [97]. To understand the connection between quantization and nonlinear laser physics more physically, one would like to associate the appearance of the gain-dependent index with local information in the optical medium. By multiplying the optical gain at each ramification boundary by an appropriate field- dependent root of unity, the electric field along this path now carries an identifier of fold-count, which returns to its original value once an entire orbit is traversed. We can consider this similar to the local choice of symplectomorphism (and hence Maslov indices), now incorporating a local mapping of the spatial manifold to the index of the ramified covering map. We would expect the principal quantum number to be another contribution to this manifold invariant, no less geometric than the Maslov indices. What is new is that we have admitted the role of a nonlinearity into defining the operation and decision of gluing. This means the structure of our optical manifold and the character of the lasing state are defined self-consistently, rather than first fixing the optical manifold, and simply choosing a lasing state from among the modes it defines.

3.5.3.8 Reconnecting with laser optics

The story above is new and interesting, but laser physics is very mature, and has its own approaches to this problem. It would be unreasonable not to consider how they differ. In most modern approaches to understanding the nonlinear portion of lasing behavior is to model the quantum mechanical behavior of the medium using a density-matrix picture for excitations in the medium coupled to optical modes defined by solutions of Maxwell’s equations through a Jaynes-Cummings type interaction [98]. It is traditional [99] to either (1) treat the medium’s quantum behavior exactly with a density matrix, but relegate the optical modes’ behavior to

80 an effective reservoir of fixed photon states using Lindblad operators and Markov- style approximations to the Jaynes-Cummings dynamics or (2) model the system as closed using a Jaynes-Cummings type Hamiltonian, treating the optical mode structure as a fixed solution to the wave-equation in linear media. It is customary to solve for an optical mode’s gain using rate equations derived from one of these two options. However, despite the fact that sustained laser oscillation cannot be described in any detail without admitting nonlinear gain, there is no known way to understand gain saturation using a Lindblad operator picture without violating the positivity of the density matrix. The traditional approach is to introduce a fudge factor to the master equations preserving the positivity and leading to Seargant and Scully’s picture of laser gain [99]. A similar problem arises in understanding parametric processes in nonlinear optical media. For nonlinear media, the origin of these problems is clear. The mode structure is not independent of the medium’s excitation, and should be solved self-consistently. Gain saturation enters from solving these systems exactly when possible, or to high order in the light-medium coupling, and has no definitive model. So the problem we are addressing with our geometrical approach to optical manifolds is an open one in laser physics.

3.5.3.9 Self-consistent lasing and topology

In the picture described above, however, we can solve for the optical mode structure and the light-medium interaction self-consistently in the gauge obtained by different gluings. This allows the mode-indexing, and spectrum itself to be directly related to the gain. The result of admitting the non-Lindblad terms is a pair of coupled nonlinear equations for the time evolution of gain and level inversion in the medium, where nonlinearity is driven by mode competition [97]. There is a simple way to introduce a more constrained form of geometry to the manifolds described above. Ultimately, those manifolds are introduced as a means to solve Maxwell’s equations in a geometric optics sense where the Eikonal is single-valued. The connection between these two approaches is through the mapping between de Rahm cohomology and harmonic functions. But there is another approach. If we consider the solutions of Maxwell’s equations to be obtained by using the method of characteristics27, the physical

27where we find a Dirac operator that solves Maxwell’s equations as a set of coupled first-order

81 optics solutions can simply be integrated. This should be largely the same as the harmonic function approach, but since we must select one Dirac operator from many available, we introduce a new degree-of-freedom to the treatment. The choices of viable Dirac operators can be represented through a type of Clifford algebra28 as discussed in Sec. 2.2.1. If the Dirac operator choice is made locally, one ascribes a local (bi-)quaternionic29 structure to the manifold [100]. With this as our perspective, the ramified manifold now comes equipped with an additional structure. In addition to the metric we began with and the connection introduced to describe non-reciprocal optical effects, we now also have a set of local complex forms that are quaternionic. In this more constrained geometric approach to the optics problem, nonlinear effects like gain saturation in the laser have an elegant, and already known [97], interpretation. A single pumping event in a laser results in a time-evolution for the optical density matrix which is just a finite translation in this space of biquaternions. This achieves much of what the heuristic approaches in laser physics above do, non-perturbatively, without fixing or approximating the mode profiles. Further, it results in an optical electric field dependent on these complex structures, which form a complex Clifford algebra. Expanding its dependence in a power-series over the Clifford generators, one obtains a super-field30 representation of the optical electric field. Squaring and integrating this electric field allows one to define a photon number operator as a volume integral over another super-field. As a result, the time evolution of the optical (reduced) density matrix depends nonlinearly on a super-field; one could attribute the time-evolution in a Hamiltonian approach to a super-potential. The shape of this super-potential defines the character of preferred optical states, and lasing transitions, through the kinds of pictures describing second-order phase- transitions in condensed matter systems. The Kuramoto transition is a dynamical version of these same transitions, so we are reformulating a model traditionally applied to systems of coupled lasers in much more constrained form. The way we have defined things, going through this transition gives behavior similar to the process of quantum measurement. equations 28this is familiar from Dirac’s construction of relativistic quantum mechanics 29biquaternions are a complexification of the quaternions 30a super-symmetric field

82 But all laser physicists will agree that the nonlinearity of gain media is completely necessary to incorporate in any complete description of laser oscillation, and also that there is no rigorous way known to do so [97]. We can take one simple mathematical cue from the story above. Understanding nonlinearity in laser media required that we introduce the action of pumping in a laser as a bi-quaternion displacement. Bi-quaternions have two forms of involution associated with them, and we can consider quadratic forms or Witt groups over that ring. This recalls the role of L-theory in surgery, whether a given ramified and glued manifold which permits the Eikonal approach of optics, is sufficient for defining differential calculus. If this is correct, we have equated the difficult technological problem of a-priori calculating lasing modes in a given pumped medium with the classification problem of manifolds in four dimensions. But since this is technology, we have a physical computation scheme for manifold classification as well. We vary the medium and identify which modes are lasing.

83 Chapter 4 | Feedback to probe excitations of an atom light system

We now want to explore the light-atom interaction dynamics by moving to a regime where there is a controllable quantum process that reveals the covariant nature of the quantization of light and atoms. An effective way to force the participation of a coherent light-atom response is by driving the atoms with light using feedback from the atoms. Earlier we introduced the analogy between a coherent state description of many-body ground states and the oscillation of a coupled set of nonlinear oscillators. Commonly, quantum many-body states are studied by cooling a quantum system and then adiabatically applying time-dependent fields to push the system towards a target Hamiltonian [101]. A challenge to implementing such protocols is keeping track of the heating or general loss of coherence of the sample [102,103]. We present an alternative way of observing coherent excitations of quantum matter in a potentially more robust manner than from the top-down construction techniques commonly utilized. Instead, much like a laser amplifying spontaneous emission, we will try to build up a quantum state by amplifying quantum noise and feeding back upon itself. To probe the existence of such collective dynamics, we position the atoms to be the connecting link between two otherwise disconnected, distinct laser paths that together form an unstable feedback loop. Those excitations, in the language of gravitational metrics introduced in Sec. 3.3, can be understood from the perspective of the light as part dielectric and part metric. Rather than cooling into a well defined state of a system to probe, we instead wish to feedback and amplify the signal to control its behavior. Similar to any other system with feedback control, we must

84 examine the stability conditions in the closed loop transfer function to determine what the coarse behavior will be. By adjusting the loop to make it oscillate, a coherent state is built up in the light atom system when the small-signal transfer function diverges for some mode. The oscillating mode becomes our ground-state in a static system similar to a treatment using the rotating wave approximation. The first experiment that we perform is one where we probe the Arnold tongue by injection locking the oscillator and sweeping the injection frequency from the lock-in frequency to the frequency it falls out of lock.

4.1 Apparatus to manipulate cold atoms

Free, neutral atoms are extraordinarily stable, high-Q resonant objects. The most accurate measurements of time are made using neutral atoms. In particular, cesium is used as the international standard for the definition of the second [104]. To perform careful experiments upon atoms, it is necessary to produce a complimentary apparatus that 1) excludes as much uncontrolled environmental noise as possible, and 2) permits the necessary probing fields that can be tailored to explore new phenomena. First, in order to extend the time with which an atom exists isolated from physical collisions with other objects, we place the atoms inside an evacuated vacuum chamber. But also in the course of running an experiment with cold atoms, it is necessary to ensure that the laser light is locked at the relevant frequencies, to control magnetic fields precisely, to stabilize the temperature loads, to manage a timing system that controls when equipment is utilized, and to control vibrations from shaking the mirror mounts. We used commercial equipment to address these issues where we could, but for many requirements, we had to custom engineer solutions ourselves. We will briefly detail each of these necessary components in running our cold atom experiment.

4.1.1 Vacuum chamber

A typical experiment probing cold atomic gasses uses the conservative potential from the AC Stark shift of a laser beam to hold and manipulate the atoms [105]. To a very good approximation, the atoms can be thought of as little oscillators sitting in a harmonic potential rocking back and forth with the frequency of their

85

Figure 4.1: This is a schematic of the parts of our cold atom chamber that interacts with the atoms; we leave off the vacuum pump components. The chamber consists of a source region on top where a MOT is used to cool and trap atoms to µK temperatures. In the lower cell, another set of coils are used to make the UHV MOT and microscope objectives are placed around the cell to allow simultaneous high resolution imaging of the atoms from four directions in a plane. rocking proportional to their temperature [106]. To be able to hold them, for example in an optical lattice, the atoms must generally be on the order of 10 µK or less. This also means that if a water molecule at 300 K desorbs off the vacuum chamber wall and collides with the cold atoms, it will supply enough kinetic energy to completely blow out all atoms that it touches directly or indirectly. Thus our vacuum chamber must create and maintain a background gas pressure sufficient to permit experiments that last ∼ 10 seconds or longer without a single background gas atom entering the region under study. We can calculate approximately what the air pressure needs to be inside the vacuum chamber to perform such long-lived experiments free of background gas collisions. The collision frequency of a molecule

86 in a gas is [107,108], s k T ν = 4nσ B (4.1) πm and for a mean free time of 10 sec, a gas of water molecules at 300 K, and a typical cross-sectional area of π × 10−18 m2, the vapor must be at a density of less than 4 × 107 cm−3. From the ideal gas law, this corresponds to a pressure below 1 nTorr. The chamber that was used for all the experiments described in this thesis was first designed and constructed in Stanford by Steven Chu’s BEC group [73]. Additional details about the construction of the “Rubidium experiment” can be found in there. The basic design is one of a double-MOT [109], with one MOT to collect, cool, and compress the atoms quickly at relatively high atom flux (above 1 nTorr) and another MOT to catch the cooled atoms a distance away where the background gas pressure is much lower than 1 nTorr. Though the ultimate lowest pressure expectable will be better than 10−12 Torr, it will not be significantly lower than that without great effort [110, 111]. Pressures of 10−14 Torr were managed in 1964 [112], but those have not been improved upon much since. For work in such ultra high vacuum (UHV) pressures, the construction of a chamber must be considered carefully. All organic materials, plastics, and even many metals such as zinc1 or cadmium2 become unsuitable due to their significant rate of evaporation, or outgassing [113]. In general, the chamber was built with only glass, electropolished 300 series stainless steel3, oxygen-free copper, and certain ceramics. The first MOT is loaded with Rb87 atoms from vapor inside a glass cell4 of dimensions 50 mm × 50 mm × 10 mm at the top of the apparatus. The outer surface is anti-reflection coated for the near IR, 700 nm - 1100 nm. Straight down from the MOT is a 10 cm long 1 cm diameter tube through which the cooled atoms are dropped, which is used to maintain a pressure differential between the source cell and the rest of the UHV chamber. The atoms fall through a region that is continuously pumped by an ion pump5 and which has windows to allow time of flight measurements on the atoms. Then they fall through another constriction6

1in brass 2commonly used to plate screws 3excluding the varieties like 303S that include sulfur or selenium which vaporize readily 4Precision Glass Blowing 540 L/s VacIon ion pump from Varian 66 mm in diameter aperture

87 into the UHV section of the chamber which is pumped by both a second ion pump7, and a titanium sublimation pump (TSP). This ensures sufficient pumping of the second UHV glass cell8 where we catch the atoms with the second MOT. The pressure in the UHV region is measured to be below 5×10−12 Torr by a hot filament ion gauge9. Measurements characterizing this vacuum [114] show that the vacuum lifetime is 137 seconds. The chamber was shipped under vacuum from Stanford to Penn State where it was mounted upon an optical table10 and the necessary MOT optics were arranged to produce our first cold atoms. Or so we thought. The chamber was not actually under vacuum at all, and, due to a comedy of misleading instrumentation errors and failure modes, we did not know this was the case until we had ruled out all other potential problems from lasers to atom sources and pumped out the top region where the atom sources were refilled. In opening the valve between the top half and the bottom half, we then discovered a hissing sound. One of the electrical connections of the TSP had cracked, and the whole apparatus needed to be baked and pumped out. To evacuate a vacuum chamber of background gasses, we first use a roughing pump11 / turbopump12 device to evacuate all of the air out of the chamber. The turbopump operates by spinning fan blades similar to a jet engine which permit molecular gas flow to be in only one direction, out of the chamber. Our turbopump can maintain this pressure differential in the chamber all the way down to 7 × 10−9 Torr. However, beginning at a pressure of about 10−5 Torr, the decline in pressure plateaus since the background gas pressure is at that point almost entirely due to water molecules slowly being thermally excited and coming off the inside of the walls. To achieve our target pressure of 10−12 Torr, we must eliminate the vast majority of atoms and molecules adsorbed onto the inside of the stainless steel walls and dissolved by osmosis into the surface matrix of the steel. To do this quickly, we raise the temperature of the chamber up to ∼ 200 C for 6 days. Details of this bake-out can be found in Jianshi Zhao’s thesis [114]. By the end of the bake-out, we achieve our target gas pressure of 10−12 Torr by obtaining a lower pressure than

775 L/s 8Killdee Scientific Glass Co. 9Varian UHV-24P; which can only measure down to 5 × 10−12 Torr 10304 SS, non-magnetic 11integrated IDP-3 Dry Scroll pump 12Agilent TPS Compact

88 our ion gauge can detect. This pressure has been kept below detectable range by the ion gauge over the years by continuously running the ion pumps, which are most efficient at pumping Nitrogen, and having a large surface area TSP which captures the more reactive gasses like water and carbon dioxide [113].

4.1.2 Atom sources

Our source of Rb87 atoms is from a small Rb getter13 which contains 4.08 mg of the element. These are created for industry usage for coating photocathodes, image intensifiers, and night vision tubes. The natural abundance of Rb87 from the total of Rb87 and Rb85 combined is 28% [115]. We will only make use of the Rb87 atoms in this experiment, though the apparatus was designed to be able to switch to the Rb85 resonances with little effort. To produce Rubidium vapor in the source cell we heat the getter by running ∼ 3 A through the metal. We create a MOT from the atoms that are liberated from the getter and set the current of the getter to a value that permits a MOT formation rate of about 100 million atoms in 500 milliseconds. We can run the experiment collecting a new MOT of 108 atoms every 5 seconds and expect that a single getter will be able to supply us with atoms for 1-2 years.

4.1.3 Laser systems

The most important method of interaction with the atoms is through a complex set of laser pulses that alternatively capture, cool, hold, and excite some or all of the atoms at precise times. The near detuned laser light is produced by homebuilt external cavity diode lasers (ECDLs) which are subsequently amplified by a tapered amplifier. The near detuned light must both be highly accurate to the absolute frequency of the atomic resonances, stable to both long term drifts and short term jitter, and also widely tunable to access atomic transitions hundreds of MHz apart. Therefore, the diode lasers are locked using a sophisticated beat note lock setup. The diode lasers are located on the opposite side of the room from the experiment chamber to reduce scatter from affecting the atoms, and are physically separated from the mechanical shutters that turn on and off the light. If the shutters were located on the same table, then the click of a shutter blade would generate a strong enough vibration so as to knock the laser out of its lock. The laser light is ported

13SAES Rb/NF/3.4/12/FT 10+10

89 around the room, from the lasers to the shutters to the amplifiers and finally to the atoms via polarization maintaining, single mode optical fibers. The far-detuned light is derived from a reliable 1064 nm turn-key NPRO laser which seeds a 40 W fiber amplifier.

4.1.3.1 Diode lasers

The lowest set of electronically excited energy levels of the Fine structure of the 87Rb atom is separated by about 780 nm and is called the D-line. This is actually 2 2 a pair of lines, the D1 which names the 5 S1/2 → 5 P1/2 transition and the D2 2 2 which names the 5 S1/2 → 5 P3/2 transition. The D2 is more useful for us in this experiment because it has a cycling transition useful for cooling and trapping the 87Rb atoms. The frequencies of the laser light are locked to a specific value by a combination of an absorption lock and a beat note lock. Output powers of the ECDLs are typically ∼ 10 mW, of which the majority is lost due to beam combinations and lossy optical components such as transmission through an AOM or optical fiber couplings. The laser diodes have a gain profile width of about 10 nm, and are centered at 785 nm. A thermoelectric cooler is used to fix (and stabilize) the ECDL’s cavity length. The center of the gain profile of the laser diodes can shift by 0.5 nm per degree C. The injection current of the ECDL can be tuned together with the external cavity to permit single-mode tuning ranges of more than 10 GHz. At the same time, the MOTs that we run use ∼ 20 mW of power and could benefit from even more power. To make up for this shortfall of light power, we use a double pass tapered amplifier to increase the power ∼ 200-fold.

4.1.3.2 Laser frequency locking

We use a lock scheme [116] based on saturated absorption spectroscopy with a beat note lock to stabilize our laser frequencies. A single dedicated laser, known as the "Reference" is locked to the most significant feature of the Rb absorption profile

close to the D2 line. Then two more lasers are locked relative to the Reference by the beat between them and the Reference. One is used for the MOT cycling light, and the other is used to pump the atoms that fall out of the cycling loop into the lower ground state manifold back up into the cycling, upper ground state manifold.

90 They are known as the "MOT" and the "Repumper" respectively. A fourth laser is built for backup, and all four are set up symmetrically which permits any of them to act as the Reference, the MOT, or the Repumper. This allows for easy diagnosis of extraordinary laser noise, and for little down-time in the event of a laser diode failure. For details, see Rene Jacome’s Master’s thesis [117].

4.1.3.2.1 Absorption lock The absolute frequency reference for the lasers is set by an absorption feature of a vapor of Rb atoms. The Reference is dedicated to maintaining a lock to the 3/4 crossover sub-Doppler absorption resonance in Rb85. The absorption lock is implemented by Modulation Transfer spectroscopy, a pump-probe sub-Doppler spectroscopy technique [118,119]. The pump beam whose frequency is modulated by an AOM modifies the number of atoms in the ground state by the degree to which the laser is on resonance. A second, unmodulated, probe beam shines through the cell at the same time. The transmitted intensity of the probe light depends on the number of atoms in the ground state and hence on the detuning of the pump laser away from resonance. Thus the intensity modulation of the probe beam comes solely from the atomic absorption profile. The laser table to implement the beat note and absorption locks are shown in Fig. 4.2.

4.1.3.2.2 Beat note locks An set of tunable locks is necessary to run the experiment in order to not merely have a stable absolute frequency lock, but also to be able to dynamically move at least two more independent laser frequencies over several hundred MHz. We use a method by Hänsch [120] where part of the light to be locked is split off to form an optical phase locked loop. The light from the MOT laser and the Repumper laser is combined with the Reference laser on a fast photodiode to produce a current which is proportional to the sum and difference frequencies of the respective lasers. The difference frequency, or beat note, is then locked to a reference oscillator using two feedback paths, one using a digital phase frequency detector (at < 50 kHz) and the other using an analog phase detector (with a bandwidth of ∼ 2 MHz). Another, long-time feedback path on the center frequency of the laser is controlled by a bi-directional thermoelectric cooler heat pump which maintains a temperature lock of the ECDL.

91 Lens Fiber Coupler λ/2 PBS AOM

Dither Oscillator λ/4 Mirror Optical Isolator Vapor Cell

To FPI

ToAbsorption PD

MOT to Switch-yard Repumper

to Switch-yard Pump beam To Beatnote PD

Reference Backup MOT ECDL ECDL ECDL

ECDL

Repumper

Probe beam

Figure 4.2: The four laser ECDLs are depicted in blue, pink, green, and orange. All four are in an identical layout so that a diode failure in one will not cause significant down time for the experiment. The light exits the ECDL and is immediately sent into an optical isolator. It is then split by a PBS, one half is coupled into an optical fiber to be sent to the switchyard and tapered amplifier, and the other is sent to the absorption cell and beat note lock. The latter light is split and recombined by another PBS where a double pass AOM has placed a distinct frequency modulation on the light. The light is then split again by another PBS into the pump and the probe beams. A last PBS splits the probe light right before the absorption cell and sends it into the beat note lock photodiode.

4.1.3.3 Double pass tapered amplifier

Since the output power of the diode lasers is too low to run the experiment by themselves, we use a tapered amplifier (TA) 14 to amplify the power necessary to run the experiment. Normally the tapered amplifier requires ∼ 50 mW of injection power to saturate the power gain and overcome the amplified spontaneous emission which pollutes the spectral content. However, since we do not have 50 mW of light available to seed the tapered amplifier, we utilized a double-pass scheme [121,122] which yields 700 mW output power after an optical isolator with only 100 µW of input power. However, after the experiment was built this double-pass setup

14EYP- TPA-0780-01000-3006-CMT03-0000

92 was the least stable part of the apparatus. We ran the TA at a lower current and output power of 1.3 A and 400 mW because the system would have a tendency to self lase from an imperfection of the AR coatings of the chip facet. The TA would mode hop and produce about 20% less overall power and much less Repumper power every ∼ 15 minutes which would decrease the loading efficiency of both the source MOT and the UHV MOT by a factor of 70% or more. We handled this by slightly adjusting the back reflection through the TA until the output spectrum and power became stable again. Next technical improvement needed for the experiment would be to lock this drift and hop away, perhaps by locking the output ratio of Repumper to MOT light.

4.1.3.4 Optical switchyard

The switchyard (Fig. 4.3) is responsible for splitting the MOT and Repumper light into several different paths each of continuously tunable intensity which are needed in different combinations and at different locations at the experiment chamber. Most of the switchyard is after the power amplification by the TA. We use double pass acoustic optical modulators (AOMs) [123] to continuously control the intensity of the light by creating a standing wave of RF power across the crystal lattice and having the light Bragg diffract off the standing wave. We take the first order diffracted light, which is also frequency shifted by one or two hundred megahertz. The advantage of this is that we can turn the light on and off on a time scale of about 25 µsec. However, if we turn the RF power to zero, there is still a non-zero amount of light that remains coupled into the optical fiber. This can be catastrophic if it couples to the atoms during the wrong times during the experiment and destroys an otherwise precise atomic superposition. To ensure zero light reaches the atoms when a particular path needs to be off, we must use a mechanical shutter to physically block the light. This much slower movement to block the light costs time, and the fastest that we can block the light is using a small aperture shutter15 which can block the beam in about 2 ms. The mechanical shutters must be vibrationally and thermally isolated from the optical table. With the shutter open, the device runs about 1 A of current steady state, and when the shutter closes it creates a significant enough vibration that it can knock the Reference laser out of lock.

15nmLaser LST200SLP

93

Figure 4.3: Mechanical shutters, and optics to control the amplitude and frequency of laser light. Once prepared, the light is launched into optical fibers that transport the light to the experiment table.

4.1.3.5 High power, 1064 nm light

The far off-resonant laser light that primarily forms the conservative potential that the atoms are suspended within is derived from a stable non-planar ring oscillator16 which is used to seed a high power fiber laser17. Nominally the Nufern amplifier produces 40 W of laser power at maximum, but we found that the level of stimulated Brillouin scattering begins to rise after about 20 W so we never ran the experiment with more light than that. Worse though, a couple years into operation the output mode became a (1,0) mode at moderate output power, and to optimize our ability to couple into single mode fibers we never ran the Nufern above 13 W during the experiments described in this thesis. The Nufern laser is split into a number of different paths depending on the particular experiment that we were performing, but for the one described next, we ultimately ported all of the light into four large-diameter multimode fibers18 which both transport the light to the atoms (Figs. 4.4 and 4.5) and scramble the Gaussian mode into a random superposition of O(100, 000) modes. The disordered light is surprisingly stable, a particular mode

16Lightwave Electronics NPRO 126 17Nufern NUA-1064-PB-0050-D0 18Thorlabs - MHP910L02, 0.22 NA, and M28L01 400 µm 0.39 NA

94 pattern remains static for many seconds unless the fiber is tapped. The light is then imaged down to the center of the vacuum chamber. There it is overlapped with another counter propagating beam, and two more coplanar beams intersect the atoms from 90 degree angles as shown in Fig. 4.5

4.1.4 Water cooling

A considerable amount of waste heat is produced by generating high electrical currents for magnetic fields, and operating high powered lasers. Since most metals expand on order of 10 µm per meter per degree C which is also about 2π of phase shift for our laser light, optical tables that heat up and cool down over the course of a day will misalign the optics and render a complicated experiment totally inoperable. Any significant source of heat in contact with optical components must be temperature controlled. Similarly, the temperature coefficients of resistors and capacitors (not to mention nonlinearities in opamps), change on order of 1/104 per degree C which means that the operation of our electronic circuits that control devices such as magnetic fields will drift in performance in accordance to the circuit’s operating temperature. The most efficient way of keeping these devices at constant temperature is through water cooling. The building supplies a chilled water loop at about 12 C and 40 psi19. We built a secondary distilled water loop which supplied utility cooling locked to a temperature of within 2 C. The entire loop only contained ∼ 20 gallons which we learned the hard way20 was not too much to cause catastrophic flooding in the case of pipe ruptures in the lab. After trying and wearing out a number of pumps, we found that we needed a pump21 that could deliver 20 feet of head in order to push water through 1/2 in PVC pipes around the lab including over a ∼ 15 foot wall. A common problem with water cooling loops in labs are air bubbles in the pipes that are beat into foam by the pump. Foam will be pushed around the pipes and is both a poor heat conductor and is turbulent as it flows thorugh laser heat sinks. We avoided this problem by having a pair of 10 in diameter, 6 ft vertical pipes in the middle of

19Though this water was neither clean, nor provided within years of schedule, nor constant pressure, nor stable against full scale temperature oscillations. 20several times 21of at least a power of 1.2 kW

95

Figure 4.4: High power, 1064 nm optical table. We obtain the high-power, far- detuned laser light used in our experiment from an 40 W fiber amplifier injected from a highly stable 1064 nm NPRO laser. The optics on the left side of the figure in the red box are the ones relevant for the experiment in this chapter. By the time the light arrives at the EOM, we have ∼ 8 W that is phase modulated before entering a SI with a lens placed early in the beam path. The lens affects one of the counter-propagating beams earlier in the beam path than the other which causes the light out the fourth port to have two different focal points making a ring interference pattern. After the SI it passes through a series of non-polarizing beam splitters and is coupled into a series of large-diameter multimode fibers that carry the light to the optical chamber. The four beams are sent into the chamber to form two intersecting pairs of counter-propagating, far-detuned light along the magnetic quadrupole axis, and non-quadrupole axis, that suspend the atoms against gravity.

96 the loop which slow the water velocity down enough for bubbles in the water to separate out. The temperature was locked to about 12 C by actuating a 3-way valve in the loop that sends a portion of the water into a heat exchanger22 where the heat is dumped into the building’s processed chilled water line. The valve actuation was controled via a digital PID control [124] with an Arduino and some thermistors. This general purpose cooling loop has a natural pole frequency of around 1/40 minutes which would make it difficult to lock the temperature of the water to within much less than tenth of a degree on experiment timescales of 10 seconds or so. A tertiary loop had to be created to stabilize the high powered fiber amplifier since the efficiency of the 808 nm pump diodes are fairly sensitive to temperature. The fiber amplifier dissipates as much as 300 W of electrical power, and we used a smaller pump23 to ensure that we get continuous water flow through the amplifier. We then lock the tertiary loop to about 24 C through the same Arduino PID setup for the bulk cooling system, but with much faster dynamics and higher gain. We controlled the temperature of the loop by leaking a small amount of cold water in from the main loop and adjusting the pulse-width modulation of a 208 V heater tape screwed across a small heat exchanger. This managed to lock the water temperate to better than 5 mK which was the limit of the sensitivity of our sensing thermistor.

4.1.5 Magnetic field controllers

Precise magnetic fields are used while trapping and cooling the atoms. Two pairs of quadrupole coils are used at the source cell and at the UHV cell to produce ∼ 10 G/cm magnetic field gradients to produce a MOT. We fine tune the background magnetic field and cancel the earth’s contribution by using 3 pairs of Helmholtz coils around both cells. The coils are placed well away from the vacuum chamber to not impede optical access. The Helmholtz field controllers are designed to run up to ∼ 3 A DC, and lock to within 10 ppm accuracy.

22ITT Standard BCF 23Laing Thermotech 6050E7042

97 4.1.6 Timing system

Rubidium atoms in free space are extraordinarily sensitive objects. Every step of an experiment from preparation to interaction to detection must be done in a way such that the atoms are not uncontrollably exposed to resonant light. A difference in timing of as little as 25 nsec is sufficient to drive half of a population of Rubidium atoms from the ground to the excited state for the D2 electronic transition at saturation intensity. Thus one great technical battle in experiments with free atoms is that when repeating an experiment, we must ensure that the timing of equipment and lasers is as consistent across repetitions as possible. For example, at the start of the MOT sequence, the MOT AOM, which is always left on for thermal drift reasons, is turned off 1 ms before the mechanical shutter for the MOT is triggered to open, and is then turned back on 3 ms afterwards. This is to reduce variations in when the light is applied to the atoms since a mechanical shutter takes ∼ 2 ms to open, but can be slowed by up to 100% due to thermal effects and high duty cycle usage. In contrast, the buildup time for RF power to create a sound wave to Bragg diffract light in an AOM crystal is ∼ 25 µs, which is also much more consistent. To handle all of the clocking we use a dedicated computer24 to run a simple Labview program that receives via TCP/IP a list of time-value pairs, or edges, and instructs an FPGA to clock a series of digital and analog boards that each update a particular channel at 25 nsec accuracy. For additional details of the integrated data/instrument/command loop that runs the experiment, see Jianshi Zhao’s thesis [114].

4.1.7 Trapping and cooling atoms

The apparatus is split in two parts. Atoms are trapped and cooled from room temperature in the source cell at a high vapor pressure to achieve a fast loading time before being passed into the lower cell for our experiment which is done at UHV pressures. The atoms are collected with a MOT in the source cell which typically makes a ball of a few hundred micrometer in diameter of 108 atoms at the Doppler temperature of 145 µK after a few hundred milliseconds. We then turn off the magnetic field and apply optical molasses to lower the temperature further

24National Instruments PXIe - 8133

98 to ∼ 4 µK. For further details see Jianshi Zhao’s thesis [114]. Now having cooled the atoms to a significant degree, we turn off the optical molasses beams and allow the atoms to drop 80 cm into the UHV region where they are caught by a second MOT, with beams 3 mm across. The UHV MOT is restricted to be a small volume because the quadrupole axis beams pass through 0.4 NA microscope objectives.

4.1.8 Imaging and probing the atoms in the UHV cell

Atoms are transferred into the UHV MOT at a rate of 107 atoms per source MOT and are imaged in one of three different ways. The atoms can be imaged by either absorption or fluorescence techniques on a low-magnification path (2.5x) or via absorption imaging on a high-resolution path (24x) with a resolution of 1.4µm. Having two paths that partially overlap allow us to switch between a large field of view but low magnification image and a high magnification, but small field of view image. The atoms can also be imaged from an orthogonal direction along the quadrupole axis of the magnetic field coils and also by retroreflecting off of the mirrors at the back of each of the imaging paths. All together, this enables us to image the cloud of atoms from each of four sides simultaneously. The imaging paths are shown schematically in Fig. 4.5. The scientific camera used is an Andor25 CCD camera. The infinity corrected microscope objectives26 are all NIR coated with an effective focal length of 1 cm. The high resolution imaging path is sufficient to image at single atom resolution. We show in Fig. 4.6 absorption images of our atoms that are averaged over tens of shots. The imaging light passes through a tight aperture in an imaging plane that is equivalent to the camera’s image plane and is projected onto the atoms for an absorption image. After passing through the atoms, the light split in two and imaged through both a high resolution telescope, and a low magnification imaging path. The two paths are overlapped on the camera, though since the aperture is small on the low resolution path, most of the CCD array is available for the high resolution image. In Figs. 4.6a and 4.6c, we show the combination of the two imaging paths with atoms, and in Figs. 4.6b and 4.6d we show the same configuration but without atoms. The color scale indicates the optical depth of the atoms. The transmitted

25iKon-M 934 BRDD 26USMC WM-020NIR

99

Figure 4.5: Red is the 1064 nm light, and blue is the 780 nm light near detuned to the D2 line. The imaging paths are denoted by the dotted lines in the top right portion of the figure. The color scale indicates the optical depth of the atoms. intensity of light through an absorbing material is given by Beer’s law,

−D It = I0e (4.2)

where I0 is the incident intensity, and D is the optical depth. The optical depth is related to the number of atoms by

D = n2dσ (4.3)

where n2d is the two dimensional density and σ is the absorption cross-section, σ σ = 0 (4.4) 1 + 4( ∆ )2 + I Γ Isat

100 0.40 0.40

0.35 0.35

0.30 0.30

0.25 0.25

0.20 0.20

0.15 0.15

0.10 0.10

0.05 0.05

0.00 0.00

(a) Quadrupole axis with atoms (b) Quadrupole axis without atoms

0.14 0.14

0.12 0.12

0.10 0.10

0.08 0.08

0.06 0.06

0.04 0.04

0.02 0.02

0.00 0.00

(c) Nonquadrupole axis with atoms (d) Nonquadrupole axis without atoms

Figure 4.6: The average over tens of absorption images of a cloud of atoms cooled into the disordered, far-detuned potential. The bright spot in the center of Figs. 4.6a and 4.6c is the low-magnification image of the cloud of atoms. The bigger region on the encompassing much of the CCD array is the same image but via the high magnification path.

where σ0 is the on-resonance cross section, Γ the linewidth, ∆, the detuning, and

Isat, the saturation intensity. To produce the absorption image we calculate,

I − I ! D = − ln wa dark (4.5) Ina − Idark where “wa” indicates an image with atoms, “na” no atoms, and “dark” means with no light at all to eliminate distortions from imperfections in the pixel-to-pixel dark

101 current and sensitivity.

4.2 Generalized Raman sideband cooling

In an earlier experiment detailed in Jianshi Zhao’s thesis [114], we develop a novel technique of Raman sideband cooling (RSC) in a disordered potential. RSC is a dark state cooling process which can in principle reduce temperatures to below a single photon recoil [73,125,126]. RSC is utilized in the source cell of our apparatus to cool the atoms another factor of 4 before dropping them into the UHV MOT. We pump the atoms into the F = 1 hyperfine ground state by applying light resonant with the F = 2 → F 0 = 3 transition without any Repumper light for 1 ms. Both a 3D optical potential and a small uniform magnetic field is applied to the atoms such that the Zeeman splitting equals the vibrational quanta, ~ω where ω is the vibrational frequency of the lattice. Raman coupling is provided by the lattice beams and optical pumping light is tuned to be mostly σ+ with a little π light. The atoms begin to spread across the vibrational and hyperfine levels of the |F = 1, mF , νi state. The optical pumping beam drives the atoms up into the 0 |F = 0, mF = 0, νi states from the |F = 1, mF = −1, νi states, and when they spontaneously emit, provided they are in the Lamb-Dicke regime, they preserve the vibrational quantum number which will place them in a lower mF state than where they began [127]. The Raman coupling provides the mechanism to make a two-photon transition and go down both a vibrational level and a mF state. After some time, the atoms accumulate in the lowest vibrational level of the mF = 1 state where the atoms are dark to the optical pumping light. In the previous experiment we demonstrated Raman sideband cooling where the far-off resonant 1064 nm light provided the Raman coupling through the small vector light shift that couples states with different magnetic sublevels. However, the potential that we created was a strongly disordered system, so it was difficult to predict whether there would be a sufficient aggregate “lattice vibrational frequency” to match with an external magnetic field to produce an obvious cooling signature. But as demonstrated, a field of about 60 mG optimized the cooling for our system and could maintain the direct laser cooling for more than 40 minutes. A rather shockingly long time for a quantum system to maintain coherence.

102 4.2.1 Characterization of the optical fiber modes

The far off-resonant disordered potential was generated by coupling the high powered laser into large numerical aperture fibers and imaging the light down through the microscope objectives onto the atom cloud. As shown in Fig 4.5, two multimode fibers27 are used for the two axes, labeled as quadrupole and non-quadrupole axes. The single mode that is sent into the optical fiber scatters from impurities and equilibrates into a maximal entropy state with all of its modes occupied. We now calculate the scaling of the information content of the mode distribution per volume. An illustration of the setup is shown in Fig. 4.7.

Microscope Objective

Aperture Plane Multimode Fiber Mode Profile Optical Pumping

Single Mode Beam Figure 4.7: Four beams of light that have been scrambled into an equilibrated distribution of optical fiber modes form four spatially disordered wavefronts which combine to form a far-off-resonant potential in a small volume (red). There, the atoms are dissipatively coupled to a single-mode optical pumping beam. The amount of information carried by the disordered wavefronts can be controlled by the relative orientation of the optical fiber launch and by the collimation of the beam coupled into the fibers. This information content can be directly observed through the altered structure in the Fourier plane (green/blue).

We want to decompose an electric field that is the result of four intersecting beams of light into an easy-to-manipulate power series in Cartesian coordinates that can be expanded to arbitrary order over an arbitrary-sized volume. Each beam is either imaged from the facet of large-diameter optical fiber that can support hundreds of thousands of modes, or from such a beam that then passes through

27core size 0.91 um, Thorlabs - MHP910L02, 0.22 NA

103 the atoms and then is retro-reflected off a mirror to create a form of a disordered standing wave. A single beam’s electric field is composed of a random linear superposition of ∼ 100, 000 modes summed over four different beams, all with a general complex phase factor, ai

X E = aiEi (4.6) i where i indexes not only all solutions to Maxwell’s equations in a step-index optical fiber, but also over the four beams, each of whose propagation direction is rotated by 90◦ and intersects at a single point as shown in Fig. 4.7. Since the electric field is imaged from a ∼ 1mm cylindrical fiber we first express the solutions in cylindrical coordinates. The solution is given explicitly in Reference [128], and we continue directly from them. Inside the core, the longitudinal field is,

jβz+jνφ Ez = AJν(κr)e (4.7) jβz+jνφ Hz = BJν(κr)e (4.8) and in the cladding, the longitudinal field is,

jβz+jνφ Ez = CKν(γr)e (4.9) jβz+jνφ Hz = DKν(γr)e (4.10) where J is the νth order Bessel function of the first kind, K is the νth order ν √ ν modified Bessel function of the second kind, and j = −1 is the imaginary unit. It is emphasized here that we are interested in a number of quantities that change depending on the particular mode designated by i. A, B, C, D, ν, κ, β, γ all depend on i, though we will suppress that index for typographical clarity. From the separation of variables of Maxwell’s equations, the longitudinal wavevectors satisfy,

2 2 κ = (n1k) − β (4.11) in the core region, and, 2 2 2 −γ = (n2k) − β (4.12) in the cladding region, where k is the magnitude of the wavevector. Roughly, ν is

104 the index that counts the number of azimuthal twists the light undergoes before coming back to the original location, and β is the longitudinal wavevector. The fields in the orthogonal directions in the core are,

j jν ! E = AβκJ 0 (κr) + Bωµ J (κr) ejβz+jνφ (4.13) r κ2 ν r ν j β ! E = A jνJ (κr) − BωµκJ 0 (κr) ejβz+jνφ (4.14) φ κ2 r ν ν j jν ! H = −Aω J (κr) + BβκJ 0 (κr) ejβz+jνφ (4.15) r κ2 1 r ν ν j β ! H = Aω κJ 0 (κr) + B jνJ (κr) ejβz+jνφ (4.16) φ κ2 1 ν r ν

For ν = 0, we have two different possibilities, the TM modes,

j E = − AβJ (κr)ejβz (4.17) r κ 1 j H = − Aω J (κr)ejβz (4.18) φ κ 1 1 jβz Ez = AJ0(κr)e (4.19)

Hz = Eφ = Hr = 0 (4.20) and the TE modes,

j H = − AβJ (κr)ejβz (4.21) r κ 1 jβz Hz = AJ0(κr)e (4.22)

Hφ = Er = Ez = 0 (4.23) j E = AωµJ (κr)ejβz (4.24) φ κ 1

For the ν 6= 0 modes, A and B are constants to be determined from the solution of the characteristic equation, which is another name for, satisfying the boundary conditions between the core and cladding. It is,

0 0 !  0 !2 0  1 Jν(κa) 1 Kν(γa) Jν(κa) n2 Kν(γa) +  +  κa Jν(κa) γa Kν(γa) κaJν(κa) n1 γaKν(γa)

105   βν 1 1 ! =  2 + 2  (4.25) n1k (κa) (γa) where a is the radius of the core of the optical fiber. Only values of κa and γa that satisfy both the characteristic equation above and the relationship between the core and cladding wavevectors,

2 2 2 2 2 (ka) (n1 − n2) = (κa) + (γa) (4.26) are possible solutions to Maxwell’s equations in a step-index fiber. There are about 100,000 solutions to this characteristic equation subject to the physical constraints of our optical fiber28 each of which we index by i. Each solution corresponds to a specific integer for νi, and real number for κi. We plot all solutions to the characteristic equation in Fig. 4.8 a. The x-axis is the azimuthal integer, ν, the number of spirals the wavevector makes before matching boundary conditions again, and the y axis is κ. The amplitude represents the degree to which the wavevector leans out towards the cladding. We assume that given sufficient time for the light to propagate through the fiber, the total power of the light would maximize entropy and distribute its power across all of the modes equally. Since the intensity is proportional to |E|2, we 0 calculate the constant of proportionality, Ei , by setting the power per each mode,

Pi, to each be equal to the same constant,

11 1 P = (4.27) i 4 N where we have 11 Watt of light split over 4 beams, and N = 168, 418 modes in each fiber. We calculate Pi by integrating the z component of the Poynting vector over the core,

Z Pi = Si · zˆ (4.28) A Z a Z 2π ! 1 ∗ ∗ Pi = r(EriHφi − EφiHri) dφ dr. (4.29) 0 0 2

0 2 which can then be inverted for |Ei | . From the above arguments, we can calculate 28Thorlabs - MHP910L02

106 c 10-2 10-1 ℓ (m)100 101 − −1 1 )

-1 (

10

log -3 80 -2 -4 40 1

) − ( 10 log 10 -12 -14 4 -16 - 0.321 3 0.01 0.1 1 10 a ν b ℓ (m) d ℓ (m)

Figure 4.8: Characterization of the optical modes that form the disor- dered potential. a, Solutions of the characteristic equation for the optical fiber for all possible modes denoted by ν, K pairs. The color scale denotes the deviation β − n1k, with n1 the core index and k = 2π/λ, for wavelength λ, of the propaga- tion constant β from that of an axial ray. b, The information content projected into a volume of dimension ` in the experiment chamber is apparent from the contraction of normalized singular values of the expansion matrix L as the scaling parameter ` is increased. All singular values of L constructed to 10th order in the power series expansion are shown for ` = 0.32µm (red), ` = 1.0µm (orange), and ` = 3.0µm(green). c, Most significant singular values for all length-scales `. The diffraction limit is visible as a closing of the gap between the largest three eigenval- ues as the length-scale is increased to O(1µm). d, Scaling of information content within a volume as determined by the eigenmode structure of the fiber, and mode κ projections into experiment chamber, demonstrating Fs ∼ ` , with κ = 0.54 ± 0.06. The power series shown contain all terms up to order 6 (squares), 8 (pluses), 10 (crosses), and 12 (circles). The red line shows the best fit between 0.04µm and 3.6µm for 12th order, which is used to extract κ. Its uncertainty corresponds to the standard deviation of pairwise slopes for all points presented here.

exactly each and every solution to Maxwell’s equations inside our optical fiber which we image onto our atoms. However, directly summing together the field from all of the separate modes each with a random unit phase factor in order to characterize the potential is computationally infeasible. Instead, we will expand

107 each of the electric field modes from each of the four laser beams in powers of a common Cartesian coordinate system.

4.2.2 Expansion of modes in Cartesian coordinates

We have many optical field modes, Ei with a known mode structure (Eqs. 4.7 -

4.24) in combination with relative phase factors, ai, with which they contribute to a total electric field,

X E = aibEib (4.30) ib

We now wish to begin to combine all of the different components to Eib together in order to expand the quantity in a common coordinate system. Let us begin with the mode structure of one particular beam, for each of the three components of the field. Let us write the Bessel functions as a sum of polynomials using [129],

∞ s X (−1) 2s+ν Jν(x) = 2s+ν x for ν > 0 (4.31) s=0 2 s!(s + ν)! along with some Bessel function identities,

ν J−ν = (−1) Jν (4.32) 1 J 0 = (J − J ) (4.33) ν 2 ν−1 ν+1 0 ν 0 J−ν = (−1) Jν (4.34)

Using the above and including the contribution of modes with ν < 0 we write the components of the electric field from one beam in cylindrical coordinates as,

∞ s 2s+|ν| X (−1) (κr) jνφ+jβz ν Ez = A 2s+(|s|!)(s+|ν|)! e (sgn(ν)) (4.35) s=0 2 ∞ s 2s+|ν|+1 j X (−1) (κr) jνφ+jβz ν Er = 2s+(|s|!)(s+|ν|)! (Aβ(2s + 1 + |ν|) + Bωµjν)e (sgn(ν)) K s=0 2 (4.36) ∞ s 2s+|ν|+1 j X (−1) (κr) jνφ+jβz ν Eφ = 2s+(|s|!)(s+|ν|)! (Aβjν − Bωµ(2s + |ν|))e (sgn(ν)) (4.37) K s=0 2

108 combining these,

δ +δ ! k,kr k,kφ ∞ s 2s+|ν|+1 j X (−1) (κr) iνφ+iβz ν Ek = 2s+|ν| e (sgn(ν)) K s=0 2 (|s|!)(s + |ν|)! !

× Aδk,kz + [Aβ(2s + |ν|) + Bωµjν] + [Aβjν − Bωµ(2s + |ν|)]δk,kφ (4.38)

where the subscript k ∈ {kr, kφ, kz}, the cylindrical coordinates, and δ is the

Kronecker delta such that δk,kr = 1 when k = kr and δk,kr = 0 when k 6= kr. Now we wish to express E in terms of a single common Cartesian coordinate systems like this, X `x `y `z ˆ E = Σ`x,`y,`z,kc x y z kc (4.39) `x,`y,`z,kc ˆ where kc ∈ {xˆ, yˆ, zˆ} indicates a Cartesian basis, x, y, and z are Cartesian coordi- nates, ` is the respective power of the Cartesian coordinate, and Σ is a coefficient. Let us set the same function equal to itself expressed in these two different sets of coordinates,

X `x `y `z ˆ X X σ ˆ Σ`x,`y,`z,kc x y z kc = aib (κirb) Eiσkb (φb, zb)kb (4.40) `x`y`zkc ib σkb

X `x `y `z X σ ˆ ˆ Σ`x,`y,`z,kc x y z = aib(κirb) Eiσkb (φb, zb)kb · kc (4.41) `x`y`zkc ibσkb where we have moved the dependence on the specific basis to the right hand side, and we note that the index b denotes each of four separate cylindrical coordinate systems but c is not an index, but merely denotes that it is in Cartesian coordinates. Now we wish to expand the right hand side in Cartesian coordinates and match coefficients. That is, there is a matrix, L that takes a coefficient with indices, (i, b), and yields (`x, `y, `z, kc). That is,

X X σ ˆ ˆ i(νiφb+βizb) Σ`x,`y,`z,kc = aib Eikbσ(κirb) (kb · kc)e (4.42) ib σkb | {z } L(`x`y`zkb;ib)

109 σ+1−|ν|−δk,kz If we let s = 2 , which defines a new index, σ in terms of s, then,

δ +δ   k,kr k,kφ j (σ+1−|ν|−δk,kz )/2 ν K (−1) (sgn(ν)) Eikbσ =   σ+1−δ σ+1+|ν|−δk z kbzb b b 2 |ν|! 2 !  

× Aδkbzb + (Aβ(σ + 1) + Bωµjν)δkbrb + (Aβjν − Bωµ(σ + 1))δkbφb

× ΘH (σ − (|ν| − 1 + δk,kz )) (4.43)

where ΘH is the Heaviside function.

We now add in the ν = 0 modes and we get for Eikbσ,

"

Eikbσ =  Aδkbzb + (Aβ(σ + 1) + Bωµjν)δkbrb

# !δk r +δk φ j b b b b + (Aβjν − Bωµ(σ + 1))δ (1 − δ ) kbφb ν0 κ  Ajωµ jβ ! + δν δm,TEδk φ + δν δm,TM(1 − δk φ )A − δk r  0 b b κ 0 b b κ b b [σ−|ν|+(1−δ )(1−δ )−δ (δ +δ δ )]/2 (−1) kbzb ν0 ν0 m,TE kbrb m,TM ×   σ+(1−δ )(1−δ ) σ+1−δk z −|ν| kbzb ν0 b b 2 |ν|! 2 !

δν0 × ΘH (σ − |ν| + (1 − δkbzb )(−1) ) (4.44) where m indicates either the TE modes or the TM modes for ν = 0. We can expand the phase factor as,

ejνiφb = ej sgn(νi)|νi|φb (4.45) = (ej|νi|φb )sgn(νi) (4.46) = ej|νi|(sgn(νi)φb) (4.47)

|νi| = [cos(sgn(νi)φb) + j sin(sgn(νi)φb)] (4.48)

|νi| = [cos(φb) + j sgn(νi) sin(φb)] (4.49) 1 = [z + j sgn(νi)(x sin θb − y cos θb)] (4.50) rb 1 !|νi| |νi| = [z + j sgn(νi) sin θb − j sgn(νi)y cos θb] (4.51) rb

110 |ν | ! 1 !|νi| X i s1 s2 s3 = z (j sgn(νi) sin θb) (−j sgn(νi)y cos θb) s1s2s3 rb s1+s2+s3=|νi| (4.52) and,

ejβizb = ej(x cos θb+y sin θb)βi (4.53) ∞ (jβ x cos θ + yβ sin θ j)f = X i b i b (4.54) f=0 f! ∞ f ! X (jβi) X f = (x cos θ )t1 (y sin θ )t2 (4.55) f! t t b b f=0 t1+t2=f 1 1

ˆ ˆ evaluating the geometric overlap from kb · kc,

2 ˆ 2 2 2 2 2 rb (z ˆb · kc) = (δk,x cos θb + sin θbδk,y)δkb,zb (z + x sin θbz sin θb cos θbxy + y cos θb) (4.56) ˆ ˆ h i rb(φb · kc) = δkx sin θbz + δky(− cos θb)z + δkz(y cos θb − x sin θb) δkb,rb (4.57) ˆ 2 2 rb(ˆrb · kc) = [δkx(sin θbx − y sin θb cos θb) + δky(cos θby − x sin θb cos θb) + zδkz]δkb,rb (4.58) and now expand rb in Cartesian coordinates,

σ 2 2 2 2 2 σ/2 rb = (z + x sin θb − 2xy sin θb cos θb + y cos θb) (4.59)

X 2u1 2u2 u4 2u3 |νi|+1 = z (x sin θb) (−2xy sin θb cos θb) (y cos θb) rb (4.60) u1+u2+u3+u4 =fl((σ−|νi|−1)/2)

These terms combine to look like,

ˆ ˆ flm jνiφb+jβizb |νi| σ−|νi|−1 σ G = (kc · kbrb)rb · e · rb · rb · κ (4.61) where G is the geometric factor, and flm is the remainder of the floor function used earlier.

111 All put together, this simplifies to,

! ! ! X X |νi| f fl((σ − |νi| − 1)/2) G = (−1)s1 (−2)u4 s1s2s3 t1t2 u1u2u3u4 s1+s2+s3=|νi| t1+t2=f f∈(0,∞) u1+u2+u3+u4 =fl((σ−|νi|−1)/2) βf × js2+f+s3 i κσsgn(ν )s2+s3 sins2+t2+2u2+u4 θ coss3+t1+u4+2u3 θ f! i b b

s2+t1+2u2+u4 s3+t2+u4+2u3 s1+2u1 ˆ ˆ flm × x y z (kc · kbrb)rb (4.62)

If flm 6= 0, then (σ − |νi| − 1) is odd and both the rb and φb terms are zero and the zb term is non-zero. If flm = 0, then (σ − |νi| − 1) is even and both the rb and φb terms are nonzero and the zb term is zero. That is,

 ˆ ˆ flm (kc · kbrb)r = δzbkb Odd[σ − |νi| − 1] (δkx cos θb + sin θbδky)  2 2 2 2 2 × (z + x sin θb − 2xy sin θb cos θb + y cos θb)  2 + δkbrb Even[σ − |νi| − 1] δkx(sin θbx − y sin θb cos θb)  2 + δky(cos θby − x sin θb cos θb) + zδkz 

+ δkbφb Even[σ − |νi| − 1] δkx sin θbz + δkyz(− cos θb)  + δkz(y cos θb − x sin θb) (4.63)

When σ = 0, ν ∈ (0, 1). In general, for |ν| = σ+1, only Er and Eφ 6= 0 while Ez = 0.

When σ = |ν|, only Ez = 0. We have computed L up to maximum power of 12. The ~ lx ly lz ˆ expansion for L was checked by confirming that Eνm,Km − Ek,lx,ly,lz x y z k = 0 with Mathematica to machine precision within a radius of 1 micron from the origin.

4.2.3 Effective number of degrees of freedom

We characterize the effective number of degrees of freedom, Fs, in the electric field by taking the exponential of the Shannon entropy29 [132] of the singular values of L , P − si ln si Fs = e i (4.64)

29sometimes known as, "numbers equivalent" in economics [130], or a measure of “diversity” in ecology [131]

112 P where si are the singular values, σi, of L normalized to their sum, si = σi/ i σi. The exponential of the Shannon entropy gives the number of equally-likely states needed to produce the given entropy. A plot of the singular values si is shown in Fig. 4.8 b and c, illustrating for distances short compared to the optical wavelength three effective degrees of freedom for the three directions of electric field. In Fig. 4.8 d we see for longer distances a sub-extensive power-law scaling due to long-range correlations in the field strength, and at larger distances, a breakdown of the power-series expansion, which occurs later for a larger number of terms in the expansion. The value of κ, the slope of the power-law scaling, is extracted from a best fit over the scaling region (Fig. 4.8 d) for a maximum term in the expansion of lx + ly + lz = 12, which consists of approximately two decades in length scale.

4.3 Synthetic thermal body

We take up the idea of experimentally transforming one spacetime metric plus material susceptibility (g, χ), into another set (g,˜ χ˜) that we introduced in Sec. 3.3.3. There, the mapping was essentially single body physics. Although the media may be pushed and pulled and accelerated, there is a linear mapping into a frame with a new moving media coupled to a different spacetime metric. However, many material interactions and many body systems are not easily understood in terms of a single linear transformation. One takes for granted that thermal equilibrium can be established between two bodies by bringing them into physical contact with one another. However, viewed externally, any statistical reservoir must interact in ways such that the exchange of conserved quantities satisfy basic constraints which define the equilibrium it and any attached bodies reach. We construct a synthetic thermal body, engineered by controlling the spatio-temporal modulation of nominally conservative optical, radio- frequency, and microwave couplings of a neutral atomic gas carrying hyperfine-spin to a spin-dependent spatially and temporally disordered bath. We measure the out-of-equilibrium response through its resultant diffusive motion, extracting drift and diffusion parameters, and making comparison to the Einstein-Smoluchowski and generalized fluctuation-dissipation relations. Using a Synthetic Thermal Body may avoid photon-recoil sized heating and density dependent loss through light-assisted collisions.

113 Q Thermal Synthetic Reservoir Thermal Body

Wide, continuous spread in DOF Engineered spectrum and coupling Figure 4.9: A synthetic thermal body is an engineered environment that allows the exchange of entropy and energy to equilibrate as if it were a thermodynamical heat reservoir. On the left we indicate the traditional depiction of heat flow from two ensembles. On the right, we depict a Rubidium atom scattering through a bath of photons exchanging energy and entropy. As it passes through the bath, it continually changes its velocity and momentum which provides a disordered media that the atom can interact with and come into statistical equilibrium.

In the early 1900’s Sutherland, Einstein, and Smoluchowski determined that the rate of diffusion of an object through a thermal bath is proportional to the temperature of the bath. When a Brownian particle in a bath is acted upon by a conservative potential, it achieves a drift velocity of [133,134],

dV u = −µ (4.65) d dx where µ is the mobility, and V is the potential. This is countered by its own random diffusion which creates a current of,

∂f(x) j(x) = −D + u f(x) (4.66) ∂x d

df where j = −D dx is Fick’s law of diffusion and where f(x) is the concentration of the particles at the position x, and D is the diffusion constant. For classical Maxwell-Boltzman particles, the concentration is,

− V f(x) ∝ e kB T (4.67)

In equilibrium, the total current is zero and thus we obtain the Einstein’s general

114 kinetic diffusion relation,

D = µkBT (4.68)

This can be generalized using linear response theory by relating the admittance to the correlation function. The admittance is [133],

1 1 Y (ω) = (4.69) m −iω + γ(ω) where γ is the Fourier transform of the frictional force, and the correlation function is, Ψ(ω) = hu(t + τ)u(t)i (4.70) so that,

Ψ(ω) = kBTY (ω) (4.71) which is the fluctuation-dissipation theorem. With the bath as photons, we can engineer the distribution of amplitudes and frequencies of the photons where the only coupling to the atoms are via two photons processes where the atom can absorb photons from a set, {S1,E1} and emit into a bath of {S2,E2}. Given that two photon transitions coherently change populations of atoms without photon-recoil sized heating and density dependent loss through light-assisted collisions, a method of thermalizing atoms with a photon bath may permit small phase space densities. We can measure the distribution of atoms using time of flight expansion of the atom cloud, and determine the effective temperature of the photon bath through Einstein’s relation above. To implement a scheme to lower the temperature of the atoms, we want to exploit the second law of thermodynamics to carry away energy. We form a bath of photons that is composed of two parts. One is a tight spectrum of photons that are red of the atomic transition, and the other which is an incoherent, extended set of frequencies to the blue shown in Fig. 4.10. When an atom interacts with the bath making a two photon transition it will on average absorb energy from the low entropy set of photons and emit a photon into the red detuned bath thereby increasing the entropy of the system as a whole.

115 4.3.1 Preliminary experiments

We load 106 atoms into the disordered potential and cool the atoms with our generalized Raman sideband cooling scheme described in Sec. 4.2. This brings the atoms down to a temperature of ∼ 100 nK [114]. Then, while cooling, we apply a noisy pertubation that slightly kicks the atoms about the disordered potential. As the atoms are moving, they see a continuously changing potential landscape with numerous opportunities to make Landau Zener crossing transitions [135,136]. See Fig. 4.11. The random kicks that can move the atoms onto different spin levels and different physical positions come during the course of Raman sideband cooling where the atoms are coherently evolving close to the ground vibrational state in the local potential. A two-photon Raman transition would then be one that conserves both momentum and energy. In the following we describe a preliminary experiment for exploring phenomena associated with such a synthetic thermal bath. In one experiment we wrap the multimode fiber that transports the 1064 nm light to the experiment chamber around a piezo cylinder with a natural resonance frequency of 32 kHz. By driving the piezo, we shake and stretch the optical fiber which stretches different modes asymmetrically. In stretching the modes, it generates new optical frequencies through the optical fiber. We show in Fig. 4.12 an example of a power spectrum of the light that was applied to the atoms. We also mix the light that was shaken

Coherent Carrier

Incoherent Spread

Bath Frequency

Figure 4.10: An atom interacts coherently through two-photon transitions with a bath of photon that is composed of a narrow spectrum of energy detuned to the red of the atomic transition and a much wider spread in energies to the blue. To maximize entropy the atom will preferentially absorb low energy photons and emit high energy photons thus lowering the energy of the system of atoms by preferentially scattering its energy into higher entropy states.

116 Figure 4.11: An atom in a disordered potential is continually cooled by the Raman sideband cooling. During this process the atom changes magnetic sub level several times. Often, this permits the atom to make both adiabatic and diabatic transitions while interacting with the light to redistribute its energy. with light that was not, and plot the beat frequency spectrum near the atomic vibrational resonances. By controlling the drive of the piezo and simultaneously reading the optical beatnote frequency back, we can optimize the drive to form optical baths that have an asymmetric distribution of photons around a central coherentOptical peak. Magnetic

Power Spectrum Heterodyne Spectrum

v

Figure 4.12: On the left is a photo of the optical fiber stretched around the piezo cylinder. As the fiber is stretched, high angle modes are lost and reformed as the TIR condition between the core and the cladding fluctuates. In the center we plot the low frequency power spectrum for two different piezo drives. The red has an extended bath that is red detuned from the coherent carrier peak, and the blue has the bath on the opposite side. This is shown clearly in the plot on the right which is the beat note frequency between the stretched light and the unmodulated light.

In a separate experiment, we take advantage of the disordered vector light shift throughout the disordered potential. We apply a disordered, noisy magnetic field with magnitude comparable to the external magnetic field used for the Raman

117 T=1.4!K

fit

Figure 4.13: The left figure shows a typical magnitude of the applied magnetic field in units of mG. The middle shows the corresponding log of the power spectrum which shows that the frequency components fall off exponentially. Here the characteristic temperature is chosen to be 1.4 µK. On the right is a plot of the distance that the center of mass of the atoms fall while the noisy field is applied. We see that it is neither diffusive nor ballistic, but instead it has an exponent of 1.4. sideband cooling. The field is formed as the sum of a hundred sinusoidal signals with random phase distributed uniformly in frequency. The amplitude of the power − hf spectrum of the magnetic field however decayed as, ∝ e kB T shown in Fig. 4.13. After applying the disordered magnetic field noise to the atoms, we measure the cloud diffusion through time of flight and determine that the rate at which atoms fall through the disordered potential is neither linear as in classical diffusion, or quadratic as with ballistic motion. The atoms fall at a rate of, ∼ t1.4. It is important that the noise that we apply is spatial and temporal noise. This is different than the typical noise that affects atomic motion in optical lattices in that to effectively thermalize the atoms with the bath, the random noise must be at all of each of the random positions, with a high range of k vectors. This implies that the perturbation to the atoms are broadband both spatially and temporally. Higher modulation strength corresponds to a faster timescale for atomic motion. Our synthetic thermal body is an engineered environment that allows the exchange of entropy and energy to equilibrate as if it were a thermodynamical heat reservoir. It is possible that with further tuning we can adiabatically ramp down the temperature of the bath and correspondingly cool the atoms to new limits on temperature and density for direct cooling by suitably engineered baths. We would do this by simultaneously avoiding the constraints of photon-recoil and density-dependent losses from light-assisted collisional processes in traditional laser

118 cooling. New avenues in quantum simulation are possible by coupling atomic gasses to statistically-generated and open environments.

4.4 Constructing a light-atom oscillator

To probe the coherent dynamics of a light-atom coupling, we perform our generalized Raman sideband cooling described in Sec. 4.2 as part of a feedback loop. We perturb the disordered potential and Raman coupling by phase modulating the multimode beam by the fluctuations of the optical pumping beam after it passed through the atoms. We position the overall loop to be at the threshold of oscillation by maintaining a second, parallel feedback path that can be tuned to push the full system across unity gain. The feedback setup can be split into three distinct sections (See Fig. 4.14): the several watt of 1064 nm light that suspends the atoms against gravity and provides Raman coupling; the 780 nm optical pumping light that also passes through the atoms and forms a Sagnac interferometer (SI); and finally the electronic backend which produces a voltage proportional to the phase of the SI. This voltage completes the feedback loop by phase modulating the 1064 nm light. A SI is a common-path interferometer where the beam of light is split and routed around to recombine on the same beamsplitter optic, following the same path as the other half of the beam in the opposite direction. Since both beams follow the same path, the interferometers are extremely phase stable and insensitive to displacements of the mirrors or the beam splitter. Their extreme stability can be shown by the absence of fluctuations even when holding a lit match under the beam path [137]. We utilize a polarizing beam splitter [138] at the entrance of our SI in order to control the amount of σ+ optical pumping that we use in the generalized RSC of the atoms. Typically the output of a SI would be a uniform light intensity which would be proportional to the angular velocity of the interferometer. However, any effect that breaks time-reversal symmetry would also cause a phase shift. In particular, atoms absorbing and spontaneously emitting light would certainly do this. If the system undergoes no mechanical rotation, then the beams in a normal SI combine at the fourth port constructively (though at orthogonal polarizations when used with a polarizing beam splitter as in our experiment). This is because when

119

Figure 4.14: Overview of the three parts of the experiment that connect together to form an oscillator. The dotted blue box encompasses the near detuned light at 780 nm. The Repumper laser light is split off as shown in Fig. 4.3 and shifted by an AOM to be resonant to the F = 1 → F 0 = 0 transition which we call the optical pumping (OP) light. The (1 µW) OP light is sent into the chamber and is part of the machinery to create RSC conditions. Further, it is wrapped around upon itself in a Sagnac interferometer (SI) configuration and is beat with several hundred µW of Repumper light to heterodyne enhance the weak OP signal. This configuration was chosen to minimize technical fluctuations of the light leaving only the dynamics of the atoms absorbing OP light. The green dotted box contains the electronic equipment used to 1) measure the phase response of the OP SI 2) beat the carrier down to DC using the AOM RF signal to eliminate AOM jitter and drifts, and 3) amplify the result to several hundred volt. This high voltage is applied across an EOM in the red dotted box. The box encompasses the 1064 nm laser light setup. The 1064 nm light is phase modulated by the EOM before passing into another SI used to construct a constricting ring pattern in the light. The light is then sent into the chamber and is used both as the conservative potential holding the atoms against gravity, and as a source of Raman coupling for the RSC. the system is rotating, the beams see different path lengths which result in a phase shift relative to each other. In a common path interferometer like the Sagnac, any time varying, reciprocal effects in the beam path which absorbs or refracts some light also modifies the counter-propagating light heading in the opposite direction

120 and leaves the output unchanged. We are instead sensitive to phase shifts of the form, 2π ∆φ = l(n − n ) (4.72) λ + − where n+ and n− are the indices of refraction for circularly polarized light of the same handedness propagating in opposite directions. Only effects that break time-reversal symmetry contribute significantly to phase fluctuations out the fourth port. The setup for the experiment began with what we had constructed for our initial measurement of the generalized Raman sideband cooling, but then we wrapped the optical pumping beam back around to form a SI with the atoms asymmetrically placed early on one path. When the interferometer is slightly misaligned, the output of the fourth port after a polarizer was made to exhibit a slight fringe. The fourth port of the SI was sent through an AOM which then can be used to lock the SI fringe pattern to stabilize the feedback gain.

4.4.1 Feedback loop - 1064 nm section

We make use of our high powered fiber amplifier introduced before in Sec. 4.1.3.5 to both provide the Raman coupling in our disordered Raman sideband cooling, and to provide the conservative potential that suspends the atoms against gravity. As shown in fig. 4.4, the 1064 nm light first passes through an AOM that controls the overall 1064 nm light level in the experiment. Downstream from the AOM is a home built electro-optic phase modulator (EOM) formed from two metallic mirrors pushed up against a slab of lithium niobate with wires attached to the metal surfaces via silver epoxy. The EOM operates as a phase modulator because lithium niobate has a signifi- cant index of refraction (∼ 2.2) and exhibits the linear electro-optic effect where an application of a transverse voltage creates an anisotropic stress on the crystal which develops a fast and slow axis to light propagating through the crystal. A half waveplate is set before the lithium niobate slab to align the input polarization to be parallel to the applied electric field (labeled as the x-axis). The optical wave at the exit of the crystal is [139,140],

Eo(t) = Ei cos(ωt − φ) (4.73)

121 where,

2π φ = (n + ∆n )L (4.74) λ x x

= φ0 + ∆φx (4.75) is the total phase shift which consists of the nominal phase propagation term,

2π φ = Ln (4.76) 0 λ x with nx being the unperturbed index of refraction in the x direction, and a electri- cally induced term, π ∆φ = Ln3rV (4.77) x λD x The voltage that produces a π phase shift in our EOM is about 600 volts. This phase modulator is useful only because of the SI that immediately follows. Our interferometer uses a polarizing beam splitter which means that a half waveplate must be placed before the interferometer oriented at 45 degrees to the PBS’s axis to ensure that the power is balanced down each of the two paths. With a lens in the interferometer, positioned away from the middle of the interferometer, the output of the SI will focus at two different points on axis. The counterclockwise path will focus at a point earlier than the beam that travels in the clockwise direction. The interference pattern between two coaxial beams of differing focal points is shaped like a ring pattern. Thus applying a phase modulation of the EOM yields a contracting shift of the circular fringe pattern which can be seen on a CCD camera after a polarizer set to 45 degrees relative to the polarization of the light. This light is then coupled into four large diameter, multi-mode optical fibers which are aligned as shown in Fig. 4.4 to form the disordered potential that suspends the atoms against gravity. The concentric ring pattern is imaged onto the fiber facet and as it propagates through the optical fiber, it reflects off the inside cladding and creates many images of the anomaly where the k-vector difference between the two beams goes to zero. Thus when driving the EOM at the half wave voltage, multiple regions of the speckle volume contract locally with a negative, non-zero divergence. One optical fiber along the quadrupole axis is passed three times through a solenoid to provide an option for modulating the 1064 nm light with a time-reversal asymmetric drive via the Faraday effect.

122 4.4.2 Feedback loop - 780 nm section

As reviewed in Sec. 4.2, we are able to hold the atoms in the disordered potential for many minutes while directly laser cooling. We take special care to exclude fluctuations in the light due to technical phase noise. We wrap the optical pumping beam around after it passes through the atoms as shown in Fig. 4.15 to form a Polarization SI with heterodyne enhanced detection. [141–143] The amount of optical pumping light in the SI is limited by the amount of Raman coupling obtained from the disordered potential and the effectiveness of the cooling when some light is counter propagating through the atoms in a direction that heats the atoms. Thus, by the time the light exits the SI, the power of the light is down to nanowatt levels. We port several hundred microwatt of the zeroth order of the Repumping AOM, 160 MHz detuned to the optical pumping light, to the experiment chamber and combine with the fourth port of the SI in a single mode optical fiber incident on a fiber coupled photodiode. The fourth port of the SI passes through the Sagnac AOM (Fig. 4.15) of which we take the first order diffraction which causes the heterodyne beat to be only 80 MHz. To keep the feedback loop at at reliable absolute value of gain, we can choose to scan the Sagnac fringe over the optical fiber and lock in a total amount of light power, fixing the gain of the feedback loop. As shown in Fig. 4.15, we added an optional second path for the optical pumping Sagnac that is looped through a solenoid nine times. We can use the Faraday effect through this solenoid to pump angular momentum into the atoms. This is explored in a later experiment.

4.4.3 Feedback loop - electronic backend

The electronics turn the fluctuation of the phase of the optical pumping in the SI into several hundred volts that drive the 1064 nm phase modulator EOM (Sec. 4.4.1). The first stage of the electronics is to remove the jitter and drift of the Sagnac and Optical Pumping AOMs with a vector analyzer. This is done by mixing the Sagnac and optical pumping AOMs and using this to drive the LO port of an IQ modulator with the optical pumping light passing into the RF port. If the resulting IQ ports are separately squared and summed, then to zeroth order, the noise from the mirror fluctuations, and any additive AOM noise is cancelled. To see this clearly, we can

123

Figure 4.15: The 780 nm Optical Pumping light is used as part of the Raman Sideband Cooling which is sent into the experiment chamber at 45 degrees from the plane of the 1064 nm laser beams. We deliberately compromise the cooling effectiveness by wrapping the light around in a common-path, SI in order to both form a topologically nontrivial tool, and to reject technical phase noise. Due to the SI beam path in reality being taken up in a complicated 3D path, we place a Dove prism in the path to rotate the beam orientation back upon itself. Also shown is an extra path which passes through a Faraday rotator (optical fiber through a solenoid) which can be driven to actively modulate the atoms with a time-reversal-asymmetric tool. This is set up for a later experiment not detailed in this chapter. The SI is slightly misaligned to produce a fringe out the fourth port to provide a spatial gradient in the phase with which we can lock the fringe to stabilize the gain of the feedback loop. The output of the SI passes through an AOM which can be used to lock the optical power of the fringe of the SI that is coupled into a single mode fiber. Before being coupled into the fiber, the OP light is heterodyne enhanced by the same laser but coupled in from further upstream.

124

Figure 4.16: Overview of the electronics that transfers the fluctuation of the optical pumping light into a modulation of the 1064 nm light. The electronics in the gray box consist of RF frequency components that mix the heterodyne-enhanced light with the involved AOMs to eliminate jitter and drift (See Fig. 4.14) For part of the data taken, "Case A", we use a vector analyzer circuit in the green box that cancels phase noise. For the data taken without the vector analyzer, Cases "B", and "C", we use a tunable single pole filter centered at 280 Hz. We include a comparator in the path to accent the small signal gain. This is then summed with a digital injection signal from a function generator which is swept across the closed loop resonance as a probe of the Arnold tongue. Before this sum is sent to the EOM, a portion is wrapped back around as a second, shortcut, feedback path to ensure that the entire system is placed close to unity gain. See Fig. 4.17 for details.

write down that the optical pumping light is 2 × 79.9 MHz (ωop) detuned from the Repumper zeroth order light, and the Sagnac AOM shifts the frequency of the fourth port by 81.3 MHz (−ωs). First, as shown in the gray box in Figs. 4.16 and 4.17, we mix together a low power copy of the RF that drives the Sagnac and Optical Pumping AOMs. We model this by multiplying a cosine at ωop with another at ωs,

cos(ωopt + φopn) cos(ωst + φsn) ≈ cos((ωop − ωs)t + (φopn − φsn)) (4.78)

125 where φopn,(φsn) is the phase noise added by the optical pumping (Sagnac) AOM, and we throw away the beat note at high frequency by using a filter. We write down the low passed beat note frequency on the photodiode between the zeroth order Repumper and the output of the Sagnac AOM which we call the “het” signal,

fhet(t) = A cos((ωop − ωs)t + (φopn − φsn) + φmn) (4.79)

where φmn is the phase noise from mirror vibrations. We mix the result from the right hand side of Eq. 4.78 with fhet via an IQ modulator. The output of the I and Q ports are,

I = fhet(t) × cos((ωop − ωs)t + (φopn − φsn)) A ∼ cos(φ ) (4.80) 2 opn and,

Q = fhet(t) × sin((ωop − ωs)t + (φopn − φsn)) A ∼ sin(φ ) (4.81) 2 opn where another low pass filter is applied to get rid of the component at twice the difference frequency. The I and Q ports are each squared and summed to yield the output of the vector analyzer, A2. Thus the vector analyzer’s action has removed the jitter and drift from the AOMs and the phase noise from the mirror mounts. Effectively the amplitude modulation that the atoms cause would show up at DC. The gain from the optical pumping light to the output of the vector analyzer is 2 × 107 V/W. After the vector analyzer, the signal is sent into a mix-up mix-down filter which operates by mixing the input with a carrier from a function generator, filtering the product, and mixing it again. This creates an easily adjustable, single-pole roll-off, filter with a center frequency and bandpass determined by the frequency of a function generator. One unfortunate by-product of this strategy is that about 0.5% of the power from the carrier of the function generator is passed along to the output of the filter. We set the center frequency to be on order of a few hundred Hz to be sensitive to low frequency, coherent oscillations of the atoms across the

126 Ground Photodiode Amplifier Amplifier Amplifier BLP-90 BHP-90 BHP-50 BLP-100 Isolation 0.33 A/W TronTech W110F ZFL-500LN ZFL-500LN 3 db Att. 1:3.6 Heterodyne Beatnote (Optical Pumping and Repumper zeroth order)

Amplifier Amplifier 6 db Att. BBP-70 BLP-100 BHP-50 3 db Att. 3 db Att. ZFL-500LN ZFL-500HLN

Sagnac AOM R I 79 MHz L

BHP-50

Optical Pumping AOM 80X2 MHz

ZFMIQ-90M Squarers

Q Phase L Attenuator Amplifier 780 Beat R Attenuator Adjust Amplifier Sum I AOM Beat Feedback Band Pass Sum Filters

Computer

Mix Up Filter Center Mix Down Filter Flip Flop XOR Frequency ÷2

1 kW Audio Amplifier Attenuator Step-up Comparator Sum Amplifier Injection Transformer To EOM

Case A: No Or

Case B: No Or Spectrum Analyzer Case C: And No

Figure 4.17: Electronic Schematic for the Closed Loop Response. The electronics in the gray box consist of RF frequency parts that begin with a resonantly enhanced 0.33 A/W reverse-biased photodiode, followed by assorted low noise amplifiers and filters to produce a 0 db beat note at 80 MHz. The other half of the parts in the gray box mix out the relative jitter and drift between the Sagnac AOM and OP AOMs (See Fig. 4.14), and is sent into the LO port of an IQ modulator. The next set of components that reside in the dotted green box form a vector analyzer that rejects phase noise at the RF port, that is, SI phase noise. The output of these components is passed through a tunable band pass filter centered at 280 Hz with a full width at half maximum bandwidth of 20 Hz, centered at 280 Hz. We include a comparator in the path to accent the small signal gain. This is then summed with a digital injection signal from a function generator which is swept across the closed loop resonance as a probe of the Arnold tongue. Before this sum is amplified by a 1 kW audio amplifier, a portion is wrapped back around as a second, shortcut, feedback path to ensure that the entire system is placed close to unity gain.

127 cloud, while being at a high enough frequency so as to reduce the time required to capture a significant number of oscillations and escape from the thermal drifts and 1/f noise. At this point the voltage response is several volt, but we want to maximize the small signal gain so we feed the signal into a comparator, which dispenses with the notion that the amplitude of the response will be in a regime that can be faithfully reproduced through linear gain. The vector analyzer squares the signal, which doubles the frequency at "DC". Therefore, in order to see an atomic response at a frequency, f, and drive the 1064 nm light at the same frequency, f, we must divide the signal by two. This is done with a flip flop after the comparator. The voltage at this point is TTL logic, 0-5 V, so it creates a logical place to injection lock the oscillator with a well known ratio of injection drive to closed loop response. Before this output is amplified to ∼ 300 V, a portion is split off and summed in after the vector analyzer that creates a smaller, all-electronic feedback path. We can set the gain separately on this path to force an oscillation of the entire system. In this way, we can precisely place the gain of the overall system to be close to unity gain where we are maximally sensitive to additional gain from the atoms connecting the 1064 nm light and the optical pumping paths. Data is recorded by sampling the XOR output between the injection and the closed loop oscillation. The XOR output is proportional to the phase difference between the two signals. See Fig. 4.17 for details on the electronics used.

4.5 Light-atom oscillator characterization

First we present data to characterize the experimental setup. The half-wave voltage of the phase modulation EOM of the 1064 nm light is 600 V. We run an experiment with Raman sideband cooling in a disordered potential with an optical pumping power of 100 nW. We can optimally run the cooling at much smaller optical powers, but we use a high power to increase the scattering rate for our ∼ 1 second interrogation time frame. After passing through several non-polarizing beamsplitter cubes, the Sagnac AOM, and the single mode optical fiber, there is 5 nW of optical pumping light added into 100 µW of Reference light. The SI is then misaligned slightly in order to produce a dark fringe and maximize sensitivity to phase shifts. We place active filters centered at 140 Hz at the output of the I and Q ports of

128 11

1

1

1 1 1 1 1

Figure 4.18: Red is the power spectrum of the closed loop response from Case A of the setup shown and explained in Fig. 4.17 averaged over 20 traces. The gain of the electronic feedback path was set high enough to cause the closed loop to cross unity gain at 138 Hz. Blue is the same condition, but with no electronic feedback. In this condition there is nothing intentionally closing the Optical Pumping - 1064 nm - photodetector/EOM feedback loop, since there is negligible 1064-OP coupling in the vacuum chamber. The broad hump in the spectrum is a result from the active band pass filters centered at 140 Hz with a 20 Hz bandwidth. Green is the same condition as the Blue curve, but with the optical pumping light blocked entirely. Both the overall power between 90 Hz and 190 Hz and the concentration of power around 140 Hz has decreased in this situation and reveals the marginal, additive noise contributions picked up in the electronics, which may be suppressed as the gain is raised and various stages begin to saturate.

the IQ modulator and record the power spectrum of the closed loop response in Fig. 4.18. We show data here for three different configurations of the heterodyne and amplification electronics. First, we show "Case A" as depicted by Fig. 4.17. We have no atoms loaded into the disordered potential so any response is a result of background couplings and amplified Johnson noise, filtered around 140 Hz with a 20 Hz bandwidth. We have placed enough gain in the loop to cause the Johnson

129 noise to begin saturating op amp stages. The curves in Fig. 4.18 are all averaged over 20 traces. The blue curve is the closed loop response, and the green is the same, but with the optical pumping light blocked. The change in qualitative profile for the green curve is due to taking stages out of saturation. The remaining structure shows pickup noise at nearby frequencies. Because of the comparator in the feedback loop, it is difficult to determine what the small signal gain is, however, we can bring the system into oscillation at 138 Hz by using the auxiliary electronic feedback path. The red curve shows the closed loop response oscillating at 138 Hz after turning up the gain of the auxiliary path. Note that the peak is sharp, and the power at other frequencies has been pushed into the oscillating frequency. Next, in Fig. 4.19 we show the result from removing the vector analyzer and using the single pole roll off filter centered at 280 Hz. Green data was taken with no auxiliary electronic feedback, note that the peak at the center is from the feedthrough from the filter. The other three curves are taken at successive levels of auxiliary electronic feedback. The blue curve was taken just below oscillation at an auxiliary gain of 50. Note that the peak of the power is away from the contribution at 280 Hz and is being pulled to higher frequencies as the gain is increased. The pink curve is with an auxiliary gain of 56.8, and the whole loop is just above the point of unity gain. The oscillation frequency is at 289.5 Hz. The purple curve was taken at a gain 10 times higher than the pink curve, at a auxiliary gain of 268. The peak is within 0.5 Hz of the peak for the pink curve, and the comb of lines comes from the beat between the oscillation frequency of 289.5 Hz and the feedthrough at 280 Hz which is clipped by saturation in the amplifiers and the comparator. For reference, Fig. 4.20 shows the power spectrum of such a clipped beat frequency.

4.5.1 Injection Locking

Our experiment is designed to maximize the sensitivity to light-atom dynamics by feeding back upon the quantum fluctuations caused by the quantization of the electromagnetic field interacting with atoms. To explore the behavior of our oscillator, we injection lock it using a frequency generator fed at a point where the response is saturated by digital logic parts at 0-5 V. The equation governing the conditions necessary for phase locking of two oscillators have been explored by several authors [19,144–147]. An oscillator can be locked to a driving signal if the

130 1

11

1

1

1

1

Figure 4.19: Power Spectrum of the closed loop response from Case B at various levels of feedback gain from the setup shown in Figs. 4.14 and 4.17, and averaged over 20 traces. The center peak at 280 Hz in the green curve is the feed-through from the filter. Details are provided in the main text. The blue curve is the same condition as the green curve, except with the feedback gain tuned just below unity gain. A new peak has formed at higher frequencies, though power from the 280 Hz feedthrough is still significant. The pink curve is with the electronic feedback path with a gain of 56.8 - just above unity gain for the entire closed loop, with an oscillation frequency of 289.5 Hz. Purple is with the electronic feedback gain of 268; the comb of lines, spaced 9.5 Hz apart, are the beat between the closed loop response and the feedthrough at 280 Hz. power of the injection is strong enough or the frequency close enough to the normal operating frequency. In our experiment, we know the ratio between the injection power and the closed loop response accurately because the amplitude of the closed loop is constrained by digital logic to be between 0 V and 5 V, and the amplitude of the injection is easily varied using an attenuated square wave. We can model injection locking with simple driven oscillators. Let us begin by modeling a damped, driven harmonic oscillator, and for physicality, assume that it

131 1

1

1

1

1

11

1

Figure 4.20: The power spectrum of 5 cos(280 × 2πt) + 10 cos(290 × 2πt) clipped at ±1. is a driven, parallel RLC circuit. We calculate the impedance across the RLC as,

R Z = (4.82) 1 − iR( 1 + 1 ) XC XL from which we obtain for the phase,

=(Z) arctan(φ) = (4.83) <(Z) 2 2 R(ω0 − ω ) = 2 (4.84) Lωω0 now approximate, 2 2 ω0 − ω ≈ 2ω0(ω0 − ω) (4.85) and set, Lω 1 = (4.86) R Q

132 Combining Eqs. 4.84 and 4.85 we get,

2Q tan(φ) ≈ (ω0 − ω) (4.87) ω0

q 2 2 Then identifying tan(φ) = Iinj/IRLC and IRLC = Iosc − Iinj, we can show that,

ω ω − ω = 0 η (4.88) 0 inj 2Q where η = Einj/Eosc. The maximum frequency difference between the injection frequency and the natural oscillation frequency of two injection locked oscillators is proportional to the ratio of the electric field strengths. Adler first derived Eq. 4.88 in 1946 [19]. We explore the phase locking region of our oscillator by driving the system at both a high gain value and a low gain value (pink and purple curves in Fig. 4.19 for reference). We scan the injected drive via a linear sweep over 25 seconds and note the general features such as locking in, phase slipping, and frequency pulling. Fig. 4.21 shows three plots from the beginning of one such sweep. The digital outputs were sampled at 192 kHz. The top plot shows the injection, which is simply the output of the function generator, combined with a thresholded version of itself. The middle plot in green is the output of an XOR chip between the injection and the closed loop response. These can be combined to yield the bottom plot which is the closed loop response. It is evident that the XOR output is proportional to the phase difference between the two oscillators modulo π. That is, for two digital signals, the output of an XOR chip, φr, is equivalent to |φi − φl| modulo π. Where φi is the phase of the injection,

φl is the phase of the closed loop response, and φr is the relative phase. Before examining the locking behavior, we remove the vector analyzer and set the center frequency of the filter to be 280 Hz. This is “Case C” in Figs. 4.16 and 4.17. We drive the injection in a linear sweep over 25 Hz from 275 Hz to 300 Hz over 25 sec which we show in Fig. 4.22. The ratio between the injection voltage to the closed loop response is, η = 0.21. For a low injection power of η = 0.25, we observe that although the lock-in region should theoretically be 3.8 Hz assuming Q = 9.5, we observe no locking. This may be due to the persistent feedthrough power at 280 Hz, or the tight band pass filter which creates a narrow window in

133 Injection 0.000 0.001 0.002 0.003 0.004 XOR 0.000 0.001 0.002 0.003 0.004

0.000 0.001 0.002 0.003 0.004 Closed Loop Closed Time [sec]

Figure 4.21: The data are the first 4 ms of a 25 second linear sweep of the injection from 275 Hz to 300 Hz. The apparatus is set up as Case C as shown in Fig. 4.17. Light blue is the function generator output itself, and light green is the output of the XOR chip. Both are sampled at 192 kHz. Dark blue and dark green show the thresholded versions of the raw data from which we can easily reconstruct the closed loop response of the system, shown in red. phase angle where the feedback loop permits oscillation. In fact there are many examples of solved models that exhibit Arnold Tongues that have infinitesimal widths at moderate η and the become significantly gapped only at large nonlinear coupling [148]. An oscillator linearly swept in frequency summed with an independent oscillator free running at a frequency roughly in the center of the swept range will produce a beat note that begins at a high frequency, and then slows to zero as the frequencies match. The beat note will then pick up again in frequency as the swept frequency becomes faster than the free running oscillator’s frequency. The relative phase difference between the two will be parabolic. The frequency of a sinusoidal function is given by,

1 dθ(t) f(t) = . (4.89) 2π dt

134 π

π/2

0 275 280 285 290 295 300 π

π/2

0 275.00 275.05 275.10 275.15 π

π/2 [rad] 0 275.0 275.5 276.0 276.5 277.0 277.5 π

π/2

0 287.0 287.5 288.0 288.5 289.0 289.5 290.0 290.5

[Hz]

Figure 4.22: The relative phase, φr = |φi − φl| modulo π, is the relative phase between the injection and the closed loop response, modulo π. Shown in the figure is the result for the duration of a 25 sec linear sweep of the injection from 275 Hz to 300 Hz at η = 0.21. The bottom three plots show zoomed in subsets of the top plot. The general pattern of oscillation shown is what is expected for two oscillators added together, one fixed in frequency, one linearly swept, where the injection amplitude is insufficient to induce phase locking. The beat frequency is large on the extremes, and slows to a minimum close to the center where the frequencies match.

135 We can invert this function to obtain a function for θ(t).

Z t θ(t) = θ0 + 2π f(τ)dτ (4.90) t0 where θ0 is a constant. The phase of a linearly swept sinusoidal frequency is,

Z t θ(t) = θ0 + 2π (f0 + kτ) dτ (4.91) t0 0 2 = θ0 + π(2f0t + kt ) (4.92)

The relative phase, φr, for two unlocked oscillators will be a (slightly curved) triangle wave of decreasing frequency until the frequency difference is close to zero, where the parabola will be most evident. As the swept frequency becomes higher than the closed loop oscillation frequency, the triangle wave becomes increasingly faster. The expected shape is that of a triangle wave instead of a saw tooth because the phase difference does not jump to zero as φi − φl rises above π. Instead, φr falls to zero at (almost) the same rate at which it rose.

In Fig. 4.23, we plot φr after raising the injection power to η = 0.52 which shows is a significant lock-in region. The response in relative phase begins the same way as with the low-η case, but as the injection approaches the free-running frequency, the phase of the closed loop response begins to slip. The phase slips are visible on the bottom plot. When the injection frequency approaches the free-running oscillation frequency and the relative phase approaches π, the closed loop response is carried forward earlier than its free running response by π radians. Then, when the relative phase changes sign, and begins to decrease in phase, the closed loop response remains locked at φr = 0 for some time. This asymmetric response in the direction of the lock in range is likely due to the fact that the injection is being swept higher in frequency. For clarity, we unwrap the relative phase to show the absolute phase difference,

φi − φl, in Fig. 4.24 at several different injection drive strengths. We plot η = 0.21, 0.244, 0.396, 0.52, 0.56, and 0.64 for the respective colors of purple, dark green, blue, yellow, red, light green. Note the phase slipping behavior on the approach into the locked region in the light green, yellow, and red data. Note that only the yellow data set shows the frequency at which the oscillator fell out of lock. In Fig. 4.25a, we plot the Arnold tongue [149] for the data taken at the feedback

136 π

π/2

0 275 280 285 290 295 300 π

π/2

0 275.00 275.05 275.10 275.15 π

π/2 [rad] 0 275.0 275.5 276.0 276.5 277.0 277.5 π

π/2

0 287.0 287.5 288.0 288.5 289.0 289.5 290.0 290.5

[Hz]

Figure 4.23: The relative phase, φr = |φi − φl| modulo π, is the relative phase between the injection and the closed loop response, modulo π. Shown in the figure is the result for the duration of a 25 sec linear sweep of the injection from 275 Hz to 300 Hz at η = 0.52. The bottom three plots show zoomed in subsets of the top plot. The first part of the sweep is the same as for the case at low injection strength, but at an injection frequency of 289.6 Hz, the closed loop locks to the injection, and remains locked until 297.6 Hz. Note that on the left side of the bottom plot, there is no longer a triangle wave, but instead the phase is slipping when φr approaches π, by π phase.

137 2

2

2 2 22 2 2 2 22 2 2

f

Figure 4.24: The phase difference between the injection and the closed loop response at varying levels of injection power. As the injection approaches the lock-in frequency, the closed loop response begins to slip 2π of phase. The purple curve represents η = 0.21, and dark green, blue, yellow, red, light green are at, η = 0.244, 0.396, 0.52, 0.56, 0.64 respectively. Note that the four curves of the largest injection power all show signs of phase locking as the injection frequency approaches that of the closed loop unity gain crossover frequency. gain level indicated by the purple curve in Fig. 4.19 and Fig. 4.25b is the result for the system at the feedback gain level indicated by the pink curve.

In Fig. 4.26, we plot the derivative of φl over the sweep indicating the frequency of the oscillator. At low η, the closed loop response is mostly flat, undisturbed by the injection. At η = 0.396, the injection just begins to phase lock the free running oscillator and pull its frequency higher. For the top three plots, the injection locks the closed loop response for increasingly larger frequency differences, and the spikes in frequency before phase lock indicate phase slips.

4.6 Atomic participation in the feedback path

The previous section explained the phenomena of phase locking oscillators between an injected drive of a linearly swept wave and a free running oscillator. Even though the entire system from 1064 nm light to the optical pumping SI, to the

138 0.7 0.7

0.6 0.6

η 0.5 η 0.5

0.4 0.4

0.3 0.3

0.2 0.2 280 290 300 280 290 300 Frequency [Hz] Frequency [Hz]

(a) Arnold tongue for the high gain system (b) Arnold tongue for the low gain system whose power spectrum is shown in purple in whose power spectrum is shown in pink in Fig. 4.19. Fig. 4.19.

Figure 4.25: The right hand side of both plots have been truncated at 300 Hz. In a closed loop response with a single pole roll-off, one would expect the unity gain crossover frequency to increase with increasing gain and in that vein, Fig. 4.25b shows an Arnold tongue with center frequency lower than that of Fig. 4.25a at injection powers higher than 0.4.

electronic amplifiers were set up for the data presented in the previous section, the loop was not actually closed through the light since there were no atoms present to mediate a light-light coupling. The next step is to actually close the loop by running the cold atom apparatus and performing Raman sideband cooling in the UHV chamber with the light that forms part of the feedback loop. The gain of the electronic auxiliary path is turned down to just below the oscillation threshold to maximize the sensitivity to the presence of the marginal gain by the atoms. The power spectrum of the closed loop response in this configuration is shown in blue in Fig. 4.19.

4.6.1 Modeling the propagator

The interferometer is transporting spin and orbital angular momentum around both directions of its Sagnac loop. This field of light circulates around the loop

139 300 = 0.64

280 300 = 0.56

280 300 = 0.52

280 300 = 0.396 [Hz] f 280 300 = 0.244

280 300 = 0.21

280 275 280 285 290 295 300

f [Hz]

Figure 4.26: Instantaneous frequency of the closed loop response calculated by taking the derivative of the red data in Fig. 4.21. The bottom plot is the frequency for the sweep with an injection strength of 0.21. The other five plots are the same but at increasing injection strength: 0.21, 0.244, 0.396, 0.52, 0.56, 0.64. The closed loop response is flat up to 0.396, where it then begins to show an effect of being pulled around by the injection.

140 until it reaches the atom cloud. Upon interaction with the atoms, a field line is cut and the momentum is transferred into the atoms. The change in the motion of the atoms can impart a change in the resonance and interaction conditions of the light during RSC. As the momentum is transferred out of one arm of one Sagnac, and transferred into the cloud, the phase of the interferometer will shift in proportion to the amount of light transferred. In this way, the dynamics of the phase of the 1064 nm Sagnac loop indicates a force on the atoms and the phase of the optical pumping interferometer indicates the system’s response to a force from the atoms on the momentum of the optical pumping field. We consider the fundamental modes of oscillation of the system, ψ(p) =

EyagEopψa where Eyag/op is the electric field amplitude from the yag light or the optical pumping light and ψa constitutes the collective interaction from the atoms. The propagator, which gives the amplitude for a mode to travel from one space- time point to another, determines the state of the system given initial conditions. It is given by, [150],

Z ψ(qf , tf ) = ψ(qi, ti)K(qf , tf ; qi, ti)dqi (4.93) in 1D where ψ is the non-relativistic state of the atom plus light system. The probability that the mode is observed at qf at time tf is,

2 P (qf tf ; qiti) = |K(qf , tf ; qi, ti)| (4.94)

The propagator can also be written as, K(qf , tf ; qi, ti) = hqf tf |qitii which can be re-written as,

" # Z DqDp i Z tf K(qf , tf ; qi, ti) = exp dt[pq˙ − H(p, q)] (4.95) h ~ ti

p2 when H = 2m + V (q), this can be simplified to,

" # Z i Z tf K(qf , tf ; qi, ti) = N Dq exp dtL(q, q˙) (4.96) ~ ti

141 From Srendicki [151] Chap. 8, we can begin with a path integral,

Z = hqf tf |qitii (4.97) " # Z i Z tf = Dψ exp dtL(q, q˙) (4.98) ~ ti

This captures the essential features of the atoms. It propagates the action of a mode of light-atom system from one point to another. We can write the Lagrangian of the first order effect of a single redistribution event of photon modes as,

L = K(p, ω)ψ(p, ω)ψ†(p, ω) (4.99) indicating the fundamental normal mode of the light-atom system, and,

1 K = (4.100) p2 − ω2 − m2 ± i is the relativistic, free space propagator [152]. We use the relativistic propagator here because we wish to deal with extremely slow light effects where the speed of light may approach atomic motion.

Z Z φ(x)eipxφ(x0)eipx0 K = d4p d4x (4.101) ω2 + p2 − m2 ± i The bottom factors into two poles which by Cauchy’s formula get placed in the exponential fixing the energy conservation. From the above we can see that the product of the yag and op sagnac phases enter into the Lagrangian at first order. We further can solve K to be,

Z φ˜(p, ω)φ˜†(p, ω) K = d4p (4.102) p2 − ω2 − m2 ± i Z R φ(p, t)eiωt1 dt R φ(p, t )eiωt2 = dωd3p 1 2 (4.103) p2 − ω2 − m2 ± i 1 Z e−ip(x−y) = d4p (4.104) (2π)4 p2 − m2 + i 1 m √ = − δ(s) + √ H(2)(m s) s ≥ 0 (4.105) 4π 8π s 1

142 with s = (x0 − y0)2 − (~x − ~y)2.

4.6.2 Adding atoms to the closed loop

We load 107 atoms into the UHV MOT and cool them with molasses into the disordered potential. Immediately after, we apply Raman sideband cooling for 1 second while simultaneously sweeping the injection from 275 Hz to 289 Hz. The photodetector and electronic amplifier is on continuously throughout the experiment, but the optical pumping light only exists for its section of the loop during the time of the RSC. Thus when the optical pumping light is turned on, a large amount of amplified noise is generated at the heterodyne enhanced photodiode which is ported around to drive the EOM phase modulator for the 1064 nm. During the 1 second where the atoms and the light complete the feedback loop, we lose 70 % of the atoms from the disordered potential. We scan η from 0.23 to 0.5. During the course of repeating the experiment with new batches of atoms, we perform a background check by not loading any atoms at all into the UHV cell but performing the experiment otherwise normally. We choose to do a background check randomly with a 50 % likelihood on any given shot. Fig. 4.27 shows the result from a typical experiment repetition. The top plot is

φr and the bottom is the unwrapped phase as a function of the injection frequency. This particular shot exhibits a phase slip of ∼ π radians at a sweep frequency of 218 Hz. Aside from the phase slip, the response shows the behavior of two oscillators that are uncoupled as in Fig. 4.22. One at a constant frequency of about 282 Hz, and the other being linearly swept as shown on the x-axis. We did not see evidence of any regions of phase lock between the injection and the closed loop with atoms. We show in Fig. 4.28 a random sampling of ten repetitions of the experiment with atoms loaded in the UHV cell (left hand side, green), and another sample of ten repetitions of the experiment with no atoms loaded (right hand side, red). The y-axis of each subplot is the unwrapped phase, like the bottom plot of Fig. 4.27, modulo 2π. Phase slipping occurs in situations of both with atoms in the loop and without atoms in the loop. A hypothesis we make about this preliminary experiment is that when there are atoms in the loop connecting the 1064 nm light to the 780 nm optical pumping light, the closed loop experiences more phase slips

143 Figure 4.27: Relative phase (top) and unwrapped phase (bottom) of the closed loop response relative to the injection with atoms closing the feedback loop between the optical pumping light and the 1064 nm light. The injection is set to η = 0.232 and is swept from 275 Hz, to 289 Hz in 1 sec. The electronic feedback path gain is set just under the point of oscillation, whose power spectrum is shown in blue in Fig. 4.18. Note the existence of a phase slip of ∼ π radians at an injection frequency of 281 Hz. than when there are no atoms. We plot in Fig. 4.29, the relative proportion of experiment repetitions at η = 0.23 that contain 0 to 4 phase slips. This sample contains nna = 123 background shots with no atoms loaded, and nwa = 136 shots with atoms loaded. As we can see qualitatively in Fig. 4.29, the majority of shots with multiple phase slips occur when there are atoms loaded into the potential. To determine how likely this distribution

144 W W ϕ ϕ

Figure 4.28: The unwrapped phase modulo 2π of a random sample of multiple repetition of the experiment with atoms on the left side, and control shots with no atoms loaded on the right in red. While we do not see evidence of phase locking between the closed loop and the injection drive, we do see several instances of phase slips of ∼ π radians. is to occur due to chance, having nothing to do with the effect of any atoms present, we perform a simple Monte Carlo permutation test [153]. A permutation test [154] is a statistical test where the distribution of the test statistic under the null hypothesis is calculated by rearranging the labels of the

145 0.6 Without Atoms With Atoms 0.4

0.2

0.0 Relative Number of Shots of Number Relative

0 Slips 1 Slips 2 Slips 3 Slips 4 Slips

Figure 4.29: We show two histograms of the number of phase slips that occur during an experiment repetition scaled to the total number of repetitions with and without atoms. We see that during experiment repetitions with atoms connecting the 1064 nm light to the 780 nm optical pumping light, there is a greater chance of inducing several phase slips as compared to when there are no atoms involved. Specifically, we calculate via a Monte Carlo permutation test that this number of additional phase slips with atoms present is statistically significant at the p < 0.05 level. See text for details. observed data points30 into all possible combinations that preserve the relative number of observations. The significance level of the experiment can then be determined by calculating the fraction of rearrangements (p-value) that yield a test statistic larger than the observed one. Since the number of rearrangements grows as the factorial, it can quickly become infeasible to actually calculate all possible permutations for hundreds of observations. However, very accurate estimates of the p-value can be obtained by a Monte Carlo method of randomly sampling possible permutations [155,156] less than a thousand times. Our experiment that yielded the two histograms shown in Fig. 4.29 can be tested this way. The test statistic is the total number of phase slips for the sample with atoms compared to the number without atoms. After noting the observed number of phase slips for each, we scramble the identifier for each experiment

30In this case the labels are simply, “with atoms” and “without atoms”

146 repetition, while preserving the total number of each, nwa and nna. We find that after 107 re-samplings, the hypothesis of the presence of atoms causing more phase slips than without atoms to be significant at a level of p ≤ 0.05. We conclude that this preliminary experiment setup is sensitive enough to capture the effect of the coherent dynamics of a light-atom oscillator.

147 Appendix A| Experimental Apparatus 2.0

The apparatus that I describe here, and in which we performed all of our experiments have a number of annoyances that could be rectified with a second round of engineering. First, the chamber is very large; the distance between the MOTs is nearly 80 cm. This distance means that the atom cloud will expand in transit, and generally clip the sides of apertures as they are transported through the chamber. The transfer efficiency is only ∼ 1% which means that several source MOTs full of atoms must be collected and caught in the UHV cell before acquiring enough atoms to begin an experiment cycle. The size of the apparatus also means that the device will be built high up into the air off the optical table inviting vibrations in the large structure1. Finally, while the apparatus is fairly general purpose and versatile, with its regions of optical access being large, totally glass cells, its optical access can be improved by placing the first surface optical lenses inside the chamber permitting numerical apertures that approach, or even exceed 1. With these criticisms and improvements in mind, we designed a new, more compact, and ideally simpler to operate, apparatus utilitzing 40K and 133Cs. The source of cold, compressed atoms would begin with a pyramid MOT [158] with a hole in the tip which would serve as a robust, achromatic version of a 2D MOT [159], spraying atoms at doppler cooled temperatures out the tip of the pMOT below. The stream of cold atoms would pass through a region ∼ 2 in long pumped by an ion pump and a TSP, and pass through a ∼ 1mm diameter hole drilled through the center of a parabolic achromatic gold mirror serving as the first optic of the imaging system in a UHV region. The atoms would be held near the focus of the mirror

1Though the deflection of a solid cylinder is by the Euler-Bernoulli equation, ∝ q4 where q is the distributed load [157]

148 by lasers projected upwards through an optically flat window which would be in a less critical part of the imaging path. These lasers can be fixed in place and tuned to the right intersection using digital mirror arrays which have been used in other labs [160] to produce extraordinarily excellent control over the light position and angle. Finally, the internal divisions of the apparatus separating the chamber into sections that are differentially pumped will be constructed by standard sized glass plates and cylinders, afixed to each other by glass bonding nano particles [161].

A.1 Experimental apparatus - Collisional Microscope

The new apparatus is specifically for general purpose projection of arbitrary lattice potentials and extremely high photon collection efficiency. The primary design goal was to enable the highest resolution and photon collection efficiency while keeping the apparatus general purpose and versatile. We chose 40K and 133Cs in order have the ability to work with a sample that we’ll probe of either bosons or fermions. A first experiment will begin with the Bose Hubbard system because it is well studied and contains everything that we need to make our studies on entanglement entropy. 133Cs was chosen for our boson because of the convenience in forming BECs and the possibility of using it to sympathetically cool our fermionic species. 40K was chosen as the fermionic species for its comparatively high mass compared to 6Li, aiding in co-trapping with Cesium, and making the photon recoil energy higher.

A.1.1 Transport and Imaging

The apparatus has three stages for the atoms to pass through during the course of the experiment. First, we intend on using alkali dispensers (getters) to create a sufficient background gas pressure and collect the vapor into a pyramid MOT. The dispensers were chosen partially for simplicity and for the lower chance of catastrophic failure, but also because 40K can be very expensive, prohibiting the use of large quantities of the enriched isotope through for example, a Zeeman slower. The getters will be attached to a linear shift stage and a bellows so that during normal operation the getters can be placed closed to the location of the MOT, but simultaneously, replacement of the atoms can be done without contaminating the entire apparatus with atmosphere.

149 Next, the atoms must transfer from the source region which is at a higher pressure (∼ 10−8 Torr) to a region of ultra high vacuum (∼ 10−12 Torr) that is required in order for background gas collisions during a typical experiment to be negligible. We plan to open a small hole ∼ 1mm in diameter at the tip of the PMOT, and blow the cold atoms through directly down into the UHV region with radiation pressure. We use a pyramid MOT to initially trap and cool the atoms from background vapor. [164] Since a reflection of light changes its helicity, it is possible to use multiple reflections to take one large σ+ light beam, even at multiple wavelengths to address each atomic species, and shine it at four mirrors oriented like the inside of a corner to create the required helicity of light in the correct direction of propagation. In order to maximize the loading rate, we made the size of the PMOT as large as is convenient with the vacuum apparatus. To achieve a high transfer efficiency between source and UHV, the opening of the PMOT was placed as close as possible to the UHV chamber, see Fig. A.1 We plan to maintain the high differential pressure by keeping an intermediate pumping stage between the source and the UHV. A second UHV MOT will be placed at the focus of the mirror collecting the stream of cold atoms created in the pyramid MOT. A set of three magnetic field coils will be placed on axis of the the MOT beams in a way that will allow a field zero at both MOTs simultaneously, and can create a Feshbach field of at least 250G with a curvature of less than 5 Hz. A minimum field of 250G was chosen because the largest convenient s-wave Feshbach resonance for 40K is at 224G. [165] The reflective optical design is highly achromatic, which is imperative for the generation of commensurate multichromatic lattices through holographic projection. The apparatus also permits addition of more conventional imaging techniques with dispersive lenses outside the vacuum chamber.

150 Figure A.1: The main experiment chamber for collision microscopy. MOT light comes in from the top 120 mm diameter viewport (A) and fills the entire pyramid MOT (B) volume. The cold atoms are pushed down through a small ∼ 1 mm diameter opening into an intermediate region (C) that is continuously pumped through D by an ion pump. This intermediate region with its two small (∼ 1 mm diameter) apertures help create a differential pressure between the source and the UHV regions of 104 Torr. The atoms are pushed through the second aperture, a small (∼ 1.8 mm diameter) hole drilled through the center of the parabolic mirror into the UHV region (F). The bottom viewport will be set in place using a copper flange with a knife edge cut into it to create the vacuum seal which will permit high quality viewports with little aberration and non magnetic glass-to-metal transitions. [162] All divisions and optics will be held either with metal clips or with a hydroxide bonding technique [163] that I have tested to make sure is UHV compatible.

151 Figure A.2: The vacuum chamber with attached pumps, an ion gauge, and electrical feedthroughs. The optical access to the atoms will be from the top and the bottom; the side ports will be primarily used to service the supporting vacuum equipment. The source atoms will be supplied via getter dispensers attached to a bellows and a linear shift stage (not shown).

152 Appendix B| Novel, UHV compatible Glass Bonding Techniques

B.1 Introduction

Reliable optical bonding techniques that are UHV compatible are difficult to find. There are only a handful of materials with a sufficiently low outgassing rate such that they can be used in vacuum chambers with ultimate pressures on order of 10−12 Torr. Metal, Glass, Ceramic, and a few specialty materials can be used. [113] Epoxies and glues [166] generally outgass a significant amount and are avoided for UHV work. Other widely used optical assembly technologies include optical contact bond- ing [167], frit bonding [168, 169], and diffusion bonding [170, 171]. However, frit bonding and diffusion bonding require high temperatures that may damage optical components, and contact bonding is difficult and requires extremely flat surfaces and atomic smoothness and cleanliness [172,173]. Since epoxies and glues are not useable, and only a few materials can be used at all, mounting optics inside a vacuum chamber at UHV pressures is not simple. However, in this paper we show that a hydroxide-catalyst bonding technique first described by Gwo [163,174,175] and then extended with a nanoparticle sol-gel [161] is suitable for UHV conditions. The hydroxide-catalyst bonding technique was originally invented by Gwo in 1998 to aid in the Gravity Probe B mission. [163] Since then it has been developed, used, and studied by other gravity wave and interferometric experiments such as LIGO, GEO 600, and LISA. [176–179] It has a controllable setting time by

153 exploiting stoichiometry which allows in-situ alignment, the bonding solution joints glass substrates with a silicon oxide interface keeping the differential thermal stress small [180], the interface is only tens of nanometers thick, can withstand high laser intensities with little absorption, and its bond strength is comparable to the strength of fused silica. [176] In hydroxide-catalyst bonding, an alkaline solution of either potassium hydroxide, sodium hydroxide, or sodium silicate [181, 182] is placed on a surface that is to be bonded. The chemistry of the bond consists of three steps: hydration and etching, polymerization, and dehydration. First, the aqueous hydroxide ions etch the fused silicon substrate liberating silicate ions in a hydration reaction. [183] This continues until the pH drops below 11 whereupon the silicate ions dissociate and form siloxane chains. The formation of siloxane chains joining the two substrates begins the formation of the bond: [184]

− − − SiO2 + OH + 2H2O → Si(OH)5 → Si(OH)4 + OH

Finally as the water evaporates out the bond or migrates into the bulk of the fused silica, the bond tightens and cures. [176,185] However, this technique requires surfaces that are flat to within λ/10, the solution must be filtered, and bonding must occur within a stringent cleanliness environment. [176] In 2007, the addition of sol-gel technology [186] by adding nano particles to the bonding solution lifted many of the stringent requirements while maintaining excellent bonding properties. [161] By using silicon nano particles suspended in a potassium hydroxide solution, the nano particles fill larger defects in the surface quality of the glass substrate and serve as general filler material so that the siloxane network linkages [187] can connect the two surfaces to bond even over surfaces that do not meet the flatness or smoothness requirements of catalyze hydroxide bonding. Further, after silylation with hydrophobic surface groups, the bonding is resistant to organic solvents. [188] Using soda-lime glass slides, the nanoparticle sol-gel bonding technique can reliably bond slides at room temperature. After 300 hours of curing, the bond strength exceeds 14 MPa. Slides bonded using this method transmitted over 99% of the light, and the setting time of the bond depends

154 of the ratio between the concentration of hydroxyl ions and nano particles. In the interest of work that needs a bonding technique which is not only reliable, but is compatible with UHV, such as for cold atoms or cavity interferometers groups, the rest of the paper describes the compatibility of the hydroxide-catalysis bonding method with UHV conditions.

B.2 Method

First the UHV compatability of the hydroxide-catalyst bonding technique without nano particles was tested. The bonding solution comprised of 0.25 mL of 1.39 g/mL 1 NaSiO2 and 1.5 mL of distilled water shaken vigorously for 1 min, centrifuged for 30 sec, chilled in ice water for 1 min, and then filtered through a 0.2 micron syringeless filter2. The cooling of the solution is necessary in order for the solution to pass through the filter. 2 µL of solution was placed on a 1/2 in diameter fused silica substrate inside a laminar flow hood with a hepa filter, and another identical piece of fused silica was placed lightly on top. After setting for 23 minutes, the bonded glass was placed in a vacuum chamber attached to a turbo pump and began to evacuate the air. After 147 hours in high vacuum, a bake was applied at no more that 1 degree per minute. The glass was above 250◦C for 5.5 hours attaining a maximum of 274◦C. After cooling, the pressure in the chamber was 6 × 10−10 Torr a total of 189 hours since initially bonded. After retrieval from the vacuum chamber after 504 hours in high vacuum the bond still held. Next the nanoparticle, hydroxide-catalyst bonding solution was tested. The nanoparticle bonding solution was made by combining 4.21 mg of solid KOH,

8.3 mL SiO2 nano particles (0.05 mol of 29.8 w/w % SiO2 suspended in an alkaline solution of a pH of 10.1, the nano particles are ∼ 7 nm in diameter), 3, and 2.5 mL of potassium silicate4 and diluting the mixture with distilled water to 25 mL. The glass substrate, 1 in diameter fused silica substrate5 with 40/20 scratch/dig and λ/10 flatness, was cleaned by sonicating in acetone, rinsing it with methanol, acetone and distilled water and then air dried in atmosphere under lens tissue.

1Sigma Aldrich p/n 338443 2Whatman Autovial AV125EORG 3The nano particles were Ludox SM from W. R. Grace & Co. 4KASIL 6 from the PQ Corporation. 5Thorlabs p/n: PF10-03

155 Seven 2.2 µL drops of the nanoparticle bonding solution was applied to the surface of the substrate and an identical substrate was placed lightly on top. The bond was allowed to cure for 31 days before being placed in a small stainless steel vacuum chamber that consisted of a cold cathode gauge6, a 20 l/s ion pump7, a gate valve8, and two tees and a 6 in nipple to connect them. The chamber was baked starting with a ramp of no more than 1◦C per minute and held at 250◦C for 57 hours. After the bake, the pressure settled at 5 × 10−11Torr. The outgassing rate was measured by switching off the ion pump and measuring the rate of rise of the pressure in the chamber. It is known that cold cathode gauges act as weak pumps when activated [189], so we fit the outgassing rise to the following model,

1 d(PV ) Q − P · S = Q = · (B.1) Outgassing A dt where Q is the gas load per unit surface area, the second term is the throughput of gas pumped by the components of the chamber per unit surface area, P is the chamber pressure, A is the surface area of the chamber, V is the volume of the chamber, and S is the net pumping speed. With the bonded glass exposed to the cathode gauge, we obtained seven measurements of the outgassing of the chamber plus the glass bond shown in the red squares, and eight measurements of the outgassing of the chamber with the glass bond valved off from the main chamber shown in the blue circles. See Fig. B.3. A weighted average over the fits to pressure rise gives an outgassing rate for the chamber plus glass bond of 3.4 × 10−13 ± 0.5 × 10−13 Torr L sec−1cm−2. With the glass valved off from the main chamber, we obtain an outgassing rate of 2.8×10−13 ±0.3×10−13 Torr L sec−1cm−2. The outgassing of the glass bond is compatible with the outgassing from the background stainless steel. After the bake, the outside of the nanoparticle bond became slightly brownish discolored, and the bond interface contained milky splotches. Both to check the accuracy of the outgassing measurement and to determine whether the nanoparticle bond can hold a seal against atmosphere, a pocket of air was trapped within a glass substrate and sealed with the nanoparticle catalyst-

6IMG-300 Inverted Magnetron Gauge from Agilent 7VacIon Plus 20 from Varian 8MDC Vacuum Products Corp. GV-1500M

156 Figure B.1: The two 1in glass substrates immediately after bonding

Figure B.2: The 1/2 in substrate with a pocket drilled into the center. Another 1/2 in glass blank was subsequently bonded over the top.

157 9e-9 4e-8

8e-9 (a)3.5e-8 (b) 7e-9 3e-8 6e-9 2.5e-8 5e-9 2e-8 4e-9 1.5e-8 3e-9 Pressure [Torr] [Torr] Pressure 1e-8 2e-9

1e-9 5e-9

0 0 −5 0 5 10 15 20 25 30 −5 0 5 10 15 20 Time [sec]

Figure B.3: Pressure versus time after shutting off the ion pump for the nanoparticle hydroxide-catalyzed bonded glass. The triangles in red are data taken with the bonded glass exposed to the main chamber, and the circles in blue are data taken with the glass valved off from the main chamber. The lines are fits (See Eq. B.1) to the data. (a) Data taken for the 1in diameter bonded substrates. The mean and standard deviation of the fit to the outgassing rate with the 1 in glass exposed is 3.4 × 10−13 ± 0.5 × 10−13 Torr L sec−1cm−2 and 2.8 × 10−13 ± 0.3 × 10−13 Torr L sec−1cm−2 with the glass valved off. (b) data taken for the 1/2 in diameter substrate where one piece of glass is sealing over a pocket drilled into the other. The mean and standard deviation of the fit to the outgassing rate with the 1/2in diameter capsule exposed is 1.4 × 10−12 ± 0.9 × 10−12 Torr L sec−1cm−2 and 1.2 × 10−12 ± 0.3 × 10−12 Torr L sec−1cm−2 with the capsule valved off. hydroxide bonding technique. We drilled a 1.8 mm diameter hole 0.1 in deep in the center of a 1/2 in diameter uncoated mirror blank.9 The drill bit was 1.8mm in diameter with a 120 electroplated diamond mesh10. The drill spindle

9Thorlabs p/n: PF05-03 10Crystalite Corporation p/n: C5210950

158 was operated at 210 RPM, drilling no more than 1 mil deep per 5-10 seconds, keeping the glass and drill bit submerged in water to reduce the thermal stress of the substrate. To protect the surface of the substrate while drilling, we coated the optic with nail polish. After drilling the hole, the nail polish was removed with acetone, and then the same cleaning procedure was followed for the previous nanoparticle hydroxide bonding. Three 1.27 µL drops of the same nanoparticle solution was placed around the hole drilled into the glass, and a second substrate was placed lightly on top. This was allowed to cure for 54 days before being placed in a vacuum chamber. We then proceeded to bake the chamber at 140 C by a ramp of no more than 1 deg C per minute, held at 140 C for 22 hours, which attained a maximum temperature of 165◦C. After the bake, we performed the same outgassing measurements as before, by turning off the ion pump and fitting the rise in pressure to our model B.1. After three measurements of the chamber with the capsule exposed to the main chamber, it was then valved off and three outgassing measurements were taken again. The mean and standard deviation of the fit to the outgassing rate with the glass exposed is 1.4 × 10−12 ± 0.9 × 10−12 Torr L sec−1cm−2 and 1.2 × 10−12 ± 0.3 × 10−12 Torr L sec−1cm−2 with the capsule valved off.

B.3 Conclusion

Hydroxide-catalysis bonding has been known as a useful alternative to such optical bonding techniques as frit and diffusion bonding, contact bonding, and epoxy bonding as it avoids high temperatures, maintains bond as strong as the fused silica substrate while thermally cycled, and is less technically difficult as contact bonding. The addition of nanoparticle sol-gel to the hydroxide-catalyst bonding solution allows even more flexibility in bonding glass substrates. In this paper we have studied the UHV compatibility of the bonding technique. The tests indicate that the contribution of gas load from a hydroxide-catalyst bond is not a significant source of outgassing for UHV work, and specifically the nanoparticle solution outgassing is compatible with the background outgassing of a baked stainless steel vacuum chamber. Further, our test shows that we see no significant outgassing even when the bond is holding against atmospheric pressure. This suggests that the technique allows the placement of optics inside and at the interfaces of vacuum equipment designed for people doing studies of atomic physics

159 and interferometry. It may also be used as a new method for constructing glass cells or other vacuum interfaces.

160 Bibliography

[1] Garrison, J. C. (2014) Quantum optics, Oxford University Press, Oxford. [2] Zauderer, E. (2006) Partial differential equations of applied mathematics, Wiley-Interscience, Hoboken, N.J.

[3] Unruh, W. G. (1976) “Notes on black-hole evaporation,” Phys. Rev. D, 14, pp. 870–892. URL https://link.aps.org/doi/10.1103/PhysRevD.14.870

[4] Leonhardt, U. (2010) Essential quantum optics : from quantum measurements to black holes, Cambridge University Press, Cambridge, UK.

[5] Fradkin, E. (2013) Field Theories of Condensed Matter Physics, Cambridge University Press, Cambridge.

[6] Dekker, H. (1975) “On the quantization of dissipative systems in the lagrange-Hamilton formalism,” Zeitschrift für Physik B Condensed Matter, 21(3), pp. 295–300. URL https://doi.org/10.1007/BF01313310

[7] Morse, P. and H. Feshbach (1953) Methods of theoretical physics, McGraw-Hill Book Company, New York, New York.

[8] Frankel, T. (2012) The geometry of physics : an introduction, Cambridge University Press, Cambridge New York.

[9] Sorokin, D. (2010) “Lagrangian formalism for fields,” Scholarpedia, 5(8), p. 10223, revision #169741.

[10] Landau, L. D. (1976) Mechanics, Pergamon Press, Oxford New York. [11] Goldstein, H. (2002) Classical mechanics, Addison Wesley, San Francisco. [12] Senator, M. (2006) “Synchronization of two coupled escapement-driven pendulum clocks,” Journal of Sound and Vibration, 291(3), pp. 566 – 603. URL http://www.sciencedirect.com/science/article/pii/S0022460X05004116

161 [13] Oliveira, H. M. and L. V. Melo (2015) “Huygens synchronization of two clocks,” Scientific Reports, 5, pp. 11548 EP –, article. URL http://dx.doi.org/10.1038/srep11548

[14] Walker, T. J. (1969) “Acoustic Synchrony: Two Mechanisms in the Snowy Tree Cricket,” Science, 166(3907), pp. 891–894, http://science.sciencemag.org/content/166/3907/891.full.pdf. URL http://science.sciencemag.org/content/166/3907/891

[15] BUCK, J. and E. BUCK (1966) “Biology of Synchronous Flashing of Fireflies,” Nature, 211, pp. 562 EP –. URL http://dx.doi.org/10.1038/211562a0

[16] Okamoto, K., A. Kijima, Y. Umeno, and H. Shima (2016) “Synchronization in flickering of three-coupled candle flames,” Scientific Reports, 6, pp. 36145 EP –, article. URL http://dx.doi.org/10.1038/srep36145

[17] Néda, Z., E. Ravasz, Y. Brechet, T. Vicsek, and A.-L. Barabási (2000) “The sound of many hands clapping,” Nature, 403, pp. 849 EP –. URL http://dx.doi.org/10.1038/35002660

[18] Wiesenfeld, K., P. Colet, and S. H. Strogatz (1996) “Synchronization Transitions in a Disordered Josephson Series Array,” Phys. Rev. Lett., 76, pp. 404–407. URL https://link.aps.org/doi/10.1103/PhysRevLett.76.404

[19] Adler, R. (1946) “A Study of Locking Phenomena in Oscillators,” Proceedings of the IRE, 34(6), pp. 351–357.

[20] York, R. A. (1993) “Nonlinear analysis of phase relationships in quasi-optical oscillator arrays,” IEEE Transactions on Microwave Theory and Techniques, 41(10), pp. 1799–1809.

[21] Feigenbaum, M. J., L. P. Kadanoff, and S. J. Shenker (1982) “Quasiperiodicity in dissipative systems: A renormalization group analysis,” Physica D: Nonlinear Phenomena, 5(2), pp. 370 – 386. URL http://www.sciencedirect.com/science/article/pii/0167278982900306

[22] Huygens, C. (1986) Christiaan Huygens’ the pendulum clock, or, Geometrical demonstrations concerning the motion of pendula as applied to clocks, Iowa State University Press, Ames.

[23] Pikovsky, A. (2001) Synchronization : a universal concept in nonlinear sciences, Cambridge University Press, Cambridge.

162 [24] Kuramoto, Y. (1975) “Self-entrainment of a population of coupled non-linear oscillators,” in International Symposium on Mathematical Problems in Theoretical Physics (H. Araki, ed.), Springer Berlin Heidelberg, Berlin, Heidelberg, pp. 420–422.

[25] Winfree, A. T. (1967) “Biological rhythms and the behavior of populations of coupled oscillators,” Journal of Theoretical Biology, 16(1), pp. 15 – 42. URL http://www.sciencedirect.com/science/article/pii/0022519367900513

[26] van der Pol, B. (1934) “The Nonlinear Theory of Electric Oscillations,” Proceedings of the Institute of Radio Engineers, 22(9), pp. 1051–1086.

[27] Acebrón, J. A., L. L. Bonilla, C. J. Pérez Vicente, F. Ritort, and R. Spigler (2005) “The Kuramoto model: A simple paradigm for synchronization phenomena,” Rev. Mod. Phys., 77, pp. 137–185. URL https://link.aps.org/doi/10.1103/RevModPhys.77.137

[28] Arnol’d, V. I. (1983) “Remarks on the perturbation theory for problems of Mathieu type,” Russian Mathematical Survey, 38, pp. 215–233. URL http://iopscience.iop.org/article/10.1070/RM1983v038n04ABEH004210/meta

[29] Zabusky, N. J. and M. D. Kruskal (1965) “Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States,” Phys. Rev. Lett., 15, pp. 240–243. URL https://link.aps.org/doi/10.1103/PhysRevLett.15.240

[30] Witthaut, D. and M. Timme (2014) “Kuramoto dynamics in Hamiltonian systems,” Phys. Rev. E, 90, p. 032917. URL https://link.aps.org/doi/10.1103/PhysRevE.90.032917

[31] Bobenko, A. I. and Y. B. Suris (2015) “Discrete Pluriharmonic Functions as Solutions of Linear Pluri-Lagrangian Systems,” Communications in Mathematical Physics, 336(1), pp. 199–215. URL https://doi.org/10.1007/s00220-014-2240-5

[32] Boll, R., M. Petrera, and Y. B. Suris (2013) “Multi-time Lagrangian 1-forms for families of Bäcklund transformations: Toda-type systems,” Journal of Physics A: Mathematical and Theoretical, 46(27), p. 275204. URL http://stacks.iop.org/1751-8121/46/i=27/a=275204 [33] ——— (2014) “What is integrability of discrete variational systems?” Proceedings of the Royal Soci- ety of London A: Mathematical, Physical and Engineering Sciences, 470(2162),

163 http://rspa.royalsocietypublishing.org/content/470/2162/20130550.full.pdf. URL http://rspa.royalsocietypublishing.org/content/470/2162/20130550

[34] Suris, Y. B. (2013) “Variational formulation of commuting Hamiltonian flows: Multi-time Lagrangian 1-forms,” Journal of Geometric Mechanics, 5, p. 365. URL http://aimsciences.org//article/id/7a98dba6-c0c8-4830-91c2-ed01aedf36fd

[35] Atkinson, J., S. B. Lobb, and F. W. Nijhoff (2012) “An integrable multicomponent quad-equation and its Lagrangian formulation,” Theoretical and Mathematical Physics, 173(3), pp. 1644–1653. URL https://doi.org/10.1007/s11232-012-0138-y

[36] Lobb, S. and F. Nijhoff (2009) “Lagrangian multiforms and multidimensional consistency,” Journal of Physics A: Mathematical and Theoretical, 42(45), p. 454013. URL http://stacks.iop.org/1751-8121/42/i=45/a=454013

[37] Lobb, S. B. and F. W. Nijhoff (2010) “Lagrangian multiform structure for the lattice Gel’fand-Dikii hierarchy,” Journal of Physics A: Mathematical and Theoretical, 43(7), p. 072003. URL http://stacks.iop.org/1751-8121/43/i=7/a=072003

[38] Lobb, S. B., F. W. Nijhoff, and G. R. W. Quispel (2009) “Lagrangian multiform structure for the lattice KP system,” Journal of Physics A: Mathematical and Theoretical, 42(47), p. 472002. URL http://stacks.iop.org/1751-8121/42/i=47/a=472002

[39] Yoo-Kong, S., S. Lobb, and F. Nijhoff (2011) “Discrete-time Calogero-Moser system and Lagrangian 1-form structure,” Journal of Physics A: Mathematical and Theoretical, 44(36), p. 365203. URL http://stacks.iop.org/1751-8121/44/i=36/a=365203

[40] Suris, Y. B. (2013) “Variational symmetries and pluri-Lagrangian systems,” ArXiv e-prints, 1307.2639.

[41] Eddington, A. (1987) Space, time, and gravitation : an outline of the general relativity theory, Cambridge University Press, Cambridge Cambridgeshire New York.

[42] Plebanski, J. (1960) “Electromagnetic Waves in Gravitational Fields,” Phys. Rev., 118, pp. 1396–1408. URL https://link.aps.org/doi/10.1103/PhysRev.118.1396

164 [43] de Felice, F. (1971) “On the gravitational field acting as an optical medium,” General Relativity and Gravitation, 2(4), pp. 347–357. URL https://doi.org/10.1007/BF00758153

[44] Pendry, J. B., D. Schurig, and D. R. Smith (2006) “Controlling Electromagnetic Fields,” Science, 312(5781), pp. 1780–1782, http://science.sciencemag.org/content/312/5781/1780.full.pdf. URL http://science.sciencemag.org/content/312/5781/1780

[45] Gordon, W. (1923) “Zur Lichtfortpflanzung nach der Relativitätstheorie,” Annalen der Physik, 377(22), pp. 421–456. URL http://dx.doi.org/10.1002/andp.19233772202

[46] Fagnocchi, S., S. Finazzi, S. Liberati, M. Kormos, and A. Trombettoni (2010) “Relativistic BoseÐEinstein condensates: a new system for analogue models of gravity,” New Journal of Physics, 12(9), p. 095012. URL http://stacks.iop.org/1367-2630/12/i=9/a=095012

[47] Girelli, F., S. Liberati, and L. Sindoni (2008) “Gravitational dynamics in Bose-Einstein condensates,” Phys. Rev. D, 78, p. 084013. URL https://link.aps.org/doi/10.1103/PhysRevD.78.084013

[48] Cropp, B., S. Liberati, and R. Turcati (2016) “Analogue black holes in relativistic BECs: Mimicking Killing and universal horizons,” Phys. Rev. D, 94, p. 063003. URL https://link.aps.org/doi/10.1103/PhysRevD.94.063003

[49] Barceló, C., S. Liberati, and M. Visser (2011) “Analogue Gravity,” Living Reviews in Relativity, 14(1), p. 3. URL https://doi.org/10.12942/lrr-2011-3

[50] Jackson, J. (1999) Classical electrodynamics, Wiley, New York.

[51] Misner, C. (2017) Gravitation, Princeton University Press, Princeton, N.J.

[52] Post, E. (1962) Formal Structure of Electromagnetics, North-Holland Publishing Company, Amsterdam.

[53] Paul, O. and M. Rahm (2012) “Covariant description of transformation optics in nonlinear media,” Opt. Express, 20(8), pp. 8982–8997. URL http://www.opticsexpress.org/abstract.cfm?URI=oe-20-8-8982

[54] Boyd, R. (2008) Nonlinear optics, Academic Press, Amsterdam Boston.

165 [55] Pendry, J. B. (2000) “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett., 85, pp. 3966–3969. URL https://link.aps.org/doi/10.1103/PhysRevLett.85.3966

[56] Leonhardt, U. and T. G. Philbin (2006) “General relativity in electrical engineering,” New Journal of Physics, 8(10), p. 247. URL http://stacks.iop.org/1367-2630/8/i=10/a=247

[57] Thompson, R. T. and J. Frauendiener (2010) “Dielectric analog space-times,” Phys. Rev. D, 82, p. 124021. URL https://link.aps.org/doi/10.1103/PhysRevD.82.124021

[58] Thompson, R. T., S. A. Cummer, and J. Frauendiener (2011) “A completely covariant approach to transformation optics,” Journal of Optics, 13(2), p. 024008. URL http://stacks.iop.org/2040-8986/13/i=2/a=024008

[59] ——— (2011) “Generalized transformation optics of linear materials,” Journal of Optics, 13(5), p. 055105. URL http://stacks.iop.org/2040-8986/13/i=5/a=055105

[60] Thompson, R. T. (2018) “Covariant electrodynamics in linear media: Optical metric,” Phys. Rev. D, 97, p. 065001. URL https://link.aps.org/doi/10.1103/PhysRevD.97.065001

[61] Tanji-Suzuki, H., W. Chen, R. Landig, J. Simon, and V. Vuletić (2011) “Vacuum-Induced Transparency,” Science, http://science.sciencemag.org/content/early/2011/08/03/science.1208066.full.pdf. URL http://science.sciencemag.org/content/early/2011/08/03/science.1208066

[62] Novello, M. (2002) Artificial black holes, World Scientific, Singapore River Edge, NJ.

[63] Leonhardt, U. (2002) “A laboratory analogue of the event horizon using slow light in an atomic medium,” Nature, 415, pp. 406 – 409. URL http://dx.doi.org/10.1038/415406a

[64] ——— (2002) “Theory of a slow-light catastrophe,” Phys. Rev. A, 65, p. 043818. URL https://link.aps.org/doi/10.1103/PhysRevA.65.043818

[65] Unruh, W. G. and R. Schützhold (2003) “On slow light as a black hole analogue,” Phys. Rev. D, 68, p. 024008. URL https://link.aps.org/doi/10.1103/PhysRevD.68.024008

166 [66] Steck, D. A. (2017) Quantum and Atom Optics, revision 0.12.0 ed., available online at http://steck.us/teaching.

[67] Brink, D. M. (1993) Angular momentum, Clarendon Press Oxford University Press, Oxford New York.

[68] Rose, M. E. (1961) Elementary theory of angular momentum, Wiley, New York u.a.

[69] Eckart, C. (1930) “The Application of Group theory to the Quantum Dynamics of Monatomic Systems,” Rev. Mod. Phys., 2, pp. 305–380. URL https://link.aps.org/doi/10.1103/RevModPhys.2.305

[70] Sobelman, I. I. (1992) Atomic spectra and radiative transitions, chap. 4, Springer-Verlag, Berlin New York, pp. 60–74.

[71] Loudon, R. (2000) The Quantum Theory of Light, OUP Oxford. URL https://books.google.com/books?id=AEkfajgqldoC

[72] Fleischhauer, M. (1999) “Electromagnetically induced transparency and coherent-state preparation in optically thick media,” Opt. Express, 4(2), pp. 107–112. URL http://www.opticsexpress.org/abstract.cfm?URI=oe-4-2-107

[73] Kerman, A. J. (2002) Raman sideband cooling and cold atomic collisions in optical lattices, Ph.D. thesis, Stanford University. URL http://ezaccess.libraries.psu.edu/login?url=https://search.proquest.com/docview/305520254?accountid=13158

[74] Deutsch, I. H. and P. S. Jessen (1998) “Quantum-state control in optical lattices,” Phys. Rev. A, 57, pp. 1972–1986. URL https://link.aps.org/doi/10.1103/PhysRevA.57.1972

[75] Breuer, H.-P. and F. Petruccione (2002) The theory of open quantum systems, Oxford University Press, Oxford New York.

[76] Travis, A. R. L., T. A. Large, N. Emerton, and S. N. Bathiche (2013) “Wedge Optics in Flat Panel Displays,” Proceedings of the IEEE, 101(1), pp. 45–60.

[77] Below, J. V. and S. Nicaise (1996) “Dynamical interface transition in ramified media with diffusion,” Communications in Partial Differential Equations, 21(1-2), pp. 255–279, https://doi.org/10.1080/03605309608821184. URL https://doi.org/10.1080/03605309608821184

167 [78] Almeida, M. P., J. S. Andrade, S. V. Buldyrev, F. S. A. Cavalcante, H. E. Stanley, and B. Suki (1999) “Fluid flow through ramified structures,” Phys. Rev. E, 60, pp. 5486–5494. URL https://link.aps.org/doi/10.1103/PhysRevE.60.5486

[79] Benamou, J.-D. (1997) Multivalued Solution and Viscosity Solutions of the Eikonal Equation, Research Report RR-3281, INRIA. URL https://hal.inria.fr/inria-00073407

[80] Curtis, L. J. and D. G. Ellis (2004) “Use of the Einstein-Brillouin-Keller action quantization,” American Journal of Physics, 72(12), pp. 1521–1523, https://doi.org/10.1119/1.1768554. URL https://doi.org/10.1119/1.1768554

[81] Whitney, H. (1944) “The Self-Intersections of a Smooth n-Manifold in 2n-Space,” Annals of Mathematics, 45(2), pp. 220–246. URL http://www.jstor.org/stable/1969265

[82] Crandall, M. G. and P.-L. Lions (1983) “Viscosity solutions of Hamilton-Jacobi equation,” Trans. Amer. Math. Soc., 277.

[83] Chow, B. and D. Knopf (2004) The Ricci flow : an introduction, American Mathematical Society, Providence, R.I.

[84] Ashby, D. E. T. F. and D. F. Jephcott (1963) “Measurement of Plasma Density Using a Gas Laser as an Infrared Interferometer,” Applied Physics Letters, 3, pp. 13–16.

[85] Taimre, T., M. Nikolić, K. Bertling, Y. L. Lim, T. Bosch, and A. D. Rakić (2015) “Laser feedback interferometry: a tutorial on the self-mixing effect for coherent sensing,” Adv. Opt. Photon., 7(3), pp. 570–631. URL http://aop.osa.org/abstract.cfm?URI=aop-7-3-570

[86] Tao, T. (2006) “Perelman’s proof of the Poincaré conjecture: a nonlinear PDE perspective,” ArXiv Mathematics e-prints, math/0610903.

[87] Chavel, I. (2006) Riemannian geometry : a modern introduction, Cambridge University Press, New York.

[88] Hamilton, R. S. (1982) “Three-manifolds with positive Ricci curvature,” J. Differential Geom., 17(2), pp. 255–306. URL https://doi.org/10.4310/jdg/1214436922

[89] Perelman, G. (2003) “Ricci flow with surgery on three-manifolds,” ArXiv Mathematics e-prints, math/0303109.

168 [90] Cao, H. (2003) “Lasing in random media,” Waves in Random Media, 13(3), pp. R1–R39, https://doi.org/10.1088/0959-7174/13/3/201. URL https://doi.org/10.1088/0959-7174/13/3/201

[91] Pimenov, A., E. A. Viktorov, S. P. Hegarty, T. Habruseva, G. Huyet, D. Rachinskii, and A. G. Vladimirov (2014) “Bistability and hysteresis in an optically injected two-section semiconductor laser,” Phys. Rev. E, 89, p. 052903. URL https://link.aps.org/doi/10.1103/PhysRevE.89.052903

[92] Ali, S. T. and M. Engliš (2005) “QUANTIZATION METHODS: A GUIDE FOR PHYSICISTS AND ANALYSTS,” Reviews in Mathematical Physics, 17(04), pp. 391–490. URL http://www.worldscientific.com/doi/abs/10.1142/S0129055X05002376

[93] Maslov, V. P. (1972) THÉORIE DES PERTURBATIONS ET MÉTHODES ASYMPTOTIQUES, Dunrod, Paris.

[94] Arnold, V. I. (2014) “On a characteristic class arising in quantization conditions,” in Vladimir I. Arnold - Collected Works: Hydrodynamics, Bifurcation Theory, and Algebraic Geometry 1965-1972 (A. B. Givental, B. A. Khesin, A. N. Varchenko, V. A. Vassiliev, and O. Y. Viro, eds.), Springer Berlin Heidelberg, Berlin, Heidelberg, pp. 85–97.

[95] Keller, J. B. (1958) “Corrected bohr-sommerfeld quantum conditions for nonseparable systems,” Annals of Physics, 4(2), pp. 180 – 188. URL http://www.sciencedirect.com/science/article/pii/0003491658900320

[96] Siegman, A. E. (1986) Lasers, University Science Books, Mill Valley, Calif.

[97] Henkel, C. (2007) “Laser theory in manifest Lindblad form,” Journal of Physics B: Atomic, Molecular and Optical Physics, 40(12), p. 2359. URL http://stacks.iop.org/0953-4075/40/i=12/a=012

[98] Jaynes, E. T. and F. W. Cummings (1963) “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proceedings of the IEEE, 51(1), pp. 89–109.

[99] III, M. S. and M. O. Scully (1964) “Physics of Laser Operation,” in Laser Handbook (F. T. Arecchi and E. O. Schultz-DuBois, eds.), chap. A2, Elsevier Science Publishing Co Inc., North-Holland, Amsterdam, pp. 45–114.

169 [100] Anastassiu, H. T., P. E. Atlamazoglou, and D. I. Kaklamani (2001) “Maxwell’s equations in bicomplex (quaternion) form: an alternative to the Helmholtz P.D.E,” in DIPED - 2001. Proceedings of 6th International Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (IEEE Cat. No.01TH8554), pp. 20–24.

[101] Galitski, V. and I. B. Spielman (2013) “Spin-orbit coupling in quantum gases,” Nature, 494, pp. 49 EP –, review Article. URL http://dx.doi.org/10.1038/nature11841

[102] Choudhury, S. and E. J. Mueller (2015) “Stability of a Bose-Einstein condensate in a driven optical lattice: Crossover between weak and tight transverse confinement,” Phys. Rev. A, 92, p. 063639. URL https://link.aps.org/doi/10.1103/PhysRevA.92.063639

[103] Genske, M. and A. Rosch (2015) “Floquet-Boltzmann equation for periodically driven Fermi systems,” Phys. Rev. A, 92, p. 062108. URL https://link.aps.org/doi/10.1103/PhysRevA.92.062108

[104] Markowitz, W., R. G. Hall, L. Essen, and J. V. L. Parry (1958) “Frequency of Cesium in Terms of Ephemeris Time,” Phys. Rev. Lett., 1, pp. 105–107. URL https://link.aps.org/doi/10.1103/PhysRevLett.1.105

[105] Metcalf, H. J. and P. van der Straten (1999) Laser cooling and trapping, Springer, New York.

[106] Grimm, R., M. Weidemüller, and Y. B. Ovchinnikov (2000) “Optical Dipole Traps for Neutral Atoms,” in Advances In Atomic, Molecular, and Optical Physics (B. Bederson and H. Walther, eds.), vol. 42, Academic Press, pp. 95 – 170. URL http://www.sciencedirect.com/science/article/pii/S1049250X0860186X

[107] Soto, R. (2016) Kinetic theory and transport phenomena, Oxford University Press, New York, NY.

[108] Chapman, S. (1970) The mathematical theory of non-uniform gases; an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases, Cambridge University Press, Cambridge, Eng.

[109] Anderson, M. H., J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell (1995) “Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor,” Science, 269(5221), pp. 198–201, http://science.sciencemag.org/content/269/5221/198.full.pdf. URL http://science.sciencemag.org/content/269/5221/198

170 [110] Redhead, P. A. (1999) Extreme high vacuum, Tech. rep., CERN Accelerator School on Vacuum Technology. [111] ——— (1993) The physical basis of ultrahigh vacuum, American Institute of Physics, New York.

[112] Hobson, J. P. (1964) “Measurements with a Modulated Bayard-Alpert Gauge in Aluminosilicate Glass at Pressures below 10−12 Torr,” Journal of Vacuum Science and Technology, 1(1), pp. 1–6, https://doi.org/10.1116/1.1491375. URL https://doi.org/10.1116/1.1491375 [113] Hanlon, J. (2003) A user’s guide to vacuum technology, Wiley-Interscience, Hoboken, NJ.

[114] Zhao, J. (2017) Dynamical Gauge Effects and Holographic Scaling of Non-Equilibrium Motion in a Disordered and Dissipative Atomic Gas, Ph.D. thesis, Penn State University.

[115] Steck, D. A. (2010), “Rubidium 87 D Line Data,” . URL http://steck.us/alkalidata [116] Vuletić, V. (1996) Bosonisches und fermionisches Lithium in einer steilen magnetischen Falle, Ph.D. thesis, Ludwig-Maximilians-Universität.

[117] Jacome, L. R. (2015) Microscopy with Ultracold Atoms, Master’s thesis, The Pennsylvania State University, Pennsylvania.

[118] McCarron, D. J., S. A. King, and S. L. Cornish (2008) “Modulation transfer spectroscopy in atomic rubidium,” Measurement Science and Technology, 19(10), p. 105601. URL http://stacks.iop.org/0957-0233/19/i=10/a=105601 [119] Shirley, J. H. (1982) “Modulation transfer processes in optical heterodyne saturation spectroscopy,” Opt. Lett., 7(11), pp. 537–539. URL http://ol.osa.org/abstract.cfm?URI=ol-7-11-537 [120] Telle, H. R., D. Meschede, and T. W. Hänsch (1990) “Realization of a new concept for visible frequency division: phase locking of harmonic and sum frequencies,” Opt. Lett., 15(10), pp. 532–534. URL http://ol.osa.org/abstract.cfm?URI=ol-15-10-532 [121] Bolpasi, V. and W. von Klitzing (2010) “Double-pass tapered amplifier diode laser with an output power of 1 W for an injection power of only 200 µW,” Review of Scientific Instruments, 81(11), p. 113108, https://doi.org/10.1063/1.3501966. URL https://doi.org/10.1063/1.3501966

171 [122] Valenzuela, V. M., L. Hernández, and E. Gomez (2012) “High power rapidly tunable system for laser cooling,” Review of Scientific Instruments, 83(1), p. 015111, https://doi.org/10.1063/1.3680612. URL https://doi.org/10.1063/1.3680612

[123] Donley, E. A., T. P. Heavner, F. Levi, M. O. Tataw, and S. R. Jefferts (2005) “Double-pass acousto-optic modulator system,” Review of Scientific Instruments, 76(6), p. 063112, https://doi.org/10.1063/1.1930095. URL https://doi.org/10.1063/1.1930095

[124] Åström, K. J. and R. M. Murray (2008) Feedback systems : an introduction for scientists and engineers, Princeton University Press, Princeton.

[125] Vuletić, V., C. Chin, A. J. Kerman, and S. Chu (1998) “Degenerate Raman Sideband Cooling of Trapped Cesium Atoms at Very High Atomic Densities,” Phys. Rev. Lett., 81, pp. 5768–5771. URL https://link.aps.org/doi/10.1103/PhysRevLett.81.5768

[126] Kerman, A. J., V. Vuletić, C. Chin, and S. Chu (2000) “Beyond Optical Molasses: 3D Raman Sideband Cooling of Atomic Cesium to High Phase-Space Density,” Phys. Rev. Lett., 84, pp. 439–442. URL https://link.aps.org/doi/10.1103/PhysRevLett.84.439

[127] Wineland, D., C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof (1998) “Experimental Issues in Coherent Quantum-State Manipulation of Trapped Atomic Ions,” Journal of Research of the National Institute of Standards and Technology, 103(3).

[128] Iizuka, K. (2002) Elements of Photonics, John Wiley & Sons.

[129] Stone, M. (2009) Mathematics for physics : a guided tour for graduate students, Cambridge University Press, Cambridge.

[130] Horowitz, A. and I. Horowitz (1968) “Entropy, Markov Processes and Competition in the Brewing Industry,” The Journal of Industrial Economics, 16(3), pp. 196–211. URL http://www.jstor.org/stable/2097560

[131] Jost, L. (2006) “Entropy and diversity,” Oikos, 113(2), pp. 363–375, https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.2006.0030-1299.14714.x. URL https://onlinelibrary.wiley.com/doi/abs/10.1111/j.2006.0030-1299.14714.x

172 [132] Shannon, C. E. (1948) “A mathematical theory of communication,” The Bell System Technical Journal, 27(3), pp. 379–423.

[133] Kubo, R. (1966) “The fluctuation-dissipation theorem,” Reports on Progress in Physics, 29(1), p. 255. URL http://stacks.iop.org/0034-4885/29/i=1/a=306

[134] Felderhof, B. U. (1978) “On the derivation of the fluctuation-dissipation theorem,” Journal of Physics A: Mathematical and General, 11(5), p. 921. URL http://stacks.iop.org/0305-4470/11/i=5/a=021

[135] Zener, C. (1932) “Non-adiabatic crossing of energy levels,” Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 137(833), pp. 696–702, http://rspa.royalsocietypublishing.org/content/137/833/696.full.pdf. URL http://rspa.royalsocietypublishing.org/content/137/833/696

[136] Wittig, C. (2005) “The Landau-Zener Formula,” The Journal of Physical Chemistry B, 109(17), pp. 8428–8430, pMID: 16851989, https://doi.org/10.1021/jp040627u. URL https://doi.org/10.1021/jp040627u

[137] Michelson, A. A. and E. W. Morley (1886) “Influence of Motion of the Medium on the Velocity of Light,” American Journal of Science, 31, pp. 377–386.

[138] Slussarenko, S., V. D’Ambrosio, B. Piccirillo, L. Marrucci, and E. Santamato (2010) “The Polarizing Sagnac Interferometer: a tool for light orbital angular momentum sorting and spin-orbit photon processing,” Opt. Express, 18(26), pp. 27205–27216. URL http://www.opticsexpress.org/abstract.cfm?URI=oe-18-26-27205

[139] Bass, M., C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, and E. Van Stryland (2010) Handbook of Optics, Third Edition Volume I: Geometrical and Physical Optics, Polarized Light, Components and Instruments(Set), 3 ed., McGraw-Hill, Inc., New York, NY, USA.

[140] Yariv, A. (2003) Optical waves in crystals : propagation and control of laser radiation, John Wiley and Sons, Hoboken, N.J.

[141] Beyersdorf, P. T., M. M. Fejer, and R. L. Byer (1999) “Polarization Sagnac interferometer with a common-path local oscillator for heterodyne detection,” J. Opt. Soc. Am. B, 16(9), pp. 1354–1358. URL http://josab.osa.org/abstract.cfm?URI=josab-16-9-1354

173 [142] Blake, J., P. Tantaswadi, and R. T. de Carvalho (1996) “In-line Sagnac interferometer current sensor,” IEEE Transactions on Power Delivery, 11(1), pp. 116–121.

[143] Spielman, S., K. Fesler, C. B. Eom, T. H. Geballe, M. M. Fejer, and A. Kapitulnik (1990) “Test for nonreciprocal circular birefringence in YBa2Cu3O7 thin films as evidence for broken time-reversal symmetry,” Phys. Rev. Lett., 65, pp. 123–126. URL https://link.aps.org/doi/10.1103/PhysRevLett.65.123 [144] Paciorek, L. J. (1965) “Injection locking of oscillators,” Proceedings of the IEEE, 53(11), pp. 1723–1727.

[145] Armand, M. (1969) “On the output spectrum of unlocked driven oscillators,” Proceedings of the IEEE, 57(5), pp. 798–799.

[146] Stover, H. L. (1966) “Theoretical explanation for the output spectra of unlocked driven oscillators,” Proceedings of the IEEE, 54(2), pp. 310–311.

[147] Kurokawa, K. (1973) “Injection locking of microwave solid-state oscillators,” Proceedings of the IEEE, 61(10), pp. 1386–1410.

[148] R E Ecke, J. D. F. and D. K. Umberger (1989) “Scaling of the Arnold tongues,” Nonlinearity, 2(2).

[149] Rasband, S. (1997) Chaotic dynamics of nonlinear systems, Wiley, New York.

[150] Ryder, L. (2008) Quantum field theory, Cambridge University Press, Cambridge New York.

[151] Srednicki, M. (2007) Quantum field theory, Cambridge University Press, Cambridge.

[152] Weinberg, S. (1995) The quantum theory of fields, Cambridge University Press, Cambridge New York.

[153] Davison, A. C. (1999) Bootstrap methods and their application, Cambridge University Press, Cambridge New York.

[154] Fisher, R. (1954) Statistical methods for research workers, Oliver and Boyd, Edinburgh.

[155] Gandy, A. (2009) “Sequential Implementation of Monte Carlo Tests With Uniformly Bounded Resampling Risk,” Journal of the American Statistical Association, 104(488), pp. 1504–1511, https://doi.org/10.1198/jasa.2009.tm08368. URL https://doi.org/10.1198/jasa.2009.tm08368

174 [156] Nichols, T. E. and A. P. Holmes (2001) “Nonparametric permutation tests for functional neuroimaging: A primer with examples,” Human Brain Mapping, 15(1), pp. 1–25, https://onlinelibrary.wiley.com/doi/pdf/10.1002/hbm.1058. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/hbm.1058

[157] Boresi, A. (2003) Advanced mechanics of materials, John Wiley & Sons, New York.

[158] Arlt, J., O. Maragó, S. Webster, S. Hopkins, and C. Foot (1998) “A pyramidal magneto-optical trap as a source of slow atoms,” Optics Communications, 157(1), pp. 303 – 309. URL http://www.sciencedirect.com/science/article/pii/S0030401898004994

[159] Dieckmann, K., R. J. C. Spreeuw, M. Weidemüller, and J. T. M. Walraven (1998) “Two-dimensional magneto-optical trap as a source of slow atoms,” Phys. Rev. A, 58, pp. 3891–3895. URL https://link.aps.org/doi/10.1103/PhysRevA.58.3891

[160] Zupancic, P., P. M. Preiss, R. Ma, A. Lukin, M. E. Tai, M. Rispoli, R. Islam, and M. Greiner (2016) “Ultra-precise holographic beam shaping for microscopic quantum control,” Opt. Express, 24(13), pp. 13881–13893. URL http://www.opticsexpress.org/abstract.cfm?URI=oe-24-13-13881

[161] Sivasankar, S. and S. Chu (2007) “Optical Bonding Using Silica Nanoparticle Sol-Gel Chemistry,” Nano Letters, 7(10), pp. 3031–3034, pMID: 17854226, http://pubs.acs.org/doi/pdf/10.1021/nl071492h. URL http://pubs.acs.org/doi/abs/10.1021/nl071492h

[162] Noble, A. and M. Kasevich (1994) “UHV optical window seal to conflata) knife edge,” Review of Scientific Instruments, 65(9), pp. 3042–3043. URL http://scitation.aip.org/content/aip/journal/rsi/65/9/10.1063/1.1144604

[163] Gwo, D.-H. (1998) “Ultraprecision bonding for cryogenic fused-silica optics,” in Cryogenic Optical Systems and Instruments VIII (J. B. Heaney and L. G. Burriesci, eds.), vol. 3435, pp. 136–142. URL http://dx.doi.org/10.1117/12.323731

[164] Lee, K. I., J. A. Kim, H. R. Noh, and W. Jhe (1996) “Single-beam atom trap in a pyramidal and conical hollow mirror,” Opt. Lett., 21(15), pp. 1177–1179. URL http://ol.osa.org/abstract.cfm?URI=ol-21-15-1177

175 [165] Regal, C. A. (2006) Experimental realization of BCS-BEC crossover physics with a Fermi gas of atoms, Ph.D. thesis, University of Colorado.

[166] Wimperis, J. R. and S. F. Johnston (1984) “Optical cements for interferometric applications,” Appl. Opt., 23(8), pp. 1145–1147. URL http://ao.osa.org/abstract.cfm?URI=ao-23-8-1145

[167] Greco, V., F. Marchesini, and G. Molesini (2001) “Optical contact and van der Waals interactions: the role of the surface topography in determining the bonding strength of thick glass plates,” Journal of Optics A: Pure and Applied Optics, 3(1), p. 85. URL http://stacks.iop.org/1464-4258/3/i=1/a=314

[168] Haisma, J. and G. Spierings (2002) “Contact bonding, including direct-bonding in a historical and recent context of materials science and technology, physics and chemistry: Historical review in a broader scope and comparative outlook,” Materials Science and Engineering: R: Reports, 37, pp. 1 – 60. URL http://www.sciencedirect.com/science/article/pii/S0927796X02000037

[169] Hagy, H. and F. Martin (1982), “Very low expansion sealing frits,” US Patent 4,315,991. URL http://www.google.com/patents/US4315991

[170] Akselsen, O. (1992) “Diffusion bonding of ceramics,” Journal of Materials Science, 27(3), pp. 569–579. URL http://dx.doi.org/10.1007/BF02403862

[171] Plößl, A. and G. Kräuter (1999) “Wafer direct bonding: tailoring adhesion between brittle materials,” Materials Science and Engineering: R: Reports, 25, pp. 1 – 88. URL http://www.sciencedirect.com/science/article/pii/S0927796X98000175

[172] Haisma, J., B. A. C. M. Spierings, U. K. P. Biermann, and A. A. van Gorkum (1994) “Diversity and feasibility of direct bonding: a survey of a dedicated optical technology,” Appl. Opt., 33(7), pp. 1154–1169. URL http://ao.osa.org/abstract.cfm?URI=ao-33-7-1154

[173] Rowan, S., S. Twyford, J. Hough, D.-H. Gwo, and R. Route (1998) “Mechanical losses associated with the technique of hydroxide-catalysis bonding of fused silica,” Physics Letters A, 246(6), pp. 471 – 478. URL http://www.sciencedirect.com/science/article/pii/S0375960198005337

176 [174] Gwo, D. (2001), “Ultra precision and reliable bonding method,” US Patent 6,284,085. URL http://www.google.com/patents/US6284085

[175] ——— (2003), “Hydroxide-catalyzed bonding,” US Patent 6,548,176. URL https://www.google.com/patents/US6548176

[176] Elliffe, E. J., J. Bogenstahl, A. Deshpande, J. Hough, C. Killow, S. Reid, D. Robertson, S. Rowan, H. Ward, and G. Cagnoli (2005) “Hydroxide-catalysis bonding for stable optical systems for space,” Classical and Quantum Gravity, 22(10), p. S257. URL http://stacks.iop.org/0264-9381/22/i=10/a=018

[177] Sneddon, P. H., S. Bull, G. Cagnoli, D. R. M. Crooks, E. J. Elliffe, J. E. Faller, M. M. Fejer, J. Hough, and S. Rowan (2003) “The intrinsic mechanical loss factor of hydroxy-catalysis bonds for use in the mirror suspensions of gravitational wave detectors,” Classical and Quantum Gravity, 20(23), p. 5025. URL http://stacks.iop.org/0264-9381/20/i=23/a=006

[178] Robertson, D., C. Killow, H. Ward, J. Hough, G. Heinzel, A. Garcia, V. Wand, U. Johann, and C. Braxmaier (2005) “LTP interferometer noise sources and performance,” Classical and Quantum Gravity, 22(10), p. S155. URL http://stacks.iop.org/0264-9381/22/i=10/a=004

[179] Smith, J. R., G. Cagnoli, D. R. M. Crooks, M. M. Fejer, S. Goßler, H. Lück, S. Rowan, J. Hough, and K. Danzmann (2004) “Mechanical quality factor measurements of monolithically suspended fused silica test masses of the GEO 600 gravitational wave detector,” Classical and Quantum Gravity, 21(5), p. S1091. URL http://stacks.iop.org/0264-9381/21/i=5/a=105

[180] Armandula, H. (2003) Observations on sapphire / silica bonds, Tech. rep., LIGO. URL https://dcc.ligo.org/LIGO-T030046/public

[181] Mackenzie, K., I. Brown, P. Ranchod, and R. Meinhold (1991) “Silicate bonding of inorganic materials,” Journal of Materials Science, 26(3), pp. 763–768. URL http://dx.doi.org/10.1007/BF03163519

[182] Green, K., J. Burke, and B. Oreb (2011) “Chemical bonding for precision optical assemblies,” Optical Engineering, 50(2), pp.

177 023401–023401–12. URL http://dx.doi.org/10.1117/1.3533034

[183] Iler, R. K. (1979) The Chemistry of Silica, Wiley-Interscience.

[184] Vigil, G., Z. Xu, S. Steinberg, and J. Israelachvili (1994) “Interactions of Silica Surfaces,” Journal of Colloid and Interface Science, 165(2), pp. 367 – 385. URL http://www.sciencedirect.com/science/article/pii/S0021979784712422

[185] Bakos, T., S. N. Rashkeev, and S. T. Pantelides (2002) “Reactions and Diffusion of Water and Oxygen Molecules in Amorphous SiO2,” Phys. Rev. Lett., 88, p. 055508. URL http://link.aps.org/doi/10.1103/PhysRevLett.88.055508

[186] Hench, L. L. and J. K. West (1990) “The sol-gel process,” Chemical Reviews, 90(1), pp. 33–72, http://pubs.acs.org/doi/pdf/10.1021/cr00099a003. URL http://pubs.acs.org/doi/abs/10.1021/cr00099a003

[187] Kobayashi, M., F. Juillerat, P. Galletto, P. Bowen, and M. Borkovec (2005) “Aggregation and Charging of Colloidal Silica Particles: Effect of Particle Size,” Langmuir, 21(13), pp. 5761–5769, pMID: 15952820, http://pubs.acs.org/doi/pdf/10.1021/la046829z. URL http://pubs.acs.org/doi/abs/10.1021/la046829z

[188] Prakash, S. S., C. J. Brinker, A. J. Hurd, and S. M. Rao (1995) “Silica aerogel films prepared at ambient pressure by using surface derivatization to induce reversible drying shrinkage,” Nature, 374(6521), pp. 439–443. URL http://dx.doi.org/10.1038/374439a0

[189] Li, D. and K. Jousten (2003) “Comparison of some metrological characteristics of hot and cold cathode ionisation gauges,” Vacuum, 70(4), pp. 531 – 541. URL http://www.sciencedirect.com/science/article/pii/S0042207X02007819

178 Vita Craig Chandler Price EDUCATION

2012 BS, University of Wisconsin-Madison, Physics (Honors), Mathematics 2018 PhD, Pennsylvania State University, Physics

PUBLICATIONS

• Gia-Wei Chern, Yuriy Sizyuk, Craig Price, Natalia B. Perkins, “Kitaev- Heisenberg model in a magnetic field: order-by-disorder and commensurate- incommensurate transitions”. Phys. Rev. B 95, 144427 (2017). • Jianshi Zhao, Craig Price, Qi Liu, Louis Rene Jacome, Nathan Gemelke, “Dynamical Gauge Effects and Holographic Scaling of Non-Equilibrium Motion in a Disordered and Dissipative Atomic Gas”. arXiv:1609.02163 (2016) • Jianshi Zhao, Louis Rene Jacome, Craig Price, Nathan Gemelke, “Lo- calization and Fractionalization in a Chain of Rotating Atomic Gases”. arXiv:1609.02199 (2016). • Yuriy Sizyuk, Craig Price, Peter Wölfle, Natalia B. Perkins, “Importance of anisotropic exchange interactions in honeycomb iridates: Minimal model for zigzag antiferromagnetic order in Na2IrO3”. Phys. Rev. B 90, 155126 (2014). • Craig Price, Natalia B. Perkins, “Finite-temperature phase diagram of the classical Kitaev-Heisenberg model”. Phys. Rev. B 88, 024410 (2013). • Craig C. Price, Natalia B. Perkins, “Critical Properties of the Kitaev- Heisenberg Model” Phys. Rev. Lett. 109, 187201 (2012). • R. Abbasi et al, “Observation of an Anisotropy in the Galactic Cosmic Ray arrival direction at 400 TeV with IceCube”. The Astrophysical Journal 746, 1 (2012). • R. Abbasi et al, “Observation of Anisotropy in the Arrival Directions of Galac- tic Cosmic Rays at Multiple Angular Scales with IceCube”. The Astrophysical Journal 740 16 (2011).

SELECTED AWARDS

• NSF Graduate Fellowship - Honorable Mention (2013) • PSU Elsbach Distinguished Graduate Fellowship (2012)