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ABSTRACT

HYBRID AND THE DYNAMICAL

by Robert A. McCutcheon

We explore various aspects of hybrid optomechanics. A hybrid optomechanical model in- volving a cavity, atom, and moving mirror is introduced as the intersection between cavity electrodynamics and cavity optomechanics. We describe how numerical simulations of such systems are performed and demonstrate a method that can reduce the computational scope of certain simulations. A method for calculating -dependent spectra is also ex- plained. Finally, we explain how the dynamical Casimir effect can be achieved using an with and without an atom and look at notable behavior of this system. HYBRID OPTOMECHANICS AND THE DYNAMICAL CASIMIR EFFECT

Thesis

Submitted to the

Faculty of Miami University

in partial fulfillment of

the requirements for the degree of

Master of Science

by

Robert A. McCutcheon

Miami University

Oxford, Ohio

2017

Advisor: Perry Rice

Reader: Samir Bali

Reader: Karthik Vishwanath

c 2017 Robert A. McCutcheon This thesis titled

HYBRID OPTOMECHANICS AND THE DYNAMICAL CASIMIR EFFECT

by

Robert A. McCutcheon

has been approved for publication by the

College of Arts and Science

and

Department of

(Perry Rice)

(Samir Bali)

(Karthik Vishwanath) Contents

1 Introduction 1 1.1 Cavity ...... 2 1.2 Cavity and Atom ...... 5 1.3 Cavity and Mirror ...... 9 1.4 Cavity, Atom, and Mirror ...... 11

2 Theoretical Frameworks 13 2.1 Wavefunctions and Density Matrices ...... 13 2.2 Correlation Functions and Emission Spectrum ...... 16 2.3 Spectrum of Squeezing ...... 18 2.4 QuTiP Implementation ...... 19

3 Steady-State Value Computational Method 21 3.1 Procedure ...... 21 3.2 Results ...... 23

4 Time-Dependent Spectrum 28

5 Dynamical Casimir Effect 31 5.1 Generation in a Cavity ...... 31 5.2 Photon Generation in a Cavity with an Atom ...... 35

6 Conclusion 40 6.1 Summary ...... 40 6.2 Future Work ...... 41

A Steady-State Value Computational Method Code 43

B Time-Dependent Spectrum Code 48

iii C Dynamical Casimir Effect - Cavity Code 51

D Dynamical Casimir Effect - Cavity with Atom Code 54

iv List of Figures

1.1 System configuration ...... 2 1.2 Driven cavity ...... 5 1.3 Rabi oscillations ...... 8 1.4 Vacuum Rabi doublet ...... 9 1.5 Cavity-mirror dynamics ...... 11

2.1 First-order correlation function ...... 17 2.2 Second-order correlation functions ...... 18

3.1 Method results for optomechanical system ...... 25 3.2 Results comparison ...... 25 3.3 Method results for hybrid optomechanical system ...... 27

4.1 Time-dependent spectrum ...... 30

5.1 Photon generation ...... 33 5.2 First-order correlations ...... 34 5.3 Second-order correlation ...... 34 5.4 Spectrum of squeezing ...... 35 5.5 Photon and atomic expectation values ...... 37 5.6 First-order correlations, emission spectra, and spectra of squeezing ...... 38 5.7 Impurity ...... 39

v List of Tables

1.1 List of common parameters...... 12

vi Acknowledgements

I would like to thank Dr. Perry Rice for the interesting and engaging classes over the last four years, for introducing me to the dynamical Casimir effect, and for advice on writing this thesis. I would also like to thank Dr. James P. Clemens for first getting me started on quantum and working with me for two and a half years, and for suggesting the topic of hybrid optomechanics.

vii Chapter 1 Introduction

The field of deals with light and its interaction with matter in a funda- mentally ”quantum” way. Light is treated as , and here matter consists of atoms whose absorb and emit photons to transition between discrete levels. These relatively simple interactions lead to rich and diverse phenomena that can give insight into a wider variety of physical systems and have applications on small and large scales. For example, the principles of quantum optics can be used to create extremely precise force and position sensors [1][2] and study the crossover into quantum behavior of macroscopic objects [3][4], and they have applications in [5]. We will look at several system configurations that are all built around a cavity created between two reflective mirrors. Photons reflect off the mirrors and can be contained in the cavity, although they occasionally ”leak” out. From here, the system can be modified in many ways to elicit new behavior. For example, a two-level atom can be added to the cav- ity. The interaction between an atom and photons in a cavity is known as cavity (cQED). Alternatively, one of the mirrors could be allowed to oscillate as if on a spring. The study of photons in a cavity interacting with this mirror, a macroscopic mechanical object, is called cavity optomechanics. Of course, we can include an atom to- gether with the moving mirror, which has become known as hybrid optomechanics [6]. A schematic of this hybrid system is shown in Fig. 1.1 [4][7]. These relatively simple hybrid systems can be used to demonstrate seemingly endless interesting behavior including many non-classical effects, such as photon anti-bunching and squeezed light [8][9][10]. Over the last three years, we have taken a somewhat exploratory approach to various cavity systems. In this thesis, we look at some of the outcomes of this approach. In the rest of this chapter, we describe the cavity, cavity-atom, cavity-mirror, and cavity-atom-mirror systems in more detail while also explaining the important characteristics and parameters that are used throughout the paper. In Chapter 2, we explain some of the methods in quantum theory that are used to calculate the quantities of interest, and describe how these are implemented in the Quantum Toolbox in Python (QuTiP), which we used for numerical

1 simulations. In Chapter 3, we explain a method for reducing the computational scope of these simulations for systems with large numbers of photons and . In Chapter 4, we explain a filter-cavity method which can be used to calculate time-dependent spectra of cavity systems. Finally, in Chapter 5, we use a cavity system to model the dynamical Casimir effect, which results in the creation of photon pairs out of a .

Figure 1.1: A hybrid optomechanical system with a cavity, two-level atom, and movable mirror. The labeled parameters are explained in subsequent sections.

1.1 Cavity

A cavity is used to keep photons localized to a region in space. Since photons reflecting off the mirrors typically make many round trips in the cavity, this strengthens the interactions with the atom and movable mirror when they are added later. The use of a cavity also limits the frequencies of photons found inside. Classically, a cavity of length L between two mirrors can contain an electric field subject to the standing wave condition for allowed frequencies,

cm f = , (1.1) m 2L

2 or πcm ω = , (1.2) m L where c is the speed of light and m is an integer index starting at 1. We assume that only one mode of the cavity electric field is filled, so that the photons present are only of frequency

ω1 ≡ ωc, each with energy

E = ~ωc. (1.3) The cavity is modeled as a quantum with discrete energy levels given by

 1 E = ω n + . (1.4) n ~ c 2 The ladder operators a and a†, which obey the commutation relation a, a† = 1, are used on number states of the oscillator to add and remove quanta (photons); the annihilation a removes one photon from the state |ni and the creation operator a† adds one, and together a†a is a number operator which gives the number n of photons in state |ni as follows:

√ a|ni = n|n − 1i, (1.5a) √ a†|ni = n + 1|n + 1i, (1.5b) a†a = hn|a†a|ni = hn|n|ni = n. (1.5c)

These are also related to the electric field; an electric field can be quantized and written in terms of a and a†. A single mode of the field in the cavity of volume V can be written as follows if the field does not vary much over the length of the cavity [11]:

r ω E(t) = i ~ ae−iωt − a†eiωt . (1.6) 20V The Hamiltonian which represents the energy of the cavity mode is

† H = ~ωca a. (1.7) The cavity can be driven by an external to add photons to it. This setup results in

3 adding a term to (1.7):

† iωLt † −iωLt H = ~ωca a + ~Y ae + a e , (1.8)

where ωL is the laser photon frequency and Y is the amplitude of the laser driving. (1.8) is in the Schr¨odingerpicture, in which states evolve with time due to the Hamilto- nian and operators are constant. We usually transform to the , in which states evolve due to interaction terms of the Hamiltonian and operators evolve due to non- interaction terms. This results in a time-independent Hamiltonian which is easier to use in simulations:

† † H = ~∆a a + ~Y a + a , (1.9) where ∆ is the detuning between the cavity frequency and the laser frequency,

ωc − ωL. We must also include some form of decay, since realistically the cavity is “leaky,” and photons occasionally pass through a mirror and leave the system. We explain more about † the system’s interaction with the environment in Section 2.1. For now, we just add −iκca a to the Hamiltonian, where κc is the cavity damping rate, and finally we have

† † † H = ~∆a a + ~Y a + a − i~κa a. (1.10) In Fig. 1.2, we show the expected photon number a†a for a driven cavity as a function † of detuning ∆. a a is a Lorentzian with a peak at δ = 0 of linewidth 2κc.

4 † Figure 1.2: a a as a function of ∆; a Lorentzian centered at ∆ = 0 with linewidth 2κc. Y = 1, κc = 2.

1.2 Cavity and Atom

Now a two-level atom is added to the cavity. The atom has a |gi and an |ei. It can absorb a photon from the cavity to transition from |gi to |ei, and it can emit a photon to fall from |ei to |gi. The resonance frequency for this transition is ωa. The operators we use to act on the atom’s state are σ− and σ+, which are combinations of Pauli operators,

1 σ = (σ − iσ ) ; (1.11a) − 2 x y 1 σ = (σ + iσ ) . (1.11b) + 2 x y

5 σ− and σ+ work as lowering and raising operators for the atom:

σ+|gi = |ei; (1.12a)

σ−|ei = |gi. (1.12b)

The energy of the atom together with the energy of the cavity give the non-interaction part of the Hamiltonian, ω H = ω a†a + ~ a σ . (1.13) O ~ c 2 z

σz is a Pauli operator which acts on the state of the atom to give ±1 with corresponding ±~ωa/2, with the point of zero energy in the middle. The zero point can be shifted to correspond to the ground state; then σz/2 can be replaced with σ+σ− in the Hamiltonian, and the possible energies of the atom are 0 and ~ωa. The energy of the interaction between the atom’s dipole moment p and the cavity mode is

HI = −p · E, (1.14) which can be written as [11]

† HI = ~ga (σ+ + σ−) a + a (1.15) † †  = ~ga aσ+ + a σ− + a σ+ + aσ− , where ga is the cavity-atom coupling strength. The first term in parentheses, aσ+, corre- sponds to the atom absorbing a photon from the cavity mode and transitioning to the excited † state; a σ− corresponds to the atom falling to the ground state and emitting a photon into † the cavity; a σ+ corresponds to the atom emitting a photon and becoming excited; and σ−a corresponds to the atom falling to the ground state and absorbing a photon. The last two terms do not conserve energy. The total Hamiltonian for this system reads

H = HO + HI (1.16) † † = ~ωca a + ~ωaσ+σ− + ~ga (σ+ + σ−) a + a This is known as the Jaynes-Cummings model in cQED. We again go to the interaction picture, in which the state of the system evolves due to the interaction terms of the Hamiltonian (HI ), and the operators evolve due to the non-

6 interaction terms (HO). The Heisenberg equation of for a gives

i a˙ = [HO, a] ~ = i ω a†a + ω σ σ , a c a + − (1.17)  †  = iωc a a, a

= −iωca, which shows that in the interaction picture, a picks up time dependence and becomes ae−iωct. † This can be done similarly for a , σ−, and σ+, so that in the interaction picture the operators become

a → ae−iωct, (1.18a) a† → a†eiωct, (1.18b)

−iωat σ− → σ−e , (1.18c)

iωat σ+ → σ+e , (1.18d) and the interaction Hamiltonian becomes

i(ωa−ωc)t † −i(ωa−ωc)t −i(ωa+ωc)t † i(ωa+ωc)t HI = ~ga aσ+e + a σ−e + aσ−e + a σ+e . (1.19)

For the sake of simplicity here and throughout the rest of the thesis, we assume that ωc = ωa, so that

† −2iωct † 2iωct HI = ~ga aσ+ + a σ− + aσ−e + a σ+e . (1.20) The last two terms are rapidly oscillating and average to zero over the time scales in which we are interested, so they may be dropped; this is called the rotating wave approximation (RWA). Finally, the Jaynes-Cummings Hamiltonian for a cavity coupled to a two-level atom under the RWA is

† †  H = ~ωca a + ~ωaσ+σ− + ~ga aσ+ + a σ− . (1.21)

7 Fig. 1.3 shows the cavity and atom expectation values as the system evolves in time according to (1.21). The cavity begins with one photon and the atom is initially in the † ground state, and over time, the cavity and atom exchange the photon; a a and hσ+σ−i oscillate out of phase by π. This is known as vacuum Rabi oscillation.

† Figure 1.3: Vacuum Rabi oscillations. a a (blue) and hσ+σ−i (black) as functions of time. Parameters are ωc = ωa = 1, ga = 2.

As in 1.1, we can add driving and decay to the Hamiltonian in the Schrodinger picture,

(1.16), then transform to a frame rotating at ωL, to arrive at

† †  † † H = ~∆a a + ~∆σ+σ− + ~ga aσ+ + a σ− + ~Y a + a − i~κca a − i~γσ+σ−, (1.22) where γ is the rate of the atom.

By sweeping the driving frequecy ωL, we see that the Lorentzian of Fig. 1.2 splits into the vacuum Rabi doublet, with peaks at ∆ = ±ga, as shown in Fig. 1.4.

8 † Figure 1.4: Vacuum Rabi doublet with peaks at ±ga. a a (blue) and hσ+σ−i (black) as functions of ∆. Parameters are ga = 3,Y = 0.1, κc = 0.75, γ = 0.75.

1.3 Cavity and Mirror

We go back to the cavity of Section 1.1, but now one of the mirrors is allowed to oscillate as if on a spring. This is modeled as a quantum harmonic oscillator, but it is a mechanical oscillator, whereas the cavity is an optical oscillator. The Hamiltonian for just the mechanical oscillator is mω2 xˆ2 pˆ2 H = m + , (1.23) 2 2m

where m is the mass of the mirror and ωm is the resonance frequency of its oscillation. This is analogous to the Hamiltonian for a classical harmonic oscillator, except thatx ˆ andp ˆ are

9 quantum operators.x ˆ andp ˆ can be written in terms of ladder operators b and b† [12]:

r xˆ = ~ b + b† ; (1.24a) 2mωm r mω pˆ = i ~ m b† − b . (1.24b) 2 b and b† work the same way on the state of the mirror as a and a† do on the cavity; b removes one quantum () from the mechanical mode and b† adds one. The mirror is coupled to the cavity via the force. Photons reflecting off the mirror can excite phonons, or quanta of vibration, in the mechanical oscillator, and they also exert forces which displace the mirror from its initial equilibrium position. A displacement x changes the length of the cavity from its original length L, which changes the allowed frequency of the cavity mode:

πc ω (x) = . (1.25) c L − x The frequency shift per length is defined as the optomechanical coupling strength:

dω G ≡ , (1.26) dx

so the cavity’s resonance frequency depends on the mirror’s displacement; the shift in ωc can be approximated by Gx. The Hamiltonian of this system is

† † H = ~ (ωc + Gx) a a + ~ωmb b. (1.27) Replacing x with the quantum operator (1.24a), (1.27) becomes [12]

† † † † H = ~ωca a + ~ωmb b + ~gma a b + b , (1.28) where the cavity-mirror coupling parameter is

r ~ gm ≡ G . (1.29) 2mωm

As in previous sections, we add driving and decay and transform to the interaction picture

10 to reach the full Hamiltonian

† † † † † † † H = ~∆a a + ~ωmb b + ~gma a b + b + ~Y a + a − i~κca a − i~κmb b. (1.30)

Fig. 1.5 shows the dynamics of the system with (1.30). Photons are added to the cavity initially, which excite phonons in the mirror and displace it (initially in the ”negative” direction), until the system eventually settles to a steady state.

Figure 1.5: Cavity-mirror dynamics. a†a (blue), b†b (green), and b + b† (yellow) as functions of time. Parameters are ∆ = 0, ωm = 2, gm = 1,Y = 5, κc = κm = 0.25.

1.4 Cavity, Atom, and Mirror

We can combine the configurations of Section 1.2 and 1.3 to include all three components, resulting in a hybrid optomechanical system with the Hamiltonian

11 † † †  † † H = ~∆a a + ~∆σ+σ− + ~ωmb b + ~ga aσ+ + a σ− + ~gma a b + b (1.31) † † † + ~Y a + a − i~κca a − i~γσ+σ− − i~κcb b.

The parameters that are commonly used in this thesis are listed in Table 1.1; some are explained in later chapters. The atom and the mirror can be directly coupled [13], but we neglect that possibility in this thesis. From here, the system can be simulated under various parameters or initial conditions, with or without driving, and with or without decay in order to model the dynamics of the system. In the next chapter, we explain some of the theoretical methods used to perform these simulations and calculate quantities of interest.

Symbol Meaning

ωc cavity resonance frequency

ωa atom resonance frequency

ωm mirror resonance frequency ∆ cavity-laser frequency detuning

ga cavity-atom coupling

gm cavity-mirror coupling

κc cavity photon mode decay rate γ atom spontaneous emission rate

κm mirror phonon mode decay rate Y laser driving amplitude

gf cavity-filter cavity coupling (Ch. 4)

∆f cavity-filter cavity frequency detuning (Ch. 4) Ω mirror driving amplitude (Ch. 5)

∆d cavity-driven mirror frequency detuning (Ch. 5)

Table 1.1: List of common parameters.

12 Chapter 2 Theoretical Frameworks

2.1 Wavefunctions and Density Matrices

In general, a system can be described by a wavefunction ψ which can have time and spatial dependence. States |ψi available to the system can be expanded in a given basis in Hilbert space, such as the position (x) basis, (p) basis, and energy (E) basis. The system can be in a superposition of these eigenstates. Measurements of an on a system ”collapse” the wavefunction to a single eigenstate, giving an eigenvalue of the operator corresponding to the observable. The time-dependent state |ψi of a closed system with a Hamiltonian H evolves according to the Schr¨odingerequation,

i |ψ˙i = − H|ψi. (2.1) ~ However, no system can be completely isolated from an external environment; it is not truly “closed” unless the entire environment is explicitly included in the model. A complete wavefunction would include the states of the system S and the environment, also called the reservoir R:

|ψi = |ψSi ⊗ |ψRi. (2.2)

In the systems described in Chapter 1, the reservoir can consist of an optical or thermal bath which can exchange photons and phonons with the system. The reservoir can be modeled as a harmonic oscillator with generally a vastly greater number of modes than the system. Because of this, it is practically impossible to follow the dynamics of the reservoir. Ideally, one would want to use a model that eliminates the need to keep track of each reservoir mode. This can be done using the Lindblad master equation which involves the rather than the wavefunction. The density matrix can be more convenient and useful than the wavefunction in describing the state of the system and obtaining information about it. It is defined as

13 X ρ = Pi|ψiihψi|, (2.3) i

where Pi is the probability of finding the system in the state |ψii. The density matrix can be used to extract expectation values of operators. The expec- tation value of an operator Oˆ is

ˆ X ˆ hOi = Pihψi|O|ψii. (2.4) i

Taking the trace of both sides, ! ˆ X ˆ T rhOi = T r Pihψi|O|ψii , i ! ˆ X ˆ hOi = T r Pi|ψiihψi|O , i   = T r ρOˆ , (2.5) since the trace is invariant under cyclic permutations. The density matrix is generally time-dependent, but we do not write this explicitly. The time evolution of ρ is X  ˙ ˙  ρ˙ = Pi |ψiihψi| + |ψiihψi| . (2.6) i Using (2.1),

X  i i  ρ˙ = Pi − H|ψiihψi| + |ψiihψi|H , i ~ ~ i = − (Hρ − ρH) , ~ i = − [H, ρ] , (2.7) ~ where the square brackets denote the of the operators inside. Like (2.1), (2.7)

applies to a closed system, so the reservoir must still be included; here ρ would be ρS ⊗ ρR and H would be the total Hamiltonian for the system, reservoir, and their interaction,

H = HS + HR + HI . Using the Lindblad master equation, the reservoir dynamics can be ignored and the system-reservoir interaction can be modeled more simply. This takes the

14 form

ρ˙ = Lρ, (2.8)

where L is the Liouvillian superoperator which contains dissipation and decay. We need the Born and Markov approximations: the Born approximation states that the reservoir is essentially unaffected by the coupling between its many modes and the system, although the system is significantly affected by the reservoir, while the Markov approximation states that the reservoir has no “memory,” or that there is no correlation between its present and past states. Following the derivations in [7], [11], or [14], one arrives at

i X γi  † † †  ρ˙ = − [H, ρ] + 2Oˆ ρOˆ − Oˆ Oˆ ρ − ρOˆ Oˆ , (2.9) 2 i i i i i i ~ i where H is now the Hamiltonian for the system only and does not include the decay terms ˆ ˆ † mentioned in Sections 1.1-1.4, and γi is the collapse rate for the operators Oi and Oi which couple the system to the environment. For example, for our hybrid optomechanical model, the cavity mode and two-level atom are coupled to an optical bath and the mechanical modes of the mirror are coupled to a thermal bath, and the master equation reads

i † † †  † † † ρ˙ = − [H, ρ] + κc (¯nc + 1) 2aρa − a aρ − ρa a + κcn¯c 2a ρa − aρa − ρaa ~ γ γ + (¯n + 1) (2σ ρσ − σ σ ρ − ρσ σ ) + (2σ ρσ − σ σ ρ − ρσ σ ) (2.10) 2 a − + + − + − 2 + − − + − + † † †  † † † + κm (¯nm + 1) 2bρb − b bρ − ρb b + κmn¯m 2b ρb − bb ρ − ρbb , wheren ¯c,n ¯a,n ¯m are the average thermal photon or phonon number available to the cavity, atom, and mirror, respectively. In this thesis, we assume that the cavity, atom, and mirror do not absorb photons or phonons from the environment, so thatn ¯c,n ¯a, andn ¯m may be set to 0. In the case of no absorption from the environment, (2.10) reduces to

i † † †  γ ρ˙ = − [H, ρ] + κc 2aρa − a aρ − ρa a + (2σ−ρσ+ − σ+σ−ρ − ρσ+σ−) ~ 2 (2.11) † † †  + κm 2bρb − b bρ − ρb b .

The first term in (2.11) describes the regular evolution of the system according to the Hamil- tonian H. The subsequent terms in parentheses describe jumps, which occur when the cavity or atom emits a photon to the environment and when the mirror loses a phonon, as well as

15 the decay of the state when no jump has occurred. This decay gives information about the probable number of quanta in the system based on how much time has passed since a jump occurred. A final note is that ifρ ˙ = 0, the system has reached a steady state, in which its statistics are stationary. A non-trivial requires the presence of jumps that can both add and remove quanta.

2.2 Correlation Functions and Emission Spectrum

We introduce two functions that give information about a state beyond calculating expec- tation values. We define them for the cavity mode, although they can be extended to the atom, mirror, and crosses between components. The first-order optical function, or first-order correlation function, for the cavity mode is [11]

ha†(t + τ)a(t)i g(1)(t, τ) = , (2.12) pha†(t + τ)a(t + τ)iha†(t)a(t)i

where τ is some delay after time t. This function gives information about the coherence of the state of the cavity mode. If |g(1) (t) | = 1, the state is fully coherent. If the cavity interacts with a thermal environment, over time the state becomes incoherent and g(1) (t) decays to 0. If the system starts in a steady state, the value of t is not important; only the delay τ matters, and (2.12) can be written as

ha†(τ)a(0)i g(1)(τ) = . (2.13) pha†(τ)a(τ)iha†(0)a(0)i An example of the first-order correlation is shown in Fig. 2.1. The initial state is a with g(1) = 1, and over time it becomes incoherent as shown by g(1) settling to 0. The second-order correlation function is given by

ha†(t)a†(t + τ)a(t + τ)a(t)i g(2)(t, τ) = , (2.14) ha†(t)a(t)i2 or, starting in a steady state,

ha†(0)a†(τ)a(τ)a(0)i g(2)(τ) = . (2.15) ha†(0)a(0)i2

16 (2.14) is a measure of how likely it is to detect a photon at time t + τ, given that one was detected at t. If g(2)(0) > 1, photons are bunched, or more likely to be detected near certain frequencies. This occurs for a thermal state. If g(2)(0) < 1, photons are anti-bunched, or less likely to appear together; this is a non-classical effect and is a characteristic of a number, or Fock, state. g(2)(0) = 1 signifies a coherent state, which displays Poissonian statistics. Over time, g(2) converges to 1 as photons separated by a delay become uncorrelated. Fig. 2.2 shows the second-order correlation for these three different initial states.

Figure 2.1: First-order correlation function g(1) for a coherent state as a function of time.

17 Figure 2.2: Second-order correlation functions g(2) for three types of initial state as functions of time. The initial coherent state is |αi = |1i and the initial is |ni = |1i.

Finally, the emission spectrum of the cavity as a function of frequency is the of G(1), the unnormalized first-order correlation function (just the numerator of (2.13)). This is defined only for a steady state and is given by

Z +∞ S(ω) = dτe−iωτ ha†(τ)a(0)i. (2.16) −∞

2.3 Spectrum of Squeezing

An electric field can be written in terms of quadratures defined as

1 X = a†eiθ + ae−iθ , (2.17a) θ 2 i Y = a†eiθ − ae−iθ . (2.17b) θ 2

These quadratures do not commute and so they have an uncertainty relation ∆Xθ∆Yθ ≤

1/4. Coherent states and number states have symmetric minimum uncertainties, ∆Xθ =

18 ∆Yθ = 1/2. In some situations the uncertainty in one quadrature may be reduced at the expense of increased uncertainty in the other; this is called squeezing. For example, an operator of the form  2 S(r) = exp ra2 − r∗a† (2.18) results in a squeezed state when it acts on a coherent state. The spectrum of squeezing can be used to find if and where squeezing occurs in a system over some bandwidth. It can be defined as in [8], [15], or [16], or with a slight modification here: Z ∞ iωt Sθ(ω) = dt e h: Xθ(t),Xθ(0) :i , (2.19) −∞ where the colons denote normal ordering and hA, Bi ≡ hABi − hAi hBi. If Sθ is less than 0

at some ω, then the quadrature Xθ has been squeezed. We use this in Chapter 5.

2.4 QuTiP Implementation

We use QuTiP for numerical simulations of our various systems and calculations of quantities of interest [17]. In this section ee give a brief overview of how we implement what we have explained in Chapters 1 and 2 in QuTiP. First, one has to decide on the size of the Hilbert space to be used. We choose to include N photon states and M phonon states, meaning that 0 up to N − 1 photons and 0 up to M − 1 phonons are allowed in the system. These numbers should be great enough to cover the actual ranges of photons and phonons that may be present in the system at one time, or the results could be inaccurate. The two-level atom requires a 2-dimensional Hilbert space, and the Hilbert space of the whole system is a direct product between the three separate spaces. This means that the length of the state vector is 2NM, and the size of the density matrix and operators is 2NM × 2NM. This limits the scope of the simulations that can be performed, since it takes longer for higher N and M, and in some cases the number of quanta needed exceeds the memory capabilities of typical computers. We address this further in Chapter 3. We define the required operators as tensor products of dimension 2NM, where, for ex- ample, a acts only on the cavity’s Hilbert space. The Hamiltonian is built in terms of these operators and the parameters which we specify. For convenience, we set ~ to be 1 in all calculations performed for this thesis. We also define the required collapse operators cor- responding to the modes of environmental loss or gain described in 2.1, such as the cavity

19 losing a photon to the environment or the mirror absorbing a phonon from it, as follows:

p c1 = κ (¯nc + 1)a, (2.20a) p c2 = γ (¯na + 1)σ−, (2.20b) p c3 = κm (¯nm + 1)b, (2.20c) √ † c4 = κcn¯ca , (2.20d) √ c5 = γn¯aσ−, (2.20e) √ c6 = κmn¯mb; (2.20f) these account for the jumps present in (2.10). Once the operators, parameters, Hamiltonian, and collapse operators are defined, QuTiP has built-in functions for common calculations, such as the following. steadystate calculates the steady state of a system if it has one. expect calculates the expectation value of an operator in a given state. mesolve evolves a given initial state over a specified time range, and automatically calculates the expectation values of specified operators at each time step. correlation 2op 2t calculates the numerator of (2.12), and correlation 4op 2t calculates the numerator of (2.14). Lastly, spectrum calculates the emission spectrum of the system, assuming it has a steady state.

20 Chapter 3 Steady-State Value Computational Method

As alluded to in Section 2.4, the size of the system’s Hilbert space can quickly become a concern. The length of the state vector is 2NM, and the density matrix and operators are of dimension 2NM × 2NM. The number of calculations performed in a simulation is roughly (2NM)3, since the primary operation is matrix multiplication which is O(n3) for a straightforward algorithm. In this chapter, we explain a method for reducing the scope of simulations that require a large range of photons or phonons, which in some cases would otherwise be impossible. This method assumes that the system has a steady state, and is motivated by previous work in optomechanics [3]; we match their results and extend the method to a hybrid model.

3.1 Procedure

The idea is first to calculate the steady-state values of the system operators, and then shift the operators for the cavity and mirror to include these average values. By calculating the averages ahead of time and using the new offset operators, we only need to keep track of the variations about the mean, which are often smaller than the full range of expectation values. The original operators we have used so far are shifted as follows:

a =a ¯ + d, (3.1a) b = ¯b + c, (3.1b) a† =a ¯∗ + d†, (3.1c) b† = ¯b∗ + c†, (3.1d)

21 where the bar denotes an average value, and d, c, and their Hermitian conjugates are ladder operators which remove or add quanta about the mean. The average values of Hermitian- † ∗ conjugate pairs of operators are complex conjugates (hai =a ¯, a =a ¯ , and so on). σ− and

σ+ are not shifted because the Hilbert space of the two-level atom is already as small as it can be. For a given set of parameters, the average value of each operator needs to be calcu- lated before using QuTiP’s steadystate function to find the steady state and calculate the expected photon number, phonon number, and atomic excitation. Although the atomic operators are not shifted, we need the mean values of σ− and σz (¯σ andz ¯, respectively) in addition toa ¯ and ¯b, because the equations used to calculate them are not independent. We use the Langevin equations for operator time evolution,

i κ a˙ = [H, a] − c a ~ 2 κ = −i∆a − ig σ − ig b + b† − iY − c a (3.2a) a − m 2 i κ b˙ = [H, b] − m b ~ 2 κ = −iω b − ig a†a − m b (3.2b) m m 2 i γ σ˙− = [H, σ−] − σ− ~ 2 γ = i∆σ σ + ig aσ − σ (3.2c) z − a z 2 − i σ˙z = [H, σz] − γσz ~ †  = 2iga a σ− − aσ+ − γσz (3.2d)

In the steady state, the operators assume constant values, so the above equations can be set equal to 0 and the operators are replaced with their steady-state values:

κ 0 = −i∆¯a − ig σ¯ − ig a¯ ¯b + ¯b∗ − iY − c a¯ (3.3a) a m 2 κ 0 = −ω ¯b − ig a†a − m ¯b (3.3b) m m 2 γ 0 = i∆¯zσ¯ + ig a¯z¯ − σ¯ (3.3c) a 2 ∗ ∗ 0 = 2iga (¯a σ¯ − a¯σ¯ ) − γz¯ (3.3d)

This system of equations must be solved fora ¯, ¯b,σ ¯, andz ¯. Because these mean values

22 are complex in general, we can make this easier by splitting each one into a real and complex part; for example,a ¯ = α + iβ. These are substituted into (3.3) and the real and imaginary terms of each equation can be separated, resulting in a system of eight real equations. From here, we need to use software that is capable of simultaneously solving a system of nonlinear equations. We use the fsolve method in Python’s numpy library. The solutions are saved for use in the simulation. We use the Hamiltonian (1.31) for a hybrid system,

† † †  † † † H = ~∆a a + ~∆σ+σ− + ~ωmb b + ~ga aσ+ + a σ− + ~gma a b + b + ~Y a + a . (3.4)

The decay terms are left off because they are automatically included by the specification of collapse operators in QuTiP. The shifted operators (3.1) are substituted into (3.4) and into the collapse operators (2.20). This results in many more terms in the Hamiltonian, which loses the intuitive sense of the original. We can drop constant terms which just shifts the point of zero energy of the system. We can save a little more computational time by looking at the equation forρ ˙ = 0, where we find certain terms which together add to 0 in the steady state. These can also be dropped from the Hamiltonian, resulting in the form we use in the simulation (with ~ = 1):

0 † † † ¯ ¯∗ † ∗ † † H = ∆d d + ∆σ+σ− + ωcc c + gmd d b + b + c + c + gm a¯ d +ad ¯ c + c (3.5) ∗ †  ∗ † + ga aσ¯ + +a ¯ σ− + dσ+ + d σ− − ga σ¯ d +σd ¯

The steady state for this Hamiltonian is calculated and expectation values are calculated over a range of detunings using the corresponding operator mean values.

3.2 Results

The first results are shown in Fig. 3.1, where the cavity-atom coupling was set to be small enough that the atom is effectively not present. Expected photon and phonon numbers are shown as functions of detuning; also, we plot the values for the previously calculated |a¯|2 = a† hai and |¯b|2 = b† hbi to compare them with the the actual expectation values a†a and b†b . The cavity-mirror coupling is decreased while driving strength is increased to test the method over a wide range of regimes. Fig. 3.1a,b required ranges of only N = 2 and M = 12, because the the expectation

23 values are low due to the weak driving. In Fig. 3.1b, b† hbi is small and hard to see, so the full range of M was still required. Fig. 3.1c,d required N = 2 and M = 32. Although the cavity-mirror coupling is reduced, we find higher maximum photon and phonon numbers because of increased driving. b† hbi is again nearly 0 as in Fig. 3.1b, which is why a larger M was required to capture the full range of phonons during the simulation. For the third row of figures, the coupling is weak and the driving is strong. Figs. 3.1e,f show peaks of around 1400 and 65, respectively, which are well beyond the required N and M values. Without using the mean-value method, we would need at least N = 1400 and M = 65, so this calculation would not have been possible without the method. In Fig. 3.1e, the photon expectation does not vary much from steady-state offset a† hai (the two curves overlap), which is why only N = 2 was needed. The shape looks close to a Lorentzian since the mirror has little effect on the cavity. In Fig. 3.1b, M = 32 was required to capture the smaller peak near ∆ = 2.

(a) cavity; gm = 2,Y = 0.01 (b) mirror; gm = 2,Y = 0.01

(c) cavity; gm = 0.5,Y = 0.5 (d) mirror; gm = 0.5,Y = 0.5

24 (e) cavity; gm = 0.01,Y = 20 (f) mirror; gm = 0.01,Y = 20

Figure 3.1: Cavity plots show a†a (blue) and a† hai (red) vs. ∆. Mirror plots show b†b † (green) and b hbi (magenta) vs. ∆. Common parameters throughout are ωm = 2, ga = −4 10 , κc = 1, κm = 0.01, γ = 1.

As a check, we ran the same simulation as in Figs. 3.1a,b but without using the offset method in order to make sure that the results matched. The results using the straightforward approach are shown in Fig. 3.2, which does match Fig. 3.1a,b.

(g) cavity; gm = 2,Y = 0.01 (h) mirror; gm = 2,Y = 0.01 Figure 3.2: The same calculation as in Figs. 3.1a,b to show that the offset method reproduces the results of a straightforward approach.

We now raise ga and use the offset method to simulate a full hybrid system starting with the Hamiltonian (1.31). Shown in Fig. 3.3, we observe the effect of increasing mechanical † † coupling gm. a a , hσ+σ−i, and b b are plotted again as functions of detuning ∆. We

25 use the parameters in [7], but with the addition of small mechanical decay. In Figs. 3.2a,b,

where gm is small, we see results similar to the vacuum Rabi doublet for the cavity and

atom. As gm is increased, this shape loses its as sidebands appear due to the interaction between the cavity and mirror. The spacing between these sidebands is ωm = 1, corresponding to the spacing between the phonon state energies. We also see that the photon and phonon numbers are proportional, while the atomic expectation value generally becomes lower where the photon number is higher and vice versa. In future work, we hope to explain the locations and relative peak heights of the sidebands.

(a) gm = 0.1 (b) gm = 0.1

(c) gm = 3 (d) gm = 3

26 (e) gm = 5 (f) gm = 5

† † Figure 3.3: (a,c,e) a a (blue) and hσ+σ−i (black); (b,d,f) b b (green) as functions of ∆ for increasing gm down the page. Common parameters are ωm = 1, ga = 3,Y = 0.01, κc = 1, κm = 0.01, γ = 1.

27 Chapter 4 Time-Dependent Spectrum

The typical spectrum given by (2.16) is defined only for the steady state, but a time- dependent spectrum can be formally formulated in various ways [18]. In this chapter, we explain a straightforward but indirect method for calculating the spectrum of the light leav- ing a cavity as a system evolves in time; the system need not have a steady state. This is done by adding a second cavity, called a filter cavity, to the original cavity, which is motivated by

work in [19] and [18]. The filter cavity has a resonance frequency ωf which can be changed by varying the length of the filter cavity. The filter cavity is coupled to the main cavity so that light can pass between the two, but this coupling is weak so that the filter cavity can receive photons from the main cavity without significantly affecting the dynamics of the original system. We define a new detuning between the main and filter cavity frequencies,

∆f = ωc − ωf . The damping rate of the filter cavity, κf , is ideally low so that photons rarely leak out of it. The main cavity may or may not be driven by a laser. The filter cavity is also treated as a single-mode quantum harmonic oscillator with ladder operators c and c†. The Hamiltonian with the addition of the filter cavity is (neglecting decay terms)

† † †  † † H = ~∆a a + ~∆σ+σ− + ~ωmb b + ~ga aσ+ + a σ− + ~gma a b + b (4.1) † † † †  + ~Y a + a + ~∆f c c + ~gf ac + a c .

The photon expectation number in the filter cavity, c†c , is ultimately calculated as we † sweep ∆f by varying ωf . For a certain value of ∆f , c c is proportional to the intensity of light leaving the main cavity and so it can be used to find the shape of its emission spectrum. For each value of ∆f , we start from some initial state and evolve the system in time, calculating c†c at each time step to create a synthesized spectrum as the filter cavity collects light from the main cavity. In Fig. 4.1, we show the results for the parameters from [19], but with the addition of weak driving on resonance; this set-up gives an excellent

28 demonstration of the dynamics of a hybrid optomechanical system. Filter cavity photon † expectation c c is plotted as a function of ∆f . The main cavity begins with one photon at t = 0. At t = 1, the spectrum is a Lorentzian, similar to Fig. 1.2. At t = 2, the initial peak has split into a Rabi doublet with peaks at

±ga due to the interaction with the atom. Then, from t = 4 to t = 20, we see the effects of the mirror as the two peaks lose their symmetry and sidebands gradually appear. As in Fig.

3.3c-f, the sidebands are spaced ωm apart.

(a) t = 1 (b) t = 2

(c) t = 4 (d) t = 7

29 (e) t = 10 (f) t = 20

† Figure 4.1: c c as a function of ∆f . Parameters: ∆ = 0, ωm = 1, ga = 4, gm = 1.2,Y = 0.05, κc = 0.5, κm = 0, κf = 0.1, γ = 0.01.

30 Chapter 5 Dynamical Casimir Effect

One of the surprising discoveries of quantum physics is that an ideal vacuum is not truly empty. Classically, a vacuum has zero energy, zero occupancy, and zero fluctuations, but in quantum theory, the vacuum state has some zero-point energy which gives rise to nonzero fluctuations; virtual constantly go in and out of existence. This is a result ~ω of the of the electromagnetic field, which gives a ground state with E0 = 2 and nonzero fluctuations [11]. It was predicted in 1948 that the nonzero field of the quantum electrodynamic vacuum between two parallel metallic plates could cause the plates to repel or attract each other; the virtual photons in the vacuum still impart a radiation pressure force on the plates. This became known as the static Casimir effect and was accurately measured in 1997 and 2001 [20][21][22]. If the vacuum fluctuations are somehow amplified, the virtual photons can manifest as real, observable photons. The creation of real photons out of vacuum in this way is called the dynamical Casimir effect which can be accomplished by rapidly changing the geometry or material properties of a system [23][24]. This was observed for the first time in 2011 using a superconducting circuit [25]. We employ changes in the boundary conditions of a cavity to generate photons by rapidly varying the cavity length. This can be viewed as an initial ground state in a with zero occupancy that is suddenly altered; the ground state becomes a superposition of higher states with nonzero occupation numbers [26]. This can be implemented using our hybrid optomechanical system with some modifications.

5.1 Photon Generation in a Cavity

We start with just a cavity between two mirrors; one is fixed and the other is no longer

allowed to move freely, but is somehow externally driven at frequency ωd with amplitude Ω. In our simulations, the cavity always starts in the vacuum state with hni = 0. A general Hamiltonian can be derived for this configuration which includes the energy due to the

31 motion of the mirror, but we limit the cavity to a single mode as before and make some approximations [24][27] to arrive at

†  2 †2 † H = −~∆da a − ~Ω a + a − i~κca a, (5.1) where ∆d is now defined as the detuning between half the mirror driving frequency and the cavity resonance frequency, ωd/2 − ωc. This Hamiltonian is in the interaction picture. The squared operators a and a† suggest that photons are created in pairs.

As an initial approach, we swept over the parameters ∆d and Ω and calculated various quantities to look for notable behavior as the system evolved in time. Fig. 5.1 below shows how the photon expectation number changes with time. Fig. 5.1a shows the results for on-resonance driving. The cavity starts with 0 photons and increases quadratically initially until it levels off around hni = 14. Fig. 5.1b increases driving strength, which causes hni to increase more rapidly and introduces some oscillation before it levels off around 14 again. In Fig. 5.1c, Ω is again 1, but now the driving is off resonance. hni has a much more oscillatory behavior and does not reach as high a steady-state value as before, suggesting that driving off-resonance is less effective for producing photons.

(a) ∆d = 0, Ω = 1 (b) ∆d = 0, Ω = 10

32 (c) ∆d = 5, Ω = 1

Figure 5.1: hni as a function of time. κc = 1.

In Fig. 5.2, the first-order correlation and corresponding spectra are shown for driving on and off resonance. On resonance, in Fig. 5.2a,c, the system starts in a coherent state since photons are created in pairs at the same frequency, then falls smoothly towards 0, demonstrating the loss of coherence with time. The spectrum in this case has a single peak centered at ω = 0. In Fig. 5.2b,d, the driving is off resonance as in Fig. 5.1c. The system still starts in a coherent state, but now g(1) oscillates with decreasing amplitude toward 0.

This is consistent with the spectrum, which now has two peaks centered at ∆d = ±5, since one is the Fourier transform of the other.

(a) ∆d = 0 (b) ∆d = 5

33 (c) ∆d = 0 (d) ∆d = 5

(1) Figure 5.2: (a,b) g (t); (c,d) corresponding spectra. Parameters: Ω = 1, κc = 1.

Fig. 5.3 plots the second-order correlation function for the same two cases as in the previous figure. For driving on resonance, g(2) starts above 1, indicating photon bunching. However, g(2) rises initially before falling to 0. This initial rise is a sign of non-classical behavior, suggesting that the photons are actually anti-bunched [28]. For off-resonance driving as in Fig. 5.3b, g(2) starts near 27 and monotonically decreases, suggesting strong bunching. Since photons are created in pairs, it is likely at first that if one is detected leaving the cavity, another will soon be detected.

(a) ∆d = 0 (b) ∆d = 5

(2) Figure 5.3: g as a function of time. Parameters: Ω = 1, κc = 1.

Finally, we looked for cases in which squeezing occurred. We used the quadrature (2.17a) with θ = 0. We calculated (2.18) but without subtracting the steady-state parts, that is,

34 (2.18) where hXθ,Xθi ≡ hXθXθi, just to see the shape of the squeezing spectrum and look for whether it became negative. Fig. 5.4 shows the occurrence of squeezing for the parameters in the off-resonance case. The spectrum has a negative peak near ω = 5, which suggests

that squeezing occurs near ω = +∆d.

Figure 5.4: The spectrum of squeezing. Parameters: ∆d = 5, Ω = 1, κc = 1.

5.2 Photon Generation in a Cavity with an Atom

We now add a two-level atom to the cavity to see how it changes the dynamics of the system from Section 5.1. Terms representing the energy of the atom and its interaction and decay must be added to the Hamiltonian (5.1), resulting in

†  2 †2 †  † H = −~∆da a − ~∆dσ+σ− − ~Ω a + a + ~ga aσ+ + a σ− − i~κca a − i~γσ+σ−, (5.2) where still the only mode of photon gain is due to the driven mirror.

35 We took a similar approach as in Section 5.1, but here there are two more parameters that can be adjusted: ga and γ. We again start with on-resonance, weak driving and show the effect of increasing cavity-atom coupling in Fig. 5.5. For weak coupling, the expected photon number and atomic excitation increase smoothly to steady-state values, but the rise in the atomic expectation is delayed. For stronger coupling, they increase more quickly to a maximum and then fall to a steady-state value. For further increased coupling, they show a faster rise, then a regular oscillation with decaying amplitude toward a stable value. This may be related to the vacuum Rabi oscillations mentioned in Section 1.2; the cavity and atom exchange photons before settling to a steady state determined by the driving and decay.

(a) ga = 0.1 (b) ga = 0.1

(c) ga = 1 (d) ga = 1

36 (e) ga = 5 (f) ga = 5

Figure 5.5: hni (blue) and hσ+σ−i (black) as functions of time. Parameters: ∆d = 0, Ω = 0.1, κc = 1, γ = 1.

Fig. 5.6 shows the first-order correlation function, emission spectrum, and spectrum of squeezing for two values of detuning. For ∆d = 5, the spectrum has three peaks and

is symmetric about ω = 0. For ∆d = 10, there are four peaks whose locations appear to be symmetric about ω = 0, but have varying heights. Some peaks are likely due to the interaction with the atom, while others are due to the detuning. In the future, we hope to explain how the detuning and coupling affect the peak locations and heights. The corresponding spectra of squeezing are shown in Fig. 5.6e,f. They have asymmetric peaks at the same locations as the emission spectra, but most are negative, which indicates that squeezing has occurred again for the quadrature with θ = 0. Squeezing occurs at multiple frequencies in each case, unlike in Fig. 5.4 which did not include the atom.

37 (a) ∆d = 5 (b) ∆d = 10

(c) ∆d = 5 (d) ∆d = 10

(e) ∆d = 5 (f) ∆d = 10

Figure 5.6: (a,b) g(1)(t); (c,d) emission spectra; (e,f) spectra of squeezing. Parameters: Ω = 0.1, ga = 5, κc = 1, γ = 1.

38 Finally, we look for evidence of entanglement between the cavity mode and atom. When they are entangled, the composite density matrix of the system can no longer be separated into two separate density matrices without losing information. A partial trace over one system component can be performed on the composite density matrix to leave a reduced density matrix for the state of the other system component. The impurity of the resulting reduced density matrix is a measure of entanglement in the system; this is defined by

2 I = 1 − T r(T rA(ρ)) , (5.3)

where T rA denotes a partial trace over the atom for the composite density matrix ρ. The impurity ranges from 0 to 1 for a completely pure or completely mixed state, respectively. In Fig. 5.7, the impurity is plotted as the system evolves in time. In both plots, the impurity starts at 0 and so the cavity mode and atom are not entangled. Over time the impurity rises; for weak driving, it reaches a small value between 0.05 and 0.06, but for strong driving, it exceeds 0.9, suggesting that strong driving and more rapid generation of pairs of photons leads to greater entanglement.

(a) Ω = 0.1 (b) Ω = 10

Figure 5.7: Impurity as a function of time. Parameters: ∆d = 0, ga = 5, κc = 1, γ = 1.

39 Chapter 6 Conclusion

6.1 Summary

We have built up a hybrid optomechanical model in which an optical cavity is coupled to a two-level atom and a movable mirror. We showed some signature behavior of the cavity by itself, the cavity with only an atom, and the cavity with only the movable mirror, and then finally combined these systems. We gave an overview of some of the theoretical tools that are used to study such systems, including the density matrix and Linblad master equation, correlation functions, emission spectrum, and spectrum of squeezing, and briefly explained how some of this is implemented in QuTiP to make performing calculations more convenient. We described a method that can be used to reduce the computational scope of numerical simulations. By calculating steady-state values of certain operators ahead of time, we can shift the operators to include these values so that a simulation has to account for only the variations about the mean values, which are often smaller than the full range of quanta in the system. We checked that results using our method match results obtained without it, and showed how this enabled us to run a simulation that would have been impossible otherwise. The original hybrid model was modified to include a filter-cavity, which was used to calculate time-dependent spectra and show the dynamics of a hybrid system and demonstrate the effects of the cavity, atom, and mirror. We then introduced the dynamical Casimir effect and explained how photons can be created in an empty cavity by driving one of the end mirrors, thus changing the boundary conditions of the cavity field. We included various results that show interesting and non- classical behavior of this phenomenon, and then added an atom to the cavity to see how the behavior was changed.

40 6.2 Future Work

The method explained in Chapter 3 can be used as needed for simulations with other pa- rameters, or can be adapted to other Hamiltonians by following the same procedure. We hope to explain more about the results in Fig. 3.3, especially analytically if possible to gain intuition into the system dynamics. Most of our focus in the future will be on the dynamical Casimir effect. We intend to take a more in-depth approach to explain the behavior we observed, such as the shapes of the emission and squeezing spectra and the oscillations in some of the expectation values and correlation functions. In addition, we would like to study the details of the photon generation process at a more fundamental level. † We could include driving by a laser by adding the usual term ~Y a + a to the Hamil- tonian in order to see how the behavior changes when photons are added to the cavity mode singly and in pairs. We could make the atom a two-photon absorber which absorbs and emits pairs of photons to transition between states. The Hamiltonian for this is

†  2 †2  2 †2  H = −~∆da a − ~∆dσ+σ− − ~Ω a + a + ~ga a σ+ + a σ− (6.1) † −i~κca a − i~γσ+σ−

We could also include the Jaynes-Cummings terms that do not conserve energy. They cannot always be neglected, so we could see if they are important here. The Hamiltonian for this reads

†  2 †2 † †  H = −~∆da a − ~∆dσ+σ−−~Ω a + a + ~ga aσ+ + a σ− + aσ− + a σ+ (6.2) † − i~κca a − i~γσ+σ−.

A final idea is to include a second two-level atom. The two atoms could be uncoupled, but their interaction with the photon pairs created in the cavity could entangle them. The Hamiltonian in this case is

41 †  2 †2 H = −~∆da a − ~Ω a + a X  j † j j j j j † (6.3) +~ga aσ+ + a σ− − ∆dσ+ σ− − iγσ+ σ− − i~κca a. j=1,2

42 Appendix A Steady-State Value Computational Method Code

We show the code in QuTiP used to generate the figures from Chapter 3. from qutip import * from scipy import * from scipy.optimize import * from pylab import * import time

N = 2 M = 12

#operators d = tensor(destroy(N), qeye(M), qeye(2)) c = tensor(qeye(N), destroy(M), qeye(2)) sm = tensor(qeye(N), qeye(M), sigmam()) ddag = d.dag() cdag = c.dag() sp = sm.dag()

#parameters wm = 1.0 ga = 3.0 gm = 0.1 kappa_c = 1.0 kappa_m = 0.01

43 gamma = 1.0 Y = 0.01 n_th = 0.0

Deltas = linspace(-10,10,500) L = len(Deltas)

#lists to hold expectation values: ada = zeros(L) bdb = zeros(L) spsm = zeros(L) cdc = zeros(L) ddd = zeros(L) abarsq = zeros(L) bbarsq = zeros(L)

#set up steady-state variables and equations def myFunction(z): alpha = z[0] beta = z[1] eta = z[2] nu = z[3] lambd = z[4] mu = z[5] tau = z[6]

F = empty((7)) F[0] = -Delta*alpha - kappa_c*beta/2.0 - 2.0*gm*eta*alpha - Y - ga*lambd F[1] = -Delta*beta + kappa_c*alpha/2.0 - 2.0*gm*eta*beta - ga*mu F[2] = -wm*eta - (kappa_m*nu)/2.0 - gm*(pow(alpha,2)+pow(beta,2)) F[3] = wm*nu - (kappa_m*eta)/2.0 F[4] = -Delta*tau*mu - ga*beta*tau - gamma*lambd/2.0 F[5] = Delta*tau*lambd + ga*alpha*tau - gamma*mu/2.0 F[6] = -4.0*ga*alpha*mu + 4.0*ga*beta*lambd - gamma*tau/2.0 - gamma/2.0

44 return F

#initialize steady-state arrays and initial guess zGuess = [-.001,-.001,-.001,-.001,-.001,-.001,-.001] alphas = zeros(L) betas = zeros(L) etas = zeros(L) nus = zeros(L) lambdas = zeros(L) mus = zeros(L) taus = zeros(L)

#solve for steady-state values for i in range(L): Delta = Deltas[i] solns = fsolve(myFunction, zGuess) alphas[i] = solns[0] betas[i] = solns[1] etas[i] = solns[2] nus[i] = solns[3] lambdas[i] = solns[4] mus[i] = solns[5] taus[i] = solns[6]

#build Hamiltonian, calculate steady state for each detuning i=0 start_time = time.time() for Delta in Deltas: #build steady-state values abar = alphas[i] + betas[i]*1j bbar = etas[i] + nus[i]*1j smbar = lambdas[i] + mus[i]*1j

45 #shift operators a = abar + d b = bbar + c

#collapse operators c_ops = [] c_ops.append(sqrt(kappa_c)*d) c_ops.append(sqrt(kappa_m*(n_th+1))*c) c_ops.append(sqrt(kappa_m*n_th)*cdag) c_ops.append(sqrt(gamma)*sm)

#Hamiltonian H = wm*cdag*c + gm*ddag*d*(c+cdag)\ + (Delta+gm*(bbar+bbar.conjugate()))*ddag*d\ + gm*(abar.conjugate()*d+abar*ddag)*(c+cdag) + Delta*sp*sm\ + ga*(abar*sp+d*sp+abar.conjugate()*sm+ddag*sm)\ - ga*(smbar.conjugate()*d+smbar*ddag)

#calculate steady state and expectation values ss = steadystate(H, c_ops) #jpc c2_ops for no substitutions ada[i] = expect(a.dag()*a, ss) bdb[i] = expect(b.dag()*b, ss) spsm[i] = expect(sp*sm, ss) #modulus squared of mean values abarsq[i] = abar*abar.conjugate() bbarsq[i] = bbar*bbar.conjugate()

#report progress if i%(L/20)==0: print (str(int(round(float(i)/L*100)))+’%’)

i+=1 print (’time elapsed = ’ + str(time.time() - start_time))

46 #plot expectation values figure(’cavity and atom’) plot(Deltas, ada, ’b’, Deltas, spsm, ’k’) xlabel(u’${\Delta}$’) ylabel(u’${, <\sigma_+\sigma_->}$’) figure(’mirror’) plot(Deltas, bdb, ’g’) xlabel(u’${\Delta}$’) ylabel(u’${}$’) figure(’cavity with abarsq’) plot(Deltas, ada, ’b’, label=u’${}$’) plot(Deltas, abarsq, ’r’, label=u’${}$’) xlabel(u’${\Delta}$’) ylabel(u’${, }$’) legend() figure(’mirror with bbarsq’) plot(Deltas, bdb, ’g’, label=u’${}$’) plot(Deltas, bbarsq, ’m’, label=u’${}$’) xlabel(u’${\Delta}$’) ylabel(u’${, }$’) legend() show()

47 Appendix B Time-Dependent Spectrum Code

We show the code in QuTiP used to generate the figures from Chapter 4. from qutip import * from scipy import * from pylab import * import time

N = 2 M = 10 F = 2

#operators a = tensor(destroy(N), qeye(M), qeye(F), qeye(2)) b = tensor(qeye(N), destroy(M), qeye(F), qeye(2)) c = tensor(qeye(N), qeye(M), destroy(F), qeye(2)) sm = tensor(qeye(N), qeye(M), qeye(F), sigmam()) adag = a.dag() bdag = b.dag() cdag = c.dag() sp = sm.dag()

#parameters wm = 1.0 ga = 4 gm = 1.2 gf = .01 kappa_c = 0.5

48 kappa_m = 0 kappa_f = 0.1 gamma= 0.01 Y = 0.05 n_th = 0.0

Deltas = linspace(-10,10,500) c_ops = [] c_ops.append(sqrt(kappa_c)*a) c_ops.append(sqrt(kappa_m*(n_th+1))*b) c_ops.append(sqrt(kappa_m*n_th)*bdag) c_ops.append(sqrt(kappa_f)*c) c_ops.append(sqrt(gamma)*sm) dt = .05 tf = 1 = linspace(0, tf, tf/dt)

#initial state psi0 = tensor(fock(N,0), fock(M,0), fock(F,0), fock(2,0)) cdagc = zeros(len(Deltas)) start_time = time.time() for i in range(len(Deltas)): Delta = Deltas[i]

#Hamiltonian H = wm*bdag*b + gm*adag*a*(b+bdag) + ga*(adag*sm+a*sp)\ + Y*(a+adag) + Delta*cdag*c + gf*(a*cdag+adag*c)

#evolve to get expectation values exp_list = mesolve(H, psi0, times, c_ops, [cdag*c]).expect[0]

49 #synthesized spectrum: add values at all times cdagc[i] = sum(exp_list[:]) print (’time elapsed = ’ + str(time.time() - start_time))

#plot figure(’filter cavity t=’+str(tf)) plot(Deltas, cdagc, ’c’) xlabel(u’${\Delta_f}$’) ylabel(u’${}$’) show()

50 Appendix C Dynamical Casimir Effect - Cavity Code

We show the code in QuTiP used to generate the figures in Section 5.1. from qutip import * from scipy import * from pylab import * from numpy import * import time

N = 32

#operators a = destroy(N) adag = a.dag()

#parameters Delta = 5.0 Omega = 1.0 kappa_c = 1.0 n_th = 0.0 wlist = linspace(-20,20,2000) theta = 0.0

#collapse operators c_ops = []

51 c_ops.append(sqrt(kappa_c*(n_th+1))*a) c_ops.append(sqrt(kappa_c*n_th)*adag) psi0 = basis(N,0) times = linspace(0,5,250) time_ss = linspace(0,15,750)

#Hamiltonian H = -Delta*adag*a - 0.5*Omega*(adag**2+a**2)

start_time = time.time()

#photon expectation value n = mesolve(H, psi0, times, c_ops, [adag*a]).expect[0]

#correlations #evolve to steady state rho_ss = mesolve(H, psi0, time_ss, c_ops, []).states[-1] #evolve again, get expectation values n2 = mesolve(H, rho_ss, times, c_ops, [adag*a]).expect[0]

G1 = correlation_2op_1t(H, rho_ss, times, c_ops, adag, a) g1 = G1/(sqrt(n2[0]*n2[-1])) G2 = correlation_4op_1t(H, rho_ss, times, c_ops, adag, adag, a, a) g2 = G2/(n2[0])**2

#emission spectrum spec = spectrum(H, wlist, c_ops, adag, a)

#spectrum of squeezing #Fourier transform of quadrature correlation <:X(t)X(0):> sqz = spectrum(H,wlist,c_ops,a,a)*exp(-2j*theta)\ + spectrum(H,wlist,c_ops,adag,a)\

52 + spectrum(H,wlist,c_ops,adag,a)\ + spectrum(H,wlist,c_ops,adag,adag)*exp(2j*theta) print(’time: ’+str(time.time()-start_time)+’s’)

#plot figure(’n’) plot(times, n, ’b’) xlabel(u’${t}$’) ylabel(u’${}$’) figure(’g1’) plot(times, g1, ’b’) xlabel(u’${t}$’) ylabel(u’${g^{(1)}}$’) figure(’spectrum’) plot(wlist, spec, ’b’) xlabel(u’${\omega}$’) ylabel(u’${S(\omega )}$’) figure(’g2’) plot(times, g2, ’b’) xlabel(u’${t}$’) ylabel(u’${g^{(2)}}$’) figure(’squeezing spectrum’) plot(wlist, sqz, ’b’) xlabel(u’${\omega}$’) ylabel(r’${S_{\theta} (\omega)}$’) show()

53 Appendix D Dynamical Casimir Effect - Cavity with Atom Code

We show the code in QuTiP used to generate the figures in Section 5.2 from qutip import * from scipy import * from pylab import * from numpy import * import time

N = 24 a = tensor(destroy(N),qeye(2)) adag = a.dag() sm = tensor(qeye(N),sigmam()) sp = sm.dag()

#parameters Delta = 5.0 Omega = 0.1 gamma = 1.0 ga = 5.0 kappa_c = 1.0 n_th = 0.0 wlist = linspace(-20,20,1000) theta = 0.0

54 #collapse operators c_ops = [] c_ops.append(sqrt(kappa_c*(n_th+1))*a) c_ops.append(sqrt(kappa_c*n_th)*adag) c_ops.append(sqrt(gamma)*sm) psi0 = tensor(basis(N,0),basis(2,1)) times = linspace(0,5,250) time_ss = linspace(0,15,750)

#Hamiltonian H = -Delta*adag*a - 0.5*Omega*(adag**2+a**2) + ga*(adag*sm+a*sp)\ - Delta*sp*sm

start_time = time.time()

#cavity and atom expectation values exp_list = mesolve(H, psi0, time_ss, c_ops, [adag*a, sp*sm]) nc = exp_list.expect[0] na = exp_list.expect[1]

#correlations #evolve to steady state rhos = mesolve(H, psi0, time_ss, c_ops, []).states rho_ss = rhos[-1] #evolve again, get expectation values exp_list2 = mesolve(H, rho_ss, times, c_ops, [adag*a, sp*sm]) nc2 = exp_list2.expect[0] na2 = exp_list2.expect[1]

G1 = correlation_2op_1t(H, rho_ss, times, c_ops, adag, a, ’me’) g1 = G1/(sqrt(nc2[0]*nc2[-1]))

55 G2 = correlation_4op_1t(H, rho_ss, times, c_ops, adag, adag, a, a, ’me’) g2 = G2/(nc2[0])**2

#emission spectrum spec = spectrum(H, wlist, c_ops, adag, a) #spectrum of squeezing sqz = spectrum(H,wlist,c_ops,a,a)*exp(-2j*theta)\ + spectrum(H,wlist,c_ops,adag,a)\ + spectrum(H,wlist,c_ops,adag,a)\ + spectrum(H,wlist,c_ops,adag,adag)*exp(2j*theta)

#impurity Icavity = [] for s in range(len(rhos)): Ic = 1 - (rhos[s].ptrace(0)**2).tr() Icavity.append(Ic) print(’time: ’+str(time.time()-start_time)+’s’)

#plot figure(’nc’) plot(time_ss, nc, ’b’) xlabel(u’${t}$’) ylabel(u’${}$’) figure(’na’) plot(time_ss, na, ’k’) xlabel(u’${t}$’) ylabel(u’${<\sigma_+\sigma_->}$’) figure(’g1’) plot(times, g1, ’b’) xlabel(u’${t}$’) ylabel(u’${g^{(1)}}$’)

56 figure(’g2’) plot(times, g2, ’b’) xlabel(u’${t}$’) ylabel(u’${g^{(2)}}$’) figure(’spectrum’) plot(wlist, spec, ’b’) xlabel(u’${\omega}$’) ylabel(u’${S(\omega )}$’) figure(’squeezing spectrum’) plot(wlist, sqz, ’b’) xlabel(u’${\omega}$’) ylabel(r’${S_{\theta} (\omega)}$’) figure(’impurity’) plot(time_ss, Icavity, ’b’) xlabel(u’${t}$’) ylabel(u’${impurity}$’) show()

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