Laser Cooling and Trapping and Application to Superfluid Hydrodynamics of Bose Gases Lecture 1: Light Forces

Atom-light interaction Light forces Dressed state picture

Laser cooling and trapping and application to superﬂuid hydrodynamics of Bose gases Lecture 1: Light forces

H´el`enePerrin

Laboratoire de physique des lasers, CNRS-Universit´eParis 13, Sorbonne Paris Cit´e

Coherent quantum dynamics

H´el`ene Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Outline of the course

Lecture 1: Light forces Lecture 2: Laser cooling and trapping Lecture 3: [tentative] Superﬂuid hydrodynamics of a Bose gas in a harmonic (or bubble) trap

H´el`ene Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Outline of the course

Lecture 1: Light forces Lecture 2: Laser cooling and trapping Lecture 3: [tentative] Superﬂuid hydrodynamics of a Bose gas in a harmonic (or bubble) trap

H´el`ene Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Introduction A little bit of history

First evidence of the mechanical effect of light on a sodium beam in 1933 by Otto Frisch Doppler cooling first suggested in 1975 by Hänsch(Nobel 2005) and Schawlow (N. 1981) for neutral atoms and independently by Wineland (N. 2012) and Dehmelt (N. 1989) for ions Zeeman slower demonstrated in 1982 by Phillips (N. 1997) and Metcalf First optical molasses in 1985 by Chu (N. 1997) et al. First dipole trap in 1986 by Ashkin (N. 2018) and Chu Explanation of sub-Doppler cooling in 1989 by Dalibard and Cohen-Tannoudji (N. 1997) Sub-recoil laser cooling in the 90’s (VSCPT, Raman cooling) Bose-Einstein condensation reached in 1995 by Cornell, Wieman and Ketterle (all N. 2001)

H´el`ene Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Introduction Estimation of the light force Atom irradiated by a resonant laser:

plane wave of frequency ω, photon momentum ~kL ~kL Each absorption changes the velocity by vrec = M . Spontaneous emissions cancel on average net acceleration along the laser alaser = Γscvrec ⇒ −1 7 −1 Typical numbers: for rubidium, vrec = 6 mm s , Γsc 2 10 s · ' ×

5 −2 4 acceleration of order alaser 10 m s 10 g! ⇒ ' · '

Hélène Perrin Laser cooling and trapping Metrology (fountains, optical clocks, cold atom interferometers...) Quantum information and quantum computation New insights in condensed matter physics and quantum simulation: Bloch oscillations, superfluid-insulator transitions, Cooper Sr optical clock pairing, Anderson localization, simulation of (Katorivortex lattice lab) (Ketterle) magnetic systems... quantumbosonic BEC/ memory fermionic (Kimble)BEC-BCS cross-over

Atom-light interaction Light forces Dressed state picture Introduction

A wide range of applications: High precision spectroscopy (Doppler-free lines) of atoms and molecules

credit: Chukman So anti-hydrogen trapping (ALPHA collaboration)

Hélène Perrin Laser cooling and trapping Quantum information and quantum computation New insights in condensed matter physics and quantum simulation: Bloch oscillations, credit: Chukman So anti-hydrogen trapping superfluid-insulator transitions, Cooper (ALPHA collaboration) pairing, Anderson localization, simulation of vortex lattice (Ketterle) magnetic systems... quantumbosonic BEC/ memory fermionic (Kimble)BEC-BCS cross-over

Atom-light interaction Light forces Dressed state picture Introduction

A wide range of applications: High precision spectroscopy (Doppler-free lines) of atoms and molecules Metrology (fountains, optical clocks, cold atom interferometers...)

Sr optical clock (Katori lab)

Hélène Perrin Laser cooling and trapping New insights in condensed matter physics and quantum simulation: Bloch oscillations, credit: Chukman So anti-hydrogen trapping superfluid-insulator transitions, Cooper Sr optical clock (ALPHA collaboration) pairing, Anderson localization, simulation of (Katorivortex lattice lab) (Ketterle) magnetic systems... bosonic BEC/ fermionic BEC-BCS cross-over

Atom-light interaction Light forces Dressed state picture Introduction

quantum memory (Kimble)

H´el`ene Perrin Laser cooling and trapping credit: Chukman So anti-hydrogen trapping Sr optical clock (ALPHA collaboration) (Katori lab) quantum memory (Kimble)

Atom-light interaction Light forces Dressed state picture Introduction

A wide range of applications: High precision spectroscopy (Doppler-free lines) of atoms and molecules Metrology (fountains, optical clocks, cold atom interferometers...) Quantum information and quantum computation New insights in condensed matter physics and quantum simulation: Bloch oscillations, superﬂuid-insulator transitions, Cooper pairing, Anderson localization, simulation of vortex lattice (Ketterle) magnetic systems... bosonic BEC/ fermionic BEC-BCS cross-over

H´el`ene Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture References

Lecture notes of this course + an easy EPJST introductive paper + a short bibliography to be found on my personal page http://www-lpl.univ-paris13.fr/bec/BEC/Team_Helene.htm General references – C. Cohen-Tannoudji and D. Guéry-Odelin: Advances in atomic physics: an overview (World Scientific, 2011) – C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg: Atom-Photon Interactions : Basic Processes and applications (Wiley, 1992). On laser cooling and trapping: – H. J. Metcalf and P. van der Straten: Laser Cooling and Trapping (Springer, New York, 1999); Atoms and molecules interacting with light (Cambrige UP 2016) – C. Cohen-Tannoudji: Atomic motion in laser light, in Fundamental systems in Quantum optics, Les Houches, Session LIII (Elsevier, 1992). On dipole traps: – R. Grimm, M. Weidemüllerand Y. Ovchinnokov: Optical dipole traps for neutral atoms, Adv. At. Mol. Opt. Phys., 42:95–170, 2000. http://arxiv.org/abs/physics/9902072

H´el`ene Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Outline

Outline of the lecture 1 Atom-light interaction Two-level model Dipolar interaction Hamiltonians 2 Light forces Deﬁnition of the mean light force Orders of magnitude. Approximations Calculation of the mean force Interpretation of the mean force and applications 3 The dressed state picture System under consideration Eigenstates for the coupled system: the dressed states Spontaneous emission Dipole force

H´el`enePerrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture 2-level Coupling Hˆ Two-level model

The atom has many electronic transitions of frequencies ωi . The laser frequency ω is close to one particular frequency ω0: detuning ~ωi δ = ω ω0 such that e Γ − δ ω0, ω, ωi ω for all i = 0. | | | − | 6 ~ω ~ω0 We can restrict the discussion to these two levels g and e: transition frequency g ω0, or wavelength λ0 = 2πc/ω0, lifetime of the excited state Γ−1.

N.B.: Not valid for a very far oﬀ resonant laser [Grimm2000].

H´el`ene Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture 2-level Coupling Hˆ Dipolar interaction

Assume that the wavelength λ0 is in the visible or near IR range. The strongest coupling for a L L + 1 (typically S P) transition is the dipolar electric→ coupling Vˆ = Dˆ Eˆ→. − · Dipolar coupling Dˆ = d e g + d∗ g e , (1) | ih | | ih | where d is the reduced dipole:

d = e Dˆ g d∗ = g Dˆ e h | | i h | | i We stay in the dipolar approximation for the rest of the lecture.

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture 2-level Coupling Hˆ Laser electric field

The laser field contains many photons, average number N¯. Poissonian number fluctuations ∆N = √N¯ N¯. A classical field describes the laser properly: ⇒ 1 −iωt −iφ(r) EL(r, t) = L(r) L(r) e e + c.c. (2) 2E n o The classical laser amplitude L, polarisation L and phase φ may depend on the position rE. All the quantum fluctuations are set apart (quantum field).

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture 2-level Coupling Hˆ Three coupled systems Laser, atom and quantum field

ΩΓ

~ω ~ω0

EL VÂL HÂ VÂR HˆR

The total Hamiltonian reads

Hˆ = HÂ + HˆR + VÂL + VÂR .

H´el`ene Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture 2-level Coupling Hˆ Hamiltonians

Atom: position Rˆ, momentum Pˆ

ˆ 2 ˆ P HA = ~ω0 e e + . (3) | ih | 2M Quantum field: If the quantum modes of the field are labelled by ` = (k, ), the energy of the quantum modes is given by ˆ † HR = ~ω` aˆàˆ` . ` X For the description of the atom motion, we will trace on the quantum field variables. Atom to quantum field coupling: VÂR is responsible for spontaneous emission and for the coupling to the laser field fluctuations. It is not necessary to give an explicit form of this term here.

H´el`ene Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture 2-level Coupling Hˆ Hamiltonians

Atom-laser coupling:

res non res VˆAL = Dˆ EL(Rˆ, t) = Vˆ + Vˆ − · AL AL res 1 −iωt −iφ(Rˆ) Vˆ = d (R)ˆ L(Rˆ) e g e e + h.c. AL −2 · E | ih | 1 h i ˆ Vˆnon res = d ∗(R)ˆ ∗(Rˆ) e g eiωt eiφ(R) + h.c. AL −2 · EL | ih | h i In the Heisenberg picture, e g evolves as eiω0t ˆres | ih | VAL describes a slow evolution (at frequency δ = ω ω0) while ⇒ˆnon res − VAL describes a fast evolution (at frequency ω + ω0).

H´el`ene Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture 2-level Coupling Hˆ Hamiltonians

We introduce the Rabi frequencyΩ 1(r) deﬁned by

~Ω1(r) = [d (r)] L(r) . (4) − · E

The time origin is chosen such that Ω1 is real. The atom-laser resonant coupling can then be written as

ˆ res ~Ω1(R) −iωt −iφ(Rˆ) Vˆ = e g e e + h.c. (5) AL 2 | ih | n o The Rabi frequency is the oscillation frequency between g and e at resonance in the strong coupling regime. | i | i

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Definition Approximations Calculation Interpretation Light forces

Light forces Calculation and interpretation of the mean light force

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Definition Approximations Calculation Interpretation Mean light force Definition Velocity operator in the Heisenberg representation:

dRˆ 1 1 Pˆ2 Pˆ = Rˆ, Hˆ = Rˆ, = . (6) dt i~ i~ " 2M # M h i Force operator:

dPˆ 1 Fˆ = = Pˆ, Hˆ = ∇VˆAL ∇VˆAR . (7) dt i~ − − Mean light force: h i

F = Fˆ = ∇VÂL ∇VÂR = ∇VÂL (8) h i −h i − h i −h i

as ∇VÂR = 0, see CCT Les Houches lectures. h i N.B.: ∇VÂR = 0 and contribute to the fluctuations of the random force. 6 Hélène Perrin Laser cooling and trapping Trace over the quantum field variables.

Calculate the average value of ∇VˆAL in the state described by the density matrix. And before starting all this, make some approximations...

Atom-light interaction Light forces Dressed state picture Deﬁnition Approximations Calculation Interpretation Mean light force Principle of the calculation

Mean light force F = ∇VˆAL −h i

Compute the density matrix.

Hélène Perrin Laser cooling and trapping Calculate the average value of ∇VÂL in the state described by the density matrix. And before starting all this, make some approximations...

Atom-light interaction Light forces Dressed state picture Deﬁnition Approximations Calculation Interpretation Mean light force Principle of the calculation

Mean light force F = ∇VˆAL −h i

Compute the density matrix. Trace over the quantum ﬁeld variables.

H´el`ene Perrin Laser cooling and trapping And before starting all this, make some approximations...

Atom-light interaction Light forces Dressed state picture Deﬁnition Approximations Calculation Interpretation Mean light force Principle of the calculation

Mean light force F = ∇VˆAL −h i

Compute the density matrix. Trace over the quantum ﬁeld variables.

Calculate the average value of ∇VˆAL in the state described by the density matrix.

Mean light force F = ∇VˆAL −h i

Compute the density matrix. Trace over the quantum ﬁeld variables.

Calculate the average value of ∇VˆAL in the state described by the density matrix. And before starting all this, make some approximations...

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Definition Approximations Calculation Interpretation Rotating wave approximation

The non-resonant part of the dipolar coupling oscillates at very large frequency (ω + ω0). Its contribution to the force will be negligible provided δ , Ω1 ω0, ωL. | | discard Vˆnon res and only keep Vˆres. ⇒ AL AL This is known as the rotating wave approximation. N.B. With a quantized laser ﬁeld approach, it corresponds to keeping processes where a photon is absorbed when the atom gets excited, and discarding those where a photon is emitted while the atom gets excited.

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Definition Approximations Calculation Interpretation Time scales

Time scale for the evolution of the internal atomic variables:

−1 tint = Γ .

Typical value in laser-cooled species: Γ a few MHz, 2π ∼ tint 100 ns. ∼ Time scale for the evolution of the external atomic variables: time for the atomic velocity to be changed by kL∆v Γ, such that the laser frequency is Doppler shifted signiﬁcantly:'

2 Γ ~kL kL∆v = kL alasertext kL Γscvrectext = × ' × 2 M

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Definition Approximations Calculation Interpretation Time scales

2M −1 text 2 = ωrec ⇒ ' ~kL 2 2 1 2 ~ kL Erec = 2 Mvrec = 2M = ~ωrec is the recoil energy, −1 ωrec the recoil frequency, trec = ωrec the recoil time.

Typical value: ωrec/(2π) a few kHz, trec 100 µs. ∼ ∼

usually text tint. ⇒

This is true if: Γ ωrec broadband condition. The internal variables always have the time to reach their steady state before the external state changes average value in the steady state to compute the light force. ⇒

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Definition Approximations Calculation Interpretation Semi-classical approximation

Can we treat the external motion of the atom as classical?

Relevant if the quantum fluctuations ∆R and ∆P are sufficiently small for the local phase φ and the laser frequency ω (sensitive to Doppler shift) to be well defined:

∆φ = kL∆R 1 Γ . ∆ω = kL∆v Γ or ∆P M . kL Recall ∆R ∆P > ~/2:

~ Γ M 2 or Γ ωrec broadband condition! 2 kL We now assume the broadband condition is fulﬁlled, and will compute the force F at position r = Rˆ for a velocity v = Pˆ /M. h i h i

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Definition Approximations Calculation Interpretation Calculation of the mean force

We assume: RWA, broadband condition.

F = ∇VˆAL steady state − st D E ˆ ~Ω1(R) −iωt −iφ(Rˆ) ∇ e g e e + h.c. RWA ' − 2 | ih | * ( )+st ∇Ω (r) = ~ 1 e g e−iωt e−iφ(r) − 2 h| ih |ist ~Ω1(r) −iωt −iφ(r) i∇φ(r) e g e e + c.c. semi-cl. − 2 h| ih |ist −iωt −iφ −iωt −iφ = ~∇Ω1 σge,st e e ~Ω1∇φ(r) σge,st e e − < − = h i h i e g = σge,st deduced from the optical Bloch equations on h| ih |ist the internal state density matrix operatorσ ˆ.

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Definition Approximations Calculation Interpretation Steady state of the internal variables

Optical Bloch equations deduced from

dσˆ ˆ ˆ ˆ i~ = [HA + VAL, σˆ] i~Γˆσ, dt − where Γˆ accounts for the relaxation from e to g , and we have ∗ | i | i σgg + σee = 1 and σeg = σge . We get:

Ω1(r) ∗ iωt iφ(r) −iωt −iφ(r) σ˙ ee = Γσee + i σ e e σge e e − 2 ge − Γ Ω1(r) iωt iφ(r) σ˙ ge = iω0 σge + i (1 2σee ) e e . − 2 2 −

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Definition Approximations Calculation Interpretation Steady state of the internal variables

We introduce 3 real independent variables

−iωt −iφ(r) u = σge e e in phase with the ﬁeld < −iωt −iφ(r)  v = σge e e in quadrature with the ﬁeld  =  1 1 w = (σee σgg ) = σee population inversion 2 − − 2  satisfying the coupled equations

u˙ = Γ u + δv , − 2  Γ v˙ = 2 v δu Ω1w ,  − − −  1 w˙ = Γ w + + Ω1v . − 2  

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Definition Approximations Calculation Interpretation Steady state of the internal variables

The stationary solution is

δ s Γ s ust = vst = Ω1 1 + s 2Ω1 1 + s 1 1 s w + = σ = , st 2 ee,st 2 1 + s where we have deﬁned the saturation parameter s(r)

2 Ω (r)/2 I (r)/Is s(r) = 1 = . (9) δ2 + Γ2/4 1 + 4δ2/Γ2

2 3 Is is the saturation intensity Is = 2π ~cΓ/(3λ0), typically a few mW cm−2. ·

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Definition Approximations Calculation Interpretation Steady state of the internal variables

N.B.: We get for the stationary dipole

iωt+φ Dˆ (r) = 2 d (r) e g = 2 d (r)(ust + iv st )e h i · < { · h| ih |ist } < · s(r) n o = 2 d.(r) (10) 1 + s(r)

δ Γ  cos [ωt + φ(r)] sin [ωt + φ(r)] . × Ω1(r) − 2Ω1(r)   in phase in quadrature    The term in phase with the laser ﬁeld yields to dephasing and to a conservative force (real part of polarizability). The term in quadrature with the laser ﬁeld yields to absorption and to a dissipative force (imaginary part of polarizability).

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Definition Approximations Calculation Interpretation Expression of the light force

We get for the force:

F(r) = ~∇Ω1ust ~Ω1∇φ(r)vst (11) − −

s(r) ∇Ω1 ~Γ F(r) = ~δ + ∇φ = Fdip + Fpr . (12) −1 + s(r) Ω (r) 2 1 Dissipative force: Fpr

~Γ s(r) Fpr = ∇φ. (13) − 2 1 + s(r)

Conservative force: Fdip

s(r) ∇Ω1 ~δ ∇s(r) Fdip = ~δ = . (14) − 1 + s(r) Ω1(r) − 2 1 + s(r)

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Definition Approximations Calculation Interpretation Interpretation of the light forces Radiation pressure force

1 −iωt+ik ·r Consider a plane wave 0 L e L + c.c. : 2 E Ω1 uniform, s(r) = s0. φ(r) = kL r ∇φ = kL. − · ⇒ − Γ s0 (13) Fpr = ~kL. (15) ⇒ 2 1 + s0

Population of the excited state σ = 1 s0 , scattering rate: ee 2 1+s0

Γ s0 Γsc = Γσee = (16) 2 1 + s0

Fpr = Γsc ~kL

Recoil transfer ~kL at the rate Γsc of the spontaneous scattering ⇒processes. Hence the term radiation pressure force.

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Definition Approximations Calculation Interpretation Radiation pressure force Dependence on the intensity

The intensity appears in s0. On resonance, s0 = I /Is . For I Is , Γ the force is maximum Fpr,max = 2 ~kL. Saturation behavior:

Dependence of Fpr/Fmax on I /Is .

The force is maximum on resonance (δ = 0) and depends on the detuning as the absorption does, with a Lorentzian shape:

Dependence on the detuning, for diﬀerent values of the intensity. From bottom to top: I /Is = 0.1, 1, 10 and 100.

FWHM: Γ 1 + I /Is 2 At large detuning,p Fpr 1/δ ∝ δ in units of Γ

Hélène Perrin Laser cooling and trapping stopped beam constant deceleration az = F /M = Γscvrec (F > 0) over L if ⇒First realization: B. Phillips and− H. Metcalf,− Phys. Rev. Lett. 48, 596 (1982).1 2 1 2 Mv(z) + 2Fz = Mv0 = FL v(z) = 2Γscv−rec1 (L z) Typically:2 L = v /(2Γsc2vrec) < 1 m for v0 500 m s . 0 ⇒ ∼ · − p which sets the magnetic field profile B(z) √L z. ∝ −

Atom-light interaction Light forces Dressed state picture Deﬁnition Approximations Calculation Interpretation Application of the radiation pressure force Zeeman slower

724 In the presence of a Doppler shift or a line shift∆ ω0, theWilliam detuning D. Phillips: Laser cooling and trapping of neutral atoms 0 is modified δ = δ kL v ∆ω0. These later solenoids were cooled with water flowing − · − over the coils. To improve the heat transfer, we filled the A Zeeman slower compensates spaces between the wires with various heat-conducting substances. One was a white silicone grease that we put the Doppler shift due to onto the wires with our hands as we wound the coil on a deceleration by an appropriate lathe. The grease was about the same color and consis- tency as the diaper rash ointment I was then using on my position dependent Zeeman shift baby daughters, so there was a period of time when, whether at home or at work, I seemed to be up to my ∆ω0(z) = kLv(z), to keep the elbows in white grease. maximum force at δ0 = 0. The grease-covered, water-cooled solenoids had the FIG. 4. Upper: Schematic representation of a Zeeman slower. annoying habit of burning out as electrolytic action at- Lower: Variation of the axial field with position. tacked the wires during operation. Sometimes it seemed that we no sooner obtained some data than the solenoid would burn out and we were winding a new one. ous schemes for avoiding optical pumping, were con- On the bright side, the frequent burn-outs provided tained in a proposal (Phillips, 1979) that I submitted to the opportunity for refinement and redesign. Soon we the Office of Naval Research in 1979. Around this time were embedding the coils in a black, rubbery resin. Hal Metcalf, from the State University of New York at While it was supposed to be impervious to water, it did Stony Brook, joined me in Gaithersburg and we began not have good adhesion properties (except to clothing to consider what would be the best way to proceed. Hal and human flesh) and the solenoids continued to burn contended that all the methods looked reasonable, but out. Eventually, an epoxy coating sealed the solenoid we should work on the Zeeman cooler because it would against the water that allowed the electrolysis, and in Hélène Perrin beLaser the most cooling fun! and Not trapping only was Hal right about the fun more recent times we replaced water with a fluorocar- we would have, but his suggestion led us to develop a bon liquid that does not conduct electricity or support technique with particularly advantageous properties. electrolysis. Along the way to a reliable solenoid, we The idea is illustrated in Fig. 4. learned how to slow and stop atoms efficiently (Phillips The atomic beam source directs atoms, which have a and Metcalf, 1982; Prodan, Phillips, and Metcalf, 1982; wide range of velocities, along the axis (z direction) of a Phillips, Prodan, and Metcalf, 1983a, 1983b, 1984a, tapered solenoid. This magnet has more windings at its 1984b, 1985; Metcalf and Phillips, 1985). entrance end, near the source, so the field is higher at The velocity distribution after deceleration is mea- that end. The laser is tuned so that, given the field- sured in a detection region some distance from the exit induced Zeeman shift and the velocity-induced Doppler end of the solenoid. Here a separate detection laser shift of the atomic transition frequency, atoms with ve- beam produces fluorescence from atoms having the cor- locity v0 are resonant with the laser when they reach the rect velocity to be resonant. By scanning the frequency point where the field is maximum. Those atoms then of the detection laser, we were able to determine the absorb light and begin to slow down. As their velocity velocity distribution in the atomic beam. Observations changes, their Doppler shift changes, but is compensated with the detection laser were made just after turning off by the change in Zeeman shift as the atoms move to a the cooling laser, so as to avoid any difficulties with hav- point where the field is weaker. At this point, atoms with ing both lasers on at the same time. Figure 5 shows the initial velocities slightly lower than v0 come into reso- velocity distribution resulting from Zeeman cooling: a nance and begin to slow down. The process continues large fraction of the initial distribution has been swept with the initially fast atoms decelerating and staying in down into a narrow final velocity group. resonance while initially slower atoms come into reso- One of the advantages of the Zeeman cooling tech- nance and begin to be slowed as they move further nique is the ease with which the optical pumping prob- down the solenoid. Eventually all the atoms with veloci- lem is avoided. Because the atoms are always in a strong ties lower than v0 are brought to a final velocity that axial magnetic field (that is the reason for the ‘‘bias’’ depends on the details of the magnetic field and laser windings in Fig. 4), there is a well-defined axis of quan- tuning. tization that allowed us to make use of the selection The first tapered solenoids that Hal Metcalf and I rules for radiative transitions and to avoid the undesir- used for Zeeman cooling of atomic beams had only a able optical pumping. Figure 6 shows the energy levels few sections of windings and had to be cooled with air of Na in a magnetic field. Atoms in the 3S1/2 (mF52) blown by fans or with wet towels wrapped around the state, irradiated with circularly polarized s1 light, must coils. Shortly after our initial success in getting some increase their mF by one unit, and so can go only to the substantial deceleration, we were joined by my first post- 3P3/2 (mF853) state. This state in turn can decay only to doc, John Prodan. We developed more sophisticated so- 3S1/2 (mF52), and the excitation process can be re- lenoids, wound with wires in many layers of different peated indefinitely. Of course, the circular polarization lengths, so as to produce a smoothly varying field that is not perfect, so other excitations are possible, and would allow the atoms to slow down to a stop while these may lead to decay to other states. Fortunately, in a remaining in resonance with the cooling laser. high magnetic field, such transitions are highly unlikely

Rev. Mod. Phys., Vol. 70, No. 3, July 1998 stopped beam First realization: B. Phillips and H. Metcalf, Phys. Rev. Lett. 48, 596 (1982). 2 −1 Typically: L = v /(2Γscvrec) < 1 m for v0 500 m s . 0 ∼ ·

Atom-light interaction Light forces Dressed state picture Deﬁnition Approximations Calculation Interpretation Application of the radiation pressure force Zeeman slower

724 In the presence of a Doppler shift or a line shift∆ ω0, theWilliam detuning D. Phillips: Laser cooling and trapping of neutral atoms 0 is modified δ = δ kL v ∆ω0. These later solenoids were cooled with water flowing − · − over the coils. To improve the heat transfer, we filled the A Zeeman slower compensates spaces between the wires with various heat-conducting substances. One was a white silicone grease that we put the Doppler shift due to onto the wires with our hands as we wound the coil on a deceleration by an appropriate lathe. The grease was about the same color and consis- tency as the diaper rash ointment I was then using on my position dependent Zeeman shift baby daughters, so there was a period of time when, whether at home or at work, I seemed to be up to my ∆ω0(z) = kLv(z), to keep the elbows in white grease. maximum force at δ0 = 0. The grease-covered, water-cooled solenoids had the FIG. 4. Upper: Schematic representation of a Zeeman slower. annoying habit of burning out as electrolytic action at- Lower: Variation of the axial field with position. tacked the wires during operation. Sometimes it seemed constant deceleration az = F /M = Γscvrec (F > 0) over L if that we no sooner obtained some data than the solenoid ⇒ − − would burn out and we were winding a new one. ous schemes for avoiding optical pumping, were con- On the bright side, the frequent burn-outs provided 1 1 tained in a proposal (Phillips, 1979) that I submitted to Mv(z)2 + Fz = Mv 2 = FL v(z) = 2Γ v (L z) the opportunity for refinement and redesign. Soon we 2 2 0 the Office of Naval Researchsc rec in 1979. Around this time were embedding the coils in a black, rubbery resin. Hal⇒ Metcalf, from the State University− of New York at While it was supposed to be impervious to water, it did Stony Brook, joinedp me in Gaithersburg and we began not have good adhesion properties (except to clothing which sets the magnetic field profileto considerB(z) what√ wouldL bez the. best way to proceed. Hal and human flesh) and the solenoids continued to burn contended that∝ all the− methods looked reasonable, but out. Eventually, an epoxy coating sealed the solenoid we should work on the Zeeman cooler because it would against the water that allowed the electrolysis, and in Hélène Perrin beLaser the most cooling fun! and Not trapping only was Hal right about the fun more recent times we replaced water with a fluorocar- we would have, but his suggestion led us to develop a bon liquid that does not conduct electricity or support technique with particularly advantageous properties. electrolysis. Along the way to a reliable solenoid, we The idea is illustrated in Fig. 4. learned how to slow and stop atoms efficiently (Phillips The atomic beam source directs atoms, which have a and Metcalf, 1982; Prodan, Phillips, and Metcalf, 1982; wide range of velocities, along the axis (z direction) of a Phillips, Prodan, and Metcalf, 1983a, 1983b, 1984a, tapered solenoid. This magnet has more windings at its 1984b, 1985; Metcalf and Phillips, 1985). entrance end, near the source, so the field is higher at The velocity distribution after deceleration is mea- that end. The laser is tuned so that, given the field- sured in a detection region some distance from the exit induced Zeeman shift and the velocity-induced Doppler end of the solenoid. Here a separate detection laser shift of the atomic transition frequency, atoms with ve- beam produces fluorescence from atoms having the cor- locity v0 are resonant with the laser when they reach the rect velocity to be resonant. By scanning the frequency point where the field is maximum. Those atoms then of the detection laser, we were able to determine the absorb light and begin to slow down. As their velocity velocity distribution in the atomic beam. Observations changes, their Doppler shift changes, but is compensated with the detection laser were made just after turning off by the change in Zeeman shift as the atoms move to a the cooling laser, so as to avoid any difficulties with hav- point where the field is weaker. At this point, atoms with ing both lasers on at the same time. Figure 5 shows the initial velocities slightly lower than v0 come into reso- velocity distribution resulting from Zeeman cooling: a nance and begin to slow down. The process continues large fraction of the initial distribution has been swept with the initially fast atoms decelerating and staying in down into a narrow final velocity group. resonance while initially slower atoms come into reso- One of the advantages of the Zeeman cooling tech- nance and begin to be slowed as they move further nique is the ease with which the optical pumping prob- down the solenoid. Eventually all the atoms with veloci- lem is avoided. Because the atoms are always in a strong ties lower than v0 are brought to a final velocity that axial magnetic field (that is the reason for the ‘‘bias’’ depends on the details of the magnetic field and laser windings in Fig. 4), there is a well-defined axis of quan- tuning. tization that allowed us to make use of the selection The first tapered solenoids that Hal Metcalf and I rules for radiative transitions and to avoid the undesir- used for Zeeman cooling of atomic beams had only a able optical pumping. Figure 6 shows the energy levels few sections of windings and had to be cooled with air of Na in a magnetic field. Atoms in the 3S1/2 (mF52) blown by fans or with wet towels wrapped around the state, irradiated with circularly polarized s1 light, must coils. Shortly after our initial success in getting some increase their mF by one unit, and so can go only to the substantial deceleration, we were joined by my first post- 3P3/2 (mF853) state. This state in turn can decay only to doc, John Prodan. We developed more sophisticated so- 3S1/2 (mF52), and the excitation process can be re- lenoids, wound with wires in many layers of different peated indefinitely. Of course, the circular polarization lengths, so as to produce a smoothly varying field that is not perfect, so other excitations are possible, and would allow the atoms to slow down to a stop while these may lead to decay to other states. Fortunately, in a remaining in resonance with the cooling laser. high magnetic field, such transitions are highly unlikely

Rev. Mod. Phys., Vol. 70, No. 3, July 1998 724 William D. Phillips: Laser cooling and trapping of neutral atoms

These later solenoids were cooled with water flowing over the coils. To improve the heat transfer, we filled the spaces between the wires with various heat-conducting substances. One was a white silicone grease that we put onto the wires with our hands as we wound the coil on a lathe. The grease was about the same color and consis- tency as the diaper rash ointment I was then using on my baby daughters, so there was a period of time when, whether at home or at work, I seemed to be up to my elbows in white grease. The grease-covered, water-cooled solenoids had the FIG. 4. Upper: Schematic representation of a Zeeman slower. annoying habit of burning out as electrolytic action at- Lower: Variation of the axial field with position. tacked the wires during operation. Sometimes it seemed constant deceleration az = F /M = Γscvrec (F > 0) over L if that we no sooner obtained some data than the solenoid ⇒ − − would burn out and we were winding a new one. ous schemes for avoiding optical pumping, were con- On the bright side, the frequent burn-outs provided 1 1 tained in a proposal (Phillips, 1979) that I submitted to Mv(z)2 + Fz = Mv 2 = FL v(z) = 2Γ v (L z) the opportunity for refinement and redesign. Soon we 2 2 0 the Office of Naval Researchsc rec in 1979. Around this time were embedding the coils in a black, rubbery resin. Hal⇒ Metcalf, from the State University− of New York at While it was supposed to be impervious to water, it did Stony Brook, joinedp me in Gaithersburg and we began not have good adhesion properties (except to clothing which sets the magnetic field profileto considerB(z) what√ wouldL bez the. best way to proceed. Hal and human flesh) and the solenoids continued to burn contended that∝ all the− methods looked reasonable, but out. Eventually, an epoxy coating sealed the solenoid we should work on the Zeeman cooler because it would against the water that allowed the electrolysis, and in be the most fun! Not only was Hal right about the fun more recent times we replaced water with a fluorocar- we would have, but his suggestion led us to develop a bon liquid that does not conduct electricity or support technique with particularly advantageous properties. electrolysis. Along the way to a reliable solenoid, we The idea is illustrated in Fig. 4. learned how to slow and stop atoms efficiently (Phillips The atomic beam source directs atoms, which have a and Metcalf, 1982; Prodan, Phillips, and Metcalf, 1982; wide range of velocities, along the axis (z direction) of a Phillips, Prodan, and Metcalf, 1983a, 1983b, 1984a, tapered solenoid. This magnet has more windings at its 1984b, 1985; Metcalf and Phillips, 1985). entrance end, near the source, so the field is higher at The velocity distribution after deceleration is mea- that end. The laser is tuned so that, given the field- sured in a detection region some distance from the exit induced Zeeman shift and the velocity-induced Doppler end of the solenoid. Here a separate detection laser shift of the atomic transition frequency, atoms with ve- beam produces fluorescence from atoms having the cor- locity v0 are resonant with the laser when they reach the rect velocity to be resonant. By scanning the frequency point where the field is maximum. Those atoms then of the detection laser, we were able to determine the absorb light and begin to slow down. As their velocity velocity distribution in the atomic beam. Observations changes, their Doppler shift changes, but is compensated with the detection laser were made just after turning off by the change in Zeeman shift as the atoms move to a the cooling laser, so as to avoid any difficulties with hav- point where the field is weaker. At this point, atoms with ing both lasers on at the same time. Figure 5 shows the initial velocities slightly lower than v0 come into reso- velocity distribution resulting from Zeeman cooling: a nance and begin to slow down. The process continues large fraction of the initial distribution has been swept with the initially fast atoms decelerating and staying in down into a narrow final velocity group. resonance while initially slower atoms come into reso- One of the advantages of the Zeeman cooling tech- nance and begin to be slowed as they move further nique is the ease with which the optical pumping prob- down the solenoid. Eventually all the atoms with veloci- lem is avoided. Because the atoms are always in a strong ties lower than v0 are brought to a final velocity that axial magnetic field (that is the reason for the ‘‘bias’’ depends on the details of the magnetic field and laser windings in Fig. 4), there is a well-defined axis of quan- tuning. tization that allowed us to make use of the selection The first tapered solenoids that Hal Metcalf and I rules for radiative transitions and to avoid the undesir- used for Zeeman cooling of atomic beams had only a able optical pumping. Figure 6 shows the energy levels few sections of windings and had to be cooled with air of Na in a magnetic field. Atoms in the 3S1/2 (mF52) blown by fans or with wet towels wrapped around the state, irradiated with circularly polarized s1 light, must coils. Shortly after our initial success in getting some increase their mF by one unit, and so can go only to the substantial deceleration, we were joined by my first post- 3P3/2 (mF853) state. This state in turn can decay only to doc, John Prodan. We developed more sophisticated so- 3S1/2 (mF52), and the excitation process can be re- lenoids, wound with wires in many layers of different peated indefinitely. Of course, the circular polarization lengths, so as to produce a smoothly varying field that is not perfect, so other excitations are possible, and would allow the atoms to slow down to a stop while these may lead to decay to other states. Fortunately, in a remaining in resonance with the cooling laser. high magnetic field, such transitions are highly unlikely

Rev. Mod. Phys., Vol. 70, No. 3, July 1998

Atom-light interaction Light forces Dressed state picture Deﬁnition Approximations Calculation Interpretation Application of the radiation pressure force Zeeman slower

In the presence of a Doppler shift or a line shift∆ ω0, the detuning 0 is modiﬁed δ = δ kL v ∆ω0. − · − A Zeeman slower compensates the Doppler shift due to deceleration by an appropriate position dependent Zeeman shift ∆ω0(z) = kLv(z), to keep the maximum force at δ0 = 0. stopped beam First realization: B. Phillips and H. Metcalf, Phys. Rev. Lett. 48, 596 (1982). 2 −1 Typically: L = v /(2Γscvrec) < 1 m for v0 500 m s . 0 ∼ ·

~δ ∇s(r) ~δ Fdip = = ∇ ln [1 + s(r)] . − 2 1 + s(r) − 2 Conservative force, derives from the dipole potential

δ U = ~ ln [1 + s(r)] (17) dip 2

Fdip = 0 for a plane wave (s = cst)

Fdip = 0 at resonance 2 ~Ω1(r) δ In the limit s 1: Udip(r) I (r) ' 4 δ2 + Γ2/4 ∝

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Definition Approximations Calculation Interpretation Dipole force Dependence on detuning

Detuning: dispersive shape (real part of the atomic polarizability).

Fdip for I = Is . Changes sign with δ. Scales as 1/δ at large δ.

δ in units of Γ

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Definition Approximations Calculation Interpretation Dipole force vs radiation pressure force

Compare Fdip with Fpr:

Fdip δ 1 | | , Fpr ' Γ kL`

`: the typical length scale for the variation of intensity kL` > 1 ⇒ For moderate δ , Fpr dominates. Dissipative force, high scattering rate.| |

For δ Γ, Fdip dominates. Conservative force. At large| | detuning, use the dipole potential to realize conservative traps.

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Definition Approximations Calculation Interpretation Examples of conservative dipole potentials: dipole traps Positive detunings

For blue detunings( δ > 0), repulsion from high intensity regions.

evanescent wave mirror atoms bouncing oﬀ the mirror J. Dalibard (1994)

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Definition Approximations Calculation Interpretation Dipole traps Positive detunings

For blue detunings( δ > 0), repulsion from high intensity regions.

(a) (b) B'

y x

70 µm (c) z x 35 µm 6 µm

0.045 5 (d) 150 0.04

10 0.035

0.03 15

0.025

20 0.02 OD

25 0.015

0.01 30 50 µm 0.005

35 0 10 20 30 40 50 60 70 0

A 3D box for ultracold atoms (Hadzibabic 2013)

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Definition Approximations Calculation Interpretation Dipole traps Negative detunings

For red detunings( δ < 0), attraction to high intensity regions.

crossed dipole trap optical lattice H. Perrin, PhD thesis D. Boiron, PhD thesis trap depth from 1 µK to several mK.

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Definition Approximations Calculation Interpretation Dipole traps Negative detunings

For red detunings( δ < 0), attraction to high intensity regions.

(a) (b) (c)  1 (a) (b) (c)LG 1 LG 2.3 GHz 2.3 GHz  G G

LG G LG G Norm. Density Norm. Density |1, -1 |1, -1 B |1, 0 20 |m1, 0 20 Bm 0 0

ring trap (Campbell/Phillips group, 2011)

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture Definition Approximations Calculation Interpretation Dipole traps Optical lattices

Optical lattices: standing waves with δ > 0 or δ < 0 E

λ/2 Important parameters: 2 2 ~ kL interband spacing ~ωosc = 2√U0Erec Erec = 2M lowest band width / tunneling: J δE e−2√U0/Erec possibly small ∝ ∝ effective mass in the lowest band: Meff 1/δE possibly large ∝ Lamb-Dicke regime: ∆x λ ~ωosc Erec ⇐⇒ Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture System Dressed states Spontaneous emission Dipole force The dressed state picture

The dressed state picture And now let’s quantize the ﬁeld...

Hélène Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture System Dressed states Spontaneous emission Dipole force Atom in a quantum laser field

Annihilation/creation operatorsa ˆ/ˆa†, such that aˆ, aˆ† = 1. ˆ † Photon number operator N =a ˆ aˆ. Basis of Fock states N , starting from vacuuma ˆ 0 = 0. | i | i aˆ N = √N N 1 aˆ† N = √N + 1 N + 1 Nˆ N = N N | i | − i | i | i | i | i Hamiltonian of a quantum laser ﬁeld:

ˆ † ˆ HL = ~ωaˆ aˆ = ~ωN.

Large number of photons: N¯ = Nˆ 1, ∆N √N¯ N¯. h i ∼ Semi-classical approach for the atom (motion not quantized) Atomic Hamiltonian: ˆ HA = ~ω0 e e . | ih |

H´el`ene Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture System Dressed states Spontaneous emission Dipole force Eigenstates with zero coupling

Eigenstates of the atom+ﬁeld system in the absence of { } coupling: g, N , e, N , N N. | i | i ∈ States can be grouped into two-level manifolds N = e, N 1 , g, N . E {| − i | i} Energies inside the manifold:

~δ Ee,N−1 = ~ω0 + (N 1)~ω = EN , − − 2 ~δ E = N ω = E + , g,N ~ N 2 ~δ EN = + N~ω average energy − 2 Spacing between manifolds: ~ω ~ δ . | |

H´el`ene Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture System Dressed states Spontaneous emission Dipole force Uncoupled manifolds

g, n + 1 e, n En+1 | i | i ¯hδ ¯hδ e, n g, n + 1 | i | i

Uncoupled states ¯hωL group into manifolds. g, n e, n 1 En | i | − i On resonance ¯hδ ¯hδ e, n 1 g, n (δ = 0), the two | − i | i states inside a ¯hω manifold are L degenerate. g, n 1 e, n 2 En 1 | − i | − i − ¯hδ ¯hδ e, n 2 g, n 1 | − i | − i δ > 0 δ < 0

H´el`ene Perrin Laser cooling and trapping ˆ ˆres ˆnon res VAL = VAL + VAL res Vˆ g, N e, N 1 stay inside N AL | i → | − i ⇒ E non res Vˆ g, N e, N + 1 goes from N to N+2 AL | i → | i ⇒ E E

Atom-light interaction Light forces Dressed state picture System Dressed states Spontaneous emission Dipole force Eigenstates with coupling

Dipole operator: Dˆ e g + g e ∝ | ih | | ih | Field operator: Eˆ aˆ +a ˆ† ∝ † Coupling: VˆAL ( e g + g e ) aˆ +a ˆ ∝ | ih | | ih | ∗ ~Ω0(r) † ~Ω0(r) † VˆAL = aˆ e g +a ˆ g e + aˆ g e +a ˆ e g . 2 | ih | | ih | 2 | ih | | ih |

H´el`ene Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture System Dressed states Spontaneous emission Dipole force Eigenstates with coupling

ˆ ˆres ˆnon res VAL = VAL + VAL res Vˆ g, N e, N 1 stay inside N AL | i → | − i ⇒ E non res Vˆ g, N e, N + 1 goes from N to N+2 AL | i → | i ⇒ E E

H´el`ene Perrin Laser cooling and trapping g, n + 2 En+2 | i ¯hδ e, n + 1 | i

g, n + 1 En+1 | i ¯hδ e, n | i ˆ nonres ¯hωL VAL ˆnon res VAL couples two manifolds g, n En | i Vˆ res split by ~(ω + ω0): ¯hδ AL e, n 1 nonresonant processes. | − i ˆ nonres ¯hωL VAL

g, n 1 En 1 | − i − ¯hδ non res neglect Vˆ within RWA. e, n 2 ⇒ AL | − i

g, n 2 En 2 | − i − ¯hδ e, n 3 | − i

g, n + 2 En+2 | i ¯hδ e, n + 1 | i Atom-light interaction Light forces Dressed state picture System Dressed states Spontaneous emission Dipole force Couling between bare states g, n + 1 En+1 | i ¯hδ e, n | i ˆ nonres ¯hωL VAL ˆres VAL couples inside a manifold: g, n En | i resonant processes. ˆ res ¯hδ VAL e, n 1 | − i ˆ nonres ¯hωL VAL

g, n 1 En 1 | − i − ¯hδ e, n 2 | − i

g, n 2 En 2 | − i − ¯hδ e, n 3 H´el`ene Perrin Laser cooling and trapping | − i g, n + 2 En+2 | i ¯hδ e, n + 1 | i

g, n + 1 En+1 | i ¯hδ e, n | i ˆ nonres ¯hωL VAL

g, n En | i ˆ res ¯hδ VAL e, n 1 | − i ˆ nonres ¯hωL VAL

g, n 1 En 1 | − i − ¯hδ e, n 2 | − i

g, n 2 En 2 | − i − ¯hδ e, n 3 | − i

Atom-light interaction Light forces Dressed state picture System Dressed states Spontaneous emission Dipole force Couling between bare states

g, n + 2 En+2 | i ¯hδ e, n + 1 | i

ˆres g, n + 1 V couples inside a manifold: En+1 | i AL ¯hδ resonant processes. e, n | i ˆ nonres ¯hωL VAL ˆnon res VAL couples two manifolds g, n En | i Vˆ res split by ~(ω + ω0): ¯hδ AL e, n 1 nonresonant processes. | − i ˆ nonres ¯hωL VAL

g, n 1 En 1 | − i − ¯hδ non res neglect Vˆ within RWA. e, n 2 ⇒ AL | − i

g, n 2 En 2 | − i − ¯hδ e, n 3 | − i H´el`ene Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture System Dressed states Spontaneous emission Dipole force Eigenstates with coupling

~Ω0(r) † Coupling within RWA: VˆAL = aˆ e g +a ˆ g e . 2 | ih | | ih | Ω0(r): single-photon Rabi frequency at position r.

Coupling within N only (RWA): E ~Ω0(r) g, N VˆAL e, N 1 = e, N 1 VˆAL g, N = √N h | | − i h − | | i 2

As ∆N N¯, the coupling is almost the same for all N with E N N¯ < ∆N. N¯-photon couplingΩ 1(r) = √N¯Ω0(r). | − | Hamiltonian inside N : E ˆ ~ δ Ω1 HN = EN + − . 2 Ω1 δ

H´el`ene Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture System Dressed states Spontaneous emission Dipole force Eigenstates with coupling

~ 2 2 Eigenenergies E±(r) = EN δ + Ω (r) ± 2 1 q 2 2 Frequency spacing Ω(r) = δ + Ω1(r) Ω is the generalized Rabi frequencyq . Eigenstates given by |±i θ θ +, N = sin g, N + cos e, N 1 (18) | i 2 | i 2 | − i θ θ , N = cos g, N + sin e, N 1 (19) |− i − 2 | i 2 | − i where the dressing angle θ(r) is deﬁned by

δ Ω1(r) cos θ(r) = , sin θ(r) = . −Ω(r) Ω(r)

H´el`ene Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture System Dressed states Spontaneous emission Dipole force Avoided crossing

Avoided crossing on resonance. The degeneracy is lifted.

H´el`ene Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture System Dressed states Spontaneous emission Dipole force Coupled manifolds for δ > 0

+, n + 1 g, n + 1 | i En+1 | i ¯hδ ¯hΩ e, n | i , n + 1 |− i The dressed levels ¯hωL +, n are shifted by the g, n | i En | i 2 2 interaction with the ¯hδ ¯hΩ =h ¯ δ + Ω1 laser. e, n 1 p | − i , n No more |− i ¯hωL degeneracy, even on g, n 1 g, n 1 | − i resonance. En 1 | − i − ¯hδ ¯hΩ e, n 2 | − i e, n 2 | − i

Ω = 0 Ω = 0 1 1 6

H´el`ene Perrin Laser cooling and trapping 0 ¨H s0s = s , N 1 eH¨g + g e 1L s, N W h − | ¨| ihH| | ih | ⊗ | i = s0, N 1 g, N 1 e, N 1 s, N h − | − ih − | i θ θ ++ = −− = cos sin W W − 2 2 = 2 ⇒  +− = sin (θ/2)  W 2  −+ = cos (θ/2). W 

Atom-light interaction Light forces Dressed state picture System Dressed states Spontaneous emission Dipole force Spontaneous emission

e can decay spontaneously to g at a rate Γ. | i | i The number of laser photons N 1 is conserved: − e, N 1 g, N 1 . | − i → | − i N is thus coupled to N−1 through this decay: E E s, N s0, N 1 , where s, s0 = . | i → | − i ± Reduced matrix element:

0 s0s = s , N 1 [( e g + g e ) 1L] s, N W h − | | ih | | ih | ⊗ | i

H´el`ene Perrin Laser cooling and trapping 0 s0s = s , N 1 [( e g + g e ) 1L] s, N W h − | | ih | | ih | ⊗ | i

Atom-light interaction Light forces Dressed state picture System Dressed states Spontaneous emission Dipole force Spontaneous emission

0 ¨H s0s = s , N 1 eH¨g + g e 1L s, N W h − | ¨| ihH| | ih | ⊗ | i = s0, N 1 g, N 1 e, N 1 s, N h − | − ih − | i θ θ ++ = −− = cos sin W W − 2 2 = 2 ⇒  +− = sin (θ/2)  W 2  −+ = cos (θ/2). W 

H´el`ene Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture System Dressed states Spontaneous emission Dipole force Spontaneous emission rates

3 frequency lines: ω, ω Ω ± ω + Ω and ω Ω emitted alternatively − Corresponding decay rates: 2 ∝ W±± Approach valid in the limit Γ Ω (resolved levels) +, N | i Γ++ , N Γs0s = Γs→s0 |− i Γ+− 2 θ 2 θ Γ++ = Γ−− = cos 2 sin 2 Γ Γ−− Γ−+ 4 θ Γ+− = sin 2 Γ

Γ = cos4 θ Γ. +, N 1 −+ 2 | − i , N 1 |− − i

H´el`ene Perrin Laser cooling and trapping

Atom-light interaction Light forces Dressed state picture System Dressed states Spontaneous emission Dipole force Steady state populations

Overall population in states: |±i

π± = π+,N π+ + π− = 1. ¯ |N−XN|<∆N Rate equations

dπ+ 4 θ 4 θ dπ− = Γ cos π+ + sin π− = . dt − 2 2 − dt Steady state

4 θ 4 θ 2 θ 2 θ sin 2 cos 2 sin 2 cos 2 π+ π− = − = − − sin4 θ + cos4 θ 1 2 sin2 θ cos2 θ 2 2 − 2 2 cos θ δΩ δΩ = − = = 1 (sin2 θ)/2 Ω2 Ω2/2 δ2 + Ω2/2 − − 1 1

H´el`ene Perrin Laser cooling and trapping Atom-light interaction Light forces Dressed state picture System Dressed states Spontaneous emission Dipole force Dipole force

Ω(r) depends on position through the intensity.

It yields an adiabatic potential E±(r) in state (r). |±i Instantaneous force in state + : | i 2 ~ ~∇Ω ~ 2 F+ = ∇E+ = ∇Ω = = ∇Ω − −2 − 4Ω −4Ω 1

In state : F− = F+. |−i − Total average force F = π+F+ + π−F− = (π+ π−)F+ − δ ∇Ω2/2 F = ( ) ~ ∇Ω2 = ~ 1 (20) π+ π− 1 2 2 − − 4Ω − 2 δ + Ω1/2

We recover F = ∇Udip in the limit Γ Ω. −

H´el`ene Perrin Laser cooling and trapping