cold atoms lecture 1: light forces€¦ · { c. cohen-tannoudji, j. dupont-roc, and g. grynberg:...

Atom-light interaction Light forces Dressed state picture

Cold atomsLecture 1: Light forces

Helene Perrin

Laboratoire de physique des lasers, CNRS-Universite Paris 13, Sorbonne Paris Cite

Cold atoms and molecules - Application to metrology

Helene Perrin Cold atoms Lecture 1: Light forces


IntroductionA little bit of history

First evidence of the mechanical effect of light on a sodiumbeam in 1933 by Otto Frisch

Doppler cooling first suggested in 1975 by Hansch (Nobel2005) and Schawlow (N. 1981) for neutral atoms andindependently 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 et al. (Chu’s group)

Explanation of sub-Doppler cooling in 1989 by Dalibard andCohen-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)



IntroductionEstimation of the light force

Atom irradiated by a resonant laser:

plane wave of frequency ω, photon momentum ~kLEach absorption changes the velocity by vrec = ~kL

M .Spontaneous emissions cancel on average

⇒ net acceleration along the laser alaser = Γscvrec

Typical numbers: for rubidium, vrec = 6 mm·s−1, Γsc ' 2× 107 s−1

⇒ acceleration of order alaser ' 105 m · s−2 ' 104g !



Introduction

A wide range of applications:

High precision spectroscopy (Doppler-freelines)

Quantum information and quantumcomputation

Metrology (fountains, optical clocks, coldatom interferometers...)

New insights in condensed matter physicsand quantum simulation: Bloch oscillations,superfluid-insulator transitions, Cooperpairing, Anderson localization, simulation ofmagnetic systems...

anti-hydrogen trapping(Hansch)

quantum memory(Kimble)micro-wave clockLNE-SYRTEClairon/Bize/Salomon

vortex lattice (Ketterle)bosonic BEC/ fermionicBEC-BCS cross-over



Introduction







quantum memory(Kimble)

micro-wave clockLNE-SYRTEClairon/Bize/Salomon




Introduction







quantum memory(Kimble)micro-wave clockLNE-SYRTEClairon/Bize/Salomon




References

Lecture notes of this course + an easy EPJST introductive paper + a shortbibliography to be found on my personal pagehttp://www-lpl.univ-paris13.fr/bec/BEC/Team_Helene.htm

General references– C. Cohen-Tannoudji and D. Guery-Odelin: Advances in atomic physics:an overview (World Scientific, 2011)

– C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg: Atom-PhotonInteractions : 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)

– C. Cohen-Tannoudji: Atomic motion in laser light, in Fundamentalsystems in Quantum optics, Les Houches, Session LIII (Elsevier, 1992).

On dipole traps:– R. Grimm, M. Weidemuller and Y. Ovchinnokov: Optical dipole trapsfor neutral atoms, Adv. At. Mol. Opt. Phys., 42:95–170, 2000.http://arxiv.org/abs/physics/9902072


http://www-lpl.univ-paris13.fr/bec/BEC/Team_Helene.htm

http://arxiv.org/abs/physics/9902072


Outline

Outline of the lecture1 Atom-light interaction

Two-level modelDipolar interactionHamiltonians

2 Light forcesDefinition of the mean light forceOrders of magnitude. ApproximationsCalculation of the mean forceInterpretation of the mean force and applications

3 The dressed state pictureSystem under considerationEigenstates for the coupled system: the dressed statesSpontaneous emissionDipole force


Atom-light interaction Light forces Dressed state picture 2-level Coupling H

Two-level model

~ωi

e Γ

~ω ~ω0

g

The atom has many electronic transitionsof frequencies ωi .

The laser frequency ω is close to oneparticular frequency ω0: detuningδ = ω − ω0 such that|δ| ω0, ω, |ωi − ω| for all i 6= 0.

We can restrict the discussion to thesetwo levels g and e: transition frequencyω0, or wavelength λ0 = 2πc/ω0, lifetimeof the excited state Γ−1.

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



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 V = −D · E.

Dipolar couplingD = d|e〉〈g |+ d∗|g〉〈e|, (1)

where d is the reduced dipole:

d = 〈e|D|g〉 d∗ = 〈g |D|e〉

We stay in the dipolar approximation for the rest of the lecture.



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:

EL(r, t) =1

2EL(r)

εL(r) e−iωt e−iφ(r) + c.c .

(2)

The classical laser amplitude EL, polarisation εL and phase φmay depend on the position r.

All the quantum fluctuations are set apart (quantum field).



Three coupled systemsLaser, atom and quantum field

Ω Γ

~ω ~ω0

EL VAL HA VAR HR

The total Hamiltonian reads

H = HA + HR + VAL + VAR .



Hamiltonians

Atom: position R, momentum P

HA = ~ω0|e〉〈e|+P

2M. (3)

Quantum field: If the quantum modes of the field are labelledby ` = (k, ε), the energy of the quantum modes is given by

HR =∑`

~ω` a†` a` .

For the description of the atom motion, we will trace on thequantum field variables.

Atom to quantum field coupling:VAR is responsible for spontaneous emission and for thecoupling to the laser field fluctuations. It is not necessary togive an explicit form of this term here.



Hamiltonians

Atom-laser coupling:

VAL = −D · EL(R, t) = V resAL + V non res

AL

V resAL = −1

2

(d · ε(R)

)EL(R) |e〉〈g | e−iωt e−iφ(R) + h.c .

V non resAL = −1

2

(d · ε∗(R)

)E∗L(R) |e〉〈g | e iωt e iφ(R) + h.c.

In the Heisenberg picture, |e〉〈g | evolves as e iω0t

⇒ V resAL describes a slow evolution (at frequency δ = ω − ω0) while

V non resAL describes a fast evolution (at frequency ω + ω0).



Hamiltonians

We introduce the Rabi frequency Ω1(r) defined by

~Ω1(r) = − (d · ε(r)) EL(r) . (4)

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

V resAL =

~Ω1(R)

2

|e〉〈g | e−iωt e−iφ(R) + h.c .

(5)

The Rabi frequency is the oscillation frequency between |g〉 and|e〉 at resonance in the strong coupling regime.


Atom-light interaction Light forces Dressed state picture Definition Approximations Calculation Interpretation

Light forces

Light forces

Calculation and interpretation of the mean light force



Mean light forceDefinition

Velocity operator in the Heisenberg representation:

dR

dt=

1

i~

[R, H

]=

1

i~

[R,

P2

2M

]=

P

M. (6)

Force operator:

F =dP

dt=

1

i~

[P, H

]= −∇VAL −∇VAR . (7)

Mean light force:

F = 〈F〉 = −〈∇VAL〉 − 〈∇VAR〉 = −〈∇VAL〉 (8)

as 〈∇VAR〉 = 0, see CCT Les Houches lectures.N.B.: ∇VAR 6= 0 and contribute to the fluctuations of the randomforce.



Mean light forcePrinciple of the calculation

Mean light forceF = −〈∇VAL〉

Compute the density matrix.

Trace over the quantum field variables.

Calculate the average value of ∇VAL in the state described bythe density matrix.

And before starting all this, make some approximations...



Rotating wave approximation

The non-resonant part of the dipolar coupling oscillates atvery large frequency (ω + ω0).

Its contribution to the force will be negligible provided|δ|,Ω1 ω0, ωL.⇒ discard V non res

AL and only keep V resAL .

This is known as the rotating wave approximation.

N.B. With a quantized laser field approach, it corresponds tokeeping processes where a photon is absorbed when the atom getsexcited, and discarding those where a photon is emitted while theatom gets excited.



Time scales

Time scale for the evolution of the internal atomic variables:

tint = Γ−1.

Typical value in laser-cooled species: Γ2π ∼ a few MHz,

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, suchthat the laser frequency is Doppler shifted significantly:

kL∆v = kL × alasertext ' kL × Γscvrectext =Γ

2

~k2L

M



Time scales

⇒ text '2M

~k2L

= ω−1rec

Erec = 12Mv2

rec =~2k2

L2M = ~ωrec is the recoil energy,

ωrec the recoil frequency, trec = ω−1rec 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 steadystate before the external state. changes⇒ average value in the steady state to compute the light force.



Semi-classical approximation

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

Relevant if the quantum fluctuations ∆R and ∆P are sufficientlysmall for the phase and the laser frequency to be well defined:

kL∆R 1 and kL∆v Γ or ∆P MΓ

kL.

Recall ∆R ∆P > ~/2:

~2 M

Γ

k2L

or Γ ωrec broadband condition!

We now assume the broadband condition is fulfilled, and willcompute the force F at position r = 〈R〉 for a velocity v = 〈P〉/M.



Calculation of the mean force

We assume: RWA, broadband condition.

F = −〈∇VAL〉 ' 〈∇(D · EL(r, t)

)〉

= ∇(〈D〉 · EL(r, t)

)= ∇

∑i=x ,y ,z

〈Di 〉ELi (r, t)

=

∑i=x ,y ,z

〈Di 〉∇ELi (r, t)

'∑

i=x ,y ,z

〈Di 〉st∇ELi (r, t)

〈Di 〉st deduced from the optical Bloch equations on the internalstate density matrix operator σ.



Steady state of the internal variables

Optical Bloch equations deduced from

i~d σ

dt= [HA + VAL, σ]− i~Γσ,

where Γ accounts for the relaxation from |e〉 to |g〉, and we haveσgg + σee = 1 and σge = σ∗eg . We get:

σee = −Γσee + iΩ1(r)

2

(σeg e

iωt e iφ(r) − σ∗eg e−iωt e−iφ(r))

σeg = −(iω0 +

Γ

2

)σeg − i

Ω1(r)

2(1− 2σee) e−iωt e−iφ(r) .




We introduce 3 real independent variablesu = <

[σeg e

iωt e iφ(r)]

v = −=[σeg e

iωt e iφ(r)]

w = 12 (σee − σgg ) = σee − 1

2

satisfying the coupled equationsu = −Γ

2u + δv ,

v = −Γ2v − δu − Ω1w ,

w = −Γ(w + 1

2

)+ Ω1v .




The stationary solution is

ust =δ

Ω1

s

1 + svst =

Γ

2Ω1

s

1 + s

wst +1

2= σee,st =

1

2

s

1 + s,

where we have defined the saturation parameter s(r)

s(r) =Ω2

1(r)/2

δ2 + Γ2/4=

I (r)/Is

1 + 4δ2/Γ2. (9)

Is is the saturation intensity Is = 2π2~cΓ/(3λ30), typically a few

mW·cm−2.




We get for the stationary dipole

〈D〉 · ε(r) = 2d.ε(r)s(r)

1 + s(r)(10)

×

δ

Ω1(r)cos [ωt + φ(r)]

in phase

− Γ

2Ω1(r)sin [ωt + φ(r)]

in quadrature

.

The term in phase with the laser field yields to dephasing and to aconservative force (real part of polarizability).The term in quadrature with the laser field yields to absorption andto a dissipative force (imaginary part of polarizability).



Expression of the light force

Expression of the light force (averaged over one laser period):

F(r) = − s(r)

1 + s(r)

(~δ

∇Ω1

Ω1(r)+

~Γ

2∇φ

)= Fdip + Fpr . (11)

Dissipative force: Fpr

Fpr = −~Γ

2

s(r)

1 + s(r)∇φ. (12)

Conservative force: Fdip

Fdip = −~δ s(r)

1 + s(r)

∇Ω1

Ω1(r)= −~δ

2

∇s(r)

1 + s(r). (13)



Interpretation of the light forcesRadiation pressure force

Consider a plane wave 12E0

εL e

−iωt+ikL·r + c .c .

:

Ω1 uniform, s(r) = s0. φ(r) = −kL · r⇒∇φ = −kL.

(12)⇒ Fpr =Γ

2

s0

1 + s0~kL. (14)

Population of the excited state σee = 12

s01+s0

, scattering rate:

Γsc = Γσee =Γ

2

s0

1 + s0(15)

Fpr = Γsc ~kL

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



Radiation pressure forceDependence on the intensity

The intensity appears in s0. On resonance, s0 = I/Is . For I Is ,the force is maximum Fpr,max = Γ

2~kL.Saturation behavior:

Dependence of Fpr/Fmax on I/Is .



Radiation pressure forceDependence on the detuning

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

Dependence on the detuning, fordifferent values of the intensity.

From bottom to top:I/Is = 0.1, 1, 10 and 100.

FWHM: Γ√

1 + I/Is

At large detuning, Fpr ∝ 1/δ2

δ in units of Γ



Application of the radiation pressure forceZeeman slower

In the presence of a Doppler shift or a line shift ∆ω0, the detuningis modified δ′ = δ−kL · v −∆ω0.

A Zeeman slower compensatesthe Doppler shift due todeceleration by an appropriateposition dependent Zeeman shift∆ω0(z) = kLv(z), to keep themaximum force at δ′ = 0.

ous schemes for avoiding optical pumping, were con-tained in a proposal (Phillips, 1979) that I submitted tothe Office of Naval Research in 1979. Around this timeHal Metcalf, from the State University of New York atStony Brook, joined me in Gaithersburg and we beganto consider what would be the best way to proceed. Halcontended that all the methods looked reasonable, butwe should work on the Zeeman cooler because it wouldbe the most fun! Not only was Hal right about the funwe would have, but his suggestion led us to develop atechnique with particularly advantageous properties.The idea is illustrated in Fig. 4.

The atomic beam source directs atoms, which have awide range of velocities, along the axis (z direction) of atapered solenoid. This magnet has more windings at itsentrance end, near the source, so the field is higher atthat end. The laser is tuned so that, given the field-induced Zeeman shift and the velocity-induced Dopplershift of the atomic transition frequency, atoms with ve-locity v0 are resonant with the laser when they reach thepoint where the field is maximum. Those atoms thenabsorb light and begin to slow down. As their velocitychanges, their Doppler shift changes, but is compensatedby the change in Zeeman shift as the atoms move to apoint where the field is weaker. At this point, atoms withinitial velocities slightly lower than v0 come into reso-nance and begin to slow down. The process continueswith the initially fast atoms decelerating and staying inresonance while initially slower atoms come into reso-nance and begin to be slowed as they move furtherdown the solenoid. Eventually all the atoms with veloci-ties lower than v0 are brought to a final velocity thatdepends on the details of the magnetic field and lasertuning.

The first tapered solenoids that Hal Metcalf and Iused for Zeeman cooling of atomic beams had only afew sections of windings and had to be cooled with airblown by fans or with wet towels wrapped around thecoils. Shortly after our initial success in getting somesubstantial deceleration, we were joined by my first post-doc, John Prodan. We developed more sophisticated so-lenoids, wound with wires in many layers of differentlengths, so as to produce a smoothly varying field thatwould allow the atoms to slow down to a stop whileremaining in resonance with the cooling laser.

These later solenoids were cooled with water flowingover the coils. To improve the heat transfer, we filled thespaces between the wires with various heat-conductingsubstances. One was a white silicone grease that we putonto the wires with our hands as we wound the coil on alathe. The grease was about the same color and consis-tency as the diaper rash ointment I was then using on mybaby daughters, so there was a period of time when,whether at home or at work, I seemed to be up to myelbows in white grease.

The grease-covered, water-cooled solenoids had theannoying habit of burning out as electrolytic action at-tacked the wires during operation. Sometimes it seemedthat we no sooner obtained some data than the solenoidwould burn out and we were winding a new one.

On the bright side, the frequent burn-outs providedthe opportunity for refinement and redesign. Soon wewere embedding the coils in a black, rubbery resin.While it was supposed to be impervious to water, it didnot have good adhesion properties (except to clothingand human flesh) and the solenoids continued to burnout. Eventually, an epoxy coating sealed the solenoidagainst the water that allowed the electrolysis, and inmore recent times we replaced water with a fluorocar-bon liquid that does not conduct electricity or supportelectrolysis. Along the way to a reliable solenoid, welearned how to slow and stop atoms efficiently (Phillipsand Metcalf, 1982; Prodan, Phillips, and Metcalf, 1982;Phillips, Prodan, and Metcalf, 1983a, 1983b, 1984a,1984b, 1985; Metcalf and Phillips, 1985).

The velocity distribution after deceleration is mea-sured in a detection region some distance from the exitend of the solenoid. Here a separate detection laserbeam produces fluorescence from atoms having the cor-rect velocity to be resonant. By scanning the frequencyof the detection laser, we were able to determine thevelocity distribution in the atomic beam. Observationswith the detection laser were made just after turning offthe cooling laser, so as to avoid any difficulties with hav-ing both lasers on at the same time. Figure 5 shows thevelocity distribution resulting from Zeeman cooling: alarge fraction of the initial distribution has been sweptdown into a narrow final velocity group.

One of the advantages of the Zeeman cooling tech-nique is the ease with which the optical pumping prob-lem is avoided. Because the atoms are always in a strongaxial magnetic field (that is the reason for the ‘‘bias’’windings in Fig. 4), there is a well-defined axis of quan-tization that allowed us to make use of the selectionrules for radiative transitions and to avoid the undesir-able optical pumping. Figure 6 shows the energy levelsof Na in a magnetic field. Atoms in the 3S1/2 (mF52)state, irradiated with circularly polarized s1 light, mustincrease their mF by one unit, and so can go only to the3P3/2 (mF853) state. This state in turn can decay only to3S1/2 (mF52), and the excitation process can be re-peated indefinitely. Of course, the circular polarizationis not perfect, so other excitations are possible, andthese may lead to decay to other states. Fortunately, in ahigh magnetic field, such transitions are highly unlikely

FIG. 4. Upper: Schematic representation of a Zeeman slower.Lower: Variation of the axial field with position.

724 William D. Phillips: Laser cooling and trapping of neutral atoms

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

stopped beam

First realization: B. Phillips and H. Metcalf, Phys. Rev. Lett. 48,596 (1982).Typically: L = v2

0 /(2Γscvrec) < 1 m for v0 ∼ 500 m·s−1.

⇒ constant deceleration az = −F/M = −Γscvrec (F > 0) over L if

1

2Mv(z)2 + Fz =

1

2Mv2

0 = FL⇒ v(z) =√

2Γscvrec(L− z)

which sets the magnetic field profile B(z) ∝√L− z .



Interpretation of the light forcesDipole force

Fdip = −~δ2

∇s(r)

1 + s(r)= −~δ

2∇ ln [1 + s(r)] .

Conservative force, derives from the dipole potential

Udip =~δ2

ln [1 + s(r)] (16)

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

Fdip = 0 at resonance

In the limit s 1: UdipI (r) '~Ω2

1(r)

4

δ

δ2 + Γ2/4∝ I (r)



Dipole forceDependence on detuning

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

Fdip for I = Is .

Changes sign withδ.

Scales as 1/δ atlarge δ.

δ in units of Γ



Dipole force vs radiation pressure force

Compare Fdip with Fpr:

Fdip

Fpr' |δ|

Γ

1

kL`,

`: the typical length scale for the variation of intensity ⇒ kL` > 1

For moderate |δ|, Fpr dominates. Dissipative force, highscattering rate.

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



Examples of conservative dipole potentials: dipole trapsPositive detunings

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

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



Dipole trapsPositive detunings

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

6 µm

(a)

(c)

(d)

(b)

10 20 30 40 50 60 70

5

10

15

20

25

30

350

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045150

0

OD

B'

z

y x

50 µm

70 µm

35 µm z

x

A 3D box for ultracold atoms (Hadzibabic 2013)



Dipole trapsNegative detunings

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

crossed dipole trap optical latticeH. Perrin, PhD thesis D. Boiron, PhD thesis

trap depth from 1µK to several mK.



Dipole trapsNegative detunings

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

(a)

0

1

|1, 0

G

(c)(b)

20 m|1, -1

2.3 GHz

LG

B

LG

G

Nor

m. D

ensi

ty

(a)

0

1

|1, 0

G

(c)(b)

20 m|1, -1

2.3 GHz

LG

B

LG

G

Nor

m. D

ensi

ty

ring trap (Campbell/Phillips group, 2011)



Dipole trapsOptical lattices

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

E

U0

λ/2Important parameters:

interband spacing ~ωosc = 2√U0Erec Erec =

~2k2L

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


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...



Atom in a quantum laser field

Annihilation/creation operators a/a†, such that[a, a†

]= 1.

Photon number operator N = a†a.

Basis of Fock states |N〉, starting from vacuum a|0〉 = 0.

a|N〉 =√N|N − 1〉 a†|N〉 =

√N + 1|N + 1〉 N|N〉 = N|N〉

Hamiltonian of a quantum laser field:

HL = ~ωa†a = ~ωN.

Large number of photons: N = 〈N〉 1, ∆N N.

Semi-classical approach for the atom (motion not quantized)

Atomic Hamiltonian:

HA = ~ω0|e〉〈e|.



Eigenstates with zero coupling

Eigenstates of the atom+field system in the absence ofcoupling: |g ,N〉, |e,N〉,N ∈ N.

States can be grouped into two-level manifoldsEN = |e,N − 1〉, |g ,N〉.Energies inside the manifold:

Ee,N−1 = ~ω0 + (N − 1)~ω = EN −~δ2,

Eg ,N = N~ω = EN −~δ2,

EN = −~δ2

+ N~ω average energy

Spacing between manifolds: ~ω ~|δ|.



Uncoupled manifolds

uncoupled states for: δ > 0 δ < 0

On resonance (δ = 0), the two states inside a manifold aredegenerate.



Eigenstates with coupling

Coupling within RWA: VAL =~Ω0(r)

2

(a|e〉〈g |+ a†|g〉〈e|

).

Ω0(r): single-photon Rabi frequency at position r.

Coupling within EN only (RWA):

〈g ,N|VAL|e,N − 1〉 = 〈e,N − 1|VAL|g ,N〉 =~Ω0(r)

2

√N

As ∆N N, the coupling is about the same for all EN with|N − N| < ∆N. N-photon coupling Ω1(r) =

√NΩ0(r).

Hamiltonian inside EN :

HN = EN +~2

(−δ Ω1

Ω1 δ

).



Eigenstates with coupling

Eigenenergies E±(r) = EN ±~2

√δ2 + Ω2

1(r)

Frequency spacing Ω(r) =√δ2 + Ω2

1(r)

Ω is the generalized Rabi frequency.

Eigenstates |±〉 given by

|+,N〉 = sinθ

2|g ,N〉+ cos

θ

2|e,N − 1〉 (17)

|−,N〉 = − cosθ

2|g ,N〉+ sin

θ

2|e,N − 1〉 (18)

where the dressing angle θ(r) is defined by

cos θ(r) = − δ

Ω(r), sin θ(r) =

Ω1(r)

Ω(r).



Avoided crossing

Avoided crossing on resonance. The degeneracy is lifted.



Spontaneous emission

|e〉 can decay spontaneously to |g〉 at a rate Γ.

The number of laser photons N − 1 is conserved:|e,N − 1〉 → |g ,N − 1〉.EN is thus coupled to EN−1 through this decay:|s,N〉 → |s ′,N − 1〉, where s, s ′ = ± .

Reduced matrix element:

Ws′s = 〈s ′,N − 1| [(|e〉〈g |+ |g〉〈e|)⊗ 1] |s,N〉

Ws′s = 〈s ′,N − 1|[(HHH|e〉〈g |+ |g〉〈e|

)⊗ 1

]|s,N〉

= 〈s ′,N − 1|g ,N − 1〉〈e,N − 1|s,N〉

=⇒

W++ =W−− = − cos

θ

2sin

θ

2W+− = sin2(θ/2)

W−+ = cos2(θ/2).



Spontaneous emission rates

3 frequency lines: ω, ω ± Ω

ω + Ω and ω − Ω emitted alternatively

Corresponding decay rates: ∝ W2±±

Approach valid in the limit Γ Ω (resolved levels)

Γs′s = Γs→s′

Γ++ = Γ−− = cos2 θ2 sin2 θ

2 Γ

Γ+− = sin4 θ2 Γ

Γ−+ = cos4 θ2 Γ.

|+,N〉|−,N〉

|+,N−1〉|−,N−1〉

Γ++

Γ+−

Γ−− Γ−+



Steady state populations

Overall population in |±〉 states:

π± =∑

|N−N|<∆N

π+,N π+ + π− = 1.

Rate equations

dπ+

dt= Γ

(− cos4 θ

2π+ + sin4 θ

2π−

)=

dπ−dt

.

Steady state

π+ =sin4 θ

2

sin4 θ2 + cos4 θ

2

π− =cos4 θ

2

sin4 θ2 + cos4 θ

2

π+ − π− =sin4 θ

2 − cos4 θ2

sin4 θ2 + cos4 θ

2

=sin2 θ

2 − cos2 θ2

1− 2 sin2 θ2 cos2 θ

2

=− cos θ

1− (sin2 θ)/2=

δΩ

Ω2 − Ω21/2

=δΩ

δ2 + Ω21/2



Dipole force

Ω(r) depends on position through the intensity.

Instantaneous force in state |+〉:

F+ = −∇E+ = −~2∇Ω = −~∇Ω2

4Ω= − ~

4Ω∇Ω2

1

In state |−〉: F− = −F+.

Total average force F = π+F+ + π−F− = (π+ − π−)F+

F = −(π+ − π−)~

4Ω∇Ω2

1 = −~δ2

∇Ω21/2

δ2 + Ω21/2

(19)

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


cold atoms lecture 1: light forces€¦ · { c. cohen-tannoudji, j. dupont-roc, and g. grynberg:...

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