ultracold atoms - icfp master coursenascimbene/enseignement/coldatoms/cours... · 2019. 2. 2. ·...

Ultracold atomsICFP Master Course

Fabrice Gerbier ([email protected])

Sylvain Nascimbene ([email protected])

Aurelien Perrin ([email protected])

Master ICFP, Ultracold atoms

January 29, 2019

Lecture slides, exercise classes and other material available at

http://www.phys.ens.fr/ nascimbene/enseignement.php

References on atomic physics

• Claude Cohen-Tannoudji and David Guery-Odelin, Advances In Atomic Physics:An Overview, World Scientific.

• Claude Cohen-Tannoudji, Jacques Dupont-Roc and Gilbert Grynberg, Processusd’interaction entre photons et atomes, CNRS Editions.

More details on laser cooling (in French): lectures notes by Jean Dalibard(http://www.phys.ens.fr/ dalibard/)

• Unpublished Master lecture notes : Atomes ultrafroids

• College de France lectures 2015: Une breve histoire des atomes froids

•

Quantum gases:

• Wolfgang Ketterle et al., arXiv:cond-mat/9904034 (1999) and arXiv:0801.2500(2008)

• Sandro Stringari and Lev Pitaevskii, Bose-Einstein condensation, OxfordUniversity Press.

Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])

A brief history of atomic physics

Atomic physics born with spectroscopy at the end of the 19th century.

Progressed hand-in-hand with quantum mechanics in the years 1900-1930.

AMO -Atomic, Molecular and Optical Physics : dilute gases (as opposed to denseliquids and solids).

Common view in the early 50’s was that AMO physics was essentially understood,with little left to discover. Sixty years later, this view has been proven wrong.

AMO Physics underwent a serie of revolutions, each leading to the next one :

• the 1960’s : the laser

• the 1970’s : laser spectroscopy

• the 1980’s : laser cooling and trapping of atoms and ions

• the 1990’s : quantum degenerate atomic gases (Bose-Einstein condensates andFermi gases)

• the 2000’s : femtosecond frequency combs


Control over the quantum state of an atom

The quantum mechanical description of an atom introduces several quantum numbersto describe its state :

• internal quantum numbers to describing the relative motion of electrons withrespect to the nuclei, e.g. electronic angular momentum J ,

• external quantum numbers, e.g. center of mass position R.

In spectroscopy, electromagnetic fields are used to probe the structure of internalstates. Extensions of the same techniques developped for spectroscopy allow one tocontrol the internal degrees of freedom coherently.

Laser cooling and trapping techniques allow one to do the same with the externaldegrees of freedom of the atom.


Milestones in ultracold atom physics

• first deflection of an atomic beam observed as early as 1933 (O. Frisch)

• revival of study of radiative forces in the lates 1970’s; first proposals for lasercooling of neutral atoms (Hansch – Dehmelt) and ions (Itano – Wineland)Why ? Rise of the LASER

Laser cooling and trapping:

• 1980 : Slowing and bringing an atomic beam to rest

• 1985 : Optical molasses

• 1988 : magneto-optical traps , sub-Doppler cooling

• 1997 : Nobel Prize for Chu, Cohen-Tannoudji, Phillips

Quantum degenerate gases:

• Bose-Einstein condensation in 1995 [Cornell, Wieman,Ketterle : Nobel 2001]

• Degenerate Fermi gases in 2001 [JILA]


Why are ultracold atoms (and molecules) still interesting today ?

Laser cooling andtrapping of neutral

atoms

Nobel Prize 1997 :S. Chu,

C. Cohen-Tannoudji,

W. D. Phillips

Ultracold atoms

Ultracoldchemistry:

From simple toexotic molecules,controlled at the

quantum level

Quantumdegenerate

gases

Atom inter-ferometry:Precision

measurementsand metrology

Quantum gases :Bose-Einsteincondensation

(1995)DegenerateFermi gases

(1999)

Nobel Prize 2001 :E. Cornell,

W. Ketterle,

C. Wieman


Ultracold atomic gases as many-body systems

Quantum degeneracy : phase space density λdB & n−1/3

λdB =√

2π~2

mkBT: thermal De Broglie wavelength

n: spatial density n−1/3: mean interparticle distance

From W. Ketterle group website, http://www.cua.mit.edu/

Interacting atoms, but dilute gas: na3 1

a : scattering length for s−wave interactions

8πa2: scattering cross-section (bosons)

a n−1/3 λdB

Typical values (BEC of 23Na atoms) :a ∼ 2 nm

n−1/3 ∼ 100 nm

λdB ∼ 1µm at T = 100 nK


Which atomic species ?


Outline of the course

1 Introduction and overview of experimental techniques (2 lectures),• Atomic structure; Magneto-optical trap• Atomic interferometry

2 Non-interacting quantum gases (1 lecture),

3 Atom-atom scattering in the ultracold limit (1/2 lecture),• Study of a model potential; scattering length

4 Bose-Einstein condensates (1/2 lecture),• Collective modes and free expansion of a Bose-Einstein condensate• Vortices in rotating Bose-Einstein condensates

5 Coherence of quantum gases (1 lecture),• First-order correlation function of a Bose gas• Hanbury-Brown-Twiss effect with cold atoms

6 Ultracold fermions ; BEC-BCS crossover (2 lectures),• Superfluid hydrodynamics; boson-fermion mixtures

7 Optical lattices ; superfluid-Mott insulator transition (3 lectures).• Bloch oscillations• Particle-hole excitations of a Mott insulator


The experimental path to quantum degerate gases

Typical experimental sequence :

• catch atoms in amagneto-optical trap

• laser cooling to ∼ 50µK

• transfer to conservative trap(no resonant light): opticaltrap or magnetic trap

• evaporative cooling to BEC

Take a picture of the cloud

Repeat


Ytterbium BEC experiment at LKB

Oven

Yb BeamYb MOT


Why is a two-step sequence necessary ?

Quantum gases : Phase-space density D = nλ3th ≥ 1

• Step 1: laser cooling in a magneto-optical trap

• Step 2 : evaporative cooling in a conservative trap

Laser cooling relies on the interaction between the atoms and a near-resonant laser.Spontaneous emission of photons is :

• essential to provide necessary dissipative mechanism to cool the motional degreesof freedom of the atoms,

• but also intrinsically random. This randomness prevents to cool the atoms belowa certain limiting temperature !

Typical MOT of Yb :

n ∼ 1010 − 1011 at/cm3, T ∼ 10µK, λdB ∼ 40 nm, D ∼ 10−6 − 10−5

To overcome the limitations of laser cooling, all experiments (with one exception)follow the same path :

• trapping in a conservative trap: optical trap or magnetic trap,

• evaporative cooling to quantum degeneracy.


1 Atom-photon interactionsOptical Bloch equations for a two-level atom

2 Radiative forcesSemi-classical theoryRadiation pressure; Zeeman slowing of an atomic beamOptical molassesFluctuations of radiative forces : momentum diffusionSub-Doppler cooling

3 Imaging cold atoms

4 The magneto-optical trap

5 Optical dipole traps

6 Evaporative cooling


Atom and photons

Photons :

Modes of the electomagnetic field, e.g.plane waves with

• momentum ~kµ

• energy ~ω = ~c|kµ|

• polarisation εµ ⊥ kµ

Quantized electric field E

E =∑

kµ,εµ⊥kµ

Eµεµakµ,εµ + h.c.

Eµ = i

√~c|kµ|2ε0L3

eikµ·r

akµ,εµ : annihilation operator of a photonin mode µ

Atoms :

• center-of mass variables

• internal variables, e.g. electronicangular momentum |J,mJ 〉

Example : 133Cs (or other alkali atoms)

6s, n = 6

6p, n = 6

Coulomb

1S1/2J = 1/2

2P1/2

J = 1/2

2P3/2J = 3/2

Fine

structure


Dipolar electric approximation

Dipolar electric approximation :

VA−em = −d · E(R)

• d : electric dipole of the atom (internal variables)

• R : center of mass of the atom (external variable)

• Approximation valid if atomic dimensions (∼ A) wavelength (UV, visible, IR,...)

• Higher-order terms in the multipolar expansion smaller by a factor ∼ α2

Electric dipole operator in the atomic basis:

d =∑gi,ej

dij |ej〉〈gi|+ h.c., dij = 〈ej |d|gi〉

|gi〉i, |ej〉j denotes the ground state and excited states manifolds (internalvariables).

Dipolar electric Hamiltonian :

VA−em = −∑

kµ,εµ⊥kµ

(d · εµ

)Eµakµ,εµ + h.c.


Conservation laws in atom-photon interactions : momentum and energy

Consider a two level atoms with ground state g, excited state e, initially in a

momentum eigenstate. We denote by ωeg =Ee−Eg

~ the Bohr frequency of thetransition.

Absorption of a photon (q, ε): atomic state changes from |g,ki〉 to |e,kf 〉

Conservation of momentum : kf = ki + q

Conservation of energy : ω = ωeg + ~q22M

+ ~Mq · ki

ER =~2q2

2M: recoil energy,

~Mq · ki = q · vi: Doppler shift

ωR =ER

~: recoil frequency

Gas in thermal equilibrium : P (v) ∝ e−Mv2

2kBT

Doppler broadening : spectral width ∆ω = q√

8kBTπM −3 −2 −1 0 1 2 3

ω

0.0

0.2

0.4

0.6

0.8

1.0

Spe

ctru

m


Conservation laws in atom-photon interactions : angular momentum

Laser polarisation ε: ε · kL = 0

Decomposition in the standard basis ε =∑q=0,±1 εqeq

e+ =−1√

2(ex + iey) , e− =

1√

2i(ex − iey) , , e0 = ez

σ±-polarized light: ε = e±, π-polarized light: ε = ez

N.B. in atomic physics one usually chooses z along the direction of a static magnetic field (if

any), not necessarily along the laser propagation direction kL.

Selection rules for allowed transitions :

∆J = 0,±1

∆mJ = 0,±1

mg

me = mg − 1 me = mg me = mg + 1

σ− π σ+


The standard model of quantum optics: two-level atom

Atom in free space:

Hat =P 2

2M+ ~ωeg |e〉〈e|

• center-of-mass momentum/position P /R

• two internal states g/e

Dipole operator :

d = dD+ + dD−

D+ = |e〉〈g| =(D−

)†g

e

hωeg

• Bohr frequency

ωeg =Ee−Eg

~


Master equation for the atomic density operator

Atom and field are entangled : the proper quantum mechanical labeling of a state is|φ〉atom ⊗ |α〉L ⊗ |η〉m 6=L. We need a purely atomic description that erases allinformation about the electromagnetic field, except for its effect on the atomicvariables.

Density matrix formalism : σ describes the state of the composite atom+field system

• Tr(σ) = 1

• Tr(σ2) ≤ 1

• 〈O〉 = Tr(σO)

• i~ ddtσ = [Hatom+field, σ]

Reduced atomic density matrix :

ρ = Trfield [σ] =

(ρgg ρgeρeg ρee

): purely atomic operator

N.B. ραβ = 〈α|ρ|β〉 is still an operator in general with respect to the external degreesof freedom of the atom.

i~d

dtρ = [Hat + VA−L, ρ] + Trfield

[VA−V , σatom+field

]VA−L corresponds to coherent driving by the classical Laser:EL = εLEL cos(ωLt+ ϕL)

VA−V corresponds to the coupling to the electromagnetic field vacuum (spontaneousemission): Dissipation.


Optical Bloch equations for a two-level system

Optical Bloch equations :

d

dtρee = −Γρee −

ΩL

2i(ρge − ρeg)

d

dtρgg = Γρee +

ΩL

2i(ρge − ρeg)

d

dtρeg =

(iδL −

Γ

2

)ρeg + i

ΩL

2(ρee − ρgg)

g

e

ΩL

Γ

hωeg

hδL

We defined: ρge = e−i(ωLt+ϕL)ρge, ρeg = ei(ωLt+ϕL)ρeg .

Natural linewidth of the transition: Γ =ω3egd

2

6πε0~c2

Laser electric field: EL = εLEL cos(ωLt+ ϕL)

Rabi frequency: ΩL = dEL2


Optical Bloch equations for a two-level system

Optical Bloch equations :

d

dtρee = −Γρee −

ΩL

2i(ρge − ρeg)

d

dtρgg = Γρee +

ΩL

2i(ρge − ρeg)

d

dtρeg =

(iδL −

Γ

2

)ρeg + i

ΩL

2(ρee − ρgg)

g

e

ΩL

Γ

hωeg

hδL

Define:

u =1

2(ρge + ρeg) ,

v =1

2i(ρge − ρeg)

u = −Γ

2u+ δLv,

v = −Γ

2v − δLu− ΩL

ρee − ρgg2

Interpretation of u and v :

〈d〉 = 2d [u cos(ωLt+ ϕL)− v sin(ωLt+ ϕL)]

• u : component of the average dipole in-phase with the field• v : component of the average dipole out-of-phase with the field


Steady-state solutions of the OBE

Stationary solutions :

ust =δL

ΩL

s

1 + s

vst =Γ

2ΩL

s

1 + s

ρee,st =1

2

s

1 + s

Saturation parameter :

s =

Ω2L2

δ2L + Γ2

4

Numerical integration with w(0) = −1/2, u(0) = v(0) = 0:

0 2 4 6 8 10

Γt

−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

u,v

ΩL/Γ = 5.0, δL/Γ = 1.0

u v w

0 2 4 6 8 10

Γt

0.0

0.2

0.4

0.6

0.8

1.0

ρgg,ρ

eeρgg ρee

Damped Rabi oscillations, damping to steady state after a few Γ−1


Low and high intensity limits

Stationary solutions :

ust =δL

ΩL

s

1 + s

vst =Γ

2ΩL

s

1 + s

ρee,st =1

2

s

1 + s

Saturation parameter :

s =

Ω2L2

δ2L + Γ2

4

High intensity s 1 : Ω2L δ2

L + Γ2

4

ust ≈δL

ΩL→ 0

vst ≈Γ

2ΩL→ 0

ρee,st ≈1

2

saturation of the resonant optical responseof the atom

Low intensity s 1 : Ω2L δ2

L + Γ2

4

ust ≈δL

ΩLs

vst ≈Γ

2ΩLs

ρee,st ≈s

2


Aside: optical intensity and saturation intensity

Optical intensity :

I =Optical power

Area=ε0c

2E2L,

Ω2L =

2d2I

~2ε0c

Saturation intensity : Isat defined suchthat s = 1 on resonance (δL = 0)

Isat =~Γck3

L

12π∼ mW/cm2

Ω2L =

Γ2

2

I

Isat

s =

Ω2L2

δ2L + Γ2

4

=

IIsat(

2δLΓ

)2+ 1

,

s

1 + s=

IIsat(

2δLΓ

)2+ 1 + I

Isat


Low intensity : linear susceptibility

Linear response to a harmonic perturbation: H = H0 − dEL cos(ωLt+ φL)

Complex susceptibility : χ = χ′ + iχ′′

〈d〉 =(χELe

i(ωLt+φL) + χ∗ELe−i(ωLt+φL)

)= (χ+ χ∗)︸︷︷︸

=2χ′

EL cos(ωLt+ φL)− (χ− χ∗)︸︷︷︸=2χ′′

EL sin(ωLt+ φL)

Compare with : 〈d〉 = 2d [u cos(ωLt+ φL)− v sin(ωLt+ φL)]In the limit where s 1, we have

χ′ =dust

EL=d2

~δLs

Ω2L

=d2

2~δL

δ2L + Γ2

4

, χ′′ =dvst

EL≈d2

~Γs

2Ω2L

=d2

2~

Γ2

δ2L + Γ2

4

Linear response of a driven electric dipole.

−20−15−10 −5 0 5 10 15 202δLΓ

−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

χ′[

d2

hΓ

]

χ′ = d2

2hδL

δ2L+Γ2

4

−20 −15 −10 −5 0 5 10 15 202δLΓ

0.0

0.2

0.4

0.6

0.8

1.0

χ′′[ d

2

hΓ

]χ′′ = d2

2h

Γ2

δ2L+Γ2

4


Mechanical effect of light : radiation pressure of a plane wave

Two-level atom irradiated by a near-resonant laser.

In a period of duration ∆t, the atom undergoes Nabs fluorescence cycles (absorptionfollowed by spontaneous emission).

Atomic momentum changes by

∆pat = Nabs~kL −∑i

~k(sp)i

On average :

Nabs = Γsp∆t∑i

~k(sp)i = 0

Γsp : spontaneous emission rate = netabsorption rate

ωL,kLk1

k2

k3

Average force on the atom :

F =∆pat

∆t= Γsp~kL

Very large forces! FmaxM∼ 105 − 106 g for most atoms.


Semi-classical theory

Atomic Hamiltonian :

Hat = ~ωeg |e〉〈e|+P 2

2M+ VA-L(R), VA-L(R) = −d · EL(R)

So far we have ignored the atomic center of mass position. How to treat it in a simpleway ?

Ehrenfest theorem :

d

dt〈R〉 =

1

i~

⟨[R, H]

⟩=〈P 〉M

d

dt〈P 〉 =

1

i~

⟨[P , H]

⟩= −

⟨∇VA-L(R)

⟩−⟨∇VA-V(R)

⟩As seen from the qualitative description in the beginning, one has

⟨∇VA-V(R)

⟩(no

net force from the electromagnetic vacuum). However, the mechanical contribution ofthe vacuum is NOT identically zero: only the mean force vanishes, not thefluctuations around the mean (to be discussed later).

Semi-classical approximation : replaces the full quantum dynamics of the atomicwavepacket by the trajectory defined by 〈R〉, 〈P 〉:

d

dt〈R〉 =

〈P 〉M

,d

dt〈P 〉 = Frad

(〈R〉

)=

∑q=σ±,π

〈dq〉∇EL,q(〈R〉

)


Validity of the semi-classical treatment: the broad line condition

Treat the atom as a wavepacket of extension ∆x in real space and ∆p in momentumspace:

• well localized in real space if ∆x λL,

• well localized in momentum space if kL∆pM Γ.

The second condition states that all relevant momentum components interact with thelaser in the same manner.

The conditions are compatible if

∆x∆p ≥ ~↔ ∆x ≥~

∆p

~kLMΓ

This leads to the broad line condition:

ER =~2k2

L

2M ~Γ

Separation of time scales :

• ωR = ER/~ : typical rate of evolution of external variables,

• Γ : typical rate of evolution of internal variables.

The fast internal variables adapt almost instantaneously to the slower external ones.


Average radiative force: general expression for an atom at rest

We write the laser electric field as

EL(r) = εLEL(r) cos[ωLt+ ϕL(r)].

The laser polarization εL is assumed to be uniform (for now).We found from the semi-classical assumption that:

Frad (r) = 〈d〉 (cos[ωLt+ ϕL(r)]∇EL(r)− EL(r) sin[ωLt+ ϕL(r)]∇ϕL(r)) .

where r = 〈R〉 and 〈d〉 = 〈d〉 · εL.

Using the separation of time scales, the average dipole is given to a goodapproximation by the stationary solution of the optical Bloch equations,

〈d〉 ≈ 〈d〉st = 2d [ust cos(ωLt+ ϕL)− vst sin(ωLt+ ϕL)] .

Putting everything together, one gets a mean radiative force (averaged in time over anoptical period)

Frad (r) = −dust~∇ΩL(r)− dvst~ΩL(r)∇φL

with the Rabi frequency ~ΩL(r) = −dEL(r).

The two components of the mean radiative force are called• dipole force ∝ ust: linked to the reactive response of the atom (dispersion), and

to the gradient of amplitude,• radiation pressure ∝ vst: linked to the dissipative response of the atom

(absorption), and to the gradient of phase.Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])

Radiation pressure for a plane wave

Frp (r) = −vst~ΩL(r)∇φL, vst =Γ

2ΩL

s

1 + s

For a plane wave : EL = cst, ϕL = −kL · r, and the radiation pressure force is givenby

Frp(r) =Γs

2(1 + s)︸︷︷︸=Γ(ρee)st

~kL = Γsp~kL

We recover the same result obtained with the heuristic reasoning at the beginning,with the rate of spontaneous emission given by Γsp = Γ(ρee)st.

Properties of the radiation pressure force :

• maximum on resonance

• saturates to Fsat = 12

Γ~kL for high intensities

• very large ! FsatM∼ 105 − 106 g for most atoms.


Radiation pressure on a moving atom ; Zeeman slowing of atomic beams

Detuning for an atom moving at velocity v:

δL = ωL − ωeg − kL · v

Issues arise when working with a beam with a thermal distribution :• all velocity class in a thermal beam (Boltzmann distribution) are not addressed• if atom with an initial velocity v0 are decelerated, they move out of resonance

due to the Doppler shift.

Solution:Adjust detuning with an inhomogeneousmagnetic field:

δL = ωL − ωeg + kLvz + µB(z)

Use the Zeeman effect to counter theDoppler effect.µ: difference of magnetic moments in theexcited and ground state.

Atomic

beam

ωL,−kL

Slowing beam

Phillips and Metcalf, Phys. Rev. Lett. 48, 596

(1982)

Design of the magnetic field profile : Maximum force, Fmax = Γ~kL2

s01+s0

, with s0the saturation parameter for δL = 0.

B(z) such that an atom entering with a velocity vi remains on resonance as it isdecelerated to vf ≈ 0 with a constant force Fmax.

B(z) =~δLµ−

~kLviµ

√1−

z

L, L =

v2i

amax

All velocities v ≤ vi are also slowed to zero.Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])

Optical molasses

First optical molasse demonstrated in 1985 (S. Chu’s group,Bell labs)

3 pairs of counter-propagating beams to provide 3D cooling

• a few thousand Sodium atoms caught for about 1second

• cooling transition in the visible (λL = 589 nm): visibleby naked eye ! S. Chu et al., Phys. Rev.

Lett. 55, 48 (1985)Temperature measurement by time of flight :

An atom initially at r0 with velocity v0 follows a ballistic trajectory, r(t) = r0 + v0t.At long time of flight, the density distribution reflects the initial velocity distribution.

Ballistic expansion, cloud radius : R(t) =√kBTM

t

Result of the measurement : T ≈ 200µK


Laser cooling and optical molasses

Cooling configuration in one dimension :

• Two counter-propagating beams(frequency ωL, wavevector kL)

• moving atoms with velocity v

vωL,+kL

δ+ = δL − kLv

ωL,−kL

δ− = δL + kLv

k1

k2

k3

Weak saturation s 1: F± ≈ Γ~kL2

s± = Γ~kL4

Ω2L

δ2±+ Γ2

4

In this limit (linear response regime), one can add independently the forces exerted byeach beam to obtain a total average force

Ftot = F+ + F−

=Γ~kL

2

Ω2L

2

(1

δ2+ + Γ2

4

−1

δ2− + Γ2

4

)

Maximum for kLv = |δL|,Fmax = Γ~kL

2IIsat

−8 −6 −4 −2 0 2 4 6 82kLv

Γ

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

F/F

max

δL = Γ2

δL = Γ

δL = 2Γ

For low velocities kLv Γ, we have

Ftot = −Mαv, α = −Γ~k2

L

M

2sδL

δ2L + Γ2

4

Viscous damping for δL < 0 (red detuning), with a friction coefficient α (units : T−1)Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])

Fluctuations of radiative forces : Momentum diffusion

So far we have computed the average radiative force Frad exerted by the laser on theatom. Fluctuations around these average are also present (due to the stochasticcharacter of the atom-light interaction). Recall our basic picture :

Atomic momentum changes by

∆pat = Nabs~kL −∑i

~k(sp)i

ωL,kLk1

k2

k3

The radiation pressure force fluctuates around its average Frp because of two factors :

• random momentum kicks due to the random direction of emission,

• random number of fluorescence cycles during ∆t.


Brownian motion and momentum diffusion

Equation of motion of a Brownian particle submitted to a deterministic force Frad andto a fluctuating force δF (t). We assume here that the fluctuating part has a shortmemory time, and is characterized by a correlation function

〈δF (t′)δF (t′′)〉 = 2Dpδ(t′ − t′′).

The equation of motion of the Brownian particle is

Mx = Frad + δF (t) =⇒ v(t) = v(0) +1

M

∫ t

0dt′(Frad(t′) +

1

MδF (t′)

)Averaging over many realizations of the fluctuating force ( 〈δF (t)〉 = 0):

〈v〉 = v(0) +1

M

∫ t

0dt′,Frad(t′)

∆v2 = 〈v2〉 − 〈v〉2 =1

M2

∫ t

0dt′∫ t

0dt′′〈δF (t′)δF (t′′)〉

This gives

∆p2 = M2∆v2 = 2Dpt

Fluctuations of the force lead to a brownian motion in momentum space characterizedby a diffusion coefficient Dp.

NB: 2Dp =∫dτ〈δF (t+ τ)δF (t)〉 if δF stationary random process.


Momentum diffusion coefficient for a plane wave

Fluctuations of the radiation pressure :

Random momentum kicks for each spontaneous emission.

Photons spontaneously emitted at a rate Γsp = Γ(ρee)st.

Random walk of step size ±~kL inmomentum space.⟨

∆p2

(~kL)2

⟩= Nstep

On average: Nstep = Γsp∆t steps per timeinterval ∆t Γ−1

0 5 10 15 20 25 30 35 40

Number of steps

−15−10−5

05

1015

p(t)

0 5 10 15 20 25 30 35 40

Number of steps

05

1015202530354045

∆p2

∆p2 = (~kL)2Γsp∆t = 2Γ(~kL)2

4

s

1 + s︸︷︷︸=(Dp)rec

∆t




Random walk of step size ±~kL in momentum space, with (on average) Γsp∆t stepsper time interval ∆t Γ−1

(Dp)rec =Γ(~kL)2

4

s

1 + s

Fluctuations of the number of fluorescence cycles per time interval :

Fluctuations in the number of absorbed photons during ∆t.

Poisson statistics : ∆Nabs = Nabs = Γ(ρee)st∆t, ∆p2 = ∆Nabs(~kL)2, so

(Dp)abs =Γ(~kL)2

4

s

1 + s

NB: The probability distribution of the incident photons is Poissonian. However there are

corrections in resonance fluorescence, expressed by the so-called Mandel Q-factor such that

∆Nabs = Nabs = (1 +Q). For resonant fluorescence, Q < 0 : after absorbing a photon, an

atom cannot absorb another one for a time ∼ Γ−1 (i.e. until the first photon has been reemitted).




Random walk of step size ±~kL in momentum space, with (on average) Γsp∆t stepsper time interval ∆t Γ−1

(Dp)rec =Γ(~kL)2

4

s

1 + s

Fluctuations of the number of fluorescence cycles per time interval :

Fluctuations in the number of absorbed photons during ∆t.

(Dp)abs = (Dp)rec =Γ(~kL)2

4

s

1 + s

Momentum diffusion coefficient:

Dp = (Dp)rec + (Dp)abs =Γ(~kL)2

2

s

1 + s


Brownian motion in an optical molasse

Optical molasses steady-state result from a balance between the cooling provided bythe mean force and the heating (momentum diffusion) associated with thefluctuations.

Langevin equation :

v = −αv +δF (t)

M=⇒ v(t) = v(0)e−αt +

1

M

∫ t

0δF (t′)e−α(t−t′)dt′

Average over different realizations of the fluctuating force :

〈v〉 = v(0)e−αt

∆v2 = 〈v2〉 − v(0)2e−2αt =1

M2

∫ t

0dt′∫ t

0dt′′〈δF (t′)δF (t′′)〉e−α(2t−t′−t′′)

Short memory : 〈δF (t′)δF (t′′)〉 = 2Dpδ(t′ − t′′)

∆v2 =Dp

M2α

(1− exp−2αt

)=

2Dpt

M2 , t α−1

DpM2α

, t α−1


Doppler temperature

We define a temperature (in the long time limit) by

kBT

2=

1

2M〈∆v2〉 =⇒ kBT =

Dp

MαEinstein’s relation

Dp

M=

Γs(~kL)2

2M

s 1

α =Γs(~kL)2

M

2|δL|δ2L + Γ2

4

δL < 0

Doppler temperature :

kBTD =~Γ

4

(2|δL|

Γ+

Γ

2|δL|

)Minimum for δL = −Γ

2, kBTD,min = ~Γ

2. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

-2δLΓ

0

1

2

3

4

5

2kBT

hΓ

NB: More sophisticated approaches (Fokker-Planck equation for the atomicdistribution) show that the velocity distribution is Gaussian, as expected for aMaxwell-Boltzmann distribution.


First optical molasses at Bell Labs

Results from S. Chu’s group:

Sodium atom : λL = 589 nm,Γ = 2π × 10 MHz

Erh≈ 25 KHz : broad line condition

fulfilled

• A few thousand Sodium atoms caughtfor about 1 second:Dx = kBTD

Mα≈ 2 mm2/s ( I

Isat= 0.1)

Beam size w ∼ 1 cm =⇒ Catchingtime ∼ w√

Dx∼ 6 s

S. Chu et al., Phys. Rev. Lett. 55, 48 (1985)

α−1 ≈ 30µs for δL = −Γ/2, I = 0.1Isat

• Experimental temperature: T ≈ 200µK

Doppler theory : TD ≈ 200µK


Experimental results for Sodium atoms

A couple of years after the first molasses, more careful measurements (Phillips’ groupat NIST) show that atoms in optical molasses are much colder than expected fromDoppler theory !

P. Lett et al., PRL (1998)

Dashed line : Doppler theory

Measurements done on Sodium atomsΓ = 2π × 10 MHz

• wrong minimal temperature

• wrong dependence with detuning

Doppler theory works fine for atoms with a simple internal structure (J = 0→ J = 1).

It does not work for alkali atoms because of additional effects linked with their morecomplex internal structure.


Experimental results for Mercury atoms

Γ = 2π × 1.3 MHz, λeg = 254 nm

Shaded curve: Doppler theory (saturationincluded heuristically)

McFerran et al., Optics letters 35 (18),

3078-3080 (2010)


Sub-Doppler cooling

Temperature:

TSD ≥ #ER,

# is a pure number, frequently afew tens (kBT ∼ 3µK for theabove experiment).

TSD ∝~Ω2

L

|δL|,

[Salomon et al., EuroPhysics Letters 1990]

For Cesium atoms : TD ≈ 130µK, ER/kB ≈ 100 nK, TSD ≈ 6µK for δL = Γ,s0 = 0.1

Explained by the interplay between the polarization of light and the internal structureof the atoms.


Conclusion on laser cooling

−3 −2 −1 0 1 2 32kLv

Γ

−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

F[ΓhkL

2]

δL = −Γ, s0 = 0.1

−0.2 0.0 0.2−0.05

0.00

0.05

Laser cooling : two successive mechanisms leading to ultralow temperatures

• Doppler cooling : catches atoms around v ∼ Γ2kL

, cools atoms down to TD,

• Polarization gradient cooling : leads to sub-Doppler temperatures (several recoil

temperatures in practice), active around v ∼ Γs02kL

.

Only Doppler cooling for atoms lacking an internal structure (alkaline-earth atoms,like Sr, Hg, Ca; or Yb).

No trapping in optical molasses: atoms slowly diffuse away from the trapping region(intersection volume of all beams in a real, 3D configuration)









Repeat


Imaging of a cold atomic gas: overview of the techniques

Basic setup :

Einc

Object plane Image plane

CCD camera

Camera placed in an image plane optically conjugate to the “plane of the atoms”What are the properties of the transmitted light ?

What informations on the trapped atoms are recorded on the camera ?


Propagation in a dielectric medium :

∇2E −1

c2∂2E

∂t2= µ0

∂2P

∂t2

Electric polarization density:

P = ε0χE = nat〈d〉

χ = χ′ + iχ′′: dipole susceptibility, nat(r): atomic density

Writing E(r, t) = 12EeiϕLei(kLz−ωLt) + c.c., and invoking a slowly-varying envelope

approximation for E and φL (terms ∝ ∆E,∆ϕL neglected) :

dEdz

= −kLnat

2χ′′E(z), : Beer-Lambert law

dϕL

dz=kLnat

2χ′ϕL(z) : dephasing

NB : the atomic density must also vary smoothly along z on the scale λL.


Absorption imaging

We rewrite the equation for E in terms of the laser intensity: I = ε0c2|E|2

dI

dz= −κI(z) : Beer-Lambert law, κ =

kLnat

2χ′′ =

3λ2L

2π

nat(2δLΓ

)2+ 1

Interpretation in terms of scattered photons :

d(Photon flux) = − (natdzdA)σ × (Photon flux), with a scattering cross-section

σ =σ0(

2δLΓ

)2+ 1

, σ0 =3λ2L

2π, κ = σnat

Maximum on resonance (σ = σ0 for δL ≈ 0).

The intensity on the camera (assuming that the focal depth of the imaging system is cloud size) gives a magnified version of the transmitted intensity,

It(x, y) = Iinc(x, y)e−σ∫nat(x,y,z)dz

One calls n(x, y) =∫nat(x, y, z)dz the column density and OD(x, y) = σn(x, y) the

optical depth.

Absorption signal :

n(x, y) =1

σln

(Iinc(x, y)

It(x, y)

)Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])

Absorption imaging of a MOT

Absorption image of a MOT of Ybatoms after a time of flight oft ∼ 10 ms, during which atom arereleased from the trap and expandfreely.

Why do we need to take pictures after a time of flight ?

• To reduce absorption. Often the optical depth is too large: OD ∼ 5− 10 for aMOT, OD ∼ 100 or more for a BEC. Images are “pitch black” and dominated bynoise.

• To avoid photon reabsorption and multiple scattering, which makes the abovedescription invalid and the interpretation of images difficult.

In-situ imaging are also possible, though easier with dispersive techniques than withabsorption imaging.


Time-of-flight imaging

Time-of-flight (t.o.f.) experiment :

• suddenly switch off the trap potential at t = 0,

• let the cloud expand for a time t.

The spatial distribution for long times is proportional to the initial momentumdistribution P0 evaluated at p = Mr

t:

nat(r, t) −→t→∞

(M

t

)3

P0

(p =

Mr

t

)

This results holds for a classical or a quantum gase, indistinctively, provided one canneglect the role of interactions during the expansion.

For non-degenerate gases, temperature can be inferred from the cloud size after t.o.f.

For a Boltzmann gas,

〈p2x〉

2M=kBT

2

and suitable generalizations for Bose or Fermi gases.


Proof for a classical gas

A classical gas is described by its phase-space density f(r,p), such that

P(p) =

∫d3r f(r,p) : momentum distribution

nat(r) =

∫d3p f(r,p) : spatial distribution.

Liouville’s theorem: a phase-space element of volume d3r0d3p0 centered on (r0,p0)is conserved along the classical trajectories, f(r,p, t) = f0(r(t),p0) with f0 the initialdistribution.Here, the particles undergo ballistic flight for t > 0 and the classical trajectoryevolving from (r0,p0) is r(t) = r0 + p0

Mt, p(t) = p0.

The spatial density reads

nat(r, t) =

∫d3p f(r,p, t) =

∫d3r0

∫d3p0 f0(r0,p0)δ

(r0 +

p0

Mt− r

)For long times, the terms ∝ t in the δ function dominate and

nat(r, t) ≈∫d3r0

∫d3p0 f0(r0,p0)δ

(p0

Mt− r

)≈(M

t

)3 ∫d3r0 f0

(r0,

Mr

~t

)=

(M

t

)3

P0

(Mr

t

).

Validity: let ∆r0 and ∆p0 be the extent of f0 in position and momentum,respectively. The asymptotic regime is reached when ∆p0t/M ∆r0.


Dispersive imaging

Dispersive imaging :The phase φL = δφL + kLz of the transmitted wave also carries information on n :

dϕL

dz= (n− 1)kL =

1

2σnat

(2δL

Γ

)n : optical index

By interfering the transmitted wave with the incident wave, this information can beretrieved (e.g. in phase-contrast imaging). Operated off resonance (δL Γ) tominimize absorption.

W. Ketterle and M. Zwierlein, arxiv0801.2500

(2008)

Phase-contrast images of a two-componentFermi gas

• a: level scheme

• b: φL > 0 (< 0) for state 1 (2),

• c: different mixtures with N2 −N1

increasing from left to right.









Repeat


The magneto-optical trap : general principle

Atomic structure :

• Jg = 0→ Je = 1 transition

• magnetic moment µ in statee

Cooling + trapping in onedimension :

• Two counter-propagatingbeams with frequency ωL,wavevector ∓kL,polarizations σ±

• inhomogeneous magneticfield B(x) = b′x

Jg = 0

Je = 1

σ+ σ−

x

mg = 0

me = +1

me = 0

me = −1

ωL

σ−σ+

Weak saturation s 1: the σ±-polarized beam (with wavevector ±kL, respectively)exerts a radiation pressure force on the atom, with a detuning

δ±kL = δL ∓(kLv +

µb′x

~

).


Doppler and sub-Doppler cooling in the magneto-optical trap

For weak saturation s 1, we add the forces exerted by each beam independently (asdone for optical molasses),

Ftot =Γ~kL

2

Ω2L

2

1

δ2+kL

+ Γ2

4

−1

δ2−kL

+ Γ2

4

For kLv + µb′x |∆L|, we have

Ftot ≈ −Mαv −Mω2MOTx

α =Γs0~k2

L

M

−2δL

δ2L + Γ2

4

ω2MOT =

Γs0~k2L

M

−2δL

δ2L + Γ2

4

µb′

~kL

• Viscous damping for δL < 0 (red detuning), with a friction coefficient α

Doppler (and sub-Doppler) cooling exactly as in optical molasses (near thecenter, where the Zeeman shift is small)

• Harmonic trapping for δL < 0 (red detuning), with a frequency ωMOT


The magneto-optical trap in 3D

MOT configuration in three dimensions :

• Three pairs of counter-propagatingbeams with frequency ωL, wavevector∓kL, polarizations σ±

• inhomogeneous magnetic field createdby two coils with opposite current

B ≈

−b′x−b′y2b′z

σ−

σ+σ−

σ+

σ+

σ−

x

y

z


Magneto-optical trap for large atom numbers : light reabsorption

Spatial distribution:

n(x) = n0e− x2

2R2MOT , RMOT =

√kBT

Mω2MOT

This predicts R ∼ 50µm, much smaller than what is experimentally observed for largeMOTs (N > 107 atoms).

Reabsorption of spontaneously emitted photons :

A photon emitted spontaneously by atom (say 1) can be be reabsorbed by another one

(say 2), with probability dσreabsdΩ

= σreabs

4πr212[with σreabs is the reabsorption

cross-section].

Mean momentum transfer duringa time ∆t Γ−1:

∆p = F1→2∆t = Γsp∆tσabs

4πr212

~kLe12

Long-range force mediated byspontaneous photons.

Atom 1

Atom 2

σabs

e12

r12


Reabsorption force: analogy with plasmas

F1→2 = Γsp~kLσabs

4πr212

e12

Equivalent to a Coulomb-like force between two charges q1 = q2 = q, with

q2

ε0→ ~ΓspkLσabs.

T. Walker, D. Sesko, C. Wieman, PRL 64, 408 (1990).A. Steane, M. Chowdhury and C. Foot, J. Opt. Soc. Am. B 9, 2142 (1992).

Electrostatic analogy in the limit of a continuous medium.

• Long-range forces also strongly affect the MOT dynamics, which is non-linear.Analogies with plasmas : self-sustained oscillations, Coulomb explosion afterrelease [L. Pruvost et al., PRA 61, 063408 (2000)] ...

• Absorption from the MOT beams also play a role: it adds another contribution tothe radiation pressure with the same form as the reabsorption force but oppositein sign [J. Dalibard, Optics Comm. 68, 203 (1988)]. The overall force is normally dominatedby the reabsorption part [Walker et al., PRL 1990].


Magneto-optical trap for large atom numbers : equilibrium shape

Equilibrium shape of a MOT in the regime of large reabsorption:

Assume uniform density in a sphere of radius R. The problem is then mapped to theone of a uniformly charged sphere with charge density ρ = qn.

Gauss theorem:

E(r) =ρr

3ε0er

Coulomb force :

FCoulomb =qρr

3ε0er =

nΓsp~kLσabsr

3er = nCrer

Equilibrium, neglecting the viscous force :

FCoulomb + FMOT = 0

=⇒ n =mω2

MOT

C=

2|δL|Γδ2L + Γ2

4

µb′

~Γσabs

Typical values :n ∼ 1010 at/cm3, R ∼ mm

A. Camarra, R. Kaiser, G. Labeyrie, PRA 90, 063404

(2014)









Repeat


Dipole force

Fdip (r) = −ust~∇ΩL(r) = −~δL2

s

1 + s

∇ΩL

ΩL,

s

1 + s=

Ω2L2

δ2L + Γ2

4+

Ω2L2

Properties of the dipole force :

• Fdip vanishes on resonance,

• Fdip negligible close to resonance as compared to radiation pressure Frp,

• conservative : Fdip = −∇U(r), with the dipole potential U given by

U(r) =~δL2

ln [1 + s(r)]

For a standing wave : EL = E0 cos(k · r), ϕL = cst, the dipole force is given by

U(r) =~δL2

ln[1 + s0 cos2(k · r)

]The potential is periodic, trapping atoms at the nodes or antinodes of the standingwave depending on the detuning (J. Dalibard’s lectures).Redistribution of photons between the ← and → propagating waves by stimulatedemission.


Another interpretation of the dipole force

Large detunings : δL Γ,ΩL or equivalently s 1

U(r) ≈~δLs(r)

2=

~Ω2L(r)

4δL

Interpretation as a position-dependent light shift of the ground state atoms:For δL > 0:

∆Eg =~(Ω− δL)

2=

~2

(√δ2L + Ω2

L − δL)≈

~Ω2L

4δLif δL ΩL

|e〉

|g〉

+ hδL2

− hδL2

+ hΩ2

− hΩ2

xFabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])

Far-off resonance optical traps

Dipole potential for large detunings : δL Γ,ΩL or equivalently s 1

U(r) ≈~δLs(r)

2=

~Ω2L(r)

4δL=

~Γ2

8δL

I(r)

Isat

Interpretation as a position-dependent light shift of |g〉:

|e〉

|g〉

+ hδL2

− hδL2

+ hΩ2

− hΩ2

x

• δL > 0 : repulsive force (atoms arerepelled from the intensity maxima),

• δL < 0 : attractive force (atoms areattracted to intensity maxima).

• Residual spontaneous emission :

Γsp =Γs

2=

Γ

δL

U

~


Gaussian beams and dipole traps

Actual laser beams have a Gaussiantransverse profile. This creates a dipolepotential energy of the form

U(r) = −U0e−2 x

2+y2

w2

1 +(zzR

)2

Harmonic trap near the bottom of the potential (particles with energy U0):

U(r) = −U0 +∑

xα=x,y,z

1

2mω2

αx2α

ωx = ωy =

√4U0

Mw2, ωz =

√2U0

Mz2R

=w√

2zRωx

Typical example for Ytterbium atoms :

• λeg = 399nm

• λL = 532nm

• δL = −2 · 1014 Hz

• P = 10 W, w = 50µm

• s ≈ 10−6

• I = 2 MW/m2 at focus

• U0 ≈ kB × 700µK

• ωx/(2π) ≈ 1.2 kHz

• ωz/(2π) ≈ 3 Hz


Loading a dipole trap

Single beam dipole trap (side view) :

Crossed dipole trap (viewed from above):









Repeat



Oven

Yb BeamYb MOT


Elastic collisions

Atoms coming close together interact (short-range interaction potential). The result ofthe collision process allows us to distinguish two types of collisions : elastic or inelastic.

Elastic collisions :

• Redistribution of momenta on a “collision sphere”, conserving total momentumand energy.

• Thermalization of trapped atoms, governed by the rate of elastic collisions :

γcoll = nσcollvth

• n: spatial density• σcoll: collision cross-section

• vth =

√8kBT

πM : mean thermal velocity.


How long does it take to thermalize ?

• Left: Numerical simulation of the Boltzmannequation.

• A few collisions are enough to relax to thequasi-equilibrium distribution.

• Thermalization time = a few collision times γ−1coll.

Snoke and Wolf, PRB 39, 4030

(1989).


Quasi-equilibrium in a trap

Ultracold atoms can be considered in equilbrium only in a kinetic sense. The truethermodynamic equilibrium state of the atomic bodies are all solid at sub-Ktemperatures. In this sense atomic gases are only metastable systems (with lifetimesof seconds or even minutes).

Consider a gas of atoms held in a trap (potential U(r)) of finite depth (say U0).Atoms with energy lower than U0 stay trapped, atoms with higher energy escape.

After a few collisions times, the gas equilibrates (in a kinetic sense) to aMaxwell-Boltzmann distribution, with phase-space density given by

f(r,p) =1

Ze− p2

2MkBT−U(r)kBT Θ

(p2

2M+ U(r)− U0

).

Verified in experiments and in numerical solutions of the Boltzmann equation [Luiten et

al., PRA 53, 381 (1996)].

Ergodicity: f depends only on the energy.

For a harmonic trap U(r) = 12Mω2r2 (approximation valid for energies U0),

1

Z=

N(2π~)3

(2π)3R3th(Mvth)3

= n(0)λ3th ∝

Nω3

T 3, Rth =

√kBT

Mω2, vth =

√kBT

M


Elastic and inelastic collisions

Atoms coming close together interact (short-range interaction potential). The resultof the collision process allows us to distinguish two types of collisions : elastic andinelastic.

Inelastic collisions :

• collision with hot atoms from the background gas (with energy ∼ 300 K)

• two-body collision between cold atoms with a change in internal state

• three-body recombination of cold atoms to a molecular state (binding energyEm < 0)

All these processes typically producesatoms with energy higher than the trapdepth, and lead to trap losses, described bya rate equation of the form

n = − γ1n︸︷︷︸background

− γ2n2︸︷︷︸

two-body

− γ3n3︸︷︷︸

three-body

γ1,2,3 γcoll to have (approximately)ergodic behavior.

Figure from Matthias Scholl’s PhD thesis


Principle of evaporative cooling

Atoms trapped in a potential of depth U0, undergoing collisions :

• two atoms with energy close to U0 collide

• result: one “cold” atom and a “hot” one with energy > U0

• rethermalization of the N − 1 atoms remaining in the trap results in a lower meanenergy per atom.

Evaporation parameter:

η =U0

kBT∼ 8− 11

Evaporated atoms have an energy∼ ηkBT : evaporation proceeds from thetails of the distribution.

x

E

U0

Rate of evaporation:

• fη : fraction of atoms with energy > U0 = ηkBT . For a Boltzmann gas,fη ∝

√ηe−η .

• atoms with energy ≈ ηkBT have a velocity ∝ √η

γev ∼ γcollηe−η γcoll

More details in: W. Ketterle and N. J. Van Druten, Evaporative cooling of trapped atoms, available from the

Ketterle group website.


Simple estimate of the efficiency of evaporative cooling

Step-like model of evaporation: each step N → N − dN, T → T − dT

We write the mean energy lost by the evaporated atoms as −dE = (η + κ)kBT , with

η =U0

kBT∼ 8− 11

and κ ∼ 1 the excess “energy” with respect to the trap depth.

Mean energy before the collision : E = 3NkBT .

Mean energy after evaporation and thermalization:

E − dE = E − (η + κ)kBTdN

= 3(N − dN)kB(T − dT ) ≈ 3NkBT − 3kBTdN − 3NkBdT

We get

(η + κ− 3)dN = 3dT =⇒(Tf

Ti

)=

(Nf

Ni

)α, α =

η + κ

3− 1 ∼ 2− 3

Losing a factor 10 on N cools the sample by a factor ∼ 100− 1000 !


Scaling laws for evaporative cooling

For a fixed trap frequency ω, the spatial density n, the phase space density D, elasticcollision rate γcoll scale as :

n =N

(2π)3/2R3th

∝Nω3

T 3/2∼ N1− 3α

2

D = nλ3th ∝

Nω3

T 3∼ N1−3α

γcoll = nσvth ∝Nω3

T∼ N1−α

α > 1/3 results in an increase of the phase space density of the sample.α > 1 results in an increase of the collision rate.

Practical problem : reach D ∼ 1 before loss processes set in.

Rate of evaporation ∝ γcollηe−η must be much faster than the lifetime of the sample.

Forced evaporation : decrease trap depth as the gas cools down to keep η roughlyconstant (∼ 8− 10).

Runaway eaporation: evaporation at ever increasing γcoll (α > 1). Feasible inmagnetic traps (see Appendix).


Evaporative cooling in optical traps

The trap depth U0 is varied by reducing the laser power, since U0 ∝ P .However, the trap frequencies also decrease, as ω ∝

√P .

The spatial density n, the phase space density D, elastic collision rate γcoll scale as :

n ∝Nω3

T 3/2∼ U

1α0

D = nλ3th ∼ U

1α− 3

20

γcoll = nσvth ∝ U1α

+ 12

0

Therefore, in optical traps, the collision rate never increases and runaway evaporationdoes not happen. Fortunately, the collisions rates after loading from a MOT are high(a few thousands per second, typically).

The rule of the game is then to optimize the initial conditions and the “trajectory”U0(t) such that one reaches D ∼ 1 before the collision rate has dropped too muchand evaporation stops.


Example of evaporative cooling “trajectory”

Figure from Matthias Scholl’s PhD thesis

Blue triangles: single beam DT; red circles : crossed dipole trap


Complement: magnetic traps


Magnetic traps

Zeeman Hamiltonian in a non-uniform magnetic field B(r):

Umag = −m ·B(r)

Assuming that the internal states can follow adiabatically the change of B as theatom move in space,

〈F,mF |Umag|F,mF 〉 = gFmF |B(r)|

Maxwell equations tell us that |B| can only have a minimum.

If gFmF > 0 (low-field seeking states), the potential energy is minimal where |B| is:this results in atom trapping near the minimum. Atoms in high field seeking stateswith gFmF < 0 are expelled from the potential.

NB : The state |F,mF 〉 is defined in a local reference frame, where the localquantization axis is given by the direction of B.

Quadrupole trap :field created by two coils with oppositecurrent

B ≈

−b′x−b′y2b′z

,

|B| = b′√x2 + y2 + 4z2 −6 −4 −2 0 2 4 6

x(au)

0

1

2

3

4

5

B(x

)


Adiabatic theorem

Quantum adiabatic theorem : Slowly evolving quantum system, with Hamiltonian H(t)

Instantaneous eigenbasis of H: H(t)|φn(t)〉 = εn(t)|φn(t)〉.Time-dependent wave function in the |φn(t)〉 basis:

|Ψ(t)〉 =∑n

an(t)e−i~

∫ t0 εn(t′)dt′ |φn(t)〉,

From Schrodinger equation, one gets [ωmn = εm − εn] :

an = −〈φn|φn〉an(t)−∑m 6=n

e−i~

∫ t0 ωmn(t′)dt′ 〈φn|φm〉am(t),

• Berry phase : 〈φn|φn〉 = −iγB is a pure phase. Wavefunction unchanged up to aphase evolution after a cyclic change.• The adiabatic theorem : for arbitrarily slow evolution starting from a particular

state n0 (an(0) = δn,n0 ), and in the absence of level crossings, an(t)→ δn,n0

(up to a global phase).

Adiabatic approximation for slow evolutions and initial condition an(0) = δn,n0 :

an(t)→ eiγB(t)δn,n0

Validity criterion :

∣∣∣〈φn|H|φm〉∣∣∣(εm − εn

)2

~.


Adiabatic following of a spin 1/2

Adiabaticity criterion for a time-dependent field B(t):

∣∣∣〈φn|H|φm〉∣∣∣ ∼ µB (εm − εn

)2

~∼

(µB)2

~= ~ω2

Larmor, orωLarmor

ω2Larmor

1

Here ωLarmor = µB/~ is the Larmor frequency quantifying the angular velocity of thespin precession around the magnetic field axis.

Adiabaticity criterion for an atom, described classically, moving in a static field B(r):

H = −m ·B[r(t)] =⇒ 〈↓ |B| ↑〉 = ·〈↓ | (v ·∇)B| ↑〉 µ|B(r)|2

~

• spin-flip transitions to untrapped states near the minimum of |B|• result in atom losses,

• mostly affect the fast atoms.

Majorana losses is a major issue in a quadrupole trap since min(|B|) = 0.


Ioffe traps

Biased quadrupole trap (Ioffe trap) :

B ≈

−b′x−b′y

B0 + 2b′z

|B| =√b′2(x2 + y2) + (B0 + 2b′z)2

≈ B0 + b′2(x2 + y2) + 2b′z +2b′2

B0z2

Harmonic trap with frequency ωx, ωy ωz

Typical values :

• B0 ∼ 1− 10 G [1 G=10−4T],

• b′ ∼ 100 G/cm,

• ωx = ωy ≈ 2π × 300 Hz,ωz ≈ 2π × 30 Hz. −6 −4 −2 0 2 4 6

z(au)

0123456

B(z

)/B

0


Gallery of magnetic traps

“Cloverleaf” trap :

Ketterle group, MIT

http://cua.mit.edu/

• Macroscopic trap :l ∼ 1− 10 cm,I ∼ 100− 1000 A

• kW of electrical power todissipate (water cooling)

• switch-off not trivial( powerelectric components, eddycurrents ...)

Atom chips:

Reichel group, LKB

http://www.lkb.ens.fr/-Atom-Chips-Group-

• “Microscopic” trap : l ∼ 0.1− 1 mm, I ∼ 1 A

• little power dissipated, compact

• chip problematic for some applications

• usually modest vacuum level

Many variants for both designs (practically one perexperiment ...)


Evaporative cooling in magnetic traps: radio-frequency evaporation

Forced evaporation: U0(t) decreases in time (in a controlled way). Optimizing theevaporation trajectory is done by controlling the atom lost by the evaporation processvia the trap depth.

In magnetic traps, this is achieved by radio-frequency evaporation : a radiofrequencyfield is applied, inducing transitions from a trapped state to a non-trapped state for agiven energy ~ωrf = U0.

x

EmF = −1

mF = 0

mF = +1

Usually η = U0/kBT ≈cst. Then, the scaling with trap depth is T ∼ U0, N ∼ U1α0 ,

and

n ∼ U1α− 3

20 , D ∼ U

1α−3

0 γcoll ∼ U1α−1

0

Optimization can be done (usually experimentally) to reach the runaway regime.Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])

Illustration of evaporation in magnetic traps

Durée d'évaporation (s)

DN

Durée d'évaporation (s)

piege magnetiqueavant compression

νrf55

32

16

85

T (µ

K)

γcol

l(1/s

)

10-6

10

-4

10

-2

151050

106

107

151050

400

300

200

100

0

151050

300

200

100

0

151050

νrf (MHz)

T (µ

K)

300

200

100

0

5040302010

F. Gerbier, PhD thesis


ultracold atoms - icfp master coursenascimbene/enseignement/coldatoms/cours... · 2019. 2. 2. ·...

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