Ultracold atoms - ICFP Master Coursenascimbene/enseignement/coldatoms/cours... · 2019. 2. 2. ·...
Transcript of 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• Unpublished Master lecture notes : Atomes ultrafroids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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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]
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
Which atomic species ?
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
Ytterbium BEC experiment at LKB
Oven
Yb BeamYb MOT
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
Ytterbium BEC experiment at LKB
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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+
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
~
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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〉
)
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
Momentum diffusion coefficient for a plane wave
Fluctuations of the radiation pressure :
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).
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
Momentum diffusion coefficient for a plane wave
Fluctuations of the radiation pressure :
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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)
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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)
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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 ?
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
~
).
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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].
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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)
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
~
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
Loading a dipole trap
Single beam dipole trap (side view) :
Crossed dipole trap (viewed from above):
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
Ytterbium BEC experiment at LKB
Oven
Yb BeamYb MOT
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
Ytterbium BEC experiment at LKB
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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).
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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 !
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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).
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
Example of evaporative cooling “trajectory”
Figure from Matthias Scholl’s PhD thesis
Blue triangles: single beam DT; red circles : crossed dipole trap
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
Complement: magnetic traps
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
)
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
~.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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.
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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 ...)
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])
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
Fabrice Gerbier ([email protected]) Sylvain Nascimbene ([email protected]) Aurelien Perrin ([email protected])