Ultra-cold Fermi gases - University of Cambridge · PDF fileUltra-cold Fermi gases F. Chevy...
Transcript of Ultra-cold Fermi gases - University of Cambridge · PDF fileUltra-cold Fermi gases F. Chevy...
Ultra-cold Fermi gasesUltra-cold Fermi gases
F. ChevyF. Chevy
Laboratoire Kastler Brossel, ParisLaboratoire Kastler Brossel, Paris
Evaporative Cooling of ultra-cold atomsEvaporative Cooling of ultra-cold atoms
kTU0
position
energ
y
To maintain an efficient cooling, U0 decreases with T;
Collisions required
The Pauli Exclusion Principle and the
evaporative cooling of ultra-cold Fermi gases
The Pauli Exclusion Principle and the
evaporative cooling of ultra-cold Fermi gases
Collision between two atoms. Effective potential in the l-wave:
At low temperature, the atomes
cannot cross the centrifugalbarrier: only s-wave collisions.
Symmetrization for identicalparticles: even l-wave collisions
forbidden for polarized fermions.
2
eff 2
( 1)( ) ( )
2V r V r
mr
+= +
� � �
Interatomic potential
(long range~-1/r6)
centrifugal potential
l=0 l>0
Suppression of elastic collisions in
a spin polarized Fermi gas
Suppression of elastic collisions in
a spin polarized Fermi gas
Spin mixture (s-wave)
Spi
n po
lariz
ed(p
-wav
e)
Use spin mixtures or several atomic species (eg 6Li-7Li, K-Rb…)
B. DeMarco, J. L. Bohn, J.P. Burke, Jr., M. Holland, and D.S. Jin, Phys.
Rev. Lett. 82, 4208 (1999).
The non-interacting Fermi gas
Time of flight absorption imaging
The non-interacting Fermi gas
Time of flight absorption imaging
T/TF<0.05
Atom number~105
Fermi-Dirac
Gaussian Fit
Attractive fermions atzero temperature
Attractive fermions atzero temperature
Deep potential : 2-body bound state.Many body ground state : BEC of molecules
0V
0V
Shallow potential (V0<V0*):
no 2-body bound state. No superfluid?
Many body effects:BCS pairing atarbitrarily low V0
(3He, superconductors)
+k
-k
The BEC-BCS crossoverThe BEC-BCS crossover
Size of the molecules ~ aBinding energy 2 2/bE ma= −�
*
0 0 0V V a> ⇔ > *
0 0 0V V a< ⇔ <3| | ~ 1n a
??????
*
0 0~V V
At low energy, real potential replaced by
Binding energy1/ | |
~ Fk a
FE e−
BEC of molecules BCS state
2
pseudo
4( )
aV
m
πδ= r
�
Scattering length a~(V0-V0*)-1
The BEC-BCS « patchwork »The BEC-BCS « patchwork »
Alkali BEC Superfluid He-4
High Tc superconductors
Superfluid He-3
Normal superconductors
Ultra-cold fermions
Condensation tem
pera
ture
Interaction strength
(Fano) - Feshbach resonancesin cold atoms
(Fano) - Feshbach resonancesin cold atoms
0.0 0.5 1.0 1.5 2.0
-200
-100
0
100
200
scattering length
[nm
]
Magnetic field [kG]
6 Li
BF~834 G
6LiBosons near a Feshbach
resonance:Bose-Nova (C. Weiman).
Losses in 6Li (fermions)
Predicted position of
the resonance
Looking for the Feshbach resonanceof Lithium
Looking for the Feshbach resonanceof Lithium
Inhibition of inelastic losses
In fermionic gases
Inhibition of inelastic losses
In fermionic gases
1000,1
1
G [1
0^-
13
cm
^3/s
]
Scattering length [nm]
Scaling lawG ~ a -2.0 +/-0.8 (theory Petrov et al. )2.55
G a−
∼
2 body (dimers) losses mainly : decay towards deeply bound states)
N G n N= −�
3/4 atom loss requires 2 atoms of
same spin close to each other.
JILA: 40K2
6Li2:Innsbruck
ENS6Li2
MIT6Li2
Also Rice, Duke, Melbourne 6Li2
Molecular BEC’saround the world
Molecular BEC’saround the world
Momentum distributionMomentum distribution
Broadening of the Fermi-Dirac distribution in the presence of
interactions (Viverit et al.PRA 69, 013607 (2004))
k/kF1
n(k)
1
a<0, T=0 ideal gas,width kF
a=¶, « unitary » gas
a>0, inner momentumdistribution, width~1/a
BCS (mean-field)
Wave function
Experiment vs theoryExperiment vs theory
-1 0 1-0,2
0,0
0,2
0,4
0,6
0,8
1,0
1,2In
teg
rate
dd
en
siit
y[a
.u]
k/kF
BCS theory
Thomas-Fermi for ideal gas
Ideal gas
Strongly correllated Fermi gas
0.46(10)JILA
0.46(5)Rice
0.41(15)ENSExperiment
0.42(1)Carlson
0.455Perali
0.42(1)Astrakharchick
0.59BCSTheory
Measurement of the zero temperature
universal equation of state of
the strongly interacting Fermi gas
Measurement of the zero temperature
universal equation of state of
the strongly interacting Fermi gas
( )2
2/326
2F
n Em
µ ξ π ξ↑
= =�
Determination of x
For a=¶, f(1/na3)=f(0), independent on density
( ) )/1(62
33/222
nafnm
πµ�
=↑Dimensional analysis:
Equation of state at finite temperatureEquation of state at finite temperature
Goal: measure the
thermodynamical equation ofstate S(U,N)
•Prepare an ideal gas andmeasure T/TF~S
• Ramp slowly the magnetic
field to a=¶•Measure the potential energy
E=mw2<x2>/2=U/6 (Virialtheorem)
J. T. Stewart, J. P. Gaebler, C. A. Regal, and D. S. Jin, Phys. Rev. Lett. 97, 220406
(2006).
L. Luo, B. Clancy, J. Joseph, J. Kinast, and J. E. Thomas, Phys. Rev. Lett. 98, 080402
(2007)
Superfluidity: vortex lattices (MIT)Superfluidity: vortex lattices (MIT)
Vortices at MIT
Ato
mic
BE
CM
ole
cula
r
BE
C
BC
S
su
pe
rflu
id
Ultra cold atoms in Optical LatticesUltra cold atoms in Optical Lattices
Optical lattices: periodic potential created by the interference of several laser
beams
Imaging the first Brillouin ZoneImaging the first Brillouin Zone
Momentum distribution after time of flight for various
filling factors.
M. Köhl, H. Moritz, T. Stöferle, K. Günter, T. Esslinger
Phys. Rev. Lett. 94, 080403 (2005).
Repulsive fermions in an optical latticeRepulsive fermions in an optical lattice
For bosons see M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch and I. Bloch
Nature 415, 39-44 (2002)
Fermi-Hubbard hamiltonian
Small U/J: conductor Large U/J: insulator
Mott
transitio
n
† † †
,
i j i i i ii j i
H J c c U c c c cσ σ
σ
↑ ↑ ↓ ↓= +∑ ∑
Mott transition in superfluid Fermi gasesMott transition in superfluid Fermi gases
R. Jördens, N. Strohmaier, K. Günter, H. Moritz and Tilman Esslinger, Nature 455,
204-207 (11 September 2008)
U. Schneider, L. Hackermuller, S. Will, Th. Best, I. Bloch, T. A. Costi, R. W. Helmes,
D. Rasch, A. Rosch, Science 322,1520 (2008)
Double occupancy spectroscopic
measurement
2 atoms persite
Isolated atom
Interaction
energy
hn hn’
High resolution imaging in optical latticesHigh resolution imaging in optical lattices
A. Itah, H. Veksler, O. Lahav, A. Blumkin, C. Moreno, C.
Gordon, J. Steinhauer, arXiv:0903.3282
Fermionic superfluid with population imbalanceFermionic superfluid with population imbalance
Polarized phase : One spin species (Carlson, PRL 95, 060401 (2005))
Chandrasekar and Clogston: robustness of the paired state :
Paired state stable for
And beyond?
µ µ↑ ↓>
µ µ↑ ↓
− < ∆
FFLO Phase (Fulde Ferrell Larkin Ovshinikov) : pairing in (C. Mora et R. Combescot, PRB 71, 214504 (2005))
0↓↑ − ≠k k
Sarma phase (internal gap) : pairing in opening of a gap in the Fermi sea of majorityspecies. (Liu, PRL 90, 047002 (2003))
0↓↑ − =k k
Experimental results at Feshbach resonanceExperimental results at Feshbach resonance
MIT: 3 phases- Fully paired superfluid core
-Intermediate mixture
-Fully polarized rim
Rice: 2 phasesFully paired superfluid core
Fully polarized rim
M.W. Zwierlein, et al., Science, 311
(2006) 492.
G.B. Partridge, W. Li , R.I. Kamar, Y.-A.
Liao, R.G. Hulet, Science, 311 (2006)
503.
Rice : 2 phase model(F. Chevy, PRL 96, 130401 (2006))
Rice : 2 phase model(F. Chevy, PRL 96, 130401 (2006))
Rice experiment: fully described by a 2 phase model, without any adjustable parameter.
External radius
Superfluid inner radius
(N�-N�)/(N�+N�)
Surface tension
(T. N. De Silva and E. Mueller, PRL., 97 070402 (2006))
Surface tension
(T. N. De Silva and E. Mueller, PRL., 97 070402 (2006))
Local density approximation:
int erfacesF Sγ=
2 4/3
2
n
mγ ε=
�
Fit of data:
With surface tension:
310ε −≈
(1st order transition)
What about other phases.What about other phases.
Fully polarized
ideal gas
?
αηβη
Fully paired
Superfluid
/η µ µ↓ ↑=
1-1
0/P P
3/5(2 ) 1
0.099
cη ξ= −
≈ −
Grand potential W=-PVflground state has the highest pressure.
Theoretical evidence for an intermediate
phase
Theoretical evidence for an intermediate
phase
Step 1: calculate the energy E of a single impurity atomimmersed in a Fermi sea (E=m�(n�=0+)).
Step 2: dP/dmσ=ns>0
General properties of a mixed branch?
/η µ µ↓ ↑=cη
P/P0cβη η<
cβη η<
: the new branch isstable
: the new branch isunstable
cβη η>
cβη η>
See A. Bulgac and M.
McNeil Forbes,
PRA 75, 031605 (2007)
A spin down particle immersed in a Fermi sea of spin
up atoms:
The Fermi-polaron problem
A spin down particle immersed in a Fermi sea of spin
up atoms:
The Fermi-polaron problem
(Dressed atom, Higgs mechanism, Landau’s Fermi liquid theory…)
Variational upper bound for hb:the Fermi swimmer
Variational upper bound for hb:the Fermi swimmer
One impurity: restrict the effect of interactions to the formation ofa single particle-hole pair.
0 ,
,
0 ,ϕ ϕΨ = +∑ k q
k q
k q
=kq
k
q
q-k0 =
For a=¶, E=-0.606 EF�flηβ<-0.606<ηc~-0.1
Comparison with exact results :hb=-0.58(1) (C. Lobo et al. PRL. 97, 200403 (2006)); hb=-0.62 (Prokof’ev and Svistunov, Phys. Rev. B 77, 020408 (2008))
Systematic expansion R. Combescot and S. Giraud, Phys. Rev. Lett. 101, 050404 (2008)
Structure of the intermediate phase (Combescot et al.PRL 98, 180402 (2007))
Structure of the intermediate phase (Combescot et al.PRL 98, 180402 (2007))
Ideal gas of fermionic impurities dressed by particle-hole pairs
2 *( ) / 2F
E p E p mβη= +
0.6βη = − * ~ 1.15m mVariational calculation:
MC
Ideal gas approx.
Comparison with experiment:
Shell radii and critical polarisation
Comparison with experiment:
Shell radii and critical polarisation
If polarisation (N�-N�)/(N�+N�) is to large : superfluid core vanishes
Rb
R1
Ra
-0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,80,2
0,4
0,6
0,8
1,0
Cri
tical p
ola
risa
tion
1/kFa
No adjustableparameter
See also Martin Zwierlein’s talk
Comparison with experiments:
Radio-frequency experiments(A. Schirotzek et al. Phys. Rev. Lett. 102, 230402 (2009))
Comparison with experiments:
Radio-frequency experiments(A. Schirotzek et al. Phys. Rev. Lett. 102, 230402 (2009))
Short term issues
Conclusion et future directionsConclusion et future directions
Ultra-cold atomic gases are wonderful tools to simulate
condensate matter systems:BEC-BCS crossover
Mott transitionClogston-Chandrasekhar limit
Dynamical studies, measurement of the effective mass of theFermi polaron (Nascimbène et al. arXiv:0907.3032)
l>0 superfluids (p-wave?)
Magnetic properties in optical lattices
What was demonstrated?