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C. Salomon
Dual Bose-Fermi Superfluids
Quantum Science SymposiumCity University Hong-Kong, November 9th, 2017
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ENS FERMI GAS GROUP
Y. Castin, F. Werner, X. Leyronas (ENS), S. Stringari (Trento), A. Recati, T. Ozawa,O. Goulko (Amherst), C. Lobo, J. Lau (Southampton), I. Danaila (Rouen)
S. JinS. Laurent
M. Rabinovic
C. Enesa
M. PierceM. Delehaye
D. Suchet
F. Chevy T. ReimannC. Salomon
I. Ferrier-Barbut
Theory:
T. Yefsah
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• Introduction to quantum gases
• Dual Bose-Fermi superfluid with 7Li-6Li isotopes
• DynamicsCenter of mass modes and link with Equation of StateMeasurement of critical velocity for superfluid counterflow
Outline
1) I. Ferrier-Barbut, M. Delehaye, S. Laurent, A. T. Grier, M. Pierce,B. S. Rem, F. Chevy, and C. Salomon, Science, 345, 1035, 20142) M. Delehaye, S. Laurent, I. Ferrier-Barbut, S. Jin, F. Chevy, C. Salomon, PRL, 115, 265303, 20153) Y. Castin, I. Ferrier-Barbut and C. SalomonComptes-Rendus Acad. Sciences, Paris, 16, 241, 20154) S. Laurent, M. Pierce, M. Delehaye, T. Yefsah, F. Chevy, C. Salomon Phys. Rev. Lett., 118, 103403, 2017
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Bose Einstein condensate Superconductivity
106 years of quantum fluids
4He
dilute gas BECFermi gas superfluid
T~ 2.2 K
High Tc77 K
100 nK
2.5 mK3He
+ polaritons and BEC of light
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1 K 100 K 104K 106K1 mK1 µK T
thisroom
sun surface
sun centerHigh Tc
SC liquid
4He
cold atomic gases
Dilute, but interacting systems
Typical density:
Interatomic distance range of interatomic potentials
quantum of motion in the trap or box
thermal energy
Equilibrium properties and dynamics are governed by interactions
Neutron stars107 K
Temperature scale of cold gases
Superfluid 3He
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Nobel prize 1997S. Chu, C. Cohen-Tannoudji, W. PhillipsLaser cooling and trapping of atoms
Nobel prize 2001E. Cornell, W. Ketterle, C. WiemanBose-Einstein condensation in dilutegases
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Bose-Einstein statistics (1924)
Bose-Einstein condensate
Bose enhancement
(0.83 N)1/3= T C kB
hω
Quantum gases in harmonic traps
Dilute gases: 1995, JILA, MIT
Fermi-Dirac statistics (1926)
EF
Fermi sea
Pauli Exclusion
T << T = (6 N)F 1/3
kB
hω
Dilute gases: 1999, JILA
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7Li (boson)
6Li (fermion)
~200µm
ENS 2001
T= 0.28 µK = 0.2(1) TC= 0.2 TF
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Searching for superfluid Bose-Fermi systems: 4He - 3He mixture
Expected Tc ~ 1 to 20 µK ?Volovik, Mineev, Khalatnikov, JETP, 42, 342 (1975): Fermi liquid theory of mixture
?
4He
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The magnitude and sign of a depend sensitively on the detailed shape of longrange potential Importance of position of last bound state
Atom-atom interactions
At low temperature, only s wave collisions, l =0
.
00
( )
tan ( )lim
ik r ikr
k
ar e er
kak
ψ
δ→
= −
= −
a: scattering length|a|~1 to 10 nm
2
1 2 1 24( ) ( )aV r r r r
mπ δ− = −
a > 0 : effective repulsive interactiona < 0 : effective attractive interaction
W(r)
r
66 /C r−
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0a ≥
EB=-h2/ma2
0,0 0,5 1,0 1,5 2,0
-200
-100
0
100
200
scat
terin
g len
gth
[nm
]
Magnetic field [kG]
0a ≤
a = ∞
Tuning interactions in quantum gases: Lithium 6
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Fermions with two spin states with attractive interaction
BEC of molecules BCS fermionic superfluid
Dilute gases: Feshbach resonance
Interaction strengthBound state No bound state
akFc
FeTT 2/π−≈
80’ Leggett, Nozieres, Schmidt-Rink, Eagles….
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N. Navon, S. Nascimbène, F. Chevy, C. Salomon, Science 328, 729-732 (2010)
Pressure equation of state
Equation of State in the BEC-BCS crossover
BCS-BEC crossoverat T~ 0
BECof pairs
BCSregime
P/P0= f(1/kF a)
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Experimental Setup
Absorption imaging of the in-situdensity distributions or Time of Flight
Magneto-optical trap of bosonic7Li and fermionic 6LiAfter evaporation in a magnetic trapwe load the atoms in a single beamoptical trap (OT) with magnetic axial confinement. T~ 40 µK
Evaporative cooling of mixture in OT
~ 4 second ramp, T~ 50-80 nK
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7Li BEC
6Li Fermi gasat unitarity
In situ density profiles
Lifetime of mixture: 7s in shallowest trap
Trap frequencies: vz=15.6 Hzfor bosons, vrad= 440 Hz
NB= 2 104
T=80 nKN0/NB> 80%T<Tc/2
NF= 2 105
T= 80 nK ~ Tc/2TF= 800 nK
Unitary 6Li Fermi gas can cool any species fulfilling the requirements to BEC See also 6Li-41K, USTC, China, PRL ’16, and 6Li-173Yb, UWash, PRL’17
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First sounds in mixture of superfluids
Superfluids have collective excitationsFor instance low energy excitations of a BEC are phonons, ie density waves
In a mixture of two superfluids, one expectstwo first sound modes Volovik, Mineev, Khalatnikov, JETP, 42, 342,(1975)
In a trap the lowest acoustic mode corresponds to center of mass oscillations of the clouds (dipole mode)
We displace the axial position of the clouds by having the waistof the dipole trap shifted from the magnetic trap minimum.
( )kε
k
( )
: /
k c k
c speed of sound m
ε
µ
=
=
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6
7
2 17.06(1)2 15.40(1)
HzHz
ω πω π
= ×= ×
Coupled Superfluids
Long-lived Oscillations of both Superfluids
6
7
2 17.14(3)2 15.63(1)
HzHz
ω πω π
= ×= ×
Single SuperfluidRatio = (7/6)1/2 =(m7/m6)1/2
Fermi Superfluid
BEC
time
400 ms
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Oscillations of both superfluids
Very small damping !Modulation of the 7Li BEC amplitude by ~30% at
Coherent energy exchange between the two oscillators
πωω 2/)~~( 76 −
BEC
Fermi SF
4 s0
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Mean field model
πωω 2/)~~( 76 −
2
667
2( ) ( ) bf
eff bf bf
aV V r g n r with g
mπ
= + =
67 6 7 6 7/ ( )m m m m m= +
1.5% down shift in 7Li BEC frequency
Weak interaction regime: kFabf<<1 and N7<<N6
Boson effective potential
LDA 0 06 6 6( ) ( ( ))n r n V rµ= −
is the Eq. of State of the stationary Fermi gas. )(6 µnWhere
BEC osc. amplitude beat at frequency
For the small BEC: 06)( µ<<rV 0
0 0 66 6 6
6
( ) ( ) ( ) ....dnn r n V rd
µµ
≈ − +Expand
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(0)6
66 0
(0) ( ) 1eff bf bfdnV g n V r gdµ
= + −
(0)
7 76 0
1 bfdngd
ω ωµ
= −
Thomas Fermi radius of BEC<< TF radius of Fermi Superfluid:
The potential remains harmonic with rescaled frequency
Boson effective potentialand link with Equation of State
From Thomas Fermi radius of 6Li superfluid, we find:very close to the measured value:
Hz43.152~7 ×= πω
7 2 15.40(1) Hzω π= ×
The equation of state at low T is known in the BEC-BCS crossoverN. Navon et al., Science, 2010
( )n µ
Example: at unitarity, 1/a=0
A new means to access properties of the EoS !
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Equation of State and Bose-Fermi Coupling in BEC-BCS crossover
From EoS in the crossoverN. Navon et al, Science 2010
MIT ’12
6.190 bf
f
aa
≅
Shift in BEC limit7 7/
F bfk aδω ω
1/ kF fa
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7 6 7cos( ) (1 )cos( )z t tε ω ε ω= − + +
6 6 7(1 )cos( ) cos( )z t tερ ω ερ ω= − +
with 7
6
NN
ρ = and
Hence a significant modulation of the amplitude of z7at the beat frequency 6 7ω ω−
7 7 7
7 6 7
2 0.21mm m
ω ωεω
−= −
=14
6K67K 7K
Two coupled oscillators
Coherent energy exchange between both gases isamplified by quasi-degeneracy of pendulum frequencies
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What is the critical velocityfor superfluid counterflow ?
Increase relative velocity
Increase initial displacement
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Critical velocity for superfluid counterflow
Time(ms)
13.1 sγ −=
Initial damping
Vc = 2 cm/sis quite high !
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V
2 2
'/ 2
Momentum Conservation :
Energy Conservation /2+ :
M MMV MV ε
= +
′= k
V V k
Motion of impurity is damped by the creation of elementary excitations if:
min kc kV V
kε ≥ =
V’
,εkk
For a linear excitation spectrum εk=kc, Vc= c, the sound velocity
2 2. / 2k k Mε= +k V kε≥kV ≥
Landau criterion
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cFcb
Critical velocities
?
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1 Excitation in the bosonic superfluid
1 Excitation in the fermionic superfluid
, , ·B B BE ε= +k k k V
Energy-momentum conservation: , ' , ' '·E ε= +k k k VF F F
, ,| | min B k FB k
kF k
εε −+ − ≥
V V
V c c= +c B FSound Modes:
, ', 0EE + =kk FB ' 0+ =k k
,, Bε kk, '', Fε kk
See also Abbad et al. EPJD 69, 126 (2015), F. Chevy, PRA 91, 063606 (2015), W. Zheng and H. Zhai, Phys. Rev. Lett. 113, 265304 (2014)
Revisiting Landau criterionfor a Bose-Fermi mixture @ T=0Y. Castin, I. Ferrier-Barbut and C. SalomonComptes-Rendus Acad. Sciences, Paris, 16, 241 (2015)
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cFcB
Counter-flow critical velocity
cB+cF
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Summary
• Dual Bose-Fermi superfluids have intriguing novel properties
• Surprisingly high critical velocity for superfluid counterflowNature of quasi-particles ? Measure ε(k)
• Role of temperature: phase locking of B-F oscillations
• What happens when abf increases ?• Demixing• Competition between fermionic pairing or Bose-Fermi pairing? • Spin-imbalanced system
FFLO phases, 1D, 2D, 3DIs there a p-wave superfluid in strongly imbalanced Fermi gas?
• 2D Fermi gas and BKT physics
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PhD Students and Postdocs are welcome !
Thank you for your attention !