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Universal Thermodynamics of a Unitary Fermi gas. Takashi Mukaiyama. University of Electro-Communications. outline. Introduction. Thermodynamics of a unitary Fermi gas - from global thermodynamics quantities to local ones - . Toward further understanding of a unitary gas - PowerPoint PPT Presentation
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Universal Thermodynamics of a Unitary Fermi gas
Takashi Mukaiyama
University of Electro-Communications
outline
Introduction
Thermodynamics of a unitary Fermi gas - from global thermodynamics quantities to local ones-
Toward further understanding of a unitary gas (Fermi liquid or pseudogap?)
Tan relation new, universal description of thermodynamics for interacting fermions
Cold atoms are• very dilute (1011~1014 cm-3),• with no impurities, no defects.
Amenable to simple theoretical description
J. R. Ensher, et al., Phys. Rev. Lett. 77, 4984 (1996).
Con
dens
ate
fract
ion
5% deviation of critical temperature from theoretical predictions
・ 3% shift due to finite size correction・ 2% shift due to interaction
BEC in a cold atom system
There are two channels corresponding to different spin states.
Feshbach resonance
bound state
E
ROpen (scattering) channel
Closed (bound) channel
Resonance occurs when open and closed channelare energetically degenerate.
S. Inouye, et al., Nature 392, 151 (1998).
Inter-atomic interaction is tunable !!
Loss near a Feshbach resonance
S. Inouye et al., Nature, 392, 151 (1998).
scatteringlength
Number ofatoms
E
R
loss due to vibrational quenching
Vibrational quenching
JILA
1999 Fermi degenerate gas
ultracold fermionic atoms
At the Feshbach resonance for and ,no loss occurs due to Pauli exclusion principle.
ultracold : s-wave is the dominant collision channel.
Identical fermions do not collide.
Identical bosons : l=0(s-wave), l=2 (d-wave), …Identical fermions: l=1 (p-wave), l=3 (f-wave), …
Collision channel
Think about two-component fermions
Therefore two-component fermions are stable even at a Feshbach resonance.
We are able to prepare an interacting (reasonably stable) two-component Fermi gas of atoms with an arbitrary interaction strength !!
Ultracold, dilute, interacting Fermi gases
n-1/3
T
・ dilute : details of the potential is much smaller than n-1/3
・ ultracold : s-wave is the dominant channel.
collide only with
The collision process can be described by a single parameter, so-called scattering length as.
as
R
Ultracold dilute Fermi gas
n-1/3
T
as
Remember the fact that as is tunable!!
|as| ∞Then, what happens when…
This situation is called unitarity limit.
n-1/3
T
as
Unitarity limit and Universality
n-1/3T
Thermodynamics depends only on the density n and temperature T.
as drops out of the description of the thermodynamics.
… like a non-interacting case.
Universal thermodynamics
According to universal hypothesis, all thermodynamics should obey the universal functions:
Internal energy :
Helmholtz free energy :
Chemical potential :
Entropy :
Dimensionless universal functions,(shape of the function is different from those for an ideal gas)
F
B
FE
k TN
fEE E
F
B
FF
k TN
fFE E
B
FF
k TE
fE
B
B
FS
k TN
fsk E
2
2/3F 6
2E n
m
System looks like a non-interacting Fermi gas.(no other dimensional parameters involved in the problem)
Bertsch’s Many-Body X challenge, Seattle, 1999
What are the ground state properties of the many-bodysystem composed of spin ½ fermions interacting via azero-range, infinite scattering-length contact interaction.
Universal thermodynamics
F
F
,
1
gsE f n m
N E
N E
pure number
value
s0 andT a
3/2
2 2 2
2 2 2, , 1x y z
x y zn x y z
3/2
2 2 2
2 2 2, , 1x y z
x y zn x y z
1/41
Fermi gas profile in a 3D harmonic trap at
ideal gas
unitary gas
T. Bourdel et al. (2004) -0.64(15)
M. Bartenstein et al. (2004) -0.68+0.13-0.10
J. Kinast et al. (2005) -0.49(4)
C. A. Regal et al. (2005) -0.62(7)
J. Stewart et al. (2006) -0.54 +0.05-0.12
Y.-I Shin et al. (2007) -0.50(7)
J. Joseph et al. (2007) -0.565(15)
(from kinetic energy)
(from sound velocity)
L. Luo et al. (2009) -0.61(2)
2 2F 2 2
* * 0*
12 2k
mm
x
x 2 2
F 2 20
112 2k
mm
x
x
At T=0
Conventional thermometry
Temperature is determined by fitting the profile.
5/2 2
5/2 2,
3 1( ) exp2F zTF z
N T zn z LiT
This scheme is good only when interaction energy << kinetic energy.
virial theorem at unitarity
J. E. Thomas et al.PRL 95, 120402 (2005)
takedelivative E
S T
L. Luo and J. E. Thomas,J Low Temp. Phys. 154, 1 (2009).
total pot2E E
potE totalE
entropy : can be measured after adiabatic magnetic field sweep to weakly-interacting regime
universal thermodynamic function(in a harmonic trap!!)
Universal thermodynamicsH. Hu, P. D. Drummond & X.-J. Liu, Nature Physics 3, 469 - 472 (2007)
trap inhomogeneity
T is constant over the cloud (thermal equilibrium).EF depends on the position (local density).
B
F 0
k TE n
B
F
k TE
is position-dependent.
B B
F 0F
k T k TE nE n
r
Global measurement only gives the integration of all the different phases.
Goal of this experiment
Measurement of local thermodynamic quantities
and
the determination of the universal thermodynamic function.
MOT ( magneto-optical trap )
T = 200 KN > 108
Deceleration and trapping
Li ovenT~700K
Optical dipole trap
Evaporative cooling in an optical dipole trap
Num
ber
Energy
Num
ber
Energy
Num
ber
Energy
6 Li原
子の
s波
散乱
長(a
0)
磁場 [G]
B = 650 - 800 G
molecular BEC
B = 0-450 Gdegenerate Fermi gas
6Li in quantum degenerate regime
B
I
I
FF
[ ]( ) (
()
)Ef T
n ET
n
r
rr
Determination of local energy (r)
density profile
n r
23
p r r
Trap( ) ( ) ( ) 0p n V r r r
・ Equation of state of unitary gas :
・ mechanical equilibrium (eq. of force balance) :
( )n r ( )p r ( ) r
Useful equations :
F[ ]Ef T T
FT T
23
p r r Trap( ) ( ) ( ) 0p n V r r rand total potential2E E
Adiabatic B-field sweep to turn off the interaction
entropy S
total vs E S
1 T S E
total vs E T
Determination of temperature T
Le Luo and J.E. Thomas, J Low Temp Phys 154, 1 (2009).
FF
[ ]( ) (
()
)Ef T
n ET
n
r
rr
FF
[ ]( ) (
()
)Ef T
n ET
n
r
rr
23
p
Trap( ) ( ) ( ) 0p n V r r r
Our scheme
F[ ]Ef T T
FT T
Experimental determination of fE [T/TF ]
M. Horikoshi, S. Nakajima, M. Ueda and T. Mukaiyama,Science, 327, 442 (2010).
2.0
1.5
1.0
0.5
0.0
f E[T
/TF]
1.21.00.80.60.40.2T/TF
2.0
1.5
1.0
0.5
0.0
f E[T
/TF]
1.21.00.80.60.40.2T/TF
Ideal2.0
1.5
1.0
0.5
0.0
f E[T
/TF]
1.21.00.80.60.40.2T/TF
Unitary
About 800 images are analyzed.
FF
( ) [ ]( ) ( ) Ef T T
n E n
r
r r
Verification of the determined fE [T/TF ]1. Energy comparison
pot internalE E
2 2pot
32 zE m z Potential energy par particle :
internal F ) d[ ]( EfE n n V N Internal energy par particle : Comparison
Epot = Eint
Nice co
nsisten
cy !!
total potential2E E
Verification of the determined fE [T/TF]2. Effective speed of the first sound
6Li
Light pulse to make density perturbation
Verification of the determined fE [T/TF ]
0.1ms
1.1ms
2.1ms
3.1ms
4.1ms
5.1ms
6.1ms
7.1ms
Propagation time
2. Effective speed of the first sound
Verification of the determined fE [T/TF ]
Unitary gas shows hydrodynamic behavior due to the large collision rate
21 1
0
d d[ , ]
d dz
n x yu n
pm n x yn
Effective speed of the first sound :
F[ ]23 Efp T T
Comparison
Experiment
[ P. Capuzzi, PRA 73, 021603(R) (2006) ]
2. Effective speed of the first sound
Verification of the determined fE [T/TF ]
Experimental values vs. calculated values from fE [ ]
u1,Meas. = u1,Calc
Nice co
nsisten
cy !!
2. Effective speed of the first sound
1F ~ 1.0k a
thermal bimodalBEC
1F ~ 0.2k a
absorption image after expansion
- strong interaction- pairing by many-body physics
JILA
signature of BEC = bimodal distribution
“Projection”
projection … convert correlated pair of atoms into tightly-bound molecules by sweeping the magnetic field toward BEC side of the resonance
W. Ketterle et al.,arXiv:0801.2500
Magnetic field sweep has to be …- slow enough to satisfy the adiabatic condition of two-body binding process- fast enough so that one can neglect collisions
Bimodal distribution of a fermion pair condensate
BEC side BCS side
650 700 750 800 834 900
Unitarity limit
Magnetic field [Gauss]
Preformed pair
Bound molecule
Bimodal distribution
Condensate fraction vs Temperature
Max( ) 1C
Tf x CFT
3.0(1)
Internal energy
Universal thermodynamic functions
Helmholtz free energy Chemical potential Entropy
][)(F
Ff
nNF
][)(F
fn
][B
SfNk
S
][][][ FFE fff 3][2][5][ FF fff ][][ FS ff
5 2 3E F F
E F
S F
f f f
f f f
f f
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.00.70.60.50.40.30.20.10.0
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.00.70.60.50.40.30.20.10.0
A. Bulgac et al.PRL, 96, 090404 (2006).
2.5
2.0
1.5
1.0
0.5
0.00.80.60.40.20.0
A. Bulgac et al.PRA, 78, 023625(2008).
d .P if T con nst d ddP n s T Gibbs-Duhem equation :
2 20 , ,
, , d d
1 2
2
2 0,0,
d
d
di ii x y z
x y
x y
n z n x
x y
y z x y
n m x
m
m
n
P z
2 dd d x yx y m
Ho’s scheme to obtain the local pressure
Different approach to obtain local thermodynamic quantities
T.-L. Ho and Q. Zhou, Nature Physics 6, 131 (2010).
2D profile
1D profile
absorption imaging
x
yz
atom cloud
0,0, zz Pn
z
S. Nascimbène et al.New Journal of Physics 12, 103026 (2010).
S. Nascimbène et al.Nature 463, 1057 (2010).
Different approach to obtain local thermodynamic quantities
0 B, , expP P hT T k T 2 2
,0 ,1 2 i ii x y z
m x
Assume second-order virial expansion is correct at high T region.
7Li (bosonic isotope) thermometer:T
0 :
0 B, , expP T P T h k T
S. Nascimbène et al.Nature 463, 1057 (2010).
- First experimental determination of virial-3 and virial-4 coefficient.
contribution of 3-body and 4-body physics to the unitary gas thermodynamics
- Observation of Fermi liquid behavior above superfluid Tc.
1.5
1.0
0.5
0.0
f E
1.21.00.80.60.40.20.0T/TF
Tokyo ENS
FF
( ) [ ]( ) ( ) Ef T T
n E n
r
r r
S. Nascimbène et al.Nature 463, 1057 (2010).
22 *
3/2 1/2 B0
5, 2 ,08n n
k TmP T Pm
Fermi liquid theory :
Observation of Fermi liquid behavior
0.51(2)n 0.56n Monte-Calro simulation
C. Lobo et al, Phys. Rev. Lett. 97, 200403 (2006)*
1.13(3)mm
c
F
0.157 15TT
c
F
0.17 1TT
by our result from projection
momentum resolved spectroscopy
RF
- RF pulse shorter than the trap period- no interaction effect (simple dispersion in the excited state)- no collision during expansion
J. Stewart et al.Nature 454, 744 (2008).
Make a transition from to with an RF pulse.Turn off the trap potential immediately after applying RF pulse.
Observation of pseudogap behavior
J. Stewart et al.Nature 454, 744 (2008).
ideal gas
on resonance(unitary gas)
BEC side(atoms + dimers)
Observation of pseudogap behavior
J. P. Gaebler et al.Nature Physics 6, 569 (2010).
back bendinguniversal behavior of fermions
Fermi liquid (no finite-energy excitation above Tc) or Pseudogap (preformed pair formation above Tc)
22 22
2kEm
momentum distribution1
k1 1/3 1
0,a n k r
4n k k
L.Viverit et. al,PRA 69, 013607 (2004)J. Stewart et al.PRL 104, 235301 (2010)
[kF]
experimental verification
C = “Contact” 4
Cn kk
Momentum distribution and “Contact”
- Energy relation
Tan relations
3 2 2 2
3 4, 2 42
d k k CE n Cm k ma
k
- Adiabatic relation
2
1 4d E C
da m
- Pressure relation
22
3 12EP CV ma
R
- density-density correlation
2 2
1 1 22 2 16
n n Cr ar
r rR R R
- Virial theorem in a harmonic trap
2
28
E E Cma
pot
- Inelastic two-body loss
2
Im
2
ad n Cdt m a
R R
universal relations : relations which hold any system consisting of two-component fermions with a large scattering length
by Shina Tan 2005
S. Tan, Ann. Phys. 323, 2952 2008). S. Tan, Ann. Phys. 323, 2971 (2008).S. Tan, Ann. Phys. 323, 2987 (2008). E. Braaten arXiv:1008.2922
Testing virial theoremTesting adiabatic sweep relation
determined directly
determined from C measured separately
J. Stewart et al.PRL 104, 235301 (2010)
Experimental verification of Tan relations
Summary• Thermodynamic functions of a unitary
Fermi gas was determined at the unitarity limit.
M. Horikoshi, S. Nakajima, M. Ueda and T. Mukaiyama, Science, 327, 442 (2010).
Ideal
Unitary
• Microscopic mechanism of superfluidity in a unitary Fermi gas is now under intense research.
• Tan relation - new universal thermodynamic formalism -