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 (10
11
~10
14
cm

3
),
•
with no impurities, no defects.
Amenable to simple
theoretical description
J. R.
Ensher
, et al.,
Phys. Rev.
Lett
.
77
, 4984 (1996).
Condensate fraction
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
R
Open (scattering) channel
Closed (bound) channel
Resonance occurs when open and closed channel
are 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).
scattering
length
Number of
atoms
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
a
s
.
a
s
R
Ultracold dilute Fermi gas
n

1/3
T
a
s
Remember the fact that
a
s
is tunable!!

a
s

∞
Then, what happens when…
This situation is called unitarity limit.
n

1/3
T
a
s
Unitarity limit and Universality
n

1/3
T
Thermodynamics depends only on the
density
n
and
temperature
T
.
a
s
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)
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

body
system composed of spin ½ fermions interacting via a
zero

range,
infinite scattering

length
contact interaction.
Universal thermodynamics
pure number
value
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)
At
T
=0
Conventional thermometry
Temperature is determined by fitting the profile.
This scheme is good only when interaction energy << kinetic energy.
virial
theorem at
unitarity
J. E. Thomas et al.
PRL
95,
120402 (2005)
take
delivative
L.
Luo
and J. E. Thomas,
J Low Temp. Phys. 154, 1 (2009).
entropy : can be measured after adiabatic magnetic field sweep
to weakly

interacting regime
universal thermodynamic function
(in a harmonic trap!!)
Universal thermodynamics
H.
Hu
, P. D. Drummond & X.

J. Liu,
Nature Physics
3
, 469

472 (2007)
trap inhomogeneity
T
is constant over the cloud (thermal equilibrium).
E
F
depends on the position (local density).
is position

dependent.
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
K
N㸠10
8
Deceleration and trapping
Li oven
T~700K
Optical dipole trap
Evaporative cooling in an
o
ptical
dipole trap
Number
Energy
Number
Energy
Number
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 G
degenerate Fermi gas
6
Li in quantum degenerate
regime
B
I
I
Determination of local energy
(r)
density profile
・
Equation of state of unitary gas :
・
mechanical equilibrium ⡥q. o映景rce balance⤠㨠
Useful equations :
and
Adiabatic B

field sweep to turn off
the interaction
entropy
Determination of temperature
T
Le
Luo
and J.E. Thomas,
J Low Temp Phys
154,
1 (2009).
Our scheme
Experimental determination of
f
E
[
T/T
F
]
M.
Horikoshi
, S. Nakajima,
M. Ueda and T.
Mukaiyama
,
Science,
327
, 442 (2010).
Ideal
Unitary
About 800 images are analyzed.
Verification of the determined
f
E
[
T/T
F
]
1.
Energy comparison
Potential energy par particle :
Internal energy par particle :
Comparison
E
pot
=
E
int
Verification of the determined
f
E
[
T/T
F
]
2. Effective speed of the first sound
6
Li
Light pulse to make
density perturbation
Verification of the determined
f
E
[
T/T
F
]
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
f
E
[
T/T
F
]
Unitary gas shows hydrodynamic behavior due to the large collision rate
Effective speed of the first sound :
Comparison
Experiment
[ P.
Capuzzi
, PRA
73
, 021603(R) (2006) ]
2. Effective speed of the first sound
Verification of the determined
f
E
[
T/T
F
]
Experimental values vs. calculated values from
f
E
[
]
u
1,Meas.
=
u
1,Calc
2. Effective speed of the first sound
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
Internal energy
Universal thermodynamic functions
Helmholtz free energy
Chemical potential
Entropy
A.
Bulgac
et al.
PRL,
96
, 090404 (2006).
A.
Bulgac
et al.
PRA,
78
, 023625(2008).
Gibbs

Duhem
equation :
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
atom cloud
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
Assume second

order
virial
expansion is correct at high
T
region.
7
Li (
bosonic
isotope) thermometer
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
T
c
.
S.
Nascimbène
et al.
Nature
463
, 1057 (2010).
Fermi liquid theory :
Observation of Fermi liquid behavior
Monte

Calro simulation
C. Lobo et al, Phys. Rev. Lett. 97, 200403 (2006)
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 bending
universal behavior of fermions
Fermi liquid (no finite

energy excitation above
T
c
)
or
Pseudogap
(preformed pair formation above
T
c
)
momentum distribution
1
k
L.Viverit
et.
al,
PRA
69,
013607 (2004)
J. Stewart
et al.
PRL
104
, 235301 (2010)
[
k
F
]
experimental verification
C
=
“Contact”
Momentum distribution and “Contact”

Energy relation
Tan relations

Adiabatic relation

Pressure relation

density

density correlation

Virial
theorem in a harmonic trap

Inelastic two

body loss
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
theorem
Testing 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
•
M
icroscopic
mechanism
of superfluidity
in a unitary
Fermi gas is now under intense research.
‡
Tan relation

new universal thermodynamic formalism

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