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Universal Thermodynamics of Strongly Interacting Fermi Gases Takashi Mukaiyama Japan Science Technology Agency, ERATO University of Electro-Communications
Universal Thermodynamics of Strongly Interacting Fermi Gases
Takashi Mukaiyama
Japan Science Technology Agency, ERATO
University of Electro-Communications
BEC in a cold atom system
1995 Realization of atomic gas Bose-Einstein
condensation
Science 275, 637 (1997).
JILA
MIT
MPQ
vorticesinterference
Mott-insulator phase
Atom laser
Bose nova
Super-radiance
MIT
JILA
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).
Condensate
fra
ction
5% deviation of critical temperature
from theoretical predictions・3% shift due to finite number 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 channel
are energetically degenerate.
S. Inouye, et al.,
Nature 392, 151 (1998).
Inter-atomic interaction is tunable !!
Loss near 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.
So, we are able to prepare an interacting
(reasonably stable) two-component Fermi gas of
atoms with an arbitrary interaction strength !!
Ideal Fermi gas
n-1/3
T
Thermodynamic behavior of an ideal Fermi gas is
described by its temperature T and density n.
Thermodynamic of an ideal Fermi gas
1
B1
1 1,
11n z e k T
z ee
Fermi-Dirac distribution
10
3/2
3/2
3/22 2
1
2
4 2
DN d
z e
V mLi z
3/2
2 2
1
10
2
4
1
1
s
s t
V mD
tLi z dt
s z e
3/2
3/2
B 3/22 2
3/2
3/2
F2 2
1 2
4 2
1 2
6
N mn k T Li z
V
mE
2
2/32
F 62
E nm
Thermodynamic of an ideal Fermi gas
3/2
B3/2
F
3
4
k TLi z
E
F F
B B F B F
exp exp expE E
z ek T k T E k T E
3/2
B F3/2
F B F
3exp
4
k T ELi
E k T E
B
F F
k Tf
E E
Thermodynamic of an ideal Fermi gas
10
5/2
B5/2
F F
1
3
4
DE d
z e
k TELi z
NE E
B
F F
E
k TEf
NE E
Other thermodynamic functions also have this similarity.
B
F F
F
k TFf
NE E
B
B F
S
k TSf
k E
Internal energy :
Helmholtz free energy :
Chemical potential :
Entropy :
,B
FF
E ideal
E
Nf
k
E
T
E
,B
FF
F ideal
F
Nf
k
E
T
E
,
FF
Bideal
E
Tf
k
E
,B
FB
S ideal
s
Nf
k
k
T
E
Dimensionless
functions
Thermodynamic of an ideal Fermi gas
Material specific parameter, such as m, is taken up by EF (TF).
(Shape of the functions do not depend on the particle’s nature.)
Universal thermodynamics
Ultracold, dilute, interacting Fermi gases
n-1/3R
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.
B t s
F
F in, ,Ef k T E EE
NEa
,B
FF
E ideal
E
Nf
k
E
T
E
Ideal Fermi gas Fermi gas with interaction
Thermodynamic of an interacting Fermions
Ultracold dilute Fermi gas
n-1/3
T
as
R
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
Thermodynamics depends only by the density n and temperature T.
as drops out of the description of the thermodynamics.
Universal thermodynamics holds again…?
Unitarity limit and Universality
n-1/3T
Thermodynamics depends only by the density n and temperature T.
as drops out of the description of the thermodynamics.
Universal thermodynamics holds again…?
B B F
F
, ,E
Ef k T k T U a
NE
,B
FF
E ideal
E
Nf
k
E
T
E
Ideal Fermi gas Fermi gas with interaction
BB B F B B F, ,
F F
, , ,E E a E a
k TEf k T k T U a f k T k T f
NE E
When the scattering length
diverges…
There is a hypothesis that the thermodynamic functions
again have the universal form.
Universal hypothesis
Thermodynamic of an interacting Fermions
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
F
E
k T
Nf
E
E E
F
B
F
F
k T
Nf
F
E E
B
FF
k T
Ef
E
B
B
F
S
k T
Nf
s
k E
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.
Neutron star
Wikipedia
Besides pure theoretical curiosity, this problem
is relevant to neutron stars!
Universal thermodynamics
F, ,gsE f N V m N E pure number
Universal thermodynamicsH. Hu, P. D. Drummond & X.-J. Liu,
Nature Physics 3, 469 - 472 (2007)
is still not known…
T is constant over the cloud (thermal equilibrium).
EF depends on the position (local density).
B
F 0
k T
E n
B
F
k T
Eis position-dependent.
B B
F 0F
k T k T
E nE n
r
Global measurement only gives the integration of
all the different phases.
B
F
E
k Tf
E
Goal of this experiment
Measurement of local thermodynamic quantities
and
the determination of the universal thermodynamic function.
: local energy density
F B FE k T
F F
E
Tf
n E n T n
r
r r rF
B
F
E
k T
Nf
E
E E
Experiment setup
原子の冷却（レーザー冷却）
Liの蒸気（450℃）
金属Li
Zeeman Slower
Zeeman Slower
共鳴光による減速ドップラーシフトをゼーマンシフトでキャンセル
MOT
吸収イメージ
T = 200 K
N > 108
MOT（磁気光学トラップ）四重極磁場と共鳴光を組み合わせた冷却
共振器光トラップ
T = 200 K
N = 5×107
Cavity enhanced
optical dipole trapMOT
T = 200 K
N > 108
l = 1064nm, P = 10 W, w = 250 m
⇒ トラップ深さ U = kBх5 K
共振器で増幅 ×100
w = 260 m トラップ深さ kBх2 mK
=
R=99.9%
R=98.3%
error signal
6Li
Glass Cell
Optical dipole trap
- No need to use multiple-coil configuration as used in a magnetic trap
- wide optical access
- trap can be turned off very quickly
6Li
Glass Cell
シングルビーム光トラップ
T = 200 K
N = 5×107
Cavity enhanced
optical dipole trapSingle beam
optical trap
T = 80 K
N = 1×107
共振器光トラップ：定在波によるトラップ
単純に絞ったビームにトラップしなおす
1 s
time
Cavity
Signal
Error
Signal
Trigger
・共振器の光強度を断熱的に下げる・フェッシュバッハ共鳴を使った散乱断面積の増強・共振器トラップに重なるようにビームを入射
トラップした原子全体が熱緩和しない調和型のトラップではない
原子状態とフェッシュバッハ共鳴6L
i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
Experimental scheme
Optical dipole trap
(1064nm)
834Gauss（ Resonance magnetic field of Feshbach resonance ）
6Li
: |F=1/2, mF=+1/2>
: |F=1/2, mF=-1/2>
N ~ 106
Equal mixture of
F
F
[ ]( ) (
(
)
)Ef T
n ET
n
r
r
r
Determination of local energy (r)
density profile
n r
2
3p 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
2
3p r r Trap( ) ( ) ( ) 0p n V r r rand total potential2E E
Adiabatic B-field sweep to turn off
the interactionentropy 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).
F
F
[ ]( ) (
(
)
)Ef T
n ET
n
r
r
r
F
F
[ ]( ) (
(
)
)Ef T
n ET
n
r
r
r
2
3p
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]
1.21.00.80.60.40.2
T/TF
2.0
1.5
1.0
0.5
0.0
f E[T
/TF]
1.21.00.80.60.40.2
T/TF
Ideal
2.0
1.5
1.0
0.5
0.0
f E[T
/TF]
1.21.00.80.60.40.2
T/TF
Unitary
About 800 images are analyzed.
F
F
( )[ ]
( ) ( )Ef T T
n E n
r
r r
Verification of the determined fE [T/TF ]
1. Energy comparison
pot internalE E
2 2
pot
3
2zE m z Potential energy par particle :
internal F ) d[ ]( EfE n n V N Internal energy par particle :
Comparison
Epot = Eint
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 ]
2. Effective speed of the first sound
0.1ms
1.1ms
2.1ms
3.1ms
4.1ms
5.1ms
6.1ms
7.1ms
Propagation time
Verification of the determined fE [T/TF ]
2. Effective speed of the first sound
Unitary gas shows hydrodynamic behavior due to the large collision rate
2
1 1
0
d d[ , ]
d dz
n x yu n
pm n x y
n
Effective speed of the first sound :
F[ ]2
3Efp T T
Comparison
Experiment
[ P. Capuzzi, PRA 73, 021603(R) (2006) ]
Verification of the determined fE [T/TF ]
2. Effective speed of the first sound
Experimental values vs. calculated values from fE [ ]
u1,Meas. = u1,Calc
Ideal
Unitary
The universal function of the internal energy fE [T/TF ]
F
F
[ ]EfnE
T T
2
3p
Trap( ) ( ) ( ) 0p n V r r r
Equation of state :
Universal hypothesis :
Mechanical equilibrium :
Epot = Eintu1,Meas. = u1,Calc
Speed of the first soundEnergy comparison
CCD
lens
absorption imaging
thermal bimodal pure BEC
momentum distribution measurement
Bosonic case
Is it condensed or not?
See the bimodal profile !!
… unfortunately this scheme does not work.
spatially
correlated pair
momentum
correlated pair
BEC limit BCS limit
BCS
BEC
B
Fermion pair condensate
G. Veeravalli et al.
Phys. Rev. Lett.
101, 250403 (2008)
m
2m
C. A. Regal et al., PRL 92,
040403 (2004)
JILA
M. W. Zwierlein et al.,
PRL 92, 120403 (2004)MIT
- slow enough to convert atom pairs into
molecules
- fast enough such that the momentum distribution
of the projected molecules reflects that of pairs
prior to the sweep
If we sweep the magnetic field
We can convert correlated pairs into tightly-bound molecules.
BCS
BEC
B
C. A. Regal et al.
Phys. Rev. Lett., 92, 040403 (2004)
magnetic field
sweep
“projection”
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 CF
T
3.0(1)
Internal energy
Universal thermodynamic functions
Helmholtz free energy Chemical potential Entropy
][)(F
FfnN
F ][
)(F
f
n ][
B
SfNk
S
][][][ FFE fff 3][2][5][ FF fff ][][ FS ff
5 2 3
E F F
E F
S F
f f f
f f f
f f
F F( ) ( ) ( ) [ ( ) ]En E f T T r r r r
In the case of unitary gas, equation of state
is available (exceptional case !!) which enable us to
measure local thermodynamic quantities.
( ) 2 ( )/3p r r
High resolution local probe
W. S. Bakr et al.Nature 462, 74 (2009).
Then, how can we determine local thermodynamic quantities without
help of equation of state ?
・local probe
・control by electric fieldion ＝
Co-trapping system of ions and neutral atoms
The system that I’m setting up in University of Electro-Communications
ion
Neutral atoms
Summary
• The universal function of the internal energy was determined at the unitarity limit
• The other thermodynamic functions were derived from the thermodynamic relationship
• The critical parameters were determined at the superfluid transition temperature
M. Horikoshi, S. Nakajima, M. Ueda and T. Mukaiyama,
Science, 327, 442 (2010).
Ideal
Unitary
Masahito Ueda
(project leader)
M. Horikoshi
(Postdoc)
S. Nakajima
(Ph.D student)
T. Mukaiyama
(Group leader )
The team (ERATO project)
Unitary gas Efimov physics
