# Universal Thermodynamics of a Unitary Fermi gas

Mechanics

Oct 27, 2013 (3 years and 10 months ago)

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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 (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

s

(
a
0
)

[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

-