Universal Thermodynamics of a Unitary Fermi gas

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 -