DYNAMICS OF TRAPPED BOSE AND FERMI GASES

Sandro Stringari

University of Trento

IHP Paris, June 2007

CNR-INFM

Lecture 1

Lecture 1:- Hydrodynamic theory of superfluid gases - expansion - collective oscillations and equation of state - collective vs s.p. excitations :Landau critical velocity - Fermi superfluids with unequal masses - example of non HD behavior: normal phase of spin polarized Fermi gas

Lecture 2:- Dynamics in rotating superfluid gases - Scissors mode - Expansion of rotating gas - vortex lattices and rotational hydrodynamics - Tkachenko oscillations

Understanding superfluid features requires theory for transport phenomena

(crucial interplay between dynamics and superfluidity)

Macroscopic dynamic phenomena in superfluids(expansion, collective oscillations, moment of inertia)are described by theory of irrotational hydrodynamics

More microscopic theories required to describe other superfluid phenomena (vortices, Landau critical velocity, pairing gap)

HYDRODYNAMIC THEORY OF SUPERFLUIDS

Basic assumptions:

- Irrotationality constraint (follows from the phase of order parameter)

- Conservation laws (equation of continuity, equation for the current)

Basic ingredient:

- Equation of state

- Equation for velocity field involves gradient of chemical potential; - Includes smoothly varying external field in local density approximation

0))(2

1(

0)(

2

extVnmvvt

m

vnnt

Hydrodynamic equationsof superfluids (T=0)Closed equations fordensity and superfluidvelocity field

)(rVext

HYDRODYNAMIC EQUATIONS AT ZERO TEMPERATURE

is atomic mass, is superfluid density is superfluid velocity, is chemical potential (equation of state) is external potential

mSnnSvv

extV

T=0

irrotationality

- Have classical form (do not depend on Planck constant) - Velocity field is irrotational- Should be distinguished from rotational hydrodynamics.- Applicable to low energy, macroscopic, phenomena - Hold for both Bose and Fermi superfluids- Depend on equation of state (sensitive to quantum correlations, statistics, dimensionality, ...) - Equilibrium solutions (v=0) consistent with LDA

)(n

KEY FEATURES OF HD EQUATIONS OF SUPERFLUIDS

What do we mean by macroscopic, low energy phenomena ?

BEC superfluids

gn

mgn

2

BCS Fermi superfluids

Fv more restrictive than in BEC

0)()( rVn ext

superfluid gap

healing length

size of Cooper pairs

WHAT ARE THE HYDRODYNAMIC EQUATIONS USEFUL FOR ?

They provide quantitative predictions for

- Expansion of the gas follwowing sudden release of the trap - Collective oscillations excited by modulating harmonic trap

Quantities of highest interest from boththeoretical and experimental point of view

- Expansion provides information on release energy, sensitive to anisotropy

- Collective frequencies are measurable with highest precision and can provide accurate test of equation of state

-Initially the gas is confined in anisotropic trap in situ density profile is anisotropic too

- What happens after release of the trap ?

2/)( 2222 rzmV zext

Non interacting gas expands isotropically

- For large times density distribution become isotropic

- Consequence of isotropy of momentum distribution

- holds for ideal Fermi gas and ideal Bose gas above

- Ideal BEC gas expands anysotropically because momentum distribution of condensate is anysotropic

CT

Expansion from anysotropic trap

- Expansion inverts deformation of density distribution, being faster in the direction of larger density gradients (radial direction in cigar traps)- expansion transforms cigar into pancake, and viceversa.

Superfluids expand anysotropically

Hydrodynamic equations can be solved after switching off the external potential

For polytropic equation of state (holding for unitary Fermi gas ( ) and in BEC gases ( )), and harmonic trapping, scaling solutions are available in the form(Castin and Dum 1996; Kagan et al. 1996)

222

01

2

1),(

)/,/,/()(),,,(

zyxtrv

bzbybxnbbbtzyxn

zyx

zyxzyx

with the scaling factors satisfying the equation

and

)(

12

zyxiii bbbb

b

n3/2 1

ib

iiii bbtb /,1)0(

Experiments probe HD nature of theexpansion with high accuracy. Aspect ratio

HDHD

ideal gas ideal gas

ZR /

Expansion of BEC gases

First experimental evidence for hydrodynamic anisotropic expansion inultra cold Fermi gas (O’Hara et al, 2002)

Hydrodynamics predicts anisotropic expansion inFermi superfluids (Menotti et al,2002)

EXPANSION OF ULTRACOLD FERMI GAS

collisionless

HD theory

Measured aspect ratio after expansion along the BCS-BEC crossover of a Fermi gas

(R. Grimm et al., 2007)

prediction of HD at unitarity

prediction of HD for BEC

- Expansion follows HD behavior on BEC side of the resonance and at unitarity. - on BCS side it behaves more and more like in non interacting gas (asymptotic isotropy)

Explanation:

- on BCS side superfluid gap becomes soon exponentially small during the expansion and superfluidity is lost.

- At unitarity gap instead always remains of the order of Fermi energy and hence pairs are not easily broken during the expansion

Collective oscillations: unique tool to explore consequence of superfluidity and test the equation of state of interacting quantum gases (both Bose and Fermi)

Experimental data for collective frequencies are available with high precision

Collective oscillations in trapped gases

ST

ET

)1(4

)1(3

04

00

z

TkF Beq

1

1)(

6),,(

0

03

2220

czTTk

zTTF ESBzES

Asymptotic behavior of the force out of thermal equilibrium(Antezza et al. PRL 95 083202, 2005)

Lifschitz result (thermal equilibrium)

First measurement of thermal effect on the Casimir force(JILA, Obrecht et al. PRL 2007)

Recent example of high precision measurement

Relative frequency shift

Results for the frequency shift of the center of mass oscillation due to the Casimir-Polder force produced by a substrate at different temperatures

theory

Propagation of sound in elongated traps

-If wave length is larger than radial size of elongated trapped gas sound has 1D character

where and n is determined by TF eq.

one finds

In uniform medium HD theory gives trivial sound wave solution

with nmccq /; 2 )( tqzien

11121 / nnmc D

dxdyn

ndxdymc D 1

21

)/(

1)()( rVn ext

2/1 bulkD cc (Zaremba, 1998)For BEC gas ( )

5/31 bulkD cc

ndxdyn1

nFor unitary Fermi gas ( )

3/2n (Capuzzi et al, 2006)

Sound wave packets propagating in a BEC (Mit 97)

velocity of sound as a function of central density

Sound wave packets propagating in an Interacting Fermi gas (Duke, 2006)behavior along the crossover

BCS mean field

QMC

Difference bewteen BCS and QMC reflects: -at unitarity: different value of in eq. of state-On BEC side different dimer-dimer scattering length

3/2)1( n

Collective oscillations in harmonic trap

When wavelength is of the order of the size of the atomic cloud sound is no longer a useful concept. Solve linearized 3D HD equations

))((2

2

nn

nnt

-Surface modes are unaffected by equation of state

- For isotropic trap one finds where is angular momentum

- surface mode is driven by external potential, not by surface tension - Dispersion law differs from ideal gas value (interaction effect)

hol l

hol

Solutions of HD equations predictsurface and compression modes

Surface modes

well understood in atomic BEC Stringari,1996here we focus on Fermi gases

where is non uniform equilibrium Thomas Fermi profile

)(rn

Quadrupole mode measured on ultracold Fermi gas along the crossover (Altmeyer et al. 2007)

HD prediction

2

Ideal gas value

Enhancement of damping

Minimum damping near unitarity

- Experiments on collective oscillations show that on the BCS side of the resonance superfluidity is broken for relatively small values of (where gap is of the order of radial oscillator frequency)

- Deeper in BCS regime frequency takes collisionless value

- Damping is minimum near resonance

akF/1

Compression modes

- Sensitive to the equation of state

-analytic solutions for collective frequencies available for polytropic equation of state

- Example: radial compression mode in cigar trap

- At unitarity ( ) one predicts universal value

- For a BEC gas one finds

n

)1(2

83.13

10

2

3/2

Equation of state along BCS-BEC crossover

- Fixed Node Diffusion MC (Astrakharchick et al., 2004)- Comparison with mean field BCS theory ( - - - - - )

On BEC side of the resonance QMC confirms result for the dimer-dimer scattering length (Petrov et al., 2005)and points out occurrence of Lee-Huang-Yang correction in equation of state

aaM 6.0

])(3

321[2 2/132

MddMd anna

1 100.01

0.1

(E -

Eb)

/ (N

3/5

F)

-1 / kFa

Fixed Node Monte Carlo

Gross Pitaevskii (aM=0.6a)

Gross Pitaevskii (aM=2a)

Lee Huang Yang (aM=0.6a)

aaM 2

LHY

In harmonic trap LHY effect results in enhancement of radial compressionfrequency according to: Pitaevskii and Stringari (1998) Braaten and Pearson (1999)

])0(256

1051[2 3

dMna

w

Frequency of radialcompression mode in cigar trap

(Stringari 2004)

BEC beyond mean field unitarity

akF/1

2

0

))(094.01(2 5/60akF 3/10

- Collective frequency has non monotonic behaviour (effect missed by BCS theory, accounted for by MC)- can be evaluated numerically in whole regime starting from knowledge of equation of state

)/91.1( 6/10hoF aNk

Radial breathing mode at Innsbruck (Altmeyer et al., 2007)

MC equation of state (Astrakharchick et al., 2005)

BCS eq. of state(Hu et al., 2004)

83.13/10

includes beyond mf effects

does not includes beyond mf effects

universal value at unitarity

Measurement of collective frequencies provides accurate test of equation of state !!

akF0/1

- Accurate confirmation of the universal HD value predicted at unitarity.

- Accurate confirmation of QMC equation of state on the BEC side of the resonance.

- First evidence for Lee Huang Lee effect.

- LHY effect (and in general enhancement of compression frequency above the BEC value) very sensitive to thermal effects (thermal fluctuations prevail on quantum fluctuations except at very low temperature)

Main conclusions concerning the m=0 radial compression mode

3/10

Landau’s critical velocity

While in BEC gas sound velocity provides critical velocity,in a Fermi BCS superfluid critical velocity is fixed by pair breaking mechanisms (role of the gap)

Landau’s critical velocity

p

pv pcr

)(min

- Landau’s criterion for superfluidity (metastability): fluid moving with velocity smaller than critical velocity cannot decay (persistent current)

- Hydrodynamic theory cannot predict value of critical velocity Needed full description of dispersion law (beyond phonon regime)

- Ideal Bose gas and ideal Fermi gas one has

- In interacting Fermi gas one predicts two limiting cases:

Dispersion law of elementary excitations

BEC (Bogoliubov dispersion)

cvcr

BCS (role of the gap)

Fcr pv /(sound velocity)

0crv

Landau’s critical velocity ishighest near unitarity !!

BEC

resonance

Sound velocity

BCS BEC

B

F

cr

v

v

BCSBEC

Critical velocity

unitarity

gap gapgap

(Combescot, Kagan and Stringari 2006)

Dispersion law along BCS-BEC crossover of Fermi gas

Healing length is smallest !!(Marini et al 1997)

crmv

Lecture 1:- Hydrodynamic theory of superfluid gases - expansion - collective oscillations and equation of state - collective vs s.p. excitations :Landau critical velocity - Fermi superfluids with unequal masses - example of non HD behavior: normal phase of spin polarized Fermi gas

Fermi superfluids with unequal masses ( )(G.Orso, L.Pitaevskii, S.S, in preparation)

NN

Hydrodynamic eqs. keep standard form:

0)~

)(2

1(

0)(

2

extVnmvvt

m

vnnt

with

mm

mm

rmV

mmm

iii

iiext

222

22

~~~

~2

1~

2/)(

- Relevant frequencies for HD oscillations

-If oscillator lengths are equal (natural conditionto cool down into superfluid phase) one has

ii mm

iii ~

What happens if one suddenly switches off ?

- Sudden change in trapping conditions:

- In the absence of interactions atoms down will fly away

- Superfluidity keeps component down confined even if - obvious in molecular regime - non trivial many-body effect at unitarity - weak effect in BCS regime

- Condition for energetic stability of superfluid with respect to separation of two components

- Gas starts oscillating with frequency

mm

mnew

22 ~~

0

1)1(

m

mm easily satisfied if mm

new~

)()~( NnewS EE(unitarity)

Example: mixture (aspect ratio 0.2)

KLi 406

Oscillation of radial sizeif one removes K-trapping

Equilibrium of superfluidin new trapping conditions

Oscillation of radial sizeif one removes Li-trapping

Will superfluid survive in conditionsof strong energetic instability?

Collective oscillations in spin polarized Fermi gas (normal phase)

Lobo, Recati, Giorgini and S.S

- Recent experiments on spin polarized Fermi gas show phase separation- between superfluid and spin polarized normal gas if

- Density profiles at unitarity well agree with theory (Lobo et al. 2006) based on QMC equation of state of superfluid and normal phases -Equation of state in normal phase well approximated by expansion holding for small concentrations

-QMC simulation yields A = 0.97, m/m* = 1.04

75.0

NN

NNP

..]1[5

3 3/5*

xm

mAxE

N

EF

N

n

nx

Axial densitiesin spin polarizedFermi gas

Exp. Shin et al. 2006Theory ideal gas Lobo et al. 2006

Critical value for normal-superfluidphase separationP = 0.75

P=0.80

P=0.58

Clogston-Chandrasekharlimit at unitarity (and in trap)

normal

normal plussuperfluid

At zero temperature oscillations in normal spin polarized phase are not governed by hydrodynamics (like in superfluid), but by Landau’s theory of Fermi liquids (collisionless regime + mean field)

Spin down feel potential generated by spin upresulting in renormalizationof oscillator frequency

If spin down atomsoscillate with frequencies

hoho

hoho

mAm

mAm

46.2/)5/31(2

23.1/)5/31(

*

*

dipole

quadrupole

Spin dipole mode difficult to excite because shift of harmonic trap resultsin excitation of center of mass motion (Kohn’s theorem)

Spin quadrupole is instead excited by quadrupole modulation of trap.

In general one expect to excite two modes (sum rule approach)

2/)(

22

/)5/31(2

21

2

*1

QQQQQ

QQQ

mmAQQ

ho

ho

Exciting Q one excites also spin mode !!

QUESTION: Will the spin mode be strongly damped? Pauli principle is expected to quench damping at small T(collaboration with Ch. Pethick)

Quadrupolemodes

NN

Lecture 2:- Dynamics in rotating superfluid gases - Scissors mode - Expansion of rotating gas - vortex lattices and rotational hydrodynamics - Tkachenko oscillations

Main conclusions

- Collective oscillations in trapped quantum gases can be measured with high precision.- They provide unique information on the consequences of superfluidity and accurate test of equation of state in Fermi superfluids

Can their measurement be considered a proof of superfluidity?