DYNAMICS OF TRAPPED BOSE AND FERMI GASES

Sandro Stringari

University of Trento

IHP Paris, June 2007

CNR-INFM

Lecture 2

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

- Is the measurement of anisotropic expansion and collective frequencies a proof of superfluidity?

- These measurement probe validity of hydrodynamic theory and predictions for equation of state

- More direct proofs of superfluidity concern - absence of viscosity (Landau critical velocity) - rotations (response to transverse field)

Hydrodynamics and superfluidity

This talk

Rotating superfluids behave very differently from classical fluids

- In classical fluid, due to viscosity, the velocity field of steady rotation is given by the rigid value

and is characterized by uniform vorticity

- Moment of inertia takes rigid value

rrv )(

- Superfluids are characterized by irrotationality constraint forvelocity field. Vorticity is hence vanishing ( ) except along lines of singularity (quantized vorticies)

- Without vortices moment of inertia takes irrotational value where is deformation of the cloud

2)(rv

0)( rv

)()(2 22 rnyxrdmrig

rig 2 2222 / yxyx

zL

Moment of inertia

Direct measurement of moment of inertia is difficult because images of atomic cloud probe density distribution (not velocity distribution)

In deformed traps rotation is however coupled to density oscillations. Exact relation, holding also in the presence of 2-body forces:

xyxyz QimLH )(],[ 22

angular momentum quadrupole operator

Response to transverse probe measurable thorugh density response function !!

In principle measurement of quadrupole excitations and strengths in deformed trap permits to determine moment of inertia !!

30

2

2222

0

2

)(

0||)(2

0||2

EE

Qnm

EELn

n

xy

nyxn

z

n

iixy yxQ

Exact sum rules

Example of coupling between angularmomentum and quadrupole excitationsis provided by scissor mode.

If confining (deformed) trap is suddenly rotated by angle the gas is no longerin equilibrium.

Behaviour of resulting oscillation depends on value of moment of inertia (irrotational vs rigid)

Scissors mode

Scissors frequencies (Guery-Odelin and Stringari, 1999)

Superfluid (T=0)With the irrotational ansatz one finds exact solution of HD equations for the scissor mode.If trap is deformed ( ) the solution corresponds to rotation of the gas around the principal axis in x,y plane

Result is independent of equation of state (surface mode)

22yx

xytvS )(

yx

yx

yx

CTCollisionless normal gas (above ). Gas is dilute and interactions can be ignored (collisionless regime). Excitations are provided by ideal gas Hamiltonian. Two frequencies:

Differently from superfluid the normal gas exhibits low frequency mode (crucial to ensure rigid value of moment of inertia)

Scissors measured at Oxford (Marago’et al, PRL 84, 2056 (2000))

Above (normal)

2 modes:

CTyx

Below (superfluid) : single mode: 22

yx

CT

0 2 4 6 8 10 12 14 16-8-6-4-202468

wait time (ms)

angl

e (d

eg)

0 2 4 6 8 10 12 14 16-8-6-4-202468

wait time (ms)

angl

e (d

eg)

below the resonance1/kfa = 0.65

on resonance1/kfa = 0

Scissors in Fermi gas, hydrodynamic regime (Innsbruck 2007, to be published)

22yxc

oscillation-frequency

0 2 4 6 8 10 12 14 16-15

-10

-5

0

5

10

15

wait time (ms)

angl

e (d

eg)

Colissionless

above the resonance1/kfa < -0.6

tAtAt coscos

yx

oscillation-frequencies

yx

Scissors in Fermi gas, collisionless (Innsbruck 2007, to be published)

Superfluidity is very fragile In BCS regime. Gap parameter is exponentially small

Is the measurement of the scissors mode at hydrodynamicfrequency a proof of superfluidity ?

- Scissors are characterized by irrotational flow

- If normal gas is deeply in collisional regime dynamics is governed by same HD equations as in the superfluid (viscosity effects are too weak to generate rotational components in the velocity field). - Innsbruck experiment shows that, near resonance, HD scissors mode persists also above Tc.

22yx

Scissors in superfluid Fermi gas

(From Rudy Grimm)

Investigation of scissors mode, near unitarity, cannot distinguish between superfluid and normal phase

(unless one considers steady rotating configurations)

Universal HD behavior

- In order to exploit superfluidity one should study collective oscillations in the presence of rotating trap !

- In this case frequency of the scissors mode in the superfluid and normal phases behaves differently even if normal gas is in collisional regime.

In the presence of deformed rotating trap the stationary velocity field behaves differently depending on whether the system is superfluid or normal.

In a superfluid the velocity velocity field is subject toconstraint of irrotationality : xyv

Steady rotation of the trap: irrotational vs rotational flow

Normal gas, in steadyconfiguration, insteadrotates in rigid way

What happens to the cloud if we suddenly stop the rotation of the trap? System will be no longer in equilibrium and will start oscillate (scissors mode).

If the gas is normal and collisional, the scissors mode will be described by the equations of rotational hydrodynamics

vvmVnmvvt

m ext

))(

21( 2

rv

absent in superfluid hydrodynamics !!

- Viscosity is crucial to bring the system into steady rotational flow: spin up time (long time scale), fixed by trap anisotropy and collisional cross section, - Viscosity in the deep HD regime can be ignored for describing the dynamics of the scissors mode (short time scale) :

Role of viscosity in rotating normal gas (two different time scales)

)/(1 22 houp

1ho

(Guery-Odelin, 2000)

Superfluid (T=0)

Normal (collisional)beating between

Normal (collisionless)beating between

Scissors mode (rotation angle after stopping trap)(Cozzini and Stringari, 2003)

2.0)/()(

2.02222

xxxx

22yx

22yx

yx

Possibility of distinguishing between irrotational (superfluid) hydrodynamics and collisional (classical) hydrodynamics is unique feature of Fermi gas near unitarity

(From Rudy Grimm)

Crucial condition to observe differences in the scissors mode between

a rotating superfluid and a rotating collisional gas

2

If angular velocity is too small (or if trap deformation is too large) the dynamics of the scissors is not sensitive to the presence

of rotational components in the velocity field

Expansion of a rotating superfluid gas : consequences of irrotationality

Rotating superfluid cannot appraoch spherical shape during the expansion because the moment of inertia would vanish and angular momentum would not be conserved. The gas starts rotating fast when approaches ,but deformation remains finite (aspect ratio ). Angular velocity eventually decreases for larger times

1ctt

In the absence of rotation the expansion of a cigar trapped gas is faster in the radial direction (consequence of HD forces) After time such that shape of the system becomes spherical (aspect ratio = 1)For longer times the density profile takes a pancake form.

What happens if the gas is rotating?

At t=0 superfluid gas carries irrotational angular momentum

ZtR c ct

rigzL 2

Rigid rotor decreasesangular velocity during the expansion

Superfluid gas increasesangular velocity during the expansion (and never reaches aspect ratio=1)Theory:Edwards et al., 2002Exp in BEC gas: Hechenblaickner et al, 2002)

Effect of rotation

Expansion of rotating Fermi gasrecently measured at Duke (Clancy et al. 2007)

Sudden tilt of the trap produces rotation in the gas

non rotating

Aspect ratio Angle of rotationSd

Increase of angular velocity

Lines are predictions of irrotational HD equaions

- System in both cases is deeply hydrodynamic and expansion time is too fast to reveal effects of viscosity.

- Expansion in both cases is described by irrotational HD equations (situation similar to the scissors mode)

Duke experiment shows same behavior both below and above critical temperature

Surprise?

What happens if the gas, before expansion, is not superfluid and rotates in a rigid way?

(collaboration with P. Pedri)

Preliminary answers:

- If condition is not satisfied initial rigid motion is soon damped out and expansion is practically indistinguishible from superfluid expansion (Duke scenario)

-If condition is satisfield rigid motion is preserved during the expansion and visible effects emerge in time dependence of aspect ratio and rotational angle

2

2

COLLECTIVE MODES IN THE PRESENCE OFQUANTIZED VORTICES

Quantized vortex is configurationcharacterized by velocity field

where azimuth angle is corresponding phase of order parameter

M=2m in a Fermi superfluid (m is atomic mass)

Mrr

Mv ˆˆ

-Velocity field gives rise to

- singular vorticity - quantization of circulation Mhldv /

zrMhv ˆ)()/( )2(

QUANTIZED VORTICES

Differences between vortices in Fermi and Bose superfluids

- Density does not vanish along vortex line, differently from order parameter

- Size of vortex core is smaller (healing lenght is smallest near unitarity)

- Quantum of circulation and angular momentum per particle of single vortex line twice smaller than for bosons

Bulgac and Yu, 2003Chien et al., 2006Sensarma et al. 2006

Reduced visibility

Factor 2

order parameter

density

Vortex lattices (Mit 2001)(Jila 2002)

BEC gases

(Mit 2005)Fermi gas

When the number of vortices is large it is useful to introduce conceptof diffused vorticity. Average vorticity per unit area is given by

kknMhv V

ˆ2ˆ

MnV

Despite the discretized nature of vortex linesthe superfluid, at a macroscopic level,behaves like a fluid rotating with velocity

It carries rigid value of angular momentum

rv

Vortex density in vortex lattice

2rNMLz

Vortex density

density of vortices is uniform (even if atomic density is not uniform)

Centrifugal effect due to vortex lattice accounted for by effective trapping potential

Yielding increase in radial size

22222

2)(

2)( zmrmrV z

effho

is upper limit for rotating gas at equilibrium in the Presence of harmomic trap (critical angular velocity)

22

2

2

2

z

zRR

Due to stiffer equation of state (quantum pressure)harmonically trapped Fermi gas can host

larger number of vortices compared to BEC gas

For example at unitarity:

6.0

~

~~)1()3(4

22

3/12/13/12

z

VV NRnNChoosing

and

One finds 140VN

3.0/

from eq. of state consequences on Tkachenko oscillations(see later)

610N

Rotational hydrodynamics

Collective oscillations in non rotating superfluids are described by irrotational hydrodynamics

When number of vortices is large one can introduce concept of diffusedvorticity and develop formalism of rotational hydrodynamics

vmvVgnmvvt

m

nvnt

ext

)21(

0)(

2

Includes, in particular, Thomas-Fermi result for stationary density profile with bulge effect

laboratoryframe

Predictions of rotational hydrodynamics:collective oscillations

Collective oscillations are obtained by looking for linearized solutions of hydrodynamic equations:

Both surface and compression oscillations are affected by the rotation

Choosing one finds analytic solution (Chevy and Stringari, 2003)

22

2 2

Degeneracy between is broken by the rotation.

vrvnnn ,0

22 )(,)( iyxviyxan

2

Surface quadrupole modes

- Measured in BEC gase (ENS, Jila)- Useful to ‘measure’ angular velocity

Splitting of quadrupole modes in BEC gas2

Quadrupole frequency measured at JILA (Haljan et al., 2001)

)2(

)2(

Coupling between radial and axial motion is affected by the vortex lattice. Choosing

HD equations provide two m=0 decoupled modes

)(, 22220 zrrvzaraan zz

- When (BEC gas) independent of angular velocity - At unitarity frequency evolves from at

to as

2

Rotational hydrodynamics: compression modes in rotating Fermi superfluids (Antezza et al., 2007)

0

2

For radial mode in cigar trap configuration one finds

22 )1(2)1(2

1

3/10

z

radial modeat unitarity3/2

Applicability of rotational hydrodynamics

Large number of vortices (or distance between vortices much smaller than radial size)

At unitarity condition is satisfied also for small angular velocities

1VN

RMdV /

Thomas-Fermi regime size of vortices should be much smaller than intervortex distanceAt unitarity:

for rotating harmonic trap ( ) one finds:

When condition is violated one enters LLL and QH regimes

Vdn 3/1 3/1)3(~ NhoF

N11 2

2

222~

Elastic modes of the vortex lattice (Tkachenko modes)

- In addition to shape and compression oscillations vortex lattices exhibit new oscillations of elastic nature (Tkachenko modes) corresponding to deformation of vortex lattice geometry. - Tkachenko modes cannot be described by hydrodynamic theory. They require inclusion of elastic effects or full microscopic approach (Baym, Sonin, Anglin, Simula et al., Mizushima et al. etc.. ).

Theory well agrees with experiments on BEC gases in TF regime 2mc

Tkachenko waves at Jila (2003)

sum rule approachCozzini et al, 2004

deviations from TF

20 4/ MR

Tkachenko modes in unitary Fermi superfluid(G. Watanabe, M. Cozzini and S.S. in preparation)

- Original Tkachenko work (1966) applies to incompressible fluids

- Generalization to compressible fluid (Baym 1983, Sonin 1987)

- Two relevant regimes

- Values of q fixed by radius of the cloud ( for incompressible Bose fluid in cylindrical geometry, Anglin and Crescimanno, 2002)

- In trapped BEC gases Tkachenko modes measured in intermediate regimes

222

422

44 qcqc

MT

holds in TF regime 2mc

qM

qc

qcM

qc

T

T

4

24

222

2222

Rq /45.5

highly compressiblefluid

incompressible fluid

- In order to achieve incompressible regime one needs to work at “small” angular velocities.- Condition easily ensured in superfluid Fermi gases where can be large due to large quantum pressure effects. - For example choosing one finds (at unitarity)

610,3.0 N

105.0~124140 2

2

222

2

qc

andNV

- Exact determination of and hence of lowest Tkachenko frequency requires imposing proper boundary conditions and knowledge of equation of state:

24

MRT

- Bose (cigar ) - Bose (disc )- Fermi (cigar )- Fermi (disc )

3/2n3/2n

n

2/1n

45.559.559.575.5

unitary

Conclusions and outlookCollective oscillations in traps: - power tool to explore superfluid features in quantum gases. - accurate test of equation of state along BEC-BCS crossover - sensitive to superfluid effect in rotating gases

Some open problems:

Theory: - transition from superfluid to normal phase at T=0 on BCS side - kinetic phenomena at unitarity (role of viscosity) - Collective oscillations as a function of T (second sound) - collective oscillations in spin polarized superfluids - ……… Exps: - collective oscillations in rotating configurations, in spin polarized gases and in unequal masses Fermi superfluids; - dynamic structure factor in Fermi gases; - critical velocity at unitarity - oollective oscillations in 2D Bose gase below and above Tc - ……..