ROTATING SUPERFLUID GASES IN HARMONIC TRAPS

1. Jerusalem, April 2007 ROTATING SUPERFLUID GASES IN HARMONIC TRAPS Sandro Stringari Università di Trento CNR-INFM

2. Main differences with respect to liquid helium: • Diluteness (range of force much smaller • than interparticle distance) • Harmonic trapping • Possibility of revealing Bose-Einstein condensation • and superfluidity in coordinate space

3. Rotating quantum gases Rotations provide crucial test of superfluidity: Superfluids rotate 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 Superfluids are characterized by irrotationality constraint, consequence of the phase of order parameter, yielding irrotational value (M=m in Bose superfluids M=2m in Fermi superfluids) Vorticity is hence vanishing ( ) except along lines of singularity where the density vanishes Difference in velocity field shows up in observable quantities

4. Moment of inertia of harmonically trapped gas When the harmonic trap confinining the gas is put in rotation with angular velocity The system acquires angular momentum Moment of inertia is defined by relationship If the system rotates classically ( ) one finds rigid value of moment of inertia • Superfluid gases in traps exhibits different behaviour: • at low drastic effect in scissors mode and expansion (this talk) • at higher two possibilities: • - formation of vortices (see talks by Fetter and Zwierlein) • - rotational symmetry breaking and spontaneous deformation (this talk)

5. Moment of inertia of a trapped gas • Questions: • can we evaluate in a trapped gas ? • - can we measure and probe superfluidity ? • Someanswers: • calculation of using hydrodynamic • theory of superfluids at T=0 • experimental information from • - scissors mode, expansion of rotating gas • - spontaneous deformation at high

6. T=0 Irrotational hydrodynamics and moment of inertia Role of interactions in a superfluid at T=0 can be investigated using equations of irrotational hydrodynamics It is convenient to write HD equations in rotating frame where trap is at rest and one can look for steady solutions. Equations are derived using Hamiltonian where is superfluid velocity (irrotational) in lab frame M=2m in Fermi superfluids

7. For harmonic trapping the hydrodynamic equations admit stationary curl free solutions of the form where is deformation of the trapped cloud Angular momentum is given by Irrotationality of moment of inertia follows from irrotationality of superfluid motion Results holdfor any value of . If (trap deformation)

8. Scissors mode Direct measurement of moment of inertia 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: angular momentum quadrupole operator Response to transverse probe measurable thorugh densityresponse function !! Example of coupling is provided by scissor mode. If confining (deformed) trap is suddenly rotated by angle the gas is no longer in equilibrium. Behaviour of resulting oscillation depends crucially on value of moment of inertia (irrotational vs rigid)

9. Qualitative estimate of scissors frequency (role of moment of inertia) deformation of harmonic trap Restoring force is proportional to (no energy cost for symmetric trap) • Mass parameter is given by • irrotational value in superfluid phase ( ) • - rigid value in non superfluid phase • As scissors frequency • approaches finite valuein superlfuid • vanishes in non superfluid phase

10. 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. Result is independent of equation of state (surface mode) 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)

11. Scissors measured in BEC’s at Oxford (Marago’et al, PRL 84, 2056 (2000)) Above (normal, collisionless) 2 modes: Below (superfluid) : single mode:

12. Expansion of a rotating superfluid gas : consequence of irrotationality In the absence of rotation the expansion of a cigar condensate is faster in the radial direction (Lecture 2). After time such that the 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 the gas carries irrotational angular momentum A 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 ).

13. Skater decreases angular velocity during expansion Superfluid gas increases angular velocity during the expansion. It cannot reach symmetric configuration (aspect ratio =1) because of angular momentum conservation. Theory: Edwards et al., 2002 Exp: Hechenblaickner et al. 2002

14. Many results holding for BEC gases can be generalized to ultracold superfluid Fermi gases. • Superfluidity in dilute Fermi gases is the result of interaction • between fermions occupying different spin states. • At low temperature and for dilute samples interaction is characetrized • by s-wave scattering length. • - Condition of diluteness is ( is range of the force) • (does not necessarily implies ). At resonance • Value of scattering length • (even sign) • can be modulated by • tuning the magnetic field • (Feshbach resonance)

15. Many-body aspects of a superfluid Fermi gas in the presence of a Feshbach resonance S-wave scattering length BEC regime of dimers BCS regime (Cooper pairs) unitary limit (universal behavior of equation of state)

16. oscillation-frequency Scissors in Fermi gas, hydrodynamic regime(Innsbruck 2006, unpublished) below the resonance 1/kfa = 0.65 on resonance 1/kfa = 0

17. oscillation-frequencies Scissors in Fermi gas, collisionless(Innsbruck 2006, unpublished) Colissionless Superfluidity is very fragile In BCS regime. Gap parameter is exponentially small above the resonance 1/kfa < -0.6

18. Scissors mode reveals the effects of superfluidity at small What happens at higher angular velocities? By increasing vortex lines become energetically favourable. System has then two possibilities: A) Lines of singular vorticity are created if system is allowed to jump into lowest energy configuration B) System keeps irrotationality without vortices (metastability, angular velocity should be increased adiabatically) Scenario B) can be explored by finding stationary solutions of irrotational HD equations as a function of angular velocity equations in rotating frame

19. One finds stationary solutions of HD equations with irrotational velocity where is deformation of the trapped cloud Deformation of trapped cloud obeys cubic equation (Recati et al. 2001) is trap deformation

20. Isotropic trapping( ) • - If one finds 3 solutions for • solutions with have lowest energy. • (for m=2quadrupole oscillation becomes • energeticallyunstable ( ). Spontaneous breaking of rotational symmetry (similar to bifurcation phenomena in rotating classical fluids) is zero but can be large

21. Role of trap deformation For one identifies two different branches: Main branch starting from . This branch can be followed adiabatically by increasing slowly the angular velocity up to critical angular velocity where the system exhibits dynamic instability (Sinha, Castin, 2001). Second branch extends up to angular velocities larger than trapping oscillator frequency (overcritical branch). overcritical main branch x dynamic instability (vortex formation)

22. Spontaneous deformation in rotating BEC’s Rotating configurations has been realized experimentally in BEC’s by ramping up adiabatically the angular velcocity (black circles). For larger angular velocities the system nucleates vortices due to dynamic instability of collective frequencies Overcritical branch can be also followed experimentally (white circles). Madison et al., 2001

23. Spontaneous deformation due to rotation exhibits peculiar features in spin polarized Fermi superfluids (I. Bausmerth, A. Recati and S. S.) Recent experiments at Mit and Rice Univ. have pointed out new superfluid features in spin polarized Fermi superfluids If P=0 system is superfluid If P=1 system is ideal Fermi gas (no interaction in s-channel) For intermediate P different scenarios are possible: - uniform spin polarized superfluid - periodic modulations (FFLO) - phase separation between superfluid and normal gas

24. Experiments at unitarity support phase separation scenario In situ density difference (phase contrast imaging) (MIT 2006) In superfluid phase In polarized normal phase

25. Quantized vortices in spin polarized Fermi gases near unitarity (MIT 2006) P(%) = 100 74 58 48 32 16 7 0

26. Basic idea (in the absence of vortical lines !) • superfluid undergoes spontaneous deformation • in the plane of rotation • normal spin polarized component feels centrifugal force • and exhibits bulge effect (increase of radial size)

27. unitarity polarized normal gas superfluid • Superfluid is sorrounded by • normal component Equilibrium between superfluid and spin polarzed normal gas obtained by imposing equal pressure and chemical potential in the two phases (first order transition)

28. unitarity is deformation in x-y plane • Superfluid is sorrounded by • normal component as at zero • angular velocity. • Superfluid exhibit rotational • symmetry breaking since • Normal part feels the centrifugal • force but remains isotropic • in the plane of rotation

29. unitarity is deformation in x-y plane X • - Superfluid does not rotate and remains practically isotropic • Normal part feels the centrifugal force • and forms a rotating ring

30. Some conclusions: - Rotating trapped gases are well suited to study the consequences of irrotationality in the superfluid regime: scissors mode, expansion, spontaneous deformation (and of course vortices, see Zwierlein, next talk) - Rich scenarios provided by spin polarized Fermi gases (superfluid vs normal component)