DYNAMICS OF TRAPPED BOSE AND FERMI GASES

1. IHP Paris, June 2007 DYNAMICS OF TRAPPED BOSE AND FERMI GASES Sandro Stringari Lecture 1 University of Trento CNR-INFM

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

3. Understanding superfluid features requires theory for transport phenomena (crucial interplay between dynamics and superfluidity) Macroscopicdynamic 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)

4. 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

5. HYDRODYNAMIC EQUATIONS AT ZERO TEMPERATURE Hydrodynamic equations of superfluids (T=0) Closed equations for density and superfluid velocity field irrotationality is atomic mass, is superfluid density is superfluid velocity, is chemical potential (equation of state) is external potential T=0 • Equation for velocity field involves gradient of chemical potential; • Includes smoothly varying external field • in local density approximation

6. KEY FEATURES OF HD EQUATIONS OF SUPERFLUIDS • 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 What do we mean by macroscopic, low energy phenomena ? size of Cooper pairs BEC superfluids BCS Fermi superfluids healing length more restrictive than in BEC superfluid gap

7. 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 both theoretical 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

8. Initially the gas is confined in anisotropic trap • in situ density profile is anisotropic too • What happens after release of the trap ? Expansion from anysotropic trap 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

9. 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) with the scaling factors satisfying the equation and - 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.

10. Expansion of BEC gases ideal gas ideal gas HD HD Experiments probe HD nature of the expansion with high accuracy. Aspect ratio

11. EXPANSION OF ULTRACOLD FERMI GAS Hydrodynamics predicts anisotropic expansion inFermi superfluids (Menotti et al,2002) HD theory First experimental evidence for hydrodynamic anisotropic expansion inultra cold Fermi gas (O’Hara et al, 2002) collisionless

12. 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

13. - 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 BCSside 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

14. Collective oscillations in trapped gases 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

15. Recent example of high precision measurement First measurement of thermal effect on the Casimir force (JILA, Obrecht et al. PRL 2007) Lifschitz result (thermal equilibrium) Asymptotic behavior of the force out of thermal equilibrium (Antezza et al. PRL 95 083202, 2005)

16. 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 Relative frequency shift

17. Propagation of sound in elongated traps In uniform medium HD theory gives trivial sound wave solution with • 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 For BEC gas ( ) (Zaremba, 1998) (Capuzzi et al, 2006) For unitary Fermi gas ( )

18. Sound wave packets propagating in a BEC (Mit 97) velocity of sound as a function of central density

19. 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

20. 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 where is non uniform equilibrium Thomas Fermi profile well understood in atomic BEC Stringari,1996 here we focus on Fermi gases Solutions of HD equations predict surface and compression modes Surface modes • 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 (interactioneffect)

21. Quadrupole mode measured on ultracold Fermi gas along the crossover (Altmeyer et al. 2007) Ideal gas value HD prediction Enhancement of damping Minimum damping near unitarity

22. - 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

23. 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

24. Equation of state along BCS-BEC crossover • - Fixed Node Diffusion MC (Astrakharchick et al., 2004) • Comparison with mean field BCS theory ( - - - - - )

25. 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 LHY w In harmonic trap LHY effect results in enhancement of radial compression frequency according to: Pitaevskii and Stringari (1998) Braaten and Pearson (1999)

26. Frequency of radial compression mode in cigar trap (Stringari 2004) beyond mean field unitarity BEC 0 • 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

27. Radial breathing mode at Innsbruck (Altmeyer et al., 2007) MC equation of state(Astrakharchick et al., 2005) includes beyond mf effects does not includes beyond mf effects BCS eq. of state (Hu et al., 2004) universal value at unitarity Measurement of collective frequencies provides accurate test of equation of state !!

28. Main conclusions concerning the m=0 radial compression mode • 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)

29. 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)

30. Landau’s critical velocity Dispersion law of elementary excitations • - 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: BEC (Bogoliubov dispersion) BCS (role of the gap) (sound velocity)

31. Dispersion law along BCS-BEC crossover of Fermi gas BCS BEC gap gap gap unitarity BEC Critical velocity (Combescot, Kagan and Stringari 2006) Sound velocity B resonance Landau’s critical velocity is highest near unitarity !! Healing length is smallest !! (Marini et al 1997) BCS BEC

32. 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

33. Fermi superfluids with unequal masses ( ) (G.Orso, L.Pitaevskii, S.S, in preparation) Hydrodynamic eqs. keep standard form: • - Relevant frequencies for • HD oscillations • If oscillator lengths are equal • (natural condition • to cool down into superfluid phase) • one has with

34. What happens if one suddenly switches off ? • - Sudden change in trapping conditions: • - In the absence of interactions atoms down will fly away • - Superfluiditykeeps 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 (unitarity) easily satisfied if

35. Example: mixture (aspect ratio 0.2) Oscillation of radial size if one removes K-trapping Equilibrium of superfluid in new trapping conditions Oscillation of radial size if one removes Li-trapping Will superfluid survive in conditions of strong energetic instability?

36. 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

37. normal Axial densities in spin polarized Fermi gas P=0.80 Exp. Shin et al. 2006 Theory ideal gas Lobo et al. 2006 normal plus superfluid P=0.58 Critical value for normal-superfluid phase separation P = 0.75 Clogston-Chandrasekhar limit at unitarity (and in trap)

38. 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 up resulting in renormalization of oscillator frequency

39. dipole If spin down atoms oscillate with frequencies quadrupole Spin dipole mode difficult to excite because shift of harmonic trap results in 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) Quadrupole modes 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)

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