Superfluidity of ultracold GSI-Darmstadt 25-27 September 2008 Superfluidity of ultracold atomic Fermi gases Sandro Stringari University of Trento CNR-INFM

Important manifestations of superfluidity - Absence of viscosity (Landau’s criterion) Hydrodynamic behavior at T=0 (irrotationality) Quenching of moment of inertia Occurrence of quantized vortices Josephson oscillations - Second sound …….. Furthermore, in Fermi gases Pairing gap (single particle excitations) and (at unitarity) - Phase separation in the presence of - Spin polarization Adiabatic rotation

of atomic Fermi gases interesting? Why is superfluidity of atomic Fermi gases interesting?

Important knobs available in atomic Fermi gases Temperature Scattering length Polarization Intensity of periodic potential Rotation

BEC-BCS Crossover in Fermi gases unitarity In the presence of Feshbach resonance different superfluid regimes can be explored by tuning the magnetic field BCS When scattering length is positive and small ( ) bound molecules of size a and binding energy are formed. At small T they condense giving rise to BEC superfluidity In the opposite regime of negative values of a the interaction gives rise to Cooper pairs and BCS superfluidity - At resonance the many body system exhibits strong correlations: unitary superfluidity (analog of high Tc superconductivity)

Unitary Fermi gas: main features diluteness (interparticle distance >> range of inetraction) strong interactions (scattering length >> interparticle distance) universality (no dependence on interaction parameters) high Tc (of the order of Fermi temperature) robust superfluidity (high critical velocity)

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 (see also Thomas’talk) More microscopic theories required to describe other superfluid phenomena (vortices, Landau critical velocity, pairing gap)

HYDRODYNAMIC THEORY OF SUPERFLUIDS AT T=0 Basic assumption: Irrotationality (follows from the phase of order parameter) Basic ingredient: Equation of state (sensitive to quantum correlations, statistics, dimensionality, ...)

What do we mean by macroscopic, low energy phenomena ? 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 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

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

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 Solutions of HD equations in harmonic trap predict both surface and compression modes (first investigated in dilute BEC gases, Stringari 96)

Minimum damping near unitarity HD theory predicts Ideal gas model Surface modes Quadrupole mode (Altmeyer et al. 2007) Ideal gas value HD prediction Enhancement of damping Minimum damping near unitarity

Experiments on surface oscillations (quadrupole, scissors) confirm HD behavior on BEC side of resonance (including unitarity !) on BCS side superfluidity is broken for relatively small values of (when gap is of the order of radial oscillator frequency) Deeper in BCS regime frequency takes collisionless value Damping is minimum near resonance At unitarity hydrodynamic behavior is observed also at high T (in normal phase). Consequence of collisions

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 Non monotonic behavior along the crossover (Stringari 2004)

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

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

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) - 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)

Dispersion law along BCS-BEC crossover gap gap unitarity gap BEC (R. Combescot, M. Kagan and S. Stringari 2006)

Results for Landau’s critical velocity theory experiment Sound velocity B resonance (Combescot et al, 2006) (Miller et al, 2007) Landau’s critical velocity is highest near unitarity !!

Measurement of Landau’s critical velocity (proof of superfluidity) Above critical value dissipative effects are observed (Miller et al, 2007)

Phase separation between superfluid and normal components Differently from BEC’s phase separation is not easily observed by imaging in situ profiles of Fermi gas (bimodal distribution is absent at unitarity as well as in BCS Phase separation can be nevertheless produced by playing with - Spin imbalance - Adiabatic rotation

Assumption: phase separation below Spin imbalance: what happens by increasing P ? 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 Clogston-Chandrasekhar limit : critical value of P above which the system becomes normal Assumption: phase separation below compare eq. of state of superfluid and normal phases determine equilibrium conditions (pressure and chem. potential)

(phase contrast imaging, MIT 2006) Occurrence of phase separation well supported experimentally at unitarity (see also Rice exp) Density difference (phase contrast imaging, MIT 2006) In superfluid phase In polarized normal phase

BEC (after ramping) and in situ density difference unitarity Critical polarization Pc=0.75 BEC (after ramping) and in situ density difference in strongly interacting Fermi gas at MIT

Phase separation at unitarity: unpolarized superfluid vs polarized normal gas at T=0 C. Lobo, A. Recati, S. Giorgini, S. Stringari, Phys. Rev. Lett. 97, 200403 (2006) Recati, C. Lobo and S. Stringari, PRA, cond-mat/0803.4419 Equilibrium between the two phases in uniform matter ensured by equal pressure : and equal chemical potential Look for equation of state of superfluid and normal phases

Equation of state of superfluid and normal phases at untarity Lobo et al. (PRL 2006) Unpolarized superfluid universal interaction parameter Polarized normal gas relative concentration of spin down and spin up Fermi motion of spin down particles binding energy of spin down particles Parameters and originate from interaction between spin up and spin down (Giorgini’s MC predictions, see also Chevy, Prokofeev, Recati, Combescot….). A=0 in BCS mean field eqs.

superfluid and (partially) polarized phase Equilibrium between superfluid and (partially) polarized phase best fit to MC fully polarized Fermi gas small x approximation -Ax energy of superfluid critical concentration - for x<0.44 gas is in normal phase - for x>0.44 equilibrium between normal and superfluid phases

By using LDA one derives equations Consequence of phase separation in harmonic trap By using LDA one derives equations for the profiles of superfluid, spin-up and spin-down components (surface effects ignored, see Rice, Stoof)

N S Shell structure in trapped spin polarized Fermi gas (Lobo et al. 2006) N S Predicted value for critical polarization (Chandrasekhar-Clogston limit) in good agreement with MIT result (0.75) (mean field BdeG eq. predicts 0.97)

Column density of polarized Fermi gas Exp: MIT Theory: Trento

(In situ) density jump at the interface Spin up density practically continuous at the interface Spin down density exhibits jump at the interface Exp: MIT Theory: Trento

Axial density difference above and below normal P=0.80 Exp. Shin et al. 2006 Theory ideal gas Lobo et al. 2006 normal plus superfluid P=0.58

Main conclusion: Some open questions: Interactions play a crucial role in both the superfluid and normal polarized phase - MC simulation + LDA provide excellent description of CC limit and density profiles (inadequacy of mf to describe CC limit at unitarity) Some open questions: Dynamic behavior of phase separated configuration Motion of single impurities (polaron vs molecule) Phase separation when

Phase separation induced by adiabatic rotation

Spin polarization is not the only way to produces phase separation at T=0 Phase separation induced by adiabatic rotation At small angular velocity superfluid is unable to rotate (consequence of irrotationality of moment of inertia) By increasing the system has two possibilities: Lines of singular vorticity are created if syste is allowed to jump into lowest energy configuration - Superfluid keeps irrotationality without vortices (metastability, should be increased adiabatically)

Proof of superfluidity Quantized vortices in Fermi gases observed along the BEC-BCS crossover (MIT, Nature June 2005, Zwierlein et al.) Scattering length is suddenly ramped to small and positive values in order to increase visibility of vortex lines

Adiabatic rotation of Fermi superfluid at unitarity I. Bausmerth, A.Recati and S.S., PRL, 77, 021602 (2008) Most relevant features - Fermi superfluidity is fragile near the border (density is small), superfluid energy gain being vanishingly small at unitarity System prefers to loose superfluidity gaining centrifugal energy Rotation produces phase separation between a non rotating superfluid core and an external rigidly rotating normal component. Phase separation is characterized by density discontinuity At higher angular velocity system exhibits spontaneous rotational symmetry breaking, due to instability of quadrupole surface modes (similar to BEC’s).

Equation of state in moving frame at T=0 Superfluid universal interaction parameter Normal gas (metastable in the absence of rotation) Parameters available from Monte Carlo simulation (Giorgini, Carlson). In the absence of rotation (v=0) superfluid is energetically favourable. However the rotation can make the normal component favourable

Hard sphere Fermi gas MC for normal phase BCS mean field Nnnn MC for superfluid phase

Effect of the rotation on the density profiles (LDA predictions) Density jump at the interface knee in column density (directly measured quantity)

Emergence of spontaneous symmetry breaking (Bausmerth et al. PRL 2008) BEC Rigidly rotating normal gas Unitary Fermi gas

Availability of normal phase of unpolarized T=0 Fermi gas at unitarity new opportunities for investigating Viscosity at small temperature Landau parameters and spin zero sound (different from liquid He3 similar to neutron matter )

Collaborators in Trento Lev Pitaevskii Stefano Giorgini Alessio Recati Ingrid Bausmerth Theory of ultracold atomic Fermi gases S.Giorgini, L.Pitaevskii and S. Stringari Rev.Mod.Phys, in press (2008) cond-mat/0706.3360