Sound velocity and multibranch Bogoliubov - Anderson modes of a Fermi superfluid along the BEC-BCS crossover Tarun Kanti Ghosh Okayama University, Japan In collaboration with Prof. K. Machida Ref.: Physical Review A 73, (2006) + unpublished results

Outline of this talk: part-I Difference between bosons & fermions What is Bose-Einstein condensation (BEC) & Bardeen-Cooper-Schriffer state (BCS) ? Two-component Fermi gases Brief introduction of scattering theory & Feshbach resonance BEC-BCS crossover

Outline of this talk: part-II Hydrodynamic equations of motion in the crossover regime Sound velocity along the crossover Comparison with ongoing experimental result at Duke univ. Dynamic structure factor calculation and discussion of Bragg spectroscopy to analyze multibranch Bogoliubov-Anderson spectrum

Bose-Einstein vs Fermi-Dirac A. EinsteinS. N. BoseE. Fermi P. A. M. Dirac bosons fermions harmonic trap potential

high temperature: classical gas intermediate temperature Experimental signature of Fermi pressure Li 7 6 Truscott et al. Science 2001 very low temperature: effect of Fermi pressure due to Pauli principle -: bosons +: fermions Boltzmann distribution

Many particle system can be described by a SINGLE PARTICLE MACROSCOPIC WAVE FUNCTION Inter particle distance d density n wave-particle duality

Why many alkali atoms are bosons? E. A. Cornell et al., Science condensate is much dilute compared to air Rubidium Rb 87 All alkali atoms have only one electron in the outer “s” shell electronic spin: S=1/2 nuclear spin: I=3/2 Total spin: I+S= 1 or 2 hence it behaves like a bosons Alkali atoms Sodium Na 23 Lithium Li 7 First experimental observation of BEC

Binding energy Critical temperature Bardeen-Cooper-Schriffer (BCS) state Phonon mediated exchange interaction induces attractive interaction between two electrons bare electron-electron interaction is repulsive

Trapped atomic Fermi gases Lithium : Potassium Duke Univ. -- J. E. Thomas MIT Cambridge -- W. Ketterle ENS Paris -- C. Salomon Rice Univ. -- R. Hulet Innsbruck Univ. – R. Grimm JILA Bouldar -- D. Jin ETH Zurich -- T. Esslinger LENS Florence -- M. Inguscio

Basic scattering theory (without spin degrees of freedom) distance r V(r) Lennard Jones potential 0 van der Waals potential Model potential Basic length scale: ~ 1-10 nm

Pethick & Smith Scattering length we have to exploit the presence of hyperfine state to make large scattering length and hence a bound state of two atoms

Spin dependent atom-atom interaction Spin dependent atom-atom interaction: total spin of two valence electrons is either 1 (triplet state ) or 0 (singlet state) Spin Hamiltonian: Zeeman energy Hyperfine interaction

Why two-component Fermi gas? At low temperature, s-wave scattering contribution is large, but it does not arise between identical fermions, it can occur between atoms with different values of Consider two hyperfine state of with equal number N, say |1/2,1/2> & |1/2,-1/2> “spin up”“spin down”

Feshbach Resonance Scattering length  : coupling between two channels T S Many molecular bound state in S channel Energy difference between T and S channels can be tuned by magnetic field When total energy of two colliding atoms in T is close to the bound state energy in S, the effective scattering length becomes very large and two colliding atoms in T channel forms a bound state in S channel when a > o, binding energy of a pair of atoms: Continuum energy in T channel falls within the bound state energy in S channel

|1/2,-1/2> These molecules are weakly bound but very stable |1/2,1/2> |1/2,-1/2> |1/2,1/2> |1/2,-1/2> |3/2,1/2> Space-Time diagram for Feshbach resonance Long life time: 1 msec – 20 sec !!! Bound state

BEC-BCS Crossover Scattering length: Regal & Jin PRL 2000 unitarity regime molecular BEC BCS From two-component Fermi system, one can go from molecular BEC to BCS state through the strongly interacting regime (unitarity regime) by changing external magnetic field The bound state in interacting Fermi gases are bosonic in nature, hence can Bose condense, just like a Bose atoms can

So far what we have learned? Take two different hyperfine states of fermions with equal number Apply magnetic field and tune the scattering length accordingly Interaction between two atoms can be either attractive or repulsive, depending on the external magnetic field For large repulsive interaction, tightly bound bosonic pairs will form and condense at very low temperature For attractive interaction, two different kind of fermions will form a loosely bound ATOMIC COOPER pair When magnitude of the scattering length is very large, the system behaves like a free Fermi gas, since the scattering length drops out from the problem Molecular BEC BCS state Strongly interacting regime external magnetic field Black box

Weak-coupling BCS regime Chemical potential Pairing energy Size of the Cooper pair in coordinate space is larger than inter atom distance Loosely bound pairs Note that we do not need any phonon mediated attractive interaction, we have already attractive interaction between two alkali atoms

Unitarity limit Relevant length scale: high temperature superfluidity behaves like a free Fermi gas How to measure b? Fermi pressure stabilizes the cloud against collapse, similar to neutron star Tabletop-Astrophysics a new kind of superfluid state

Molecular BEC regime Molecular scattering length: Chemical potential: Molecular density: Petrov et al. PRL 2004 Tightly bound pairs

Are 2-component fermions really superfluid? Ketterle et al., Nature 2005(MIT) Hallmark of superfluidity, be it bosonic or fermionic, is the presence of quantized vortices

Theoretical approaches Eagles (1969) – Leggett (1980): BCS state at T=0, Cooper pairs molecules Nozieres, Schmitt-Rink (1985) – Randeria et al.: finite T, Simplest crossover theory Qualitatively correct Quantitatively wrong: in BEC regime Unitarity limit

Equation of State MC:Giorgini et al. PRL 2005 Fit:Manini and Salasnich PRA 2005 Ground state energy per particle along the crossover

Chemical potential: Manini & Salasnich PRA 2005

Hydrodynamic Equations of Motion Schrodinger equation of a Fermi superfluid along the BEC-BCS crossover order parameter of the composite bosons Long cigar shaped trap: Phase  density (n) representation: Continuity equation: Euler equation: superfluid velocity

Power-law form of the chemical potential: y << -1 y ~ 0 y >> 1

Linearizing around the equilibrium: Equilibrium density profile:

Quantum numbers Wave equation for the density fluctuations

Energy spectrum: Density fluctuation: : Jacobi polynomial of order n Dipole ( n=0, m =1): Independent of interaction strength It satisfy Kohn’s theorem

Matrix elements: Each discrete radial modes are propagating along the symmetry axis 2) Multibranch Bogoliubov-Anderson modes: similar to electromagnetic wave propagation in a waveguide 1) Sound velocity

Sound velocity Sound velocity in non-uniform system: Uniform system: uniform non-uniform Smooth crossover on resonance!

BCSUnitarityBEC uniform non-uniform Sound velocity in atomic (Bose/Fermi) system: mm/sec ~ cm/sec Sound velocity in Helium 4 ~ 220 m/sec Comparison of sound velocity (in units of Fermi velocity) Atomic systems are really dilute!!

Crossover: Sound propagation at T/T F < 0.1 A. Turlapov & John Thomas

Sound: Excitation by a pulse of repulsive potential Trapped atoms Slice of green light (pulsed) Sound excitation: Observation: hold, release & image t hold =0

Speed of sound, u 1 in the BEC-BCS crossover Mean-field theory of Ghosh & Machida PRA 2006 system becomes very hot during sound propagation Also supports

Multibranch Bogoliubov-Anderson spectrum BA modes are absent in usual electronic superconductors due to long-range interaction

Dynamic structure factor Weight factors:

Density fluctuations

Weight factors Weight factors determine how many modes are excited for a given value of k

Dynamic structure factors (DSF) Location of the peak determines the excitation energy

Bragg spectroscopy z axis superfluid Bragg potential: Time duration of the Bragg pulses:

Bragg spectroscopy of a weakly interacting BEC Davidson et al. PRL 2003 Wizemann Institute of Science, Israel

Future plans Apply optical lattices into the fermionic superfluid and study the dynamical instability phenomena in this new kind of superfluid Atom Lasers & Atom Chips Quantum Hall effect in Graphene Unequal populations of two kind of hyperfine states. Bose-Fermi mixture. i) Phase separation between superfluid and normal component ii) Phase transition from superfluid to normal component when the difference between two components are increased. (Pauli limited phase transition) Finite temperature: superfluid + normal components, study the first and second sound velocity

Conclusions Brief overview of current experiments on ultra- cold atomic gases Mechanism of Feshbach resonance BEC-BCS crossover Compared predicted sound velocity with the ongoing experimental results Complete excitation spectrum of an elongated Fermi superfluid along the crossover Results of dynamic structure factors and Bragg spectroscopy to measure MBA modes