Tarun Kanti Ghosh Okayama University, Japan In collaboration with Prof. K. Machida

1. 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 K. Machida Ref.: Physical Review A 73, 013613 (2006) + unpublished results

2. 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

3. 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

4. Bose-Einstein vs Fermi-Dirac fermions bosons harmonic trap potential S. N. Bose A. Einstein E. Fermi P. A. M. Dirac 1926 1924

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

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

7. Why many alkali atoms are bosons? Alkali atoms All alkali atoms have only one electron in the outer “s” shell 87 Rubidium Rb electronic spin: S=1/2 nuclear spin: I=3/2 Total spin: I+S= 1 or 2 23 Sodium Na hence it behaves like a bosons 7 Lithium Li First experimental observation of BEC condensate is much dilute compared to air E. A. Cornell et al., Science 1995.

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

9. 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

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

11. 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

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

13. 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”

14. Feshbach Resonance Scattering length D: coupling between two channels T S • Many molecular bound state in S channel • Continuum energy in T channel falls within the bound state energy 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:

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

16. BEC-BCS Crossover Regal & Jin PRL 2000 molecular BEC Scattering length: unitarity regime BCS • The bound state in interacting Fermi gases are bosonic in nature, hence • can Bose condense, just like a Bose atoms can • 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

17. 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 Black box Molecular BEC Strongly interacting regime BCS state external magnetic field

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

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

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

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

22. 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

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

24. Chemical potential: Manini & Salasnich PRA 2005

25. 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: superfluid velocity Phase ()-density (n) representation: Continuity equation: Euler equation:

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

27. Linearizing around the equilibrium: Equilibrium density profile:

28. Wave equation for the density fluctuations Quantum numbers

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

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

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

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

33. Crossover:Sound propagationat T/TF < 0.1 A. Turlapov & John Thomas

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

35. Mean-field theory of Ghosh & Machida PRA 2006 Speed of sound, u1in the BEC-BCS crossover system becomes very hot during sound propagation Also supports

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

37. Dynamic structure factor Weight factors:

38. Density fluctuations

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

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

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

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

43. Future plans • Finite temperature: superfluid + normal components, • study the first and second sound velocity • 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) • 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

44. 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

45. Thank you very much for your kind attention