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Trento, June 4, 2009

Trento, June 4, 2009. Fermionic superfluidity in optical lattices . Gentaro Watanabe, Franco Dalfovo, Giuliano Orso, Francesco Piazza, Lev P. Pitaevskii, and Sandro Stringari. Summary.

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Trento, June 4, 2009

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  1. Trento, June 4, 2009 Fermionicsuperfluidity in optical lattices Gentaro Watanabe, Franco Dalfovo, Giuliano Orso, Francesco Piazza, Lev P. Pitaevskii, and Sandro Stringari

  2. Summary • Equation of state and effective mass of the unitary Fermi gas in a 1D periodic potential [Phys. Rev. A 78, 063619 (2008)] • Critical velocity of superfluid flow [work in progress]

  3. We were stimulated by: 6Li Feshbach res. @ B=834G N=106 weak lattice

  4. vc is maximum around unitarity Superfluidity is robust at unitarity

  5. What we have done first: Calculation of energy density chemical potential Key quantities for the characterization of the collective properties of the superfluid inverse compressibility effective mass sound velocity

  6. The theory that we have used: Bogoliubov – de Gennes equation Order parameter Refs: A. J. Leggett, in Modern Trends in the Theory of Condensed Matter, edited by A. Pekalski and R. Przystawa (Springer-Verlag, Berlin, 1980);M. Randeria, in Bose Einstein Condensation, edited by A. Griffin, D. Snoke, and S. Stringari (Cambridge University Press, Cambridge, England, 1995).

  7. We numerically solve the BdG equations. We use a Bloch wave decomposition. From the solutions we get the energy density: Caveat:a regularization procedure must be used to cure ultraviolet divergence (pseudo-potential, cut-off energy) as discussed by Randeria and Leggett. See also: G. Bruun, Y. Castin, R. Dum, and K. Burnett, Eur. Phys. J D 7, 433 (1999) A. Bulgac and Y. Yu, Phys. Rev. Lett. 88, 042504 (2002).

  8. Results: compressibility and effective mass

  9. Results: compressibility and effective mass when EF << ER : The lattice favors the formation of molecules (bosons). The interparticle distance becomes larger than the molecular size. In this limit, the BdG equations describe a BEC of molecules. The chemical potential becomes linear in density. s=5 s=0 Two-body results by Orso et al., PRL 95, 060402 (2005)

  10. Results: compressibility and effective mass when EF << ER : The lattice favors the formation of molecules (bosons). The interparticle distance becomes larger than the molecular size. In this limit, the BdG equations describe a BEC of molecules. The chemical potential becomes linear in density. The system is highly compressible. The effective mass approaches the solution of the two-body problem. The effects of the lattice are larger than for bosons! Two-body results by Orso et al., PRL 95, 060402 (2005)

  11. Results: compressibility and effective mass when EF>> ER : Both quantities approach their values for a uniform gas. Analytic expansions in the small parameter (sER/EF)

  12. Results: compressibility and effective mass when EF~ ER : Both quantities have a maximum, caused by the band structure of the quasiparticle spectrum.

  13. Sound velocity Significant reduction of sound velocity the by lattice !

  14. Density profile of a trapped gas From the results for μ(n) and using a local density approximation, we find the density profile of the gas in the harmonic trap + 1D lattice Thomas-Fermi for fermions ! Bose-like TF profile aspect ratio = 1 ħω/ER = 0.01 N=5105 s=5

  15. Summary of the first part • We have studied the behavior of a superfluid Fermi gas at unitarity in a 1D optical lattice by solving the BdG equations. • The tendency of the lattice to favor the formation of molecules results in a significant increase of both the effective mass and the compressibility at low density, with a consequent large reduction of the sound velocity. • For trapped gases, the lattice significantly changes: • the density profile • the frequency of the collective oscillations [See Phys. Rev. A 78, 063619 (2008)]

  16. Equation of state and effective mass of the unitary Fermi gas in a 1D periodic potential [Phys. Rev. A 78, 063619 (2008)] • Critical velocity of superfluid flow [work in progress]

  17. The concept of critical currentplays a fundamental role in the physics of superfluids. • Examples: • Landau critical velocityfor the breaking of superfluidity and the onset of dissipative effects. This is fixed by the nature of the excitation spectrum: • phonons, rotons, vortices in BEC superfluids • single particle gap in BCS superfluids • Critical current for dynamical instability • (vortex nucleation in rotating BECs, disruption • of superfluidity in optical lattices) • Critical current in Josephson junctions. • This is fixed by quantum tunneling

  18. In ultracold atomic gases: • Motion of macroscopic impurities has revealed the onset of heating effect (MIT 2000) • Quantum gases in rotating traps have revealed the occurrence of both energetic and dynamic instabilities (ENS 2001) • Double well potentials are well suited to explore Josephson oscillations (Heidelberg 2004) • Moving periodic potentials allow for the investigation of Landau critical velocity as well as for dynamic instability effects (Florence 2004, MIT 2007)

  19. Questions: • Can weobtain a unifyingview of criticalvelocityphenomenadriven by differentexternalpotentials and for different quantum statistics (Bose vs. Fermi)? • Can wetheoreticallyaccount for the observed values of the criticalvelocity?

  20. NOTE: The mean-field calculations by Spuntarelli et al. [PRL 99, 040401 (2007)] for fermions through a single barrier show a similar dependence of the critical velocity on the barrier height.

  21. Our goal: establishing an appropriate framework in which general results can be found in order to compare different situations (bosons vs. fermions and single barrier vs. optical lattice) and extract useful indications for available and/or feasible experiments. L Vmax d Vmax

  22. The simplest approach: Hydrodynamics in Local Density Approximation (LDA) Assumption: the system behaves locally as a uniform gas of density n, with energy densitye(n) and local chemical potential, μ(n). The density profile of the gas at rest in the presence of an external potential is given by the Thomas-Fermi relation If the gas is flowing with a constant current densityj=n(x)v(x), the Bernoulli equation for the stationary velocity field v(x) is

  23. The simplest approach: Hydrodynamics in Local Density Approximation (LDA) This equation gives the density profile, n(x), for any given current j, once the equation of state μ(n) of the uniform gas of density n is known.

  24. The simplest approach: Hydrodynamics in Local Density Approximation (LDA) The system becomes energetically unstable when the local velocity, v(x), at some point x becomes equal to the local sound velocity, cs[n(x)]. For a given current j, this condition is first reached at the point of minimum density, where v(x) is maximum and cs(x) is minimum. here the density has a minimum and the local velocity has a maximum !

  25. The simplest approach: Hydrodynamics in Local Density Approximation (LDA) The same happens in a periodic potential here the density has a minimum and the local velocity has a maximum !

  26. The simplest approach: Hydrodynamics in Local Density Approximation (LDA) To make calculations, one needs the equation of state μ(n) of the uniform gas! We use a polytropic equation of state:

  27. The simplest approach: Hydrodynamics in Local Density Approximation (LDA) To make calculations, one needs the equation of state μ(n) of the uniform gas! We use a polytropic equation of state: Unitary Fermions Bosons (BEC) α= (1+β)(3π2)2/3ħ2/2m α= g = 4πħ2as/m

  28. The simplest approach: Hydrodynamics in Local Density Approximation (LDA) To make calculations, one needs the equation of state μ(n) of the uniform gas! We use a polytropic equation of state: Unitary Fermions Bosons (BEC) α= (1+β)(3π2)2/3ħ2/2m α= g = 4πħ2as/m Local sound velocity:

  29. The simplest approach: Hydrodynamics in Local Density Approximation (LDA) Inserting the critical condition into the Bernoulli equation one gets an implicit relation for the critical current: Universal !! Bosons and Fermions in any 1D potential Note: for bosons through a single barrier see also Hakim, and Pavloff et al.

  30. LDA Fermions through a barrier Bosons through a barrier fermions bosons Fermions in a lattice Bosons in a lattice

  31. LDA The limit Vmax << μ corresponds to the usual Landau criterion for a uniform superfluid flow in the presence of a small external perturbation, i.e., a critical velocity equal to the sound velocity of the gas.

  32. LDA The limit Vmax << μ corresponds to the usual Landau criterion for a uniform superfluid flow in the presence of a small external perturbation, i.e., a critical velocity equal to the sound velocity of the gas. the critical velocity decreases because the density has a local depletion and the velocity has a corresponding local maximum

  33. LDA The limit Vmax << μ corresponds to the usual Landau criterion for a uniform superfluid flow in the presence of a small external perturbation, i.e., a critical velocity equal to the sound velocity of the gas. the critical velocity decreases because the density has a local depletion and the velocity has a corresponding local maximum When Vmax = μ the density vanishes and the critical velocity too.

  34. LDA Question: when is LDA reliable?

  35. LDA Question: when is LDA reliable? Answer: the external potential must vary on a spatial scale much larger than the healing length of the superfluid. L >> ξ

  36. LDA Question: when is LDA reliable? Answer: the external potential must vary on a spatial scale much larger than the healing length of the superfluid. L >> ξ For a single square barrier, L is just its width. For an optical lattice, L is of the order of the lattice spacing (we choose L=d/2). For bosons with density n0, the healing length is ξ=ħ/(2mgn0)1/2. For fermions at unitarity, one has ξ ≈ 1/kF, where kF = (3π2n0)1/3

  37. LDA Question: when is LDA reliable? Answer: the external potential must vary on a spatial scale much larger than the healing length of the superfluid. • Quantum effects beyond LDA become important when • - ξis of the same order or larger than L; they cause a smoothing of both density and velocity distributions, as well as the emergence of solitonic excitations (and vortices in 3D). • Vmax > µ ; in this case LDA predicts a vanishing current, while quantum tunneling effects yield Josephson current. • Quantitative estimates of the deviations from the predictions of LDA can be obtained by using quantummany-body theories, like Gross-Pitaevskii theory for dilute bosons and Bogoliubov-de Gennes equations for fermions.

  38. LDA Fermions through a barrier Bosons through a barrier Bosons in a lattice Fermions in a lattice

  39. LDA vs. GP/BdG Fermions through a barrier Bosons through a barrier L/ξ=1 LkF=4 [Spuntarelli et al.] 5 10 bosons fermions Bosons in a lattice Fermions in a lattice L/ξ=0.5 LkF=0.5 1 0.89 1.57 1.11 1.92 3 1.57 5 2.5 10

  40. Bosons (left) and Fermions (right) through single barrier bosons fermions L/ξ=1 LkF=4 5 10

  41. Bosons (left) and Fermions (right) through single barrier bosons fermions L/ξ=1 LkF=4 5 10 L >> ξ : Hydrodynamic flow in LDA

  42. Bosons (left) and Fermions (right) through single barrier L < ξ : Macroscopic flow with quantum effects beyond LDA bosons fermions L/ξ=1 LkF=4 5 10 L >> ξ : Hydrodynamic flow in LDA

  43. Bosons (left) and Fermions (right) through single barrier Vmax> μ : quantum tunneling between weakly coupled superfluids (Josephson regime) bosons fermions L/ξ=1 LkF=4 5 10 L >> ξ : Hydrodynamic flow in LDA

  44. Bosons (left) and Fermions (right) in a periodic potential The periodic potential gives results similar to the case of single barrier bosons fermions L/ξ=0.5 LkF=0.5 1 0.89 1.57 1.11 1.92 3 1.57 5 2.5 10

  45. Bosons (left) and Fermions (right) in a periodic potential L < ξ : Macroscopic flow with quantum effects beyond LDA bosons fermions L/ξ=0.5 LkF=0.5 1 0.89 1.57 1.11 1.92 3 1.57 5 2.5 10 Vmax>> μ : quantum tunneling between weakly coupled supefluids (Josephson current regime) L >> ξ : Hydrodynamic flow in LDA

  46. Differences between single barrier and periodic potential (bosons) Bosons through a barrier L/ξ=1 5 10 Bosons in a lattice L/ξ=0.5 1 1.57 3 5 10

  47. Single barrier (bosons) For a single barrier, μ and ξ are fixed by the asymptotic density n0 only. They are unaffected by the barrier. All quantities behave smoothly when plotted as a function of L/ ξ or Vmax/ μ. For L/ ξ >> 1: vc/cs 1 – const  (Vmax )1/2 For L/ ξ << 1: vc/cs 1 – const  (LVmax )2/3 L/ξ=1 5 10

  48. Periodic potential (bosons) In a periodic potential the barriers are separated by distance d. The energy density, e, and the chemical potential, µ, are not fixed by the average density n0only, but they depend also on Vmax. They exhibit a Bloch band structure. L/ξ=0.5 1 1.57 3 5 10

  49. Periodic potential (bosons) Bloch band structure. p = quasi-momentum pB= Bragg quasi-momentum ER= p2B/2m = recoil energy Vmax=sER = lattice strength Energy density vs. quasi-momentum Lowest Bloch band for the same gn0=0.4ERand different s

  50. Periodic potential (bosons) Bloch band structure. p = quasi-momentum pB= Bragg quasi-momentum ER= p2B/2m = recoil energy Vmax=sER = lattice strength Energy density vs. quasi-momentum

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