Dipolar Quantum Gases: Bosons and Fermions

1. Dipolar Quantum Gases: Bosons and Fermions Han Pu 浦晗 Rice University, Houston, TX, USA • Dipolar interaction in quantum gases • Dipolar BEC • Rotons and charge density wave • Dipolar Fermions $$: NSU, Singapore

2. Current cold atom research at Rice Univ. • Hulet group (exp.) • BEC (Anderson localization, soliton) • Fermi superfluid (Feshbach resonance, BEC-BCS crossover) • Strongly correlated fermions in optical lattice • Killian group (exp.) • Ultracold plasma • Neutral Sr (photoassociation, condensation) • Pu group (th.) • BEC (vortex and vortex lattice, spinor condensate, dipolar condensate, coupled atom-molecule condensate) • Fermi gas (dipolar, BEC-BCS crossover, Fermi superfluid) • Bose-Fermi mixture • Quantum simulation (optical lattice systems) • Quantum optics (interaction between light and cold atoms) http://rlup.rice.edu

3. Gross-Pitaevskii Equation Two-body collisions at ultracold temperature(current paradigm) • Dominated by the s-wave (isotropic) • Short ranged (~1/R6) • Effective interaction --- pseudopotential

4. Dipolar system • Atoms: magnetic dipole moment • Polar molecules: large electric dipole moment Besides the short-range interaction, atoms or molecules may interact with each other via long-rangeanisotropic dipole-dipole interaction.

5. Dipolar interaction between two atoms (Polarized dipole) • Long ranged (~1/R3) • Anisotropic

6. Computational method Two problems: • How to calculate efficiently the convolution integrals. • Dipolar interaction looks singular. One solution: Convolution theorem → Fourier Transform (FFT). Fourier transform of dipolar interaction is non-singular.

7. Chromium BEC realized Dipolar effects on expansion dynamics Griesmaier et al., Phys. Rev. Lett. 95, 150406 (2005) Stuhler et al., Phys. Rev. Lett. 95, 150406 (2005)

8. Vortex States in Dipolar condensate (and its connection with roton) Yi and Pu, PRA 73, 061602(R) (2006)

9. (Ketterle, MIT) Vortex studies in BEC • Formation and decay • Multiply charged vortices • Tkachenko and Kelvin mode excitation • Vortex state in combined harmonic and quartic trap • Vortex in spinor condensate • Vortex rings • Scissors mode excitation • Vortex lattice Bragg spectroscopy • Quantum melting • Vortex pinning

10. Quasi-2D dipolar condensate • Trapping potential • Wave function decomposition Trap aspect ratio: Axial wave function:

11. y x Dipoles polarized along z-axis Isotropic (azimuthal symmetry in the xy-plane Effective 2D dipolar interaction potential

12. Higher rotating freq. Vortex lattice: hexagonal Single vortex state: Vortex state for z-polarized dipoles What causes the density ripples?

13. Vortex line as roton emitter? calculated vortex core structure for He II, at different pressure Excitation spectrum of He II Dalfovo, PRB 46, 5482 (1992)

14. Single vortex state: Excitation spectrum (homogeneous system) Santos et al., PRL 90 250403 (2003) Vortex structure vs. roton excitation in dipolar BEC

15. Rotons can be ‘emitted’ by other impurities • Vortex line, alien atoms, container wall A qualitative plot of superfluid density of HeII confined between two rigid walls. --- Rasetti and Regge, 1978 Difficult to observe in superfluid helium as the roton wavelength is only a few angstrons.

16. Quasi-2D BEC in a box Like the vortex line, here the wall may become the roton emitter. Roton instability induces charge density wave: Square CDW → triangular CDW → collapse

17. Beyond quasi-2D Charge density wave damped inside the bulk.

18. Dipolar BEC in 3D harmonic trap Ronen, Bortolotti, Bohn  PRL 98 , 030406 (2007)

19. Conclusion for dipolar BEC • Dipolar interaction: long-range, anisotropic, • New ground state structure • Roton excitation and CDW (crystallization, supersolid?)

20. Dipolar fermions (spin polarized) Miyakawa, Sogo and Pu, arXiv:0710.5223

21. Fermions vs. Bosons (T=0) • Bosons • Condensed (all atoms occupy the same state, macroscopic wave function) • Symmetric many-body wave function: Hartree direct energy • Real space and momentum space distribution: Fourier transform • Fermions • Pauli principle • Anti-symmetric many-body wave function: Hartree direct and Fock exchange energy • Real space and momentum space distribution: no direct relation • Trapped non-interacting fermions: isotropic momentum distribution.

22. Model N spin polarized (along z-axis) fermions Interacting with each other via the dipolar interaction

23. Total energy Goal: minimize the total energy Strategy: treat the Wigner function variationally

24. Homogeneous case: Wigner function

25. Homogeneous case: energies Kinetic energy favors an isotropic Fermi surface (α =1); Fock energy tends to stretch the Fermi surface along z-axis (α =0); Competition b/w the two results in a prolate Fermi surface (0< α <1).

26. Inhomogeneous case: Wigner function

27. Inhomogeneous case: energies Interaction energy is not bounded from below (dipolar interaction is partially attractive). The system is not absolutely stable against collapse (λ→∞). A local minimum may exist: the system may sustain a metastable state.

28. Results (I): density profiles Real space: Momentum space:

29. Results (II): density profiles in real space Solid lines: interacting Dashed lines: non-interacting

30. Results (III): deformation Momentum space Real space

31. Results (IV): stability Sufficiently large dipolar strength will collapse the system.

32. Conclusion and outlook • Dipolar interaction deforms the quantum Fermi gas in both real and momentum space. • Dipolar effects can be observed in TOF image. • Ideal gas: isotropic expansion • Dipolar gas: anisotropic expansion • Importance of the Fock exchange term. • Near future: • Collective excitation • Dipolar induced superfluid pairing

33. Collaborators: • Hong Lu (Rice) • Su Yi (ITP, CAS) • Takahiko Miyakawa (Tokyo Univ. of Sci.) • Takaaki Sogo (Kyoto Univ.)