Excitations in Bose-Einstein condensates

1. Trento, 2 May 2006 Excitations in Bose-Einstein condensates

2. Excitations in Bose-Einstein condensates……a long story • Collective excitations and hydrodynamic equations • Collective vs. single-particle • Excitations in low dimensions • Collapse, expansion and nonlinear dynamics • Solitons

3. Excitations in Bose-Einstein condensates……the first 2 years @ • Response of a condensate to a Bragg pulse • Evaporation of phonons in a free expansion • Landau damping of collective excitations

4. Excitations in Bose-Einstein condensates……the most recent results @ • Parametric resonances in optical lattices • Pattern formation in toroidal condensates • Stability of solitons in 2D

5. Excitations in Bose-Einstein condensates……the most recent results @ • Parametric resonances in optical lattices • Pattern formation in toroidal condensates • Stability of solitons in 2D Parametric excitation of a Bose-Einstein condensate in a 1D optical lattice M. Kraemer, C. Tozzo and F. Dalfovo, Phys. Rev. A 71, 061602(R) (2005) Stability diagram and growth rate of parametric resonances in Bose-Einstein condensates in 1D optical lattices C. Tozzo, M. Kraemer, and F. Dalfovo, Phys. Rev. A 72, 023613 (2005)

6. Starting point: experiments by Esslinger et al. T.Stoeferle, et al., PRL 92, 130403 (2004); M.Koehl et al., JLTP 138, 635 (2005); C.Schori et al., PRL 93, 240402 (2004).

7. Starting point: experiments by Esslinger et al. T.Stoeferle, et al., PRL 92, 130403 (2004); M.Koehl et al., JLTP 138, 635 (2005); C.Schori et al., PRL 93, 240402 (2004).

8. Gross-Pitaevskii simulations

9. Axial width after expansion Fraction of atoms with q close to resonance

10. Simulations vs. experiments expt GP

11. What kind of resonance? ωq= Ω /2

12. ωq= Ω /2 It’s a parametric resonance Classical example: the vertically driven pendulum. Stationary solutions:  φ = 0 and φ = 180°. In the undriven case, these solutions are always stable and unstable, respectively. But vertical driving can change stability into instability and vice versa. The dynamics is governed by the Mathieu equation:  (t) = c(t)exp(  t), where c(t+1/f) = c(t). Floquet exponent. If is is real and positive, then the oscillator is parametrically unstable.

13. Parametric resonances Very general phenomenon (classical oscillators, nonlinear optics, systems governed by a Non-Linear Schroedinger Equation, Hamiltonian chaotic systems, etc.) Previously mentioned in the context of BEC by Castin and Dum, Kagan and Maksimov, Kevrekidis et al., Garcia-Ripoll et al., Staliunas et al., Salasnich et al., Salmond et al., Haroutyunyan and Nienhuis, Rapti et al.). Very recent experiments: Parametric Amplification of Matter Waves in Periodically Translated Optical Lattices N. Gemelke, E. Sarajlic, Y. Bidel, S. Hong, and S. Chu Phys. Rev. Lett. 95, 170404 (2005) Parametric Amplification of Scattered Atom Pairs Gretchen K. Campbell, Jongchul Mun, Micah Boyd, Erik W. Streed, Wolfgang Ketterle, and David E. Pritchard Phys. Rev. Lett. 96, 020406 (2006) Important remark: in order to be parametrically amplified, the “resonant” mode must be present at t=0 (seed excitation). The parametric amplification is sensitive to the initial quantum and/or thermal fluctuations.

14. A deeper theoretical analysis in a simpler case: no axial trap, infinite condensate, Bloch symmetry GP equation: with Ground state + fluctuations: j = band index k = quasimomentum

15. Using Bloch theorem: Bogoliubov quasiparticle amplitudes: Bogoliubov equations: with and

16. Bogoliubov spectrum in a stationary lattice

17. Dynamics in a periodically modulated lattice is small. Assume the order parameter to be still of the form where at time t, is the solution of the stationary GP equation for s(t) is small. and Linearized GP gives This term is a source of excitations in the linear response regime. It is negligible in the range of Ω we are interested in.

18. Linearized GP gives Bloch wave expansion: Floquet analysis: Look for unstable regions in the (Ω,k)-plane. Calculate the growth rate The lattice modulation enters here (this equation is the analog of Mathieu equation of classical oscillators)

19. Stability diagram

20. Remarks on thermal and quantum seed In GP simulations the seed is numerical noise or some extra noise added by hand to simulate the actual noise. In the experimental BECs, the seed can be: Excitations due to non-adiabatic loading of BEC in the lattice Imprinted ad-hoc excitations Thermal fluctuations Quantum fluctuations

21. Remarks on thermal and quantum seed GP theory Excitations due to non-adiabatic loading of BEC in the lattice yes Imprinted ad-hoc excitations yes Thermal fluctuations no Quantum fluctuations no

22. Remarks on thermal and quantum seed Possible approach beyond GP: use the full Bogoliubov expansion with operators, not c-numbers. Use the Wigner representation of quantum fields. In this way, the dynamics is still governed by “classical” Bogoliubov-like equations; the depletion is included through a stochastic distribution of the coefficients cjk. Exact results can be obtained by averaging over many different realizations of the condensate in the same equilibrium conditions. One has Thermal fluctuations Quantum fluctuations

23. Remarks on thermal and quantum seed Two limiting cases: Thermal fluctuations. Possible measurement of T, even when the thermal cloud is not visible (selective amplification of thermally excited modes). Amplification of quantum fluctuations. Analogous to parametric down-conversion in quantum optics. Source of entangled counter-propagating quasiparticles. example: Dynamic Casimir effect: the environment in which quasiparticles live is periodically modulated in time and this modulation transforms virtual quasiparticles into real quasiparticles (as photons in oscillating cavities).

24. Excitations in Bose-Einstein condensates……the most recent results @ • Parametric resonances in optical lattices • Pattern formation in toroidal condensates • Stability of solitons in 2D Detecting phonons and persistent currents in toroidal Bose-Einstein condensates by means of pattern formation M. Modugno, C.Tozzo and F.Dalfovo, to be submitted (today!)

25. Bose-Einstein condensates have recently been obtained with ultracold gases in a ring-shaped magnetic waveguide (Stamper-Kurn et al.) Other groups are proposing different techniques to get toroidal condensates. Main purpose: create a system in which fundamental properties, like quantized circulation and persistent currents, matter-wave interference, propagation of sound waves and solitons in low dimensions, can be observed in a clean and controllable way. An important issue concerns also the feasibility of high-sensitivity rotation sensors. Our approach: Parametric resonances as a tool to measure the excitation spectrum and rotations. Advantage of toroidal geometry: Clean response; nonlinear mode-mixing suppressed; periodic pattern formation.

26. Procedure: • The condensate is initially prepared in the torus. • The transverse harmonic potential is periodically modulated in time. • Both the trap and the modulation are switched off and the condensate expands. We solve numerically the time dependent GP equation, using the Wigner representation for fluctuations at equilibrium at step (i).

27. GP simulations (with seed) no modulation modulation in trap

28. Pattern visibility (in trap)

29. GP simulations (with seed) no modulation modulation in trap

30. GP simulations (with seed) no modulation modulation in trap after expansion

31. Mean-field effects in the expansion without with

32. Sensitive rotation sensor

33. a periodic modulation of the confining potential of a toroidal condensate induces a spontaneous pattern formation through the parametric amplification of counter-rotating Bogoliubov excitations. • This can be viewed as a quantum version of Faraday's instability for classical fluids in annular resonators. • The occurrence of this pattern in both density and velocity distributions provides a tool for measuring fundamental properties of the condensate, such as the excitation spectrum, the amount of thermal and/or quantum fluctuations and the presence of quantized circulation and persistent currents.

34. Excitations in Bose-Einstein condensates……the most recent results @ • Parametric resonances in optical lattices • Pattern formation in toroidal condensates • Stability of solitons in 2D Work in progress Shunji Tsuchiya, L.Pitaveskii, F. Dalfovo, C.Tozzo

35. Starting point: Motion in a Bose condensate: Axisymmetric solitary waves Jones and Roberts, J. Phys. A 15, 2599 (1982) Numerical solutions of GP equation. A continuous family of solitary waves solutions is obtained. At small velocity: a pair of antiparallel vortices, mutually propelling in obedience to Kelvin’s theorem. At large velocity: rarefaction pulse of increasing size and decreasing amplitude.

36. U = 0.2

37. U = 0.5

38. U = 0.2 U = 0.5 ( Sound speed: 1/√2 ) U = 0.7

41. In 3D: Crow instability of antiparallel vortex pairs Berloff and Roberts, J. Phys. A 34, 10057 (2001) 1D soliton in 2D: Instability against transverse modulations (self-focusing) Kuznetsov and Turitsyn , Zh. Eksp. Teor. Fiz. 94, 119 (1988)

42. Stability or instability of Jones-Roberts soliton in 2D Our approach: calculate the real and imaginary (if any) eigenfrequencies of the linearized GP equation (Bogoliubov spectrum)

43. When U approaches the speed of sound: Kadomtsev-Petviashvili equation Linearized Kadomtsev-Petviashvili equation

44. Localized excited states Work in progress …

45. Thank you

46. (instantaneous) Bogoliubov quasiparticle basis Multi-mode coupling, induced by s(t) Coupling parameters: j-j’bands, same k Nk constant j-j’bands, opposite k Exponential growth of Nk

47. Assumption: coupling by pairs. Two-mode approximation Replace sum over j’ with a single j’ and keep leading terms (small A): with and growth rate Resonance condition: Growth rate on resonance: with Seed: