Spontaneous topological defects in the formation of a Bose-Einstein condensate

Transcription

1 Spontaneous topological defects in the formation of a Bose-Einstein condensate Matthew Davis 1, Ashton Bradley 1,, Geoff Lee 1, Brian Anderson 2 1 ARC Centre of Excellence for Quantum-Atom Optics, University of Queensland, Brisbane, Australia. 2 College of Optical Sciences, University of Arizona, Tuscon, USA. Now at Jack-Dodd Centre for Quantum Technology, University of Otago, Dunedin, New Zealand. Funding: Australian Research Council, National Science Foundation, University of Queensland.. p.1

2 UQ theory 2 talks: Andrew Sykes: Force on a slow moving impurity due to quantum fluctuations in a 1D BEC (Wed 1250). Joel Corney: Quantum dynamics of ultracold Fermions (Thu 830). Ashton Bradley : Scale invariant thermodynamics of a toroidally Bose gas (Fri 1720). Simon Haine: Generating number squeezing in a BEC through nonlinear interaction (Fri 1740). UQ theory 2 posters: Andy Ferris: Detection of continuous variable entanglement without a coherent local oscillator Geoff Lee: Coherence properties of a continuous-wave atom laser at finite temperature Sarah Midgley: A comparative study of simulation methods for the dissociation of molecular BECs Tania Haigh: Macroscopic superpositions in small double well condensates Jacopo Sabbatini: Topological defect formation in 87 Rb Bose Ferromagnet with quantum noise Michael Garrett: Bose-Einstein Condensation in a Dimple Trap Chao Feng: Mean-field study of superfluid critical velocity in a trapped Bose-Einstein condensate UQ experiment: Erik van Ooijen: Macroscopic superpositions in small double well condensates Leif Humbert: Towards an all-optical BEC in optical toroidal traps Sebastian Schnelle: Ultra-cold atoms in a time averaged optical potential. p.2

3 Overview 1. Bose-Einstein condensation phase transition in a trap. 2. Finite temperature Bose gases. 3. Simulating condensate formation. 4. Observations of spontaneous vortices in BEC. 5. What causes spontaneous vortices in BEC? 6. Condensate formation in flat / elongated systems.. p.3

4 What is a Bose-Einstein condensate (BEC)? It is a mesoscopic quantum many-body system: Typically atoms in single translational quantum state. Matter equivalent of a single mode laser. Confined by lasers / magnetic fields in a vacuum. 100,000 times less dense than atmosphere. Form at temperatures around 100 nk. Size 100 µm. To a good approximation: All atoms in a BEC share the same wave function. How does it arise from a boring old incoherent thermal gas?. p.4

5 The BEC phase transition. p.5

6 The BEC phase transition N 0 / N T / T c. p.6

7 Non-equilibrium finite temperature BEC Challenge: is it possible to develop a practical non-equilibrium formalism for finite temperature Bose gases? Desirable features: Can deal with inhomogeneous potentials. Can treat interactions non-perturbatively. Calculations can be performed on a reasonable time scale (say under one week). Application to condensate formation. Possibilities: Positive-P method, ZGN formalism, quantum kinetic theory, 2PI.... p.7

8 Energy Stochastic Gross-Pitaevskii method Procedure: Split field operator into low- and high-energy pieces. ˆΨ(x) = ˆψ(x) + ˆφ(x). Treat high-energy region using quantum kinetic theory. Treat low-energy region in trun- E S { Beyond S-Wave cated Wigner approximation highly occupied, classical field modes. R NC E R R C { Non-Condensate Band Condensate Band Essentially GPE approximation: ψ(x) = ˆψ(x). Position. p.8

9 Stochastic Gross-Pitaevskii equation (SGPE) dψ(x) = i L GPψ(x)dt + P { } G(x) k B T (µ L GP)ψ(x)dt + dw G (x,t). First term standard GPE (but with energy cutoff). Second term growth: coupling to thermal cloud described by chemical potential µ and temperature T. G(x) γk B T : collision rate (from quantum Boltzmann integral). dw G (x,t) γk B T : driving noise associated with growth. Could solve a quantum Boltzmann equation for thermal cloud but instead we model bath dynamics with time dependent µ, T. P : Project dynamics onto the low-energy basis (e.g. SHO states.). p.9

10 Thermal equilibrium Growth term causes SGPE to evolve any initial condition to thermal equilibrium for given µ,t. Example: above T c in an oblate harmonic trap. Density slice (z=0) of classical region 6 Column density of classical region y 0 y x x Anderson lab parameters: (ν,ν z ) = (7.8, 17.3) Hz, µ 0, T 60 nk.. p.10

11 Simulating BEC formation 1. Begin with µ i,t i above critical point for BEC. 2. Sample initial state using ergodic evolution of SGPE. 3. Model evaporative cooling: Ramp µ i µ f (up) and T i T f (down). 4. Watch the condensate band rethermalise to new equilibrium. 5. Repeat a number of times (200). 6. Analysis: e.g. average over many trajectories to determine condensate number N 0 via Penrose-Onsager criterion.. p.11

12 Results Vortex probability N 0 (10 5 atoms) Expt measurements SGPE Expt result range Expt observation times SGPE Time (s). p.12

13 Experimental and numerical column densities a b c. p.13

14 . p.14

15 Effect of ramp of µ and T Previous results for an instant change in thermal cloud parameters: µ : 0 25 ω T : 54 nk 38 nk What happens for a finite time ramp? Dotted lines indicate the end of the ramp. Vortex prob. fairly insensitive for ramps up to t = 4T. a N 0 (10 5 atoms) Vortex probability b Time (s). p.15

16 What is special about these simulations/experiments? Nothing in particular! The evaporative cooling procedure is a standard rf ramp. However: Final cooling is in a very weakly trapped pancake Bose gas many classical modes shorter correlation length at T c. Extra-long time-of-flight imaging (59 ms) with magnetic levitation. Imaging axis is parallel to symmetry axis of the TOP trap. David Hall also reports seeing spontaneous vortices in BECs. Jean Dalibard said that occasionally they saw holes in condensates in a TOP trap. What happens in a spherical trap?. p.16

17 What causes vortices to appear? Initial noise? (Fluctuations in the initial condition.) Dynamical noise? (Atoms entering/leaving the low-energy region.) SGPE: dψ(x) = i L GPψ(x)dt + P {γ(µ L GP )ψ(x)dt + dw G (x,t)}. Noise correlations: dw G (x,t) dw G (x,t ) = 2γk B Tδ(x x )δ(t t ). Numerical experiment: 1. Initial state noise only set final temperature T f = Dynamical noise only start with 132 atoms in SHO ground state.. p.17

18 Results Vortex probability N 0 (10 5 atoms) Expt measurements Full SGPE SGPE init. noise SGPE growth noise Expt result range Expt observation times Full SGPE SGPE init. noise SGPE growth noise Time (s). p.18

19 Testing Kibble-Zurek mechanism Kibble-Zurek mechanism predicts the scaling of the vortex density with the speed of the phase transition. Experimentally: we need to make the BEC form faster. How? Squash the pancake increases collision rate. Early results suggest a factor of 20 increase is possible. Example movie.. p.19

20 What about cigar-shaped systems? Regions merge to form dark solitons. These are stable in a quasi-1d system. Images from Peter Engels (Washington State Univ.) (ν,ν z ) = (400, 7) Hz.. p.20

21 Summary [C. N. Weiler et al., Nature, 455, 938 (2008).] Spontaneous vortices have been observed in condensate formation, and have been modelled with a stochastic GPE. We find vortex statistics in broad agreement with experiment. Simulations suggest defects occur due to initial thermal fluctuations. Outlook More vortices in flat pancake systems. Solitons occur in elongated cigar systems. Test of the Kibble-Zurek mechanism in a Bose gas.. p.21

22 Matt s arxiv selection Weekly that filters arxiv.org for you! Includes all abstracts with relevance to the physics of ultra-cold atoms. Typically abstracts / week (from about 300). Currently about 450 subscribers. See to sign up.. p.22