Quantum dynamics of atomic Bose-Einstein condensates using the positive-P method N.G. Parker, D. H. J. O'Dell and E. Sorensen Department of Physics and Astronomy, McMaster University, Canada ABSTRACT The coherence properties and controllability of ultra-cold atomic gases make them well suited to testing quantum mechanics in novel regimes where macroscopic many-body quantum effects become dominant. Experiments with ultra-cold atoms have already probed such regimes, for example, in the collisions of BoseEinstein condensates and quasi-condensation in reduced dimensional systems. As such, it is increasingly important to have sophisticated theoretical approaches to treat the full quantum dynamics. Recently, Drummond et al. [1] demonstrated the maturity of the positive-P method as an exact approach to describe the quantum dynamics of the density matrix. We discuss the positive-P approach to quantum dynamics and outline one of many possible applications, that of Josephson oscillations between Bose-Einstein condensates (BECs) in double well potentials [2-5].

MANY-BODY QUANTUM SIMULATIONS • For N atoms distributed among M modes, Hilbert space size ~ MN → intractable for large N • Approaches include exact solution, path integral and Monte-Carlo, quantum computing, truncated-Wigner methods and positive-P • Of these, the positive-P method offers an exact, tractable and versatile approach to quantum dynamics

POSITIVE-P METHOD Methodology • Coherent state basis for fields α and β

Features • Reduces to simple dynamical equations ☺ • Scales as M → tractable for large N! ☺ • Stochastically-averaged over multiple phase space trajectories S.  • Exact: gives true expectation values for S→ ∞ ☺ • Versatile (bosons/fermions, open/closed systems, quantum optics) ☺ • Limited simulation time due to growth of noise 

• Recast density matrix as integration positive off-diagonal probability state projection over modes M

• Probabilistic sampling of off-diagonal state projectors. • Separable over each subsystem (lattice point)

POSITIVE-P REPRESENTATION FOR A BEC • Local interaction Hamilton

Density matrix

with commutation relation

Positive-P off-diagonal projection

Master equation Stochastic equations with quantum noise ξ

• Relatively simple stochastic coupled equations • Key success for colliding BECs: • Quantum dynamics of 150,000 atoms with 1×106 modes (one of the largest Hilbert spaces ever simulated) • Scattering halos and velocity correlations

FUTURE APPLICATION: QUANTUM DYNAMICS IN A DOUBLE WELL Overview Single, phase coherent BEC Global phase φ

V

Coupled twin BECs Local phases φ1 and φ 2

|ψ|2

Applicability??? • Limited simulation time due to growth of sampling noise [6] 2.5h (∆x) D / 3 ≈ 1 − 5ms •τ sim ≈ for D=3 and typical parameters g n02 / 3 2 • Josephson period τ J ≈ 2π / NEC E J + E J ≈ 1 − 50ms [2,4] (EC =on-site interaction energy; EJ =Josephson overlap energy) • Extend τsim: • Inclusion of damping or reduced dimensionality (τsim >1s) • Hybrid positive-P/truncated Wigner approach [7]

• Josephson junction with BECs [2-4] is ideal for study of fluctuations [5] • Josephson oscillations in relative phase ∆φ and relative number ∆N • Symmetric (number and phase) splitting → no Josephson oscillations • Fluctuations induce φ1≠ φ2 → Josephson oscillations • Thermal fluctuations probed experimentally [3] • Quantum fluctuations! • occur at zero temperature • sensitive to dimensionality • ultimate limit for matter wave interferometry

REFERENCES [1] P. Deuar and P. D. Drummond, PRL 98, 120402 (2007). [2] M. Albiez et al., PRL 95, 010402 (2005). [3] R. Gati et al,. PRL 96, 130404 (2006). [4] S. Levy et al., Nature 449, 579 (2007). [5] A. J. Leggett, RMP 73, 307 (2001). [6] P. Deuar and P. D. Drummond, J. Phys. A 39, 1163 (2006). [7] S. E. Hoffman et al., arXiv:0803.1887 (2008).