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Chaos Solitons & Fractals

On the full quantum description of nonlinear dynamic systems in the phase-space representation

Serguei Maximov, L. S. Kuz’menkov

Published Nov 16, 2006• 1 citation
Publisher
DOI
10.1016/j.chaos.2006.09.040

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We reformulate the full quantum dynamics of spin systems using a phase-space representation based on SU(2) coherent states which generates an exact mapping of the dynamics of any spin system onto a set of stochastic differential equations. This representation is superior in practice to an earlier phase-space approach based on Schwinger bosons, with the numerical effort scaling only linearly with system size. By also implementing extrapolation techniques from quasiclassical equations to the full quantum limit, we are able to extend useful simulation times severalfold. This approach is applicable in any dimension including cases where frustration is present in the spin system. The method is demonstrated by simulating quenches in the transverse-field Ising model in one and two dimensions.
A method of simulating the full quantum field dynamics of multimode, multi-component Bose-Einstein condensates is developed. The truncated Wigner representation is used to obtain a probabilistic theory. The representation is extended to to use a functional calculus and can be applied to quantum fields in the arbitrary number of dimensions. Detailed proofs of the corresponding theorems are given. This approach produces c-number stochastic equations which may be solved using conventional stochastic integration methods, with the limitation that it is only valid for large mode occupation numbers. The equations describe the spatial evolution of spinor components and properly account for nonlinear interactions and losses. The method is further used to model several experiments where quantum effects are significant, and cannot be calculated with simpler approximations. It is shown to describe leading quantum corrections accurately, including effects such as phase noise, quantum squeezing, entanglement, and interactions with engineered nonlinear reservoirs. In addition to this approach, other positive phase-space methods are also analysed for sampling highly nonclassical states. In conclusion, the phase-space representations developed in this thesis provide a relatively fast and accurate method of simulating dynamics of large nonlinear quantum systems, such as trapped Bose-Einstein condensates. The required stochastic equations are easily and scalably integrated in parallel. Algorithms for efficient computation on modern graphical processing unit hardware are presented.