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Journal of Contemporary Physics (Armenian Academy of Sciences)

Ergodicity in computer simulation of quantum dissipative processes

Hayk H. Adamyan, T. V. Gevorgyan, Gagik Yu. Kryuchkyan

Published Jan 1, 2007• 1 citation
Publisher

Abstract

The ergodicity principle in quantum theory is employed for elaboration of a new quantum trajectory technique which is used for numerical simulation of quantum dissipative systems. With this purpose the density matrix of a quantum system is represented as a sum over an ensemble of quantum states in time intervals. The method is employed for computations of a quantum anharmonic oscillator.

DOI
10.3103/s106833720701001x

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The ergodicity principle in quantum theory is employed for elaboration of a new quantum trajectory technique which is used for numerical simulation of quantum dissipative systems. With this purpose the density matrix of a quantum system is represented as a sum over an ensemble of quantum states in time intervals. The method is employed for computations of a quantum anharmonic oscillator.
The principle of ergodicity of the quantum theory has been used for elaboration of a new technique for numerical simulation of the Wigner function of open dissipative quantum systems. With this purpose the density matrix of a quantum system is represented via averaging over the ensemble of quantum states in time intervals instead of averaging over the ensemble of stochastic variables. It is shown that this approach leads to new approximate expressions for quantum distributions in the phase space, in particular, Wigner functions for systems localized in the region of classical phase trajectories. As an application, the Wigner functions are calculated for the process of intracavity second harmonic generation in the region of Hopf bifurcations.
Using the methods of quantum trajectories we investigate the effects of dissipative decoherence in a quantum computer algorithm simulating dynamics in various regimes of quantum chaos including dynamical localization, the quantum ergodic regime, and quasi-integrable motion. As an example we use the quantum sawtooth algorithm which can be implemented in a polynomial number of quantum gates. It is shown that the fidelity of quantum computation decays exponentially with time and that the decay rate is proportional to the number of qubits, number of quantum gates, and per gate dissipation rate induced by external decoherence. In the limit of strong dissipation the quantum algorithm generates a quantum attractor which may have a complex or simple structure. We also compare the effects of dissipative decoherence with the effects of static imperfections.
Stochastic simulations of quantum trajectories have been proposed as an alternative to density matrix calculations for some open quantum systems. In this paper it is shown that Monte Carlo wavefunction simulations of open atomic systems produce quantum trajectories in a discrete Markov process. This process can be simulated with a discrete Markov chain, and this is demonstrated for several example systems. Statistical properties for these atomic systems are produced more rapidly and with greater accuracy using this method than by calculating an ensemble of individual trajectories.
Evolution with energy dissipation can be used to obtain the ground state of a quantum-mechanical system. This dissipation is introduced in the quantum trajectory framework by adding an empirical friction force to the equations of motion for the trajectories, which, as an ensemble, represent a wave function. The quantum effects in dynamics are incorporated via the quantum force derived from the properties of this ensemble. For scalability to large systems, the quantum force is computed approximately yet with sufficient accuracy to describe the strongly anharmonic ground state of solid (4)He represented by a simulation cell of 180 atoms.
Generic open quantum systems are notoriously difficult to simulate unless one looks at specific regimes. In contrast, classical dissipative systems can often be effectively described by stochastic processes, which are generally less computationally expensive. Here, we use the paradigmatic case of a dissipative quantum oscillator to give a pedagogic introduction into the modelling of open quantum systems using quasiclassical methods, i.e. classical stochastic methods that use a ‘quantum’ noise spectrum to capture the influence of the environment on the system. Such quasiclassical methods have the potential to offer insights into the impact of the quantum nature of the environment on the dynamics of the system of interest whilst still being computationally tractable.