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International Journal of Modern Physics B

The Behaviour of Individual Ultra-Small Capacitance Quantum Devices

Timothy P. Spiller, J. R. Ralph, Terrence D. Clark, Robert J. Prance, Helen Prance

Published Mar 10, 1997• 1 citation
Publisher

Abstract

We apply quantum state diffusion theory to an open angular system, such as a quantum capacitor or rotor. We consider various physically interesting Hamiltonian and environment terms, such as those which give rise to tunnelling, and dissipation. Quantum state diffusion (QSD) models the irreversible behaviour of an individual open system, underpinned by the statistics corresponding to a Bloch type of master equation evolution. Our work therefore provides examples of individual quantum device behaviour for cases where the ensemble evolution is Markovian. Since the Markovian approximation is decent for many experiments and since quantum device experiments usually address just one device, QSD should prove to be a useful tool for future device research.

DOI
10.1142/s0217979297000447

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We apply quantum state diffusion theory to an open angular system, such as a quantum capacitor or rotor. We consider various physically interesting Hamiltonian and environment terms, such as those which give rise to tunnelling, and dissipation. Quantum state diffusion (QSD) models the irreversible behaviour of an individual open system, underpinned by the statistics corresponding to a Bloch type of master equation evolution. Our work therefore provides examples of individual quantum device behaviour for cases where the ensemble evolution is Markovian. Since the Markovian approximation is decent for many experiments and since quantum device experiments usually address just one device, QSD should prove to be a useful tool for future device research.
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An open quantum system with multiple levels coupled to a bosonic environment at zero temperature is investigated systematically using the non-Markovian quantum-state-diffusion (QSD) method [W. T. Strunz, L. Di\'osi, and N. Gisin, Phys. Rev. Lett. 82, 1801 (1999)]. We have established exact time-local QSD equations for a set of interesting multilevel open systems, including high-spin systems, multiple-transition atomic models, and multilevel atomic models driven by time-dependent external fields. These exact QSD equations have paved the way to evaluate the dynamics of open multilevel atomic systems in the general non-Markovian regimes without any approximation.
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A nonlinear stochastic Schr\"odinger equation for pure states describing non-Markovian diffusion of quantum trajectories and compatible with non-Markovian master equations is presented. This provides an unraveling of the evolution of any quantum system coupled to a finite or infinite number of harmonic oscillators without any approximation. Its power is illustrated by several examples, including measurementlike situations, dissipation, and quantum Brownian motion. Some examples treat this environment phenomenologically as an infinite reservoir with fluctuations of arbitrary correlation. In other examples the environment consists of a finite number of oscillators. In such a quasiperiodic case we see the reversible decay of a macroscopic quantum-superposition (``Schr\"odinger cat''). Finally, our description of open systems is compatible with different positions of the ``Heisenberg cut'' between system and environment.
We analyze the quantum evolution of a particle moving in a potential in interaction with an environment of harmonic oscillators in a thermal state, using the quantum state diffusion (QSD) picture of Gisin and Percival. The QSD picture exploits mathematical connection between the usual Markovian master equation for the evolution of the density operator and a class of stochastic nonlinear Schr\"odinger equations (Ito equations) for a pure state \ensuremath{\Vert}\ensuremath{\psi}〉, and appears to supply a good description of individual systems and processes. We find approximate stationary solutions to the Ito equation (exact, for the case of quadratic potentials). The solutions are Gaussians, localized around a point in phase space undergoing classical Brownian motion. We show, for quadratic potentials, that every initial state approaches these stationary solutions in the long time limit. We recover the density operator corresponding to these solutions, and thus show, for this particular model, that the QSD picture effectively supplies a prescription for approximately diagonalizing the density operator in a basis of phase space localized states. We show that the rate of localization is related to the decoherence time, and also to the time scale on which thermal and quantum fluctuations become comparable. We use these results to exemplify the general connection between the QSD picture and the decoherent histories approach to quantum mechanics, discussed previously by Di\'osi, Gisin, Halliwell, and Percival. \textcopyright{} 1995 The American Physical Society.
Using a newly introduced connection between the local and non-local description of open quantum system dynamics, we investigate the relationship between these two characterisations in the case of quantum semi-Markov processes. This class of quantum evolutions, which is a direct generalisation of the corresponding classical concept, guarantees mathematically well-defined master equations, while accounting for a wide range of phenomena, possibly in the non-Markovian regime. In particular, we analyse the emergence of a dephasing term when moving from one type of master equation to the other, by means of several examples. We also investigate the corresponding Redfield-like approximated dynamics, which are obtained after a coarse graining in time. Relying on general properties of the associated classical random process, we conclude that such an approximation always leads to a Markovian evolution for the considered class of dynamics.