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From quantum trajectories to classical orbits

Brun, T A, Gisin, Nicolas, Patrick F. O’Mahony, Rigo, M

Published Aug 26, 1996• 19 citations
OpenAlex

Abstract

Recently it has been shown that the evolution of open quantum systems may be ``unraveled'' into individual ``trajectories,'' providing powerful numerical and conceptual tools. In this letter we use quantum trajectories to study mesoscopic systems and their classical limit. We show that in this limit, Quantum Jump (QJ) trajectories approach a diffusive limit very similar to the Quantum State Diffusion (QSD) unraveling. The latter follows classical trajectories in the classical limit. Hence, both unravelings show the rise of classical orbits. This is true for both regular and chaotic systems (which exhibit strange attractors).

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Quantum trajectories generalized

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Summary form only given. Quantum trajectories of state vectors can be used to describe open systems that otherwise can only be described using a density matrix. Besides the numerical advantage, quantum trajectories provide insight in the actual evolution of a system under continuous observation. Two distinct methods are usually considered, the quantum-jump method, associated with direct photodetection, and the quantum state diffusion (QSD) method, associated with homodyne or heterodyne mixing. We derive a general stochastic equation for quantum trajectories.

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Classical from Quantum

Timothy J. Hollowood

We consider the quantum-to-classical transition for macroscopic systems coupled to their environments. By applying Born's Rule, we are led to a particular set of quantum trajectories, or an unravelling, that describes the state of the system from the frame of reference of the subsystem. The unravelling involves a branch dependent Schmidt decomposition of the total state vector. The state in the subsystem frame, the conditioned state, is described by a Poisson process that involves a non-linear deterministic effective Schrodinger equation interspersed with quantum jumps into orthogonal states. We then consider a system whose classical analogue is a generic chaotic system. Although the state spreads out exponentially over phase space, the state in the frame of the subsystem localizes onto a narrow wave packet that follows the classical trajectory due to Ehrenfest's Theorem. Quantum jumps occur with a rate that is the order of the effective Lyapunov exponent of the classical chaotic system and imply that the wave packet undergoes random kicks described by the classical Langevin equation of Brownian motion. The implication of the analysis is that this theory can explain in detail how classical mechanics arises from quantum mechanics by using only unitary evolution and Born's Rule applied to a subsystem.

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We introduce jump-time unraveling as a distinct description of open quantum systems. As our starting point, we consider quantum jump trajectories, which emerge, physically, from continuous quantum measurements, or, formally, from the unraveling of Markovian quantum master equations. If the stochastically evolving quantum trajectories are ensemble-averaged at specific times, the resulting quantum states are solutions to the associated quantum master equation. We demonstrate that quantum trajectories can also be ensemble-averaged at specific jump counts. The resulting jump-time-averaged quantum states are then solutions to a discrete, deterministic evolution equation, with time replaced by the jump count. This jump-time evolution represents a trace-preserving quantum dynamical map if and only if the associated open system does not exhibit dark states. In the presence of dark states, on the other hand, jump-time-averaged states may decay into the dark states and the jump-time evolution may eventually terminate. Jump-time-averaged quantum states and the associated jump-time evolution are operationally accessible in continuous measurement schemes, when quantum jumps are detected and used to trigger the readout measurements. We illustrate the jump-time evolution with the examples of a two-level system undergoing relaxation or dephasing, a damped harmonic oscillator, and a free particle exposed to collisional decoherence.

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Classical mechanics from quantum state diffusion - a phase-space approach

Walter T. Strunz, Ian Colin Percival

Quantum state diffusion (QSD) provides a natural unravelling of a mixed-state open quantum system into component pure states. We investigate the semiclassical limit of QSD in a phase-space approach using the Wigner function. As , QSD exhibits two very different dynamical regimes, depending on the volume of phase space covered by the quantum state. For large volumes there is a localization regime represented by classical nonlinear and nonlocal diffusion processes. For small volumes, comparable in size with a Planck cell, there is a wavepacket regime . Here, the centroid of the wavepacket follows a classical Langevin equation, obtained through the adiabatic elimination of the dynamics of the second-order moments of the wavepacket. The corresponding Fokker-Planck equation is identical to the one obtained from the classical limit of the original mixed-state dynamics. In the companion paper we present an axiomatic approach to a classical theory of quantum localization without using the underlying QSD theory.

Journal of Physics A Mathematical and General•Feb 20, 1998•31 citations
Recently it has been shown that the evolution of open quantum systems may be unraveled'' into individual trajectories,'' providing powerful numerical and conceptual tools. In this letter we use quantum trajectories to study mesoscopic systems and their classical limit. We show that in this limit, Quantum Jump (QJ) trajectories approach a diffusive limit very similar to the Quantum State Diffusion (QSD) unraveling. The latter follows classical trajectories in the classical limit. Hence, both unravelings show the rise of classical orbits. This is true for both regular and chaotic systems (which exhibit strange attractors).
Summary form only given. Quantum trajectories of state vectors can be used to describe open systems that otherwise can only be described using a density matrix. Besides the numerical advantage, quantum trajectories provide insight in the actual evolution of a system under continuous observation. Two distinct methods are usually considered, the quantum-jump method, associated with direct photodetection, and the quantum state diffusion (QSD) method, associated with homodyne or heterodyne mixing. We derive a general stochastic equation for quantum trajectories.
We consider the quantum-to-classical transition for macroscopic systems coupled to their environments. By applying Born's Rule, we are led to a particular set of quantum trajectories, or an unravelling, that describes the state of the system from the frame of reference of the subsystem. The unravelling involves a branch dependent Schmidt decomposition of the total state vector. The state in the subsystem frame, the conditioned state, is described by a Poisson process that involves a non-linear deterministic effective Schrodinger equation interspersed with quantum jumps into orthogonal states. We then consider a system whose classical analogue is a generic chaotic system. Although the state spreads out exponentially over phase space, the state in the frame of the subsystem localizes onto a narrow wave packet that follows the classical trajectory due to Ehrenfest's Theorem. Quantum jumps occur with a rate that is the order of the effective Lyapunov exponent of the classical chaotic system and imply that the wave packet undergoes random kicks described by the classical Langevin equation of Brownian motion. The implication of the analysis is that this theory can explain in detail how classical mechanics arises from quantum mechanics by using only unitary evolution and Born's Rule applied to a subsystem.
We introduce jump-time unraveling as a distinct description of open quantum systems. As our starting point, we consider quantum jump trajectories, which emerge, physically, from continuous quantum measurements, or, formally, from the unraveling of Markovian quantum master equations. If the stochastically evolving quantum trajectories are ensemble-averaged at specific times, the resulting quantum states are solutions to the associated quantum master equation. We demonstrate that quantum trajectories can also be ensemble-averaged at specific jump counts. The resulting jump-time-averaged quantum states are then solutions to a discrete, deterministic evolution equation, with time replaced by the jump count. This jump-time evolution represents a trace-preserving quantum dynamical map if and only if the associated open system does not exhibit dark states. In the presence of dark states, on the other hand, jump-time-averaged states may decay into the dark states and the jump-time evolution may eventually terminate. Jump-time-averaged quantum states and the associated jump-time evolution are operationally accessible in continuous measurement schemes, when quantum jumps are detected and used to trigger the readout measurements. We illustrate the jump-time evolution with the examples of a two-level system undergoing relaxation or dephasing, a damped harmonic oscillator, and a free particle exposed to collisional decoherence.
Non-Markovian evolution of an open quantum system can be ``unraveled'' into pure state trajectories generated by a non-Markovian stochastic (diffusive) Schr\"odinger equation, as introduced by Di\'osi, Gisin, and Strunz. Recently we have shown that such equations can be derived using the modal (hidden variable) interpretation of quantum mechanics. In this paper we generalize this theory to treat jumplike unravelings. To illustrate the jumplike behavior we consider a simple system: a classically driven (at Rabi frequency $\ensuremath{\Omega}$) two-level atom coupled linearly to a three mode optical bath, with a central frequency equal to the frequency of the atom, ${\ensuremath{\omega}}_{0}$, and the two side bands have frequencies ${\ensuremath{\omega}}_{0}\ifmmode\pm\else\textpm\fi{}\ensuremath{\Omega}$. In the large $\ensuremath{\Omega}$ limit we observed that the jumplike behavior is similar to that observed in this system with a Markovian (broad band) bath. This is expected as in the Markovian limit the fluorescence spectrum for a strongly driven two level atom takes the form of a Mollow triplet. However, the length of time for which the Markovian-like behavior persists depends upon which jumplike unraveling is used.
We apply the QSD to the quantum version of Duffing oscillator and discuss the QCC and dissipative quantum chaos in this system. We calculate the constant phase map and the pseudo-Lyapunov exponent, varying Planck constant effectively and show there is a certain clear critical stage in which a crossover from classical to quantum behavior occurs. Furthermore, we discuss the effect of dissipation in the dissipative quantum chaos by means of a degrees of localization introduced in the QSD formalism[5]. Unfortunately, the pseudo-Lyapunov exponent is not a quantity characterized a dissipative system. Therefore, it will be necessary to discuss the connection between the classical trajectory (strange attractor) and the quantum observable which characterize a dissipative quantum system.
Quantum state diffusion (QSD) provides a natural unravelling of a mixed-state open quantum system into component pure states. We investigate the semiclassical limit of QSD in a phase-space approach using the Wigner function. As , QSD exhibits two very different dynamical regimes, depending on the volume of phase space covered by the quantum state. For large volumes there is a localization regime represented by classical nonlinear and nonlocal diffusion processes. For small volumes, comparable in size with a Planck cell, there is a wavepacket regime . Here, the centroid of the wavepacket follows a classical Langevin equation, obtained through the adiabatic elimination of the dynamics of the second-order moments of the wavepacket. The corresponding Fokker-Planck equation is identical to the one obtained from the classical limit of the original mixed-state dynamics. In the companion paper we present an axiomatic approach to a classical theory of quantum localization without using the underlying QSD theory.