Quantum state diffusion provides a dynamics for the localization or reduction of a quantum-mechanical wavepacket during a measurement or similar physical process. The essence of this localization dynamics is captured in the classical theory presented here, which applies to the common situation where the localization takes place over macroscopic distances and the wave properties of the system are no longer relevant. It provides a picture of the localization process in classical phase space, and a practical aid for computations on open quantum systems. The theory is developed from classical Hamiltonian dynamics in phase space and the known properties of localization. As an example it is used to illustrate how absorption by a screen leads to quantum jumps. A derivation from quantum state diffusion using the Wigner function is given in the companion paper.
Quantum state diffusion provides a dynamics for the localization or reduction of a quantum-mechanical wavepacket during a measurement or similar physical process. The essence of this localization dynamics is captured in the classical theory presented here, which applies to the common situation where the localization takes place over macroscopic distances and the wave properties of the system are no longer relevant. It provides a picture of the localization process in classical phase space, and a practical aid for computations on open quantum systems. The theory is developed from classical Hamiltonian dynamics in phase space and the known properties of localization. As an example it is used to illustrate how absorption by a screen leads to quantum jumps. A derivation from quantum state diffusion using the Wigner function is given in the companion paper.
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.
In a wide-open quantum system, the effect of the Hamiltonian is negligible by comparison with the effect of the environment. For open systems, this is the opposite limit to closed or isolated systems. The quantum state diffusion model provides equations for the localization or reduction of quantum states of wide-open systems. The ensemble localization of an operator is defined, and it is proved that the rate of selflocalization of a selfadjoint environment operator towards one of its eigenstates is no less than 2. A bound is also obtained for the rate of selflocalization of some non-selfadjoint operators, which localize to minimum indeterminacy wave packets. The theory is presented for quasiclassical systems. For a sufficient number of independent environment operators, the states localize asymptotically to wave packets with Heisenberg indeterminacy products close to the minimum, which look to classical eyes like phase space points. To zeroth order in h(cross), the time-dependent WKB theory of quantum state diffusion due to a single operator shows localization or reduction within and between fixed classical sheets or Lagrangian manifolds. To first order, the sheets themselves diffuse. The rate of localization in an ensemble is determined by commutation terms with either sign and by correlation terms which always increase the localization. For the quasiclassical case the latter dominate, and this leads to a purely classical theory of localization, with a picture based on the diffusion of phase space densities. This means that state diffusion dynamics, like Hamiltonian dynamics, has a purely classical form, in which Planck's constant plays no role.
Except for the universe, all quantum systems are open, and according to quantum state diffusion theory, many systems localize to wavepackets in the neighbourhood of phase space points. This is due to decoherence from the interaction with the environment, and makes the quasiclassical limit of such systems both more realistic and simpler in many respects than the more familiar quasiclassical limit for closed systems. A linearized version of this theory leads to the correct classical dynamics in the macroscopic limit, even for nonlinear and chaotic systems. We apply the theory to the forced, damped Duffing oscillator, comparing the numerical results of the full and linearized equations, and argue that this can be used to make explicit calculations in the decoherent histories formalism of quantum mechanics.
For all but the simplest open quantum systems, quantum trajectory Monte Carlo methods, including quantum jump and quantum state diffusion (QSD) methods, have besides their intuitive insight into the measurement process a numerical advantage over direct solutions for the density matrix, especially where many degrees of freedom are involved. For QSD the trajectories are continuous, and often localize to small near-minimum uncertainty wave packets which follow approximately classical paths in phase space. The mixed representation discussed here takes advantage of this localization to reduce computing space and time by a further significant factor, using a quantum oscillator representation that follows a classical path. The classical part of this representation describes the time evolution of the expectation values of position and momentum in classical phase space, while the quantal part determines the degree of localization of the quantum mechanical state around this phase space point. The method can be applied whether or not the localization is produced by a measuring apparatus.
Numerical simulation of individual open quantum systems has proven advantages over density operator computations. Quantum state diffusion with a moving basis (MQSD) provides a practical numerical simulation method which takes full advantage of the localization of quantum states into wavepackets occupying small regions of classical phase space. Following and extending the original proposal of Percival, Alber and Steimle (1995), we show that MQSD can provide a further gain over ordinary QSD and other quantum trajectory methods of many orders of magnitude in computational space and time. Because of these gains, it is even possible to calculate an open quantum system trajectory when the corresponding isolated system is intractable. MQSD is particularly advantageous where classical or semiclassical dynamics provides an adequate qualitative picture but is numerically inaccurate because of significant quantum effects. The principles are illustrated by computations for the quantum Duffing oscillator and for second-harmonic generation in quantum optics. Potential applications in atomic and molecular dynamics, quantum circuits and quantum computation are suggested.
The decay of coherence when a quantum system interacts with a much larger environment is usually described by a master equation for the system reduced density matrix and emphasizes the evolution of an entire ensemble. We consider two methods that have been developed recently to simulate the evolution of single realizations. Quantum-state diffusion involves both diffusion, where the individual quantum trajectory fluctuates through a Wiener process deriving from the environment, and localization to a coherent state, an eigenstate of the relevant Lindblad operator describing the coupling of the system to the environment. We demonstrate the localization process for different initial states and utilize the Wigner function to depict this localization in phase space. We concentrate on quantum states that can be expressed as a superposition of appropriate coherent states. For an initial superposition of two coherent states (a Schr\"odinger ``cat''), one of the two components will dominate the evolution. For initial Fock states, which can be described as a continuous superposition of coherent states on a ring, localization takes place when one coherent state is selected from that ring where each component has nearly the same energy as the original Fock state. We also consider the localization from a nonclassical squeezed ground state, which can be expressed as a superposition of coherent states along a line in phase space. The second simulation method considered is the state vector Monte Carlo, or ``quantum jump,'' approach, which relates to the direct counting of decay quanta. In the case of an initial Schr\"odinger ``cat,'' we find that when no quantum is detected the ``cat'' shrinks, but when a quantum is detected, the Schr\"odinger ``cat'' ``jumps'' from one type of ``cat'' to another with different internal phase. For an initial squeezed state we show how quantum jumps lead to individual realizations which are superpositions of two squeezed states.