Beyond their use as numerical tools, quantum trajectories can be ascribed a degree of reality in terms of quantum measurement theory. In fact, they arise naturally from considering continuous observation of a damped quantum system. A particularly useful form of quantum trajectories is as linear (but non-unitary) stochastic Schrodinger equations. In the limit where a strong local oscillator is used in the detection, and where the system is not driven, these quantum trajectories can be solved. This gives an alternative derivation of the probability distributions for completed homodyne and heterodyne detection schemes. It also allows the previously intractable problem of real-time adaptive measurements to be treated. The results for an analytically soluble example of adaptive phase measurements are presented, and future developments discussed.

https://iopscience.iop.org/article/10.1088/1355-5111/8/1/015/pdf

Quantum trajectories and quantum measurement theory

Several high-precision physics experiments are approaching a level of sensitivity at which the intrinsic quantum nature of the experimental apparatus is the dominant source of fluctuations limiting the sensitivity of the measurements. This quantum limit is embodied by the Heisenberg uncertainty principle, which prohibits arbitrarily precise simultaneous measurements of two conjugate observables of a system but allows one-time measurements of a single observable with any precision. The dynamical evolution of a system immediately following a measurement limits the class of observables that may be measured repeatedly with arbitrary precision, with the influence of the measurement apparatus on the system being confined strictly to the conjugate observables. Observables having this feature, and the corresponding measurements performed on them have been named quantum nondemolition or back-action evasion observables. In a previous review (Caves et al., 1980, Rev. Mod. Phys. 52, 341) a quantum-mechanical analysis of quantum nondemolition measurements of a harmonic oscillator was presented. The present review summarizes the experimental progress on quantum nondemolition measurements and the classical models developed to describe and guide the development of practical implementations of quantum nondemolition measurements. The relationship between the classical and quantum theoretical models is also reviewed. The concept of quantum nondemolition and back-action evasion measurements originated in the context of measurements on a macroscopic mechanical harmonic oscillator, though these techniques may be useful in other experimental contexts as well, as is discussed in the last part of this review. [S0034-6861(96)00103-1]

On the measurement of a weak classical force coupled to a harmonic oscillator: experimental progress

A stochastic mean-field (SMF) approach to nonadiabatic molecular simulations is introduced. Based on the quantum-classical mean-field approximation, SMF extents the classical model of the environment to incorporate its quantum properties. SMF differs from the ordinary mean-field method by the presence of additional terms in the Schrodinger equation that are due to the system-environment interaction. SMF resolves the two major drawbacks of mixed quantum-classical models. First, decoherence effects in the quantum subsystem are rigorously included. Present in all open systems, decoherence is crucial for nonadiabatic transitions taking place in condensed media. Second, the correct branching of the quantum-classical trajectories is achieved. In earlier approaches, the correct branching of the trajectories was attained via ad hoc surface hopping procedures, which experienced the hop rejection problem and could produce unfavorable classical trajectories in regions of nonadiabatic transitions depending on the qua...

Mean field approximation for the stochastic Schrödinger equation

In earlier papers the authors introduced a quantum state diffusion model for the evolution of an individual open quantum system and proved localization theorems based on this model. This paper shows in more detail how the diffusion leads to localization in position and phase space, and to symmetry breaking for chiral molecules. The theory of radioactive decay of absorbers and detectors is described in the state diffusion picture. The Mott and Gurney theory of latent image formation in photography is presented in its state diffusion version. It is an example of quantum detection without significant amplification.

https://iopscience.iop.org/article/10.1088/0305-4470/26/9/019/pdf

The quantum state diffusion picture of physical processes

The quantum state diffusion model introduced in an earlier paper represents the evolution of an individual open quantum system by an Ito diffusion equation for its quantum state. The diffusion and drift terms in this equation are derived from interaction with the environment. In this paper two localization theorems are proved. The dispersion entropy theorem shows that under special conditions, which are commonly satisfied to a good approximation, the mean quantum dispersion entropy, which measures the mean dispersion or delocalization of the quantum states, decreases at a rate equal to a weighted sum of effective interaction rates, so that the localization always increases in the mean, except when the effective interaction with the environment is zero. The general localization theorem provides a formula for more general conditions.

https://iopscience.iop.org/article/10.1088/0305-4470/26/9/018/pdf

Ian C. Percival

Papers

Wave-function approach to dissipative processes: are there quantum jumps?