Quantum Continual Measurements and a Posteriori Collapse on CCR

TL;DR

This paper develops a quantum theory for continuous unsharp measurements using CCR stochastic methods, deriving equations for posterior state evolution and demonstrating exponential collapse to pure states in certain systems.

Contribution

It introduces a new CCR-based stochastic framework for quantum continual measurements and describes the posterior collapse dynamics of open quantum systems.

Findings

Derived stochastic equations for posterior quantum state evolution.

Solved quantum Langevin equations for linear quasi-free Hamiltonians.

Demonstrated exponential collapse of mixed states to pure Gaussian states under continuous measurement.

Abstract

A quantum theory for the Markovian dynamics of an open system under the unsharp observation which is continuous in time, is developed within the CCR stochastic approach. A stochastic classical equation for the posterior evolution of quantum continuously observed system is derived and the spontaneous collapse (stochastically continuous reduction of the wave packet) is described. The quantum Langevin evolution equation is solved for the general linear case of a quasi--free Hamiltonian in the initial CCR algebra with a fixed output observable field, and the posterior Kalman dynamics coresponding to an initial Gaussian state is found. It is shown for an example of the posterior dynamics of quantum unstable open system that any mixed state under a complete nondemolition measurement collapses exponentially to a pure Gaussian one.