Quantum continuous measurements: The stochastic Schroedinger equations and the spectrum of the output

TL;DR

This paper reviews the connection between quantum and classical stochastic Schrödinger equations for continuous quantum measurements, analyzing the output spectrum, quantum bounds, and phenomena like squeezing using a two-level atom model.

Contribution

It provides a detailed link between quantum and classical descriptions of continuous measurements and analyzes the spectral properties of measurement outputs, including quantum bounds and squeezing effects.

Findings

Derived the classical stochastic Schrödinger equation from the quantum version.

Analyzed the spectrum of the measurement output and its quantum bounds.

Illustrated quantum phenomena like squeezing and the Mollow triplet in a two-level atom model.

Abstract

The stochastic Schr\"odinger equation, of classical or quantum type, allows to describe open quantum systems under measurement in continuous time. In this paper we review the link between these two descriptions and we study the properties of the output of the measurement. For simplicity we deal only with the diffusive case. Firstly, we discuss the quantum stochastic Schr\"odinger equation, which is based on quantum stochastic calculus, and we show how to transform it into the classical stochastic Schr\"odinger equation by diagonalization of suitable quantum observables, based on the isomorphism between Fock space and Wiener space. Then, we give the a posteriori state, the conditional system state at time $t$ given the output up to that time and we link its evolution to the classical stochastic Schr\"odinger equation. Finally, we study the output of the continuous measurement, which is a stochastic process with probability distribution given by the rules of quantum mechanics. When the output process is stationary, at least in the long run, the spectrum of the process can be introduced and its properties studied. In particular we show how the Heisenberg uncertainty relations give rise to characteristic bounds on the possible spectra and we discuss how this is related to the typical quantum phenomenon of squeezing. We use a simple quantum system, a two-level atom stimulated by a laser, to discuss the differences between homodyne and heterodyne detection and to explicitly show squeezing and anti-squeezing and the Mollow triplet in the fluorescence spectrum.