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

The stochastic Schrodinger 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 Schrodinger equation, which is based on quantum stochastic calculus, and we show how to transform it into the classical stochastic Schrodinger 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 Schrodinger 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.

Entanglement protection and generation under continuous monitoring

Entanglement between two quantum systems is a resource in quantum information, but dissipation usually destroys it. In this article we consider two qubits without direct interaction and we show that, even in cases where the open system dynamics destroys any initial entanglement, the mere monitoring of the environment can preserve or create the entanglement, by filtering the state of the qubits. While the systems we study are very simple, we can show examples with entanglement protection or entanglement birth, death, rebirth due to monitoring.