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

• Authors: Alberto Barchielli , Matteo Gregoratti
• Languages of publication: EN

Abstracts

EN

The stochastic Schrödinger 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 Schrödinger equation, which is based on quantum
stochastic calculus, and we show how to transform it into
the classical stochastic Schrödinger equation by diagonalization
of suitable commuting quantum observables. 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 Schrödinger equation.
Moreover, the relation with quantum filtering theory is shortly
discussed. 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
of the homodyne spectrum and the Mollow triplet in the fluorescence
spectrum.

Keywords

EN

Quantum continuous measurements ; Quantum trajectories ; Homodyne detection ; Heterodyne detection ; Spectrum of the squeezing ; Power spectrum ; Uncertainty relations in continuous measurements

• Publisher: Versita
• Journal: Quantum Measurements and Quantum Metrology
• Year: 2013
• Volume: 1
• Pages: 34-56

Dates

accepted

12 - 6 - 2013

online

13 - 08 - 2013

received

16 - 01 - 2013

Contributors

author

Alberto Barchielli

• Politecnico di Milano, Department of Mathematics,
  Piazza Leonardo da Vinci 32, 20133 Milano, Italy
• Istituto Nazionale di Fisica Nucleare, Sezione di Milano,
  Via Celoria 16, 20133 Milano, Italy
• Istituto Nazionale di Alta Matematica, GNAMPA
  (Gruppo Nazionale per l’Analisi Matematica,
  la Probabilità e le loro Applicazioni)

author

Matteo Gregoratti

• Istituto Nazionale di Fisica Nucleare, Sezione di Milano,
  Via Celoria 16, 20133 Milano, Italy

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Identifiers

DOI

10.2478/qmetro-2013-0005