A tutorial introduction to quantum stochastic master equations based on the qubit/photon system

TL;DR

This tutorial introduces quantum stochastic master equations for a qubit/photon system, illustrating their derivation from discrete to continuous time, including measurement imperfections, decoherence, and numerical integration methods that preserve physical properties.

Contribution

It provides a comprehensive, step-by-step derivation of quantum SME for qubit/photon systems, emphasizing the Kraus-map structure and numerical schemes that maintain positivity and trace.

Findings

Derivation of discrete-time SME using explicit interaction propagators

Transition from discrete to continuous-time SME with measurement signals

Development of positivity-preserving numerical integration schemes

Abstract

From the key composite quantum system made of a two-level system (qubit) and a harmonic oscillator (photon) with resonant or dispersive interactions, one derives the corresponding quantum Stochastic Master Equations (SME) when either the qubits or the photons are measured. Starting with an elementary discrete-time formulation based on explicit formulae for the interaction propagators, one shows how to include measurement imperfections and decoherence. This qubit/photon quantum system illustrates the Kraus-map structure of general discrete-time SME governing the dynamics of an open quantum system subject to measurement back-action and decoherence induced by the environment. Then, on the qubit/photon system, one explains the passage to a continuous-time mathematical model where the measurement signal is either a continuous real-value signal (typically homodyne or heterodyne signal) or a discontinuous and integer-value signal obtained from a counter. During this derivation, the Kraus map formulation is preserved in an infinitesimal way. Such a derivation provides also an equivalent Kraus-map formulation to the continuous-time SME usually expressed as stochastic differential equations driven either by Wiener or Poisson processes. From such Kraus-map formulation, simple linear numerical integration schemes are derived that preserve the positivity and the trace of the density operator, i.e. of the quantum state.