Quantum Stratonovich Stochastic Calculus and the Quantum Wong-Zakai Theorem

Abstract

We extend the Ito -to- Stratonovich analysis or quantum stochastic differential equations, introduced by Gardiner and Collett for emission (creation), absorption (annihilation) processes, to include scattering (conservation) processes. Working within the framework of quantum stochastic calculus, we define Stratonovich calculus as an algebraic modification of the Ito one and give conditions for the existence of Stratonovich time-ordered exponentials. We show that conversion formula for the coefficients has a striking resemblance to Green's function formulae from standard perturbation theory. We show that the calculus conveniently describes the Markov limit of regular open quantum dynamical systemsin much the same way as in the Wong-Zakai approximation theorems of classical stochastic analysis. We extend previous limit results to multiple-dimensions with a proof that makes use of diagrammatic conventions.

24 pages

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• Feedback control of quantum state reduction
• All-optical versus electro-optical quantum-limited feedback
• Bellman equations for optimal feedback control of qubit states
• Hamilton-Jacobi-Bellman equations for Quantum Filtering and Control
• The Langevin equation from Markovian Quantum Central Limits
• Quantum Flows as Markovian Limit of Emission, Absorption and Scattering Interactions
• Asymptotic Stochastic Transformations for Nonlinear Quantum Dynamical Systems
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• From repeated to continuous quantum interactions
• Quantum White Noise and the Master Equation for Gaussian Reference States
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Cited by in corpus (24)

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• A Quantum Extended Kalman Filter
• Singular perturbation of quantum stochastic differential equations with coupling through an oscillator mode
• Dynamics of photosynthetic light harvesting systems interacting with N-photon Fock states
• Phase-preserving linear amplifiers not simulable by the parametric amplifier
• Exact calculation of stimulated emission driven by pulsed light
• Quantum filter for a class of non-Markovian quantum systems
• Optimal Zeno Dragging for Quantum Control: A Shortcut to Zeno with Action-based Scheduling Optimization
• The Series Product and Its Application to Quantum Feedforward and Feedback Networks
• Quantum filter for a non-Markovian single qubit system
• The Global versus Local Hamiltonian Description of Quantum Input-Output Theory
• On The Scattering Process in Quantum Optics
• Symplectic Noise & The Classical Analog of the Lindblad Generator
• Characteristic Operator Functions for Quantum Input-Plant-Output Models & Coherent Control
• Classical and Quantum Stochastic Models of Resistive and Memristive Circuits
• A Trotter-Kato Theorem for Quantum Markov Limits
• The Chebotarev-Gregoratti Hamiltonian as singular perturbation of a nonsemibounded operator
• The Stratonovich Formulation of Quantum Feedback Network Rules
• The Gough-James Theory of Quantum Feedback Networks in the Belavkin Representation