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Quantum Stochastic Calculus with Unbounded Coefficient

Ricardo Castro Santis

Published Aug 12, 2010• 1 citation
OpenAlex

Abstract

Quantum Open System Theory is nowadays a highly developed field which involves, on the mathematical side, many mathematical concepts such as operator theory, semigroups of operators on certain Banach spaces (algebras), classical and quantum stochastic differential equations. Inside Quantum Open System Theory there is the theory of quantum continual measurements: a quantum system is monitored with continuity in time. This is surely an open system because, apart from possible interactions with the external environment like a thermal bath, the system is interacting with the measuring apparatus. The typical field in which a theory of continual measurements is needed is quantum optics: when light emitted by some system is observed (by counting of photons - direct detection - or through some interference mechanism - heterodyne and homodyne detection) a continual measurement is performed.

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Quantum Open System Theory is nowadays a highly developed field which involves, on the mathematical side, many mathematical concepts such as operator theory, semigroups of operators on certain Banach spaces (algebras), classical and quantum stochastic differential equations. Inside Quantum Open System Theory there is the theory of quantum continual measurements: a quantum system is monitored with continuity in time. This is surely an open system because, apart from possible interactions with the external environment like a thermal bath, the system is interacting with the measuring apparatus. The typical field in which a theory of continual measurements is needed is quantum optics: when light emitted by some system is observed (by counting of photons - direct detection - or through some interference mechanism - heterodyne and homodyne detection) a continual measurement is performed.
Multi-photon models are theoretically and experimentally important because in them quantum properly phenomena are verified; as well as squeezed light and quantum entanglement also play a relevant role in quantum information and quantum communication (see Refs. 18–20). In this paper we study a generic model of a multi-photon system with an arbitrary number of pumping and subharmonics fields. This model includes measurement on the system, as could be direct or homodyne detection and we demonstrate the existence of dynamics in the context of Continuous Measurement Theory of Open Quantum Systems (see Refs. 1–11) using Quantum Stochastic Differential Equations with unbounded coefficients (see Refs. 10–15).
In this thesis we study continuously observed open quantum systems from the point of view of quantum filtering theory. The conditioned state of the open system obeys a non-commutative or quantum analogue of the Kushner-Stratonovich equation in classical filtering theory. The thesis consists out of four chapters. The first chapter is a brief introduction to quantum probability theory and quantum stochastic calculus. The second chapter is a description of a photon counting experiment in continuous time within the framework of Davies and Srinivas. The third chapter focusses on the derivation of the quantum filtering or Belavkin equation from the quantum stochastic differential equation governing the interaction between the open system and its environment, the electromagnetic field. The fourth chapter shows how the quantum filtering equation can be used to control quantum systems.
The theory of measurements continuous in time in quantum mechanics (quantum continual measurements) has been formulated by using the notions of instrument and positive operator valued measure [ 1 , 2 , 3 , 4 , 5 , 6 ], arisen inside the operational approach [ 1 , 7 ] to quantum mechanics, by using functional integrals [ 8 , 9 , 10 ], by using quantum stochastic differential equations [ 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 ] and by using classical stochastic differential equations (SDE’s) [ 12 , 13 , 14 , 15 , 16 , 17 , 18 , 4 , 19 , 20 , 21 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 6 ]. Various types of SDE’s are involved, and precisely linear and non linear equations for vectors in Hilbert spaces and for trace-class operators. All such equations contain either a diffusive part, or a jump one, or both.
Abstract Quantum stochastic differential equations are shown to naturally arise because of the mathematical background of the correctness of the Schrödinger equation, which describes the interaction of an open quantum system with broadband quantized fields in the Markov approximation. The main classical and quantum stochastic processes appear in the notation in the Markov approximation of the operator of the electric broadband quantized field strength and the effective Hamiltonian of the interaction of a quantum system with it obtained by the methods of algebraic resonance perturbation theory. All basic stochastic quantum processes, including the counting and two-quantum Lévy processes, are constructed based on quantum white noise, which differs significantly from the classical case. At the same time, the apparatus of stochastic differential equations demonstrates consistency and parallelism.