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Lecture notes series, Institute For Mathematical Sciences

QUANTUM STOCHASTIC DIFFERENTIAL EQUATIONS ASSOCIATED WITH SQUARE OF ANNIHILATION AND CREATION PROCESSES

Un Cig Ji, Kalyan B. Sinha

Published Aug 25, 2017• 2 citations
Publisher

Abstract

We develop the quantum stochastic calculus associated with the square of annihilation and creation processes. We study the quantum stochastic integral and quantum stochastic differential equation associated with the square of annihilation and creation processes. Under a sufficient condition the solution of the quantum stochastic differential equation becomes unitary and then the unitary solution gives a unitary dilation of an evolution system of which the Hamiltonian is depending on time.

DOI
10.1142/9789813225466_0011

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We develop the quantum stochastic calculus associated with the square of annihilation and creation processes. We study the quantum stochastic integral and quantum stochastic differential equation associated with the square of annihilation and creation processes. Under a sufficient condition the solution of the quantum stochastic differential equation becomes unitary and then the unitary solution gives a unitary dilation of an evolution system of which the Hamiltonian is depending on time.
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We consider the quantum stochastic differential equation introduced by Hudson and Parthasarathy to describe the stochastic evolution of an open quantum system together with its environment. We study the (unbounded) Hamiltonian operator generating the unitary group connected, as shown by Frigerio and Maassen, to the solution of the equation. We find a densely defined restriction of the Hamiltonian operator; in some special cases we prove that this restriction is essentially self-adjoint and in one particular case we get the whole Hamiltonian with its full domain.
Kurzweil or generalized differential equations associated with Lipschitzian quantum stochastic differential equations (QSDEs) are introduced and studied. This is accomplished within the framework of the Hudson-Parthasarathy formulations of quantum stochastic calculus. Results concerning the equivalence of these classes of equations satisfying the Caratheodory conditions are presented. It is further shown that the associated Kurzweil equation may be used to obtain a reasonably high accurate approximate solutions of the QSDEs. This generalize analogous results for classical initial value problems to the noncommutative quantum setting involving unbounded linear operators on a Hilbert space. Numerical examples are given.