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Infinite Dimensional Analysis Quantum Probability and Related Topics

ON THE HAMILTONIAN OPERATOR ASSOCIATED TO SOME QUANTUM STOCHASTIC DIFFERENTIAL EQUATIONS

Matteo Gregoratti

Published Dec 1, 2000• 17 citations
Publisher

Abstract

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.

DOI
10.1142/s0219025700000327

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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.
We consider a quantum stochastic evolution in continuous time defined by the quantum stochastic differential equation of Hudson and Parthasarathy. On one side, such an evolution can also be defined by a standard Schrödinger equation with a singular and unbounded Hamiltonian operator K. On the other side, such an evolution can also be obtained as a limit from Hamiltonian repeated interactions in discrete time. We study how the structure of the Hamiltonian K emerges in the limit from repeated to continuous interactions. We present results in the case of 1-dimensional multiplicity and system spaces, where calculations can be explicitly performed, and the proper formulation of the problem can be discussed.
The main object of this paper is to establish the Hamiltonian of the Hudson–Parthasarathy quantum stochastic differential equation [Formula: see text] As its solution forms a cocycle with respect to the time shift, its product with the time shift establishes a strongly continuous one-parameter unitary group W(t)= exp (-itH) and H is called its Hamiltonian. Our main result is that H is the closure of the restriction of the singular operator [Formula: see text] on an appropriate domain that shall be explicitly described. The symmetric differentiation [Formula: see text] is a generalization of the generator of the time shift, [Formula: see text] and [Formula: see text] are annihilation and creation operators and [Formula: see text] is the symmetric Dirac δ-function. In one dimension the symmetric differentiation might be used to establish a quantum stochastic calculus without Ito term.
If the Hamiltonian of the system is a bounded selfadjoint operator, we give a simple alternative characterization of the infinitesimal generator iC of the unitary group associated with a Quantum Stochastic Differential Equation (QSDE). By using the defect indices of the symmetric operator C, we give a sufficient condition that assures that iC is the restriction of a generator of a strongly continuous semigroup of isometries in the case when the Hamiltonian of the system is an unbounded symmetric operator.
In the calculus of Hudson and Parthasarathy we show the existence of a solution to the quantum stochastic evolution equation U ( t ) = I + ∫ 0 t U ( s ) ( L 1 d Λ ( s ) + L 2 d A ( s ) + L 3 d A † ( s ) + L 4 d s ) for a class of unbounded operators L1, …, L4 in the initial space, and show that the necessary and sufficient condition that U be a unitary process is the same as in the case that L1, …, L4 are bounded. Applications of the vacuum conditional expectation to U give rise to strongly continuous semigroups and their generators in the initial space, and to ultraweakly continuous completely positive semigroups and their generators in the von Neumann algebra of the initial space.