ON THE HAMILTONIAN OPERATOR ASSOCIATED TO SOME QUANTUM STOCHASTIC DIFFERENTIAL EQUATIONS

2000 ◽  
Vol 03 (04) ◽  
pp. 483-503 ◽  
Author(s):  
M. GREGORATTI
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Unitary Group ◽  
Open Quantum System ◽  
Hamiltonian Operator ◽  
Stochastic Evolution ◽  
Quantum Stochastic Differential Equation ◽  
Special Cases ◽  
Densely Defined ◽  
Full Domain

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.

The Hamiltonian Generating Quantum Stochastic Evolutions in the Limit from Repeated to Continuous Interactions

2015 ◽  
Vol 22 (04) ◽  
pp. 1550022
Author(s):  
Matteo Gregoratti
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Discrete Time ◽  
Continuous Time ◽  
Schrodinger Equation ◽  
Hamiltonian Operator ◽  
The Other ◽  
Stochastic Evolution ◽  
Quantum Stochastic Differential Equation ◽  
Proper Formulation

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.

scholarly journals Pure Gaussian state generation via dissipation: a quantum stochastic differential equation approach

2012 ◽  
Vol 370 (1979) ◽  
pp. 5324-5337 ◽  
Author(s):  
Naoki Yamamoto
Keyword(s):  
Differential Equation ◽  
Steady State ◽  
Stochastic Differential Equation ◽  
Cluster State ◽  
Dissipative System ◽  
Complete Characterization ◽  
Gaussian State ◽  
Quantum State Transfer ◽  
Quantum Stochastic Differential Equation ◽  
Gaussian System

Recently, the complete characterization of a general Gaussian dissipative system having a unique pure steady state was obtained. This result provides a clear guideline for engineering an environment such that the dissipative system has a desired pure steady state such as a cluster state. In this paper, we describe the system in terms of a quantum stochastic differential equation (QSDE) so that the environment channels can be explicitly dealt with. Then, a physical meaning of that characterization, which cannot be seen without the QSDE representation, is clarified; more specifically, the nullifier dynamics of any Gaussian system generating a unique pure steady state is passive. In addition, again based on the QSDE framework, we provide a general and practical method to implement a desired dissipative Gaussian system, which has a structure of quantum state transfer.

Quantum Stochastic Differential Equation is Unitarily Equivalent to a Symmetric Boundary Value Problem in Fock Space

1998 ◽  
Vol 01 (02) ◽  
pp. 175-199 ◽  
Author(s):  
Alexander M. Chebotarev
Keyword(s):  
Differential Equation ◽  
Schrödinger Equation ◽  
Boundary Value Problem ◽  
Stochastic Differential Equation ◽  
Schrodinger Equation ◽  
Fock Space ◽  
Boundary Value ◽  
Weak Limit ◽  
Quantum Stochastic Differential Equation ◽  
Markov Evolution

We show a new remarkable connection between the symmetric form of a quantum stochastic differential equation (QSDE) and the strong resolvent limit of the Schrödinger equations in Fock space: the strong resolvent limit is unitarily equivalent to QSDE in the adapted (or Ito) form, and the weak limit is unitarily equivalent to the symmetric (or Stratonovich) form of QSDE. We also prove that QSDE is unitarily equivalent to a symmetric boundary value problem for the Schrödinger equation in Fock space. The boundary condition describes standard jumps in phase and amplitude of components of Fock vectors belonging to the range of the resolvent. The corresponding Markov evolution equation (the Lindblad or Markov master equation) is derived from the boundary value problem for the Schrödinger equation.

On the theory and application of one-step numerical schemes for solving quantum stochastic differential equation (QSDE)

2014 ◽  
Vol 07 (03) ◽  
pp. 1450037
Author(s):  
T. O. Akinwumi ◽  
B. J. Adegboyegun
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Stochastic Differential Equation ◽  
Numerical Experiments ◽  
Numerical Schemes ◽  
Quantum Stochastic Differential Equation ◽  
One Step ◽  
Definition Of ◽  
Convergent Sequences ◽  
Solution Techniques

This paper presents one-step numerical schemes for solving quantum stochastic differential equation (QSDE). The algorithms are developed based on the definition of QSDE and the solution techniques yield rapidly convergent sequences which are readily computable. As well as developing the schemes, we perform some numerical experiments and the solutions obtained compete favorably with exact solutions. The solution techniques presented in this work can handle all class of QSDEs most especially when the exact solution does not exist.

scholarly journals THE STABILITY OF QUANTUM MARKOV FILTERS

2009 ◽  
Vol 12 (01) ◽  
pp. 153-172 ◽  
Author(s):  
RAMON VAN HANDEL
Keyword(s):  
Differential Equation ◽  
Stochastic Model ◽  
Stochastic Differential Equation ◽  
Initial State ◽  
Rank Condition ◽  
Quantum Stochastic Differential Equation ◽  
Absolutely Continuous ◽  
Finite Dimensional ◽  
Initial States ◽  
The Stability

When are quantum filters asymptotically independent of the initial state? We show that this is the case for absolutely continuous initial states when the quantum stochastic model satisfies an observability condition. When the initial system is finite dimensional, this condition can be verified explicitly in terms of a rank condition on the coefficients of the associated quantum stochastic differential equation.

scholarly journals Simulation of Quantum Dynamics Based on the Quantum Stochastic Differential Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Ming Li
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Quantum Dynamics ◽  
Dynamical Behavior ◽  
Computational Cost ◽  
Quantum Master Equation ◽  
Level System ◽  
Quantum Stochastic Differential Equation ◽  
State Diffusion ◽  
Quantum State Diffusion

The quantum stochastic differential equation derived from the Lindblad form quantum master equation is investigated. The general formulation in terms of environment operators representing the quantum state diffusion is given. The numerical simulation algorithm of stochastic process of direct photodetection of a driven two-level system for the predictions of the dynamical behavior is proposed. The effectiveness and superiority of the algorithm are verified by the performance analysis of the accuracy and the computational cost in comparison with the classical Runge-Kutta algorithm.

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