Abstract
The aim of this paper is to study the problem of controlling the stochastic evolution of a two-level quantum system in the presence of two randomly fluctuating electromagnetic fields, given by a Wiener process. The system is modeled by the stochastic Schr¨odinger equation dependent on time. We set up the quantum optimal control problem by choosing a cost functional type Bolza. By applying the Pontryagin Maximum Stochastic Principle to an extended Hamiltonian, we express the stochastic optimal controls in terms of the co-state of the system. To solve numerically the resulting stochastic differential equations we propose an iterative algorithm using the Euler-Maruyama method. Finally, we obtain the optimal trajectories on the Bloch sphere.
Highlights
Studying the motion of a-spin particle in a two-level system and the interaction with its surronding, sometimes is necessary to introduce a stochastic model
We consider the Brownian motion of the spin represented by a randomly fluctuating electromagnetic field [Kubo and Hashitsume, 1970]
Many of the models correspondig to these phenomena, which are described by stochastic differential equations, don’t have analytic solutions
Summary
-spin particle in a two-level system and the interaction with its surronding, sometimes is necessary to introduce a stochastic model. Maruyama approximation is the simplest method which can be implemented and which converges to an appropriate step size, it is sometimes inefficient and gives poor stability properties In our case, this numerical scheme, by its simplicity, allowed us to develop a simple program in MATLAB to simulate sample paths close to those of the solution of the deterministic system. 1 2 spin particle interacting with a electromagnetic field, neglecting other interactions with the system This quantum control system describes the dynamics of a system like a 2-level quantum system, governed by the Schrodinger equation for a pure state (we set = 1). An adequate projection of the space of the pure states over the Bloch sphere is given by the Hopf projection Π : S3 ⊂ C2 → SB, Π:
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
