The effect of classical noise on a quantum two-level system

Jean-Philippe Aguilar and Nils Berglund
J. Math. Phys. 49:102102 (23 pp) (2008)

We consider a quantum two-level system perturbed by classical noise. The noise is implemented as a stationary diffusion process in the off-diagonal matrix elements of the Hamiltonian, representing a transverse magnetic field. We determine the invariant measure of the system and prove its uniqueness. In the case of Ornstein--Uhlenbeck noise, we determine the speed of convergence to the invariant measure. Finally, we determine an approximate one-dimensional diffusion equation for the transition probabilities. The proofs use both spectral-theoretic and probabilistic methods.

Mathematical Subject Classification: 60h10, 35P15 (primary), 81Q15, 93E03 (secondary)

Keywords and phrases: spin 1/2, noise, heat bath, open systems, stochastic differential equations, Ornstein-Uhlenbeck process, transition times, transition probabilities, spectral gap, diffusions on Lie groups, Stroock-Varadhan theorem, averaging, renewal equations, Laplace transforms.

 

Journal Homepage Published article: 10.1063/1.2988180 MR2464598 Zbl1152.81306

PDF file (420 Kb) hal-00277789 arXiv/0805.0869