Riccati equation and the problem of decoherence

Abstract

The block operator matrix theory is used to investigate the problem of a single qubit. We will establish a connection between the Riccati operator equation and the possibility of obtaining an exact reduced dynamics for the qubit in question. The model of the half spin particle in the rotating magnetic field coupling with the external environment is discussed. We show that the model defined in such a way can be reduced to a time independent problem.Comment: 4 pages, accepted by Journal of Mathematical Physics

Evaluation of spin-flip scattering in chirality-induced spin selectivity using the Riccati equation

“…This nonlinear equation is independent of the number of electrons N. It describes the change of the spin polarisation as a function of propagation distance through the organic film due to loss and spin-flip processes. The equation is equivalent to Riccatis equation which is also used, e.g., to model the propagation of light in thin films for ellipsometry [27] or to describe the loss of coherence in qubits with time [28]. With the substitution P = P * + 1/ P…”

Exact reduced dynamics for a qubit in a precessing magnetic field and in contact with a heat bath

“…This problem was addressed in Ref. [8]. The results of this article as well as the analysis we carried out suggest that this problem is at least as difficult as resolving the Riccati equation associated with the total Hamiltonian.…”

“…The procedure that allows us to do so is as follows (for details see Ref. [8] and references therein)…”

“…In order to accomplish this, we use a recently proven theorem that says if (for more details see Appendix A and also Ref. [8]) (i) the total system is defined by the Hamiltonian (12); (ii) the interaction term H int between the systems takes the form f (σ z ) ⊗ V , where f is an analytical function of σ z and V is a Hermitian operator; and (iii) ρ t (β) represents the solution of the system defined by the Hamiltonian H QE (0,β) (note that Hamiltonian H QE depends on β, hence so is ρ t ), then the reduced dynamics η t for the model governed by the Hamiltonian (12) can be obtained using the following simple formula:…”

“…Recently, the efforts to answer this question were made [7,8]. It was shown that from the mathematical perspective this task is at least as complicated as the problem of solving the Riccati algebraic equation associated with the Hamiltonian defining the model in question.…”