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

Abstract

The two-level quantum system (qubit) in a precessing magnetic field and in contact with a heat bath is investigated. The exact reduced dynamics for the qubit in question is obtained. We apply the approach based on the block operator matrices theory, in contrast with the standard methods provided by the theory of the open quantum systems. We also present the solution of the Riccati operator equation associated with the Hamiltonian of the system. Next, we study the adiabatic approximation for the system under consideration using quantum fidelity as a way to measure validity of the adiabatic theory. We find that in the weak-coupling domain the standard condition that ensures adiabatic behavior of the spin in the magnetic field also guarantees its adiabatic evolution in the open-system variant of this model. Therefore, we provide the explicit example of the open quantum system that satisfies the adiabatic approximation first formulated for the closed quantum systems.

Stationary states of two-level open quantum systems

“…Examples of such systems occur in the literature, e.g. [42][43][44]. The block operator matrix representation of ( 24) is given by (19) with H ± = H 0 ± H 1 and D(H) = H E ⊕ H E .…”

Exact solution of the Schrödinger equation with the spin-boson Hamiltonian

“…It can be found (see e.g., [26][27][28]) that the problem of diagonalization on the Hilbert space C 2 ⊗ F B can be mapped to the problem of resolving the Riccati equation [29]. This new approach was recently successfully applied to the time-dependent spin-spins model [30]. As a result, the exact reduced dynamics of the qubit in contact with a spin environment and in the presence of a precessing magnetic field has been obtained.…”