Landau-Zener problem with decay and dephasing

Abstract

Two aspects of the classic two-level Landau-Zener (LZ) problem are considered. First, we address the LZ problem when one or both levels decay, i.e., εj(t) → εj(t) − iΓj /2. We find that if the system evolves from an initial time −T to a final time +T such that |ε1(±T ) − ε2(±T )| is not too large, the LZ survival probability of a state |j can increase with increasing decay rate of the other state |i = j . This surprising result occurs because the decay results in crossing of the two eigenvalues of the instantaneous non-Hermitian Hamiltonian. On the other hand, if |ε1(±T ) − ε2(±T )| → ∞ as T → ∞, the probability is independent of the decay rate. These results are based on analytic solutions of the time-dependent Schrödinger equations for two cases: (a) the energy levels depend linearly on time, and (b) the energy levels are bounded and of the form ε1,2(t) = ±ε tanh(t/T ). Second, we study LZ transitions affected by dephasing by formulating the Landau-Zener problem with noise in terms of a Schrödinger-Langevin stochastic coupled set of differential equations. The LZ survival probability then becomes a random variable whose probability distribution is shown to behave very differently for long and short dephasing times. We also discuss the combined effects of decay and dephasing on the LZ probability.

Three-level Landau-Zener dynamics

“…There are several methods for modelling dynamics of open systems, including master equations [10,27], the Monte Carlo wavefunction approach [28], and stochastic differential equation techniques [27]. Here we model dephasing and decoherence using a von-Neumann Liouville equation for the density matrix of the system with Lindblad operators [10, 26,27]. For systems that are coupled to Gaussian white noise, the stochastic dynamics can be described using the Schrödinger-Langevin equation [27].…”

On Landau–Zener Transitions for Dephasing Lindbladians

“…We mention [10,14] and [17][18][19], as general references and references to Landau-Zener transitions respectively. For further references see [3]; more recent works include [1,20].…”

Investigation of Temperature Effect in Landau-Zener Avoided Crossing

“…It is obvious that this equation at zero temperature would be reduced to the conventional Landau-Zener equation [10].…”

“…where H 12 describes the strength of the interstate interaction and α is the rate of unperturbed energy change [10]. It is evident that the Hamiltonians at different times do not 58 commute so the coupled equation should be solved to find the solution.…”

“…Up to now, many works are devoted to consider different aspects or extensions to this classic problem, e.g. considering different noise models, decay and dissipation or dephasing issues [8][9][10][11][12][13]. It should be noted that most of the exact solutions to this problem are at zero temperature, however there are studies to investigate the effect of the finite temperature in LZ solution; e.g.…”