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LANDAU-ZENER EFFECT GENERALIZED FOR SUPERFLUID SYSTEMS

M. Mirea

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

The Landau-Zener effect explains the promotion mechanism from one adiabatic intrinsic energy level to another one when the generalized coordinates vary. The two levels have the same spin. In the case of single-particle models without including residual interactions, it is possible to quantify the Landau-Zener effect by mean of a time dependent system of coupled channel equations. At this conference, a generalization of the Landau-Zener equations for seniority one superfluid nuclear systems is presented. A new set of time dependent coupled channel equations are derived in detail. The particular cases of these equations are the classical time dependent Landau-Zener equations and the time dependent Hartree-Fock-Bogoliubov equations for even systems.

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The Landau-Zener effect explains the promotion mechanism from one adiabatic intrinsic energy level to another one when the generalized coordinates vary. The two levels have the same spin. In the case of single-particle models without including residual interactions, it is possible to quantify the Landau-Zener effect by mean of a time dependent system of coupled channel equations. At this conference, a generalization of the Landau-Zener equations for seniority one superfluid nuclear systems is presented. A new set of time dependent coupled channel equations are derived in detail. The particular cases of these equations are the classical time dependent Landau-Zener equations and the time dependent Hartree-Fock-Bogoliubov equations for even systems.
If the Hartree‐Fock‐Bogoliubov equations of motion are solved in the case of seniority one nuclear systems, the unpaired nucleon remains on the same orbital. The blocking effect hinders the possibility that the unpaired nucleon skips from one orbital to another. A new set of coupled channel equations based on the time dependent Hartree‐Fock‐Bogoliubov formalism is deduced. This system includes a mechanism similar to the Landau‐Zener effect, allowing that the unpaired nucleon jumps from one level to another.
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., ${\ensuremath{\varepsilon}}_{j}(t)\ensuremath{\rightarrow}{\ensuremath{\varepsilon}}_{j}(t)\ensuremath{-}i{\ensuremath{\Gamma}}_{j}/2$. We find that if the system evolves from an initial time $\ensuremath{-}T$ to a final time +T+T+T such that $|{\ensuremath{\varepsilon}}_{1}(\ifmmode\pm\else\textpm\fi{}T)\ensuremath{-}{\ensuremath{\varepsilon}}_{2}(\ifmmode\pm\else\textpm\fi{}T)|$ is not too large, the LZ survival probability of a state $|j\ensuremath{\rangle}$ can increase with increasing decay rate of the other state $|i\ensuremath{\ne}j\ensuremath{\rangle}$. 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 $|{\ensuremath{\varepsilon}}_{1}(\ifmmode\pm\else\textpm\fi{}T)\ensuremath{-}{\ensuremath{\varepsilon}}_{2}(\ifmmode\pm\else\textpm\fi{}T)|\ensuremath{\rightarrow}\ensuremath{\infty}$ as $T\ensuremath{\rightarrow}\ensuremath{\infty}$, the probability is independent of the decay rate. These results are based on analytic solutions of the time-dependent Schr\"odinger equations for two cases: (a) the energy levels depend linearly on time, and (b) the energy levels are bounded and of the form ${\ensuremath{\varepsilon}}_{1,2}(t)=\ifmmode\pm\else\textpm\fi{}\ensuremath{\varepsilon}tanh(t/\mathcal{T})$. Second, we study LZ transitions affected by dephasing by formulating the Landau-Zener problem with noise in terms of a Schr\"odinger-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.
During the adiabatic time evolution levels crossing violates the adiabaticity and makes transitions between levels possible. Conventionally only two energy levels cross simultaneously. The transition probabilities for this case were found by Landau and Zener. However, the multilevel crossing happens systematically rather than occasionally if the Hamiltonian possesses a special symmetry. The simplest physical realization of the multilevel crossing are the Zeeman multiplet in a varying magnetic field and an electron in one-dimensional chain driven by the time-dependent electric field. We present asymptotics of the transition amplitudes for these kinds of the multilevel crossing. They are based on an exact solution for a model n-state, time-dependent Schrödinger equation.
The degenerate Landau–Zener–Majorana–Stückelberg model consists of two degenerate energy levels whose energies vary with time and in the presence of an interaction which couples the states of the two levels. In the adiabatic limit, it allows for the populations transfer from states of one level to the states of the other level. The presence of an interaction with the environment influences the efficiency of the process. Nevertheless, identification of possible decoherence-free subspaces permits to engineer coupling schemes for which the effects of quantum noise can be made negligible.
We propose a simple ansatz that allows us to generate new exactly solvable multistate Landau-Zener models. It is based on a system of two decoupled two-level atoms whose levels vary with time and cross at some moments. Then we consider multiparticle systems with Heisenberg equations for annihilation operators having a similar structure as the Shr\"odinger equation for amplitudes in multistate Landau-Zener models and show that the corresponding Shr\"odinger equation in the multiparticle sector belongs to the multistate Landau-Zener class. This observation allows us to generate new exactly solvable models from already known ones. We discuss possible applications of the new solutions in the problem of the driven charge transport in quantum dots.
We present a rigorous analysis of the generalized Landau-Zener problem for the two-level interacting Bose-Einstein condensates. We show that the dynamics of the system is accurately, in detail, described by a two-term variational ansatz that is valid for the whole time domain and is applicable for any set of involved parameters. Applying an exact third-order nonlinear differential equation we construct an advanced fifth-order polynomial equation for the final transition probability serving as a highly accurate generalized Landau-Zener formula.