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arXiv:quant-ph/0607221 · Journal of Physics A Mathematical and General

Nonadiabatic transitions for a decaying two-level system: geometrical and dynamical contributions

R. Schilling, Mark Vogelsberger, Dmitry A. Garanin

Published Oct 17, 2006• 25 citations
arXivHTMLPDF

Abstract

We study the Landau–Zener problem for a decaying two-level system described by a non-Hermitian Hamiltonian, depending analytically on time. Use of a super-adiabatic basis allows us to calculate the survival probability P in the slow-sweep limit, without specifying the Hamiltonian explicitly. It is found that P consists of a 'dynamical' and a 'geometrical' factor. The former is determined by the complex adiabatic eigenvalues E ± ( t ), only, whereas the latter solely requires the knowledge of α ± ( t ), the ratio of the components of each of the adiabatic eigenstates. Both factors can be split into a universal one, depending only on the complex level crossing points, and a nonuniversal one, involving the full time dependence of E ± ( t ) and α ± ( t ). This general result is applied to the Akulin–Schleich model where the initial upper level is damped with damping constant γ. For analytic power-law sweeps we find that Stückelberg oscillations of P exist for γ smaller than a critical value γ c and disappear for γ > γ c . A physical interpretation of this behaviour will be presented by use of a damped harmonic oscillator.

DOI
10.1088/0305-4470/39/44/008

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We study the Landau–Zener problem for a decaying two-level system described by a non-Hermitian Hamiltonian, depending analytically on time. Use of a super-adiabatic basis allows us to calculate the survival probability P in the slow-sweep limit, without specifying the Hamiltonian explicitly. It is found that P consists of a 'dynamical' and a 'geometrical' factor. The former is determined by the complex adiabatic eigenvalues E ± ( t ), only, whereas the latter solely requires the knowledge of α ± ( t ), the ratio of the components of each of the adiabatic eigenstates. Both factors can be split into a universal one, depending only on the complex level crossing points, and a nonuniversal one, involving the full time dependence of E ± ( t ) and α ± ( t ). This general result is applied to the Akulin–Schleich model where the initial upper level is damped with damping constant γ. For analytic power-law sweeps we find that Stückelberg oscillations of P exist for γ smaller than a critical value γ c and disappear for γ > γ c . A physical interpretation of this behaviour will be presented by use of a damped harmonic oscillator.
We investigate the Landau–Zener (LZ) like dynamics of decaying two- and three-level systems with decay rates Γ 1 and Γ 2 for levels with minimum and maximum spin projection. Non-adiabatic and adiabatic transition probabilities are calculated from diabatic and adiabatic bases for two- and three-level systems. We extend the familiar two-level model of atoms with decay from the excited state out of the system into the hierarchy of three-level models which can be solved analytically or computationally in a non-perturbative manner. Exact analytical solutions are obtained within the framework of an extended form of the proposed procedure which enables to take into account all possible initial moments rather than large negative time t 0 = − ∞ as in standard LZ problems. We elucidate the applications of our results from a unified theoretical basis that numerically analyzes the dynamics of a system as probed by experiments.
Non-Hermitian quantum systems with explicit time dependence are of ever-increasing importance. There are only a handful of models that have been analytically studied in this context. Here, a PT-symmetric non-Hermitian N-level Landau-Zener type problem with two exceptional points of N-th order is introduced. The system is Hermitian for asymptotically large times, far away from the exceptional points, and has purely imaginary eigenvalues between the exceptional points. The full Landau-Zener transition probabilities are derived, and found to show a characteristic binomial behaviour. In the adiabatic limit the final populations are given by the ratios of binomial coefficients. It is demonstrated how this behaviour can be understood on the basis of adiabatic analysis, despite the breakdown of adiabaticity that is often associated with non-Hermitian systems.
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.
Non-Hermitian quantum systems with explicit time dependence are of ever increasing importance. There are only a handful of models that have been analytically studied in this context. Here, a PT\mathcal{PT}PT-symmetric non-Hermitian NNN-level Landau-Zener type problem with two exceptional points of NthN\mathrm{th}Nth order is introduced. The system is Hermitian for asymptotically large times, far away from the exceptional points, and has purely imaginary eigenvalues between the exceptional points. The full Landau-Zener transition probabilities are derived, and found to show a characteristic binomial behavior. In the adiabatic limit the final populations are given by the ratios of binomial coefficients. It is demonstrated how this behavior can be understood on the basis of adiabatic analysis, despite the breakdown of adiabaticity that is often associated with non-Hermitian systems.
The time evolution of non-Hermitian systems is generally nonunitary. Dynamics governed by time-dependent non-Hermitian Hamiltonians lead to a variety of novel phenomena, one of which is state amplification or suppression induced by the complex Berry phase. Here, we extend the framework of geometric modulation to multi-level systems and show that both transition and survival intensities can be modulated. We apply our theory to the non-Hermitian Landau-Zener (LZ) problem. First, we show that, in the half-LZ problem, both the transition and survival probabilities exhibit nonreciprocity due to the complex Berry phase. In the non-Hermitian standard LZ problem, only the survival intensity is known to exhibit nonreciprocity, whereas the transition intensity does not. However, the physical origin of this nonreciprocal behavior remains unclear. In this work, we show that the nonreciprocity originates from the complex Berry phase.
Using a comparison equation method rooted in the Landau-Zener model, we report results for the probability of transition and survival in two-state models for a crossing of levels having some width, to account for dissipation. The theory allows arbitrary forms for the state energies, their rate of decay, and coupling, subject only to the limitation inherent in the Landau-Zener class (coupling with infinite ``pulse area''). We show how the strong decay result of adiabatic elimination can be recovered, but with important restrictions as to its usability. We also confirm the validity of the well-known Dykhne formula for crossings of real energy levels, and use the new theory to extend the Dykhne result to a class of dissipative crossings we call pseudoreal. Finally, we examine the overall probability for transition at two successive crossings, as occurs in atomic collisions. Here, interference of the crossing states leads to oscillations in the transition probability with collision time. For slow collisions we find that even a modest level of dissipation heavily damps these oscillations, but at the same time enhances the average probability of transition.