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arXiv:cond-mat/0408346 · Journal of Experimental and Theoretical Physics Letters

Landau-Zener problem for energies close to potential crossing points

Viktor A. Benderskii, E. V. Vetoshkin, Efim I Kats

Published Sep 1, 2004• 2 citations
arXivHTMLPDF

Abstract

We examine a previously overlooked aspect of the well-known Landau-Zener (LZ) problem, namely, the behavior in the intermediate, i.e., close to a crossing point, energy region, when all four LZ states are coupled and should be taken into account. We calculate the 4×4 connection matrix in this intermediate energy region, possessing the same block structure as the known connection matrices for the tunneling and in the over-barrier regions of the energy and continuously matching those in the corresponding energy regions. Applications of the results may concern various systems of physics, chemistry, or biology, ranging from molecular magnets and glasses to Bose condensed atomic gases.

DOI
10.1134/1.1830664

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arXiv:cond-mat/0303275•Journal of Experimental and Theoretical Physics•Aug 1, 2003•8 citations
We examine a previously overlooked aspect of the well-known Landau-Zener (LZ) problem, namely, the behavior in the intermediate, i.e., close to a crossing point, energy region, when all four LZ states are coupled and should be taken into account. We calculate the 4×4 connection matrix in this intermediate energy region, possessing the same block structure as the known connection matrices for the tunneling and in the over-barrier regions of the energy and continuously matching those in the corresponding energy regions. Applications of the results may concern various systems of physics, chemistry, or biology, ranging from molecular magnets and glasses to Bose condensed atomic gases.
Based on the achievements of the previous three papers of this series, the best working formulas for scattering matrix are obtained for both the Landau–Zener (LZ) and the nonadiabatic tunneling (NT) case: two formulas valid at b2≥0 and b2≤0 in the LZ case, and three formulas valid atb2≤−1, −1≤b2≤1 and b2≥1 in the NT case, where b2 represents the effective energy. Simple and compact formulas which work far better than the LZ formula are proposed for nonadiabatic transition probability by one passage of crossing point for both the LZ and NT cases. Furthermore, compact expressions are derived, for the first time, for the nonadiabatic tunneling probability at b2≤1, i.e., at energies lower than the bottom of the upper adiabatic potential. All the formulas proposed here can be usefully utilized at any coupling strength, namely the validity range has been very much expanded compared to the previous formulas by employing certain empirical corrections. Besides, these formulas have convenience to enable an extension to general curved potentials.
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.
Landau-Zener (LZ) transitions of a two-level system (e.g., electronic spin in molecular magnets) coupled to one or many environmental spins (e.g., nuclear spins) are studied. For rather general interactions the LZ problem is reduced to that of a Landau-Zener grid. It is shown analytically that environmental spins initially in their ground state do not influence the staying probability PPP. This changes if they are prepared in a statistical ensemble. For a more specific model with environmental spins in a transverse field, LZ transitions are studied in the case of well-separated resonances in the LZ grid. The full evolution of the system is described as a succession of elementary transitions at avoided crossings and free evolution between them. If the environmental spins are strongly coupled to the central spin, their effect on PPP is weak. In other cases LZ transitions are strongly suppressed and PPP is decreasing very slowly with the sweep-rate parameter $\ensuremath{\epsilon}\ensuremath{\propto}1/v$, vvv being the energy sweep rate.
The compact analytical complete solutions recently obtained for the two-state Landau-Zener-Stueckelberg problems are reviewed and explained. The theory covers both Landau-Zener (LZ) type in which the two diabatic potential curves cross with the same sign of slopes and the nonadiabatic tunneling (NT) type in which the potentials cross with different signs of slopes. The theory is applicable virtually in the whole range of energy and coupling strength and is convenient for practical use. The new theory for time-dependent nonadiabatic transition can be formulated from the time-independent theory of the LZ-type. The utilizability of the theory to various multi-channel problems and also to multi-dimensional problems is demonstrated and explained. The intriguing phenomenon of complete reflection which appears in the NT-case is explicitly utilized to propose a new type of molecular switching and to control molecular processes such as molecular photodissociation. Also proposed is a new way of controlling molecular processes by using time-dependent external fields. Finally, a trial to formulate a unified analytical theory to cover both Landau-Zener-Stueckelberg and Rosen-Zener-Demkov types of nonadiabatic transitions is briefly touched upon.
Abstract Three analytic solutions to the Schrödinger equation for the time-dependent Landau–Zener (LZ) Hamiltonian are presented. They correspond to specific finite-time driving paths in a bounded parameter space of a two-level system. Two of these paths go through the avoided crossing of levels, either with a constant speed or with variable speed that decreases in the region of reduced energy gap, the third path bypasses the crossing such that the energy gap remains constant. The solutions yield exact time dependencies of the excitation probability for the system evolving from the ground state of the initial Hamiltonian. The LZ formula emerges as an approximation valid within a certain interval of driving times for the constant-speed driving through the avoided crossing. For long driving times, all solutions converge to the prediction of the adiabatic perturbation theory. The excitation probability vanishes at some discrete time instants.
Different theoretical approaches to the famous two-state Landau-Zener problem are briefly discussed. Apart from traditional methods of the adiabatic perturbation theory, the Born-Oppenheimer approximation with geometric phase effects, the two-level approach, and the momentum space representation, the problem is treated semiclassically in the coordinate space. In the framework of the instanton approach, we present a full and unified description of the 1D Landau-Zener problem of level crossing. The method enables us to accurately treat all four transition points (appearing at two-level crossing), while the standard WKB approach takes only two of them into account. The latter approximation is adequate for calculating the transition probability or for studying scattering processes, but it does not work in finding the corresponding chemical reaction rates, in which all four transition points can often be relevant in the typical range of parameters. Applications of the method and of the results may concern various systems in physics, chemistry, and biology.