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arXiv:cond-mat/0303275 · Journal of Experimental and Theoretical Physics

Instanton versus traditional WKB approach to the Landau-Zener problem

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

Published Aug 1, 2003• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1134/1.1608992

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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.
Within the framework of the instanton approach, we present analytical results for the following model problems: (i) particle penetration through a parabolic potential barrier, where the instanton solution practically coincides with the exact (quantum) one; (ii) descriptions of highly excited states in anharmonic potentials of two types: double-well X 4 and decay X 3 . For the former potential, the instanton method accurately reproduces not only single-well and double-well quantization, but also a crossover region (in contrast to the standard WKB approach that fails to describe the crossover behavior); for the latter potential, the instanton method allows studying the resonance broadening and collapse phenomena. We also investigate resonance tunneling that plays a relevant role in many semiconducting devices. We show that the instanton approach gives exact (quantum) results in a broad range of energies. Applications of the method and of the results are applicable to various systems in physics, chemistry, and biology exhibiting double-level behavior and resonance tunneling.
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.
They have analytically tested several quantum transitions in molecular dynamics. In addition, they have introduced applications of theoretical formalism in physics to economics. First, they have completed the semiclassical theory of nonadiabatic transitions in the exponential potential model. They treated a two-state quantum system, which commonly appears as a subsystem in many-state problems. They were concerned with one-dimensional Schroedinger equation, although the type or semiclassical approach which they used can be applied to higher-dimension potential surfaces by means of Fourier transform. In particular, they solved the case of repulsive exponentially growing potentials, that is the forward-back scattering. They re-expressed the nonadiabatic transition matrix, which describes the transition between the two states for one-direction passage, in terms of contour integrals in the complex coordinate plane. The integrands are adiabatic momenta derived from the eigenvalues of potential matrix. Such type of concept has the advantage that it can be applied to other models than just the exponential potential model. Based on this, they have obtained an analytical tool for describing one-transition point class of potential curve systems. Examples are Rosen-Zener-Demkov model and Landau-Zener-Stueckelberg model. The formalism, which they contributed to, is useful in describing physical processes such as the charge transfer, and it is suitable for shaping external fields or modulating their frequencies in order to achieve a desired state or a product with the theoretical 100% probability. Second, they have derived the first exact analytical solution for a diabatically avoided crossing model. The physical system is described with one-dimensional coupled potential well and barrier which do not intersect in the whole coordinate range. On the basis of special mathematical functions they derived the complete quantum mechanical solution for the wave functions and the transmission coefficient. They also analyzed this model semiclassically and compared it with the exact results. This comparison yields interesting implications on the validity and accuracy of semiclassical approximation. In particular, they solved the case of dramatically avoided crossing with exponentially steep potential curves, and obtained the probability of one-way passage through the interaction region. The overall behavior corresponds to the probability of tunneling through the single potential barrier. In addition, there are complete reflection resonances at discrete energies, resulting from the trapping of a wave in the potential well. The formalism of diabatically avoided crossing model is useful in laser control of molecular reactions. This is because the potential energy curves in realistic molecular systems can be shifted by the photon energy; the curves are coupled by the field intensity via the dipole moment. That is why the new topology of potential energy curves, which was so far broadly omitted, becomes practically interesting due to the development in lasers techniques. Third, they treated several two-state quantum systems common in molecular physics. The semiclassical conditions for the complete transmission and the complete reflection in one-dimensional scattering were given analytically. They discovered the possibility of these phenomena in energy ranges in which the complete reflection or the complete transmission are not commonly expected to occur. In particular, they have made use of the broadly shared assumption that the transitions between the states are localized phenomena. This enables them to separate the whole coordinate region into distinct subsets and to analyze the physics in each subset separately. Applying this approach, they obtained a fully algebraic problem, which can be treated e.g. by diagrammatic techniques. Let them also mention that the complete transmission or complete reflection conditions simplify the wave function, and hence the mathematical solution. The formalism which they used while deriving the semiclassical conditions for the resonant phenomena mentioned above can be further generalized. This will probably yield a whole spectrum of tools in the laser control of molecular processes. Fourth, they developed a powerful technique for solving integral equations with singular kernel. One of the physical formulations of scattering is the time-independent approach based on the Lippmann-Schwinger equation for the Green operator. They have developed a detailed technique how to treat the singularity in this equation. In particular, any scattering problem can be divided into two parts. The first one is the geometry of the system and the static interaction of its components. The second one is more dynamical: the singular equation for the Green operator including the variable energy of the scattering. They fix the most costly calculations in the static part. Then they solve the singularity analytically and thus obtain the cross section as a function of energy. The so far common approach is quite opposite: to fix the singularity first and then to recalculate heavily all the static part for any new value of energy. Obviously, their approach saves a considerable amount of computational time; therefore it is a good standpoint for treating large systems. The formalism above was proved to be extremely efficient taking the example of dissociative recombination H2+ + e- → H + H* because this is a diatomic system, quantum chemical data are easily available in the literature and they could demonstrate the superiority of their approach to the other approaches. Fifth, they suggested applications of basic physical concepts to economics. They introduced aggregate quantities in financial markets and interpreted these quantities as potential energy surfaces. They also formalized interactions between cyclic processes in terms of state to state transitions. Summarizing, they treated the exponential potential model of nonadiabatic transitions, introduced a new exactly solvable model of diabatically avoided crossing, discussed the semiclassical conditions for complete transmission and complete reflection from the viewpoint of laser control, contributed to scattering calculations, and introduced some interdisciplinary applications.
We show that a simple approximation based on concepts underlying the Kibble-Zurek theory of second order phase-transition dynamics can be used to treat avoided level crossing problems. The approach discussed in this paper provides an intuitive insight into quantum dynamics of two-level systems, and may serve as a link between the theory of dynamics of classical and quantum phase transitions. To illustrate these ideas we analyze dynamics of a paramagnet-ferromagnet quantum phase transition in the Ising model. We also present exact unpublished solutions of the Landau-Zener-like problems.
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.
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.