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arXiv:2407.06832 · Physical Review A

Perturbative approach to time-dependent quantum systems and applications to one-crossing multistate Landau-Zener models

Rongyu Hu, Chen Sun

Published Nov 21, 2024• 0 citations
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Abstract

We formulate a perturbative approach for studying a class of multilevel time-dependent quantum systems with constant off-diagonal couplings and diabatic energies being odd functions of time. Applying this approach to a general multistate Landau-Zener (MLZ) model with all diabatic levels crossing at one point (named the one-crossing MLZ model), we derive analytical formulas of all its transition probabilities up to fourth order in the couplings. For one-crossing MLZ models it is difficult to obtain such analytical results by other kinds of approximation methods; thus, these perturbative results can serve as reliable benchmarks for future studies of any one-crossing MLZ models that have not been exactly solved.

DOI
10.1103/physreva.110.052219

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We formulate a perturbative approach for studying a class of multilevel time-dependent quantum systems with constant off-diagonal couplings and diabatic energies being odd functions of time. Applying this approach to a general multistate Landau-Zener (MLZ) model with all diabatic levels crossing at one point (named the one-crossing MLZ model), we derive analytical formulas of all its transition probabilities up to fourth order in the couplings. For one-crossing MLZ models it is difficult to obtain such analytical results by other kinds of approximation methods; thus, these perturbative results can serve as reliable benchmarks for future studies of any one-crossing MLZ models that have not been exactly solved.
Abstract The concept of quantum integrability has been introduced recently for quantum systems with explicitly time-dependent Hamiltonians (Sinitsyn et al 2018 Phys. Rev. Lett . 120 190402). Within the multistate Landau–Zener (MLZ) theory, however, there has been a successful alternative approach to identify and solve complex time-dependent models (Sinitsyn and Chernyak 2017 J. Phys. A: Math. Theor . 50 255203). Here we compare both methods by applying them to a new class of exactly solvable MLZ models. This class contains systems with an arbitrary number of interacting states and shows quick growth with N number of exact adiabatic energy crossing points, which appear at different moments of time. At each N , transition probabilities in these systems can be found analytically and exactly but complexity and variety of solutions in this class also grow with N quickly. We illustrate how common features of solvable MLZ systems appear from quantum integrability and develop an approach to further classification of solvable MLZ problems.
We study the SSS matrix for the transitions at an avoided crossing of several energy levels, which is a multilevel generalization of the Landau-Zener problem. We demonstrate that, by extending the Schr\"odinger evolution to complex time, one can obtain an exact answer for some of the transition amplitudes. Similar to the Landau-Zener case, our result covers both the adiabatic (slow evolution) and the diabatic (fast evolution) regimes. The form of the exact transition amplitude coincides with that obtained in a sequential pairwise level crossing approximation, in accord with the conjecture of Brundobler and Elser [J. Phys. A 26, 1211 (1993)].
Abstract Recently, integrability conditions (ICs) in mutistate Landau-Zener (MLZ) theory were proposed [1]. They describe common properties of all known solved systems with linearly time-dependent Hamiltonians. Here we show that ICs enable efficient computer assisted search for new solvable MLZ models that span complexity range from several interacting states to mesoscopic systems with many-body dynamics and combinatorially large phase space. This diversity suggests that nontrivial solvable MLZ models are numerous. In addition, we refine the formulation of ICs and extend the class of solvable systems to models with points of multiple diabatic level crossing.
A dynamic four-state system is considered within the context of multistate Landau-Zener theory. It is shown that the theory accounts very well for the time-dependent state populations and final transition probabilities even in cases when multiple crossings appear in close vicinity of each other. This is also true for multiple paths systems when the phases are appropriately accounted for. It is found that transitions may take place also between diabatic states that do not couple directly and that the dynamics of such crossings may be accurately described within the multichannel Landau-Zener theory.
Abstract All currently known exactly solvable multistate Landau–Zener (MLZ) models are associated with families of operators that commute with the MLZ Hamiltonians and depend on time linearly. There can also be operators that satisfy the integrability conditions with the MLZ Hamiltonians but depend on time quadratically. We show that, among the MLZ systems, such time-quadratic operators are much more common. We demonstrate then that such operators generally lead to constraints on the independent variables that parametrize the scattering matrix. Such constraints lead to asymptotically exact expressions for the transition probabilities in the adiabatic limit of a three-level MLZ model. New more complex fully solvable MLZ systems are also found.