Nanoscale systems have sizes intermediate between atomic and macroscopic ones. Therefore their treatment often requires a combination of methods from atomic and condensed matter physics. The conventional Landau-Zener theory, being a powerful tool in atomic physics, often fails to predict correctly nonadiabatic transition probabilities in various nanostructures because it does not include many-body effects typical for mesoscopics. In this research project the generalizations of the Landau-Zener theory that solve this problem were studied. The multistate, multiparticle and nonunitary extensions of the theory have been proposed and investigated. New classes of exactly solvable models have been derived. I discuss their applications in problems of the molecular condensate dissociation and of the driven charge transport. In application to the physics of nanomagnets new approaches in modeling the influence of the environment on the Landau-Zener evolution are proposed and simple universal formulas are derived for the extensions of the theory that include the coupling to noise and the nuclear spin bath.
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.
The search for exactly solvable models is an evergreen topic in theoretical physics. In the context of multistate Landau-Zener models --
N-state quantum systems with linearly time-dependent Hamiltonians -- the theory of integrability provides a framework for identifying new solvable cases. In particular, it was proved that the integrability of a specific class known as the multitime Landau-Zener (MTLZ) models guarantees their exact solvability. A key finding was that an
N-state MTLZ model can be represented by data defined on an
N-vertex graph. While known host graphs for MTLZ models include hypercubes, fans, and their Cartesian products, no other families have been discovered, leading to the conjecture that these are the only possibilities. In this work, we conduct a systematic graph-theoretical search for integrable models within the MTLZ class. By first identifying minimal structures that a graph must contain to host an MTLZ model, we formulate an efficient algorithm to systematically search for candidate graphs for MTLZ models. Implementing this algorithm using computational software, we enumerate all candidate graphs with up to
N=13 vertices and perform an in-depth analysis of those with
N≤11. Our results corroborate the aforementioned conjecture for graphs up to
11 vertices. For even larger graphs, we propose a specific family, termed descendants of ``
(0,2)-graphs'', as promising candidates that may violate the conjecture above. Our work can serve as a guideline to identify new exactly solvable multistate Landau-Zener models in the future.
Abstract We discuss a class of models that generalize the two-state Landau–Zener Hamiltonian to both the multistate and multitime evolution. It is already known that the corresponding quantum mechanical evolution can be understood in great detail. Here, we present an approach to classify such solvable models, namely, to identify all their independent families for a given number N of interacting states and prove the absence of such families for some types of interactions. We also discuss how, within a solvable family, one can classify the scattering matrices, i.e., the system’s dynamics. Due to the possibility of such a detailed classification, the multitime Landau–Zener model defines a useful special function of theoretical physics.
We develop a framework to solve a large class of linearly driven non-Hermitian quantum systems. Such a class of models in the Hermitian scenario is commonly known as multistate Landau-Zener models. The non-Hermiticity is due to the anti-Hermitian couplings between the diabatic levels. We find that there exists a conservation law, unique to this class of models, that describes the simultaneous growth of the unnormalized wave functions. These models have practical applications in Bose-Einstein condensates, and they can describe the dynamics of multispecies bosonic systems. The conservation law relates to a pair-production mechanism that explains the dissociation of diatomic molecules into atoms. We provide a general framework for both solvable and semiclassically solvable non-Hermitian Landau-Zener models. Our findings will open alternative avenues for a number of diverse emergent phenomena in explicitly time-dependent non-Hermitian quantum systems.
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.