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Physical Review A

Evolution of superpositions of quantum states through a level crossing

Boyan T. Torosov, Nikolay V. Vitanov

Published Dec 9, 2011• 14 citations
Publisher

Abstract

The Landau-Zener-St\"uckelberg-Majorana (LZSM) model is widely used for estimating transition probabilities in the presence of crossing energy levels in quantum physics. This model, however, makes the unphysical assumption of an infinitely long constant interaction, which introduces a divergent phase in the propagator. This divergence remains hidden when estimating output probabilities for a single input state insofar as the divergent phase cancels out. In this paper we show that, because of this divergent phase, the LZSM model is inadequate to describe the evolution of pure or mixed superposition states across a level crossing. The LZSM model can be used only if the system is initially in a single state or in a completely mixed superposition state. To this end, we show that the more realistic Demkov-Kunike model, which assumes a hyperbolic-tangent level crossing and a hyperbolic-secant interaction envelope, is free of divergences and is a much more adequate tool for describing the evolution through a level crossing for an arbitrary input state. For multiple crossing energies which are reducible to one or more effective two-state systems (e.g., by the Majorana and Morris-Shore decompositions), similar conclusions apply: the LZSM model does not produce definite values of the populations and the coherences, and one should use the Demkov-Kunike model instead.

DOI
10.1103/physreva.84.063411

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The Landau-Zener-St"uckelberg-Majorana (LZSM) model is widely used for estimating transition probabilities in the presence of crossing energy levels in quantum physics. This model, however, makes the unphysical assumption of an infinitely long constant interaction, which introduces a divergent phase in the propagator. This divergence remains hidden when estimating output probabilities for a single input state insofar as the divergent phase cancels out. In this paper we show that, because of this divergent phase, the LZSM model is inadequate to describe the evolution of pure or mixed superposition states across a level crossing. The LZSM model can be used only if the system is initially in a single state or in a completely mixed superposition state. To this end, we show that the more realistic Demkov-Kunike model, which assumes a hyperbolic-tangent level crossing and a hyperbolic-secant interaction envelope, is free of divergences and is a much more adequate tool for describing the evolution through a level crossing for an arbitrary input state. For multiple crossing energies which are reducible to one or more effective two-state systems (e.g., by the Majorana and Morris-Shore decompositions), similar conclusions apply: the LZSM model does not produce definite values of the populations and the coherences, and one should use the Demkov-Kunike model instead.
We analyze the effect of a dissipative bosonic environment on the Landau-Zener-St\"uckelberg-Majorana (LZSM) level crossing model by using a microscopic approach to derive the relevant master equation. For an environment at zero temperature and weak dissipation, our microscopic approach confirms the independence of the survival probability on the decay rate that has been predicted earlier by the simple phenomenological LZSM model. For strong decay the microscopic approach predicts a notable increase of the survival probability, which signals dynamical decoupling of the initial state. Unlike the phenomenological model, our approach makes it possible to study the dependence of the system dynamics on the temperature of the environment. In the limit of very high temperature we find that the dynamics is characterized by a very strong dynamical decoupling of the initial state---the temperature-induced quantum Zeno effect.
A driven quantum system can experience Landau-Zener-St\"uckelberg-Majorana (LZSM) transitions between its states when the respective energy levels quasicross. If this quasicrossing is traversed repeatedly under periodic driving, the trajectories can interfere either constructively or destructively. In the latter case, known as coherent destruction of tunneling, the transition between the energy states is suppressed. Even for the double-passage case, the accumulated phase difference (also referred to as the St\"uckelberg phase) can lead to destructive interference, resulting in no transition. In this paper, we discuss a similar process for the single-passage dynamics. We study the LZSM single-passage problem starting from a superposition state. The phase difference of this initial state results in interference. When this results in either a zero or a unit transition probability, such a situation can be called single-passage complete localization in a target state. The phase can be chosen so that the occupation probabilities do not change after the transition, which is analogous to the problem of transitionless driving. We demonstrate how varying the system parameters (driving velocity, initial phase, initial detuning) can be used for quantum coherent control.
The exact analytical solution of the degenerate Landau-Zener model, wherein two bands of degenerate energies cross in time, is presented. The solution is derived by using the Morris-Shore transformation, which reduces the fully coupled system to a set of independent nondegenerate two-state systems and a set of decoupled states. Due to the divergence of the phase of the off-diagonal element of the propagator in the original Landau-Zener model, not all transition probabilities exist for infinite time duration. In general, apart from some special cases, only the transition probabilities between states within the same degenerate set exist, but not between states of different sets. An illustration is presented for the transition between the magnetic sublevels of two atomic levels with total angular momenta J=2J=2J=2 and 1.
Since the pioneering works by Landau, Zener, Stückelberg, and Majorana (LZSM), it has been known that driving a quantum two-level system results in tunneling between its states. Even though the interference between these transitions is known to be important, it is only recently that it became both accessible, controllable, and useful for manipulating a growing number of quantum systems. Here, we systematically study various aspects of LZSM physics and review the relevant literature, significantly expanding the review article in Ref. [504].
While the Hamiltonians used in standard quantum mechanics are Hermitian, it is also possible to extend the theory to non-Hermitian Hamiltonians. Particularly interesting are non-Hermitian Hamiltonians satisfying parity–time ( P T ) symmetry, or more generally pseudo-Hermiticity, since such non-Hermitian Hamiltonians can still exhibit real eigenvalues. In this article, we present a quantum simulation of the time-dependent non-Hermitian non- P T -symmetric Hamiltonian used in a pseudo-Hermitian extension of the Landau–Zener–Stückelberg–Majorana (LZSM) model. The simulation is implemented on a superconducting processor by using Naimark dilation to transform a non-Hermitian Hamiltonian for one qubit into a Hermitian Hamiltonian for a qubit and an ancilla; postselection on the ancilla state ensures that the qubit undergoes nonunitary time evolution corresponding to the original non-Hermitian Hamiltonian. We observe properties such as the dependence of transition probabilities on time and the replacement of conservation of total probability by other dynamical invariants in agreement with predictions based on a theoretical treatment of the pseudo-Hermitian LZSM system. Published by the American Physical Society 2024
During the adiabatic time evolution levels crossing violates the adiabaticity and makes transitions between levels possible. Conventionally only two energy levels cross simultaneously. The transition probabilities for this case were found by Landau and Zener. However, the multilevel crossing happens systematically rather than occasionally if the Hamiltonian possesses a special symmetry. The simplest physical realization of the multilevel crossing are the Zeeman multiplet in a varying magnetic field and an electron in one-dimensional chain driven by the time-dependent electric field. We present asymptotics of the transition amplitudes for these kinds of the multilevel crossing. They are based on an exact solution for a model n-state, time-dependent Schrödinger equation.