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arXiv:1708.05483 · Physical review. B./Physical review. B

Landau-Zener transition in a two-level system coupled to a single highly excited oscillator

Rajesh K. Malla, M. É. Raǐkh

Published Jan 22, 2018• 22 citations
arXivHTMLPDF

Abstract

Two-level system strongly coupled to a single resonator mode (harmonic oscillator) is a paradigmatic model in many subfields of physics. We study theoretically the Landau-Zener transition in this model. Analytical solution for the transition probability is possible when the oscillator is highly excited, i.e., at high temperatures. Then the relative change of the excitation level of the oscillator in the course of the transition is small. The physical picture of the transition in the presence of coupling to the oscillator becomes transparent in the limiting cases of slow and fast oscillator. A slow oscillator effectively renormalizes the drive velocity. As a result, the transition probability either increases or decreases depending on the oscillator phase. The net effect is, however, the suppression of the transition probability. On the contrary, a fast oscillator renormalizes the matrix element of the transition rather than the drive velocity. This renormalization makes the transition probability a nonmonotonic function of the coupling amplitude.

DOI
10.1103/physrevb.97.035428

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Two-level system strongly coupled to a single resonator mode (harmonic oscillator) is a paradigmatic model in many subfields of physics. We study theoretically the Landau-Zener transition in this model. Analytical solution for the transition probability is possible when the oscillator is highly excited, i.e., at high temperatures. Then the relative change of the excitation level of the oscillator in the course of the transition is small. The physical picture of the transition in the presence of coupling to the oscillator becomes transparent in the limiting cases of slow and fast oscillator. A slow oscillator effectively renormalizes the drive velocity. As a result, the transition probability either increases or decreases depending on the oscillator phase. The net effect is, however, the suppression of the transition probability. On the contrary, a fast oscillator renormalizes the matrix element of the transition rather than the drive velocity. This renormalization makes the transition probability a nonmonotonic function of the coupling amplitude.
We consider the Landau-Zener problem for a two-level system (or qubit) when this system interacts with one harmonic oscillator mode that is initially set to a finite-temperature thermal equilibrium state. The oscillator could represent an external mode that is strongly coupled to the qubit, e.g., an ionic oscillation mode in a molecule, or it could represent a prototypical uncontrolled environment. We analyze the qubit's occupation probabilities at the final time in a number of regimes, varying the qubit and oscillator frequencies, their coupling strength, and the temperature. In particular, we find a surprising nonmonotonic dependence on the coupling strength and temperature.
An analytical study of the dissipative Landau-Zener model is presented. The model where two energy levels at constant speed are brought to cross is a standard model used to describe a large variety of phenomena. In many cases of interest the presence of coupling of the two-state system to an environment is of importance, the accounting for which shall be done here from first principles. Analytical results for the excitation transition from the ground state of the two-state system at large negative times to the excited state at large positive times (as well as the opposite, decay, transition) are obtained in terms of the speed by which the two levels approach each other, the energy gap between the adiabatic energies, the coupling strength to the environment, and its temperature. For the excitation transition we find the following results: In the slow-passage limit of small sweeping speed it is shown that adiabaticity is limited to low temperatures and the quantitative adiabatic criterion is established. Particularly, at zero temperature there is no influence of the environment on the transition probability as a consequence of a compensation property shown to be peculiar to the linear-sweep model. The transition is at low temperatures due to quantum tunneling and, with an increase in temperature, an intermediate region appears where the transition is dominated by thermally assisted transitions across the energy gap before finally at high temperatures a saturated regime is reached with equal population of the levels. In contrast to the dependence on the temperature the dependence of the transition probability as a function of coupling strength is nonmonotonic with maximum influence at intermediate strength. In the fast-passage limit with rapid sweep speed there is no influence of the environment on the transition probability. For the decay transition the adiabatic limit does not exist for the linear-sweep model for physically relevant spectra of the environment and the decay transition is dominated by spontaneous emission, except in the fast-passage limit and the high-temperature limit where the decay transition probability equals the excitation probability.
The Landau-Zener transition in a two level system can be suppressed or enhanced by coupling to an environment, depending on the temperature and the environment spectral function. We consider the marginal spectral function, when the dissipation effects are important for arbitrarily slow motion. Landau-Zener transition rate demonstrates a non-trivial dependence of the on the ``bias'', i. e. on the rate of the two energy levels relative motion. The Landau-Zener transition is fully suppressed for the values of the bias below a threshold bias set by the coupling strength. Above the threshold, the transition rate for zero temperature is found using the instanton method. At finite temperature, the Landau-Zener transition rate has a non-monotonic dependence on the coupling strength, being suppressed at the strong coupling.
We study Landau-Zener transitions in a fermionic dissipative environment where a two-level (up and down states) system is coupled to two metallic leads kept with different chemical potentials at zero temperature. The dynamics of the system is simulated by an iterative numerically exact influence functional path integral method. In the pure Landau-Zener problem, two kinds of transition (from up to down state and from down to up state) probability are symmetric. However, this symmetry is destroyed by coupling the system to the bath. In addition, in both kinds of transitions, there exists a nonmonotonic dependence of the transition probability on the sweep velocity; meanwhile nonmonotonic dependence of the transition probability on the system-bath coupling strength is only shown in one of them. As in the spin-boson model, these phenomena can be explained by a simple phenomenological model.
The time-dependent behavior of a two-level system interacting with a quantum oscillator system is analyzed in the case of a coupling larger than both the energy separation between the two levels and the energy of the quantum oscillator ($\ensuremath{\Omega}<\ensuremath{\omega}<\ensuremath{\lambda}$, where $\ensuremath{\Omega}$ is the frequency of the transition between the two levels, $\ensuremath{\omega}$ is the frequency of the oscillator, and $\ensuremath{\lambda}$ is the coupling between the two-level system and the oscillator). Our calculations show that the amplitude of the expectation value of the oscillator coordinate decreases as the two-level system undergoes the transition from one level to the other, while the transfer probability between the levels is staircaselike. This behavior is explained by the interplay between the adiabatic and nonadiabatic regimes encountered during the dynamics with the system acting as a quantum counterpart of the Landau-Zener model. The transition between the two levels occurs as long as the expectation value of the oscillator coordinate is driven close to zero. On the contrary, if the initial conditions are set such that the expectation values of the oscillator coordinate are far from zero, the system will remain locked on one level.
We consider the Landau-Zener problem for a multilevel quantum system that is coupled to an external environment. In particular, we consider a number of cases of three-level systems coupled to a harmonic oscillator that represents the external environment. We find that, similar to the case of the Landau-Zener problem with a two-level system, when the quantum system and the environment are both initially in their ground states the probability that the system remains in the same quantum state is not affected by the coupling to the environment. The final occupation probabilities of the other states are well described by a common general principle: the coupling to the environment turns each Landau-Zener transition process in the closed system into a sequence of smaller transitions in the combined Hilbert space of the system and environment, and this sequence of transitions lasts a total duration that increases with increasing system-environment coupling strength. These results provide an intuitive understanding of Landau-Zener transitions in open multilevel quantum systems.