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. 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, fast oscillator renormalizes the matrix element of the transition rather than the drive velocity. This renormalization makes the transition probability a non-monotonic function of the coupling amplitude.

We consider the simplest non-integrable model of multistate Landau-Zener transition. In this model two pairs of levels in two tunnel coupled quantum dots are swept passed each other by the gate voltage. Although this 2 × 2 model is non-integrable, it can be solved analytically in the limit when the inter-level energy distance is much smaller than their tunnel splitting. The result is contrasted to the similar 2 × 1 model, in which one of the dots contains only one level. The latter model does not allow interference of the virtual transition amplitudes, and it is exactly solvable. In 2 × 1 model, the probability for a particle, residing at time t → −∞ in one dot, to remain in the same dot at t → ∞ falls off exponentially with tunnel coupling. By contrast, in 2 × 2 model, this probability grows exponentially with tunnel coupling. The physical origin of this growth is the formation of the tunneling-induced collective states in the system of two dots. This can be viewed as manifestation of the Dicke effect. PACS numbers: v 2 v 2 J J J J arXiv:1706.09503v1 [cond-mat.mes-hall] 28 Jun 2017 0 J 0 ∆ − vt 2 J J J vt 2 −∆ − vt 2 0 J J 0 ∆ − vt 2 J J J J −∆ + vt 2

When the drive, which causes the level crossing in a qubit, is slow, the probability, PLZ, of the Landau-Zener transition is close to 1. In this regime, which is most promising for applications, the noise due to the coupling to the environment, reduces the average PLZ. At the same time, the survival probability, 1 − PLZ, which is exponentially small for a slow drive, can be completely dominated by noise-induced correction. Our main message is that the effect of a weak classical noise can be captured analytically by treating it as a perturbation in the Schrödinger equation. This allows us to study the dependence of the noise-induced correction to PLZ on the correlation time of the noise. As this correlation time exceeds the bare Landau-Zener transition time, the effect of noise becomes negligible. On the physical level, the mechanism of enhancement of the survival probability can be viewed as an absorption of the "noise quanta" across the gap. With characteristic energy of the quantum governed by the noise spectrum, the slower is the noise, the less is the number of quanta for which the absorption is allowed energetically. We consider two conventional realizations of noise: gaussian noise and telegraph noise.

We demonstrate that the general model of a linearly time-dependent crossing of two energy bands is integrable. Namely, the Hamiltonian of this model has a qaudratically time-dependent commuting operator. We apply this property to four-state Landau-Zener (LZ) models that have previously been used to describe the Landau-Stückelberg interferometry experiments with an electron shuttling between two semiconductor quantum dots. The integrability then leads to simple but nontrivial exact relations for the transition probabilities. In addition, the integrability leads to a semiclassical theory that provides analytical approximation for the transition probabilities in these models for all parameter values. The results predict a dynamic phase transition, and show that similarly-looking models belong to different topological classes.

The Harper equation arising out of a tight-binding model of electrons on a honeycomb lattice subject to a uniform magnetic field perpendicular to the plane is studied. Contrasting and complementary approaches involving von Neumann entropy, fidelity, fidelity susceptibility, and multifractal analysis are employed to characterize the phase diagram. Remarkably even in the absence of the quasi-periodic on-site potential term, the Hamiltonian allows for a metal-insulator transition. The phase diagram consists of three phases: two metallic phases and an insulating phase. A variant model where next nearest neighbor hopping is included, exhibits a mobility edge and does not allow for a simple single phase diagram characterizing all the eigenstates.