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Journal of Physics A Mathematical and Theoretical

The finite time multi-levelSU(2) Landau–Zener problems: exact analytical results

S. E. Mkam Tchouobiap, M. B. Kenmoe, Lukong Cornelius Fai

Published Sep 1, 2015• 5 citations
Publisher

Abstract

The multi-level SU (2) Landau–Zener problem is analytically solved at finite time within the framework of the Bloch tensor formalism and with the help of the disentanglement Wei–Norman ordering technique. A generalized and exact analytical solution is achieved that accounts not only for all projections of an arbitrary spin S along the Zeeman field direction but also for non-adiabatic and adiabatic evolutions.

DOI
10.1088/1751-8113/48/39/395301

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The multi-level SU (2) Landau–Zener problem is analytically solved at finite time within the framework of the Bloch tensor formalism and with the help of the disentanglement Wei–Norman ordering technique. A generalized and exact analytical solution is achieved that accounts not only for all projections of an arbitrary spin S along the Zeeman field direction but also for non-adiabatic and adiabatic evolutions.
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)].
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.
We propose a universal approach to the Landau-Zener problem in a three-level system. The problem is formulated in terms of Gell-Mann operators which generate SU (3) algebra and map the Hamiltonian on the effective anisotropic pseudospin 1 model. The vector Bloch equation for the density matrix describing the temporal evolution of the three-level crossing problem is also derived and solved analytically for the case where the diabatic states of the SU (3) Hamiltonian form a triangle. This analytic solution is in excellent quantitative agreement with the numerical solution of the Schrödinger equation for a 3-level crossing problem. The model demonstrates oscillation patterns which radically differ from the standard patterns for the two-level Landau-Zener problem. The triangle works as an interferometer and the interplay between two paths results in formation of "beats" and "steps" pattern in the time-dependent transition probability. The characteristic time scales describing the "beats" and "steps" depend on a dwell time in the triangle. These scales are related to the geometric size of the interferometer. The possibilities of the experimental realization of this effect in triple quantum dots and in two-well traps for cold gases are discussed.
The tunneling between the 2S+12S+12S+1 Zeeman multiplets of an arbitrary spin SSS using the SU(2)SU(2)SU(2) Lie group exponential ordering Wei-Norman technique is investigated. A spin subjected to a regular time-dependent magnetic field and a transverse colored noise is considered and the regimes of fast and slow noise examined.
We formulate and approximately solve a specific many body generalization of the Landau-Zener problem. Unlike with the single particle Landau-Zener problem, our system does not abide in the adiabatic ground state, even at very slow driving rates. The structure of the theory suggests that this finding reflects a more general phenomenon in the physics of adiabatically driven many particle systems. Our solution can be used to understand, for example, the behavior of two-level systems coupled to an electromagnetic field, as realized in cavity QED experiments.
We investigate the Landau–Zener (LZ) like dynamics of decaying two- and three-level systems with decay rates Γ 1 and Γ 2 for levels with minimum and maximum spin projection. Non-adiabatic and adiabatic transition probabilities are calculated from diabatic and adiabatic bases for two- and three-level systems. We extend the familiar two-level model of atoms with decay from the excited state out of the system into the hierarchy of three-level models which can be solved analytically or computationally in a non-perturbative manner. Exact analytical solutions are obtained within the framework of an extended form of the proposed procedure which enables to take into account all possible initial moments rather than large negative time t 0 = − ∞ as in standard LZ problems. We elucidate the applications of our results from a unified theoretical basis that numerically analyzes the dynamics of a system as probed by experiments.