Demkov-Kunike Models with Decay

TL;DR

This paper derives exact analytical solutions for dissipative two-state quantum systems with decay, driven by complex pulses, unifying models like Demkov-Kunike, Rabi, and Landau-Zener, and exploring different detuning regimes.

Contribution

It introduces a comprehensive analytical framework for decaying Demkov-Kunike models with arbitrary initial conditions, encompassing various pulse shapes and detuning limits.

Findings

Exact solutions for decaying DK models with arbitrary initial conditions.

Reduction of DK models to Rabi and Landau-Zener models in specific limits.

Analytical approximations that closely match exact solutions.

Abstract

Exact analytical solutions to the dissipative time-dependent Schr\"odinger equation are obtained for a decaying two-state system with decay rates $\Gamma_{1}$ and $\Gamma_{2}$ for levels with extremal spin projections. The system is coherently driven with a pulse whose detuning is made up of two parts: a time-dependent part (chirp) of hyperbolic-tangent shape and a static part with real and imaginary terms. This gives us a wide range of possibilities to arbitrarily select the interaction terms. We considered two versions which led to decaying Demkov-Kunike (DK) models; the version in which the Rabi frequency (interaction) is a time-dependent hyperbolic-secant function (called decaying DK1 model) and the case when it is constant in time and never turns off (decaying DK2 model). Our analytical solutions account for all possible initial moments instead of only $t_{0}=0$ or $t_{0}=-\infty$ as for non-decaying models and may be useful for experiments on level crossings. Two complementary limits of the pulse detuning are considered and explored: the limit of fast (i) and slow rise (ii). In the case (i), the coupling between level positions in the first DK model collapses while the second DK model reduces to a Rabi model (constant Hamiltonian), in the case (ii), both DK models reduce to the LZ model. In both cases (i) and (ii), analytical approximated solutions which conveniently approach the exact solutions are derived.