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Physical Review A

Adiabatic evolution amidst dephasing

Peter A. Ivanov, Nikolay V. Vitanov

Published Jun 21, 2005• 17 citations
Publisher

Abstract

An analytic description of adiabatic evolution of a two-state system in the presence of dephasing processes is presented. The adiabatic solution of the Bloch equation is derived in the most general case of a mixed initial state. Examples of the general analytic solution are given in the particular cases of energy-level crossing, no crossing, and half crossing.

DOI
10.1103/physreva.71.063407

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An analytic description of adiabatic evolution of a two-state system in the presence of dephasing processes is presented. The adiabatic solution of the Bloch equation is derived in the most general case of a mixed initial state. Examples of the general analytic solution are given in the particular cases of energy-level crossing, no crossing, and half crossing.
We discuss adiabatic quantum phenomena at a level crossing. Given a path in the parameter space which passes through a degeneracy point, we find a criterion which determines whether the adiabaticity condition can be satisfied. For paths that can be traversed adiabatically we also derive a differential equation which specifies the time dependence of the system parameters, for which transitions between distinct energy levels can be neglected. We also generalize the well-known geometric connections to the case of adiabatic paths containing arbitrarily many level-crossing points and degenerate levels.
Abstract We consider adiabatic interaction of quantum systems with electromagnetic field in the presence of various dissipation processes. As it is known that in the presence of large intermediate detunings any n-level system can be reduced to an effective two-level system, we chose the two-level model as the basic model for a detailed analytical study. We demonstrate the possibility of reducing losses due to dephasing and non-adiabatic corrections by choosing an appropriate design of time-dependent interaction parameters. Simple analytical expressions are derived for both cold and hot atomic ensembles. The results obtained for the two-level system are applied to a three-level system by using the method of adiabatic elimination. Efficient population transfer is shown despite the relatively high dephasing rates.
Two aspects of the classic two-level Landau-Zener (LZ) problem are considered. First, we address the LZ problem when one or both levels decay, i.e., ${\ensuremath{\varepsilon}}_{j}(t)\ensuremath{\rightarrow}{\ensuremath{\varepsilon}}_{j}(t)\ensuremath{-}i{\ensuremath{\Gamma}}_{j}/2$. We find that if the system evolves from an initial time $\ensuremath{-}T$ to a final time +T+T+T such that $|{\ensuremath{\varepsilon}}_{1}(\ifmmode\pm\else\textpm\fi{}T)\ensuremath{-}{\ensuremath{\varepsilon}}_{2}(\ifmmode\pm\else\textpm\fi{}T)|$ is not too large, the LZ survival probability of a state $|j\ensuremath{\rangle}$ can increase with increasing decay rate of the other state $|i\ensuremath{\ne}j\ensuremath{\rangle}$. This surprising result occurs because the decay results in crossing of the two eigenvalues of the instantaneous non-Hermitian Hamiltonian. On the other hand, if $|{\ensuremath{\varepsilon}}_{1}(\ifmmode\pm\else\textpm\fi{}T)\ensuremath{-}{\ensuremath{\varepsilon}}_{2}(\ifmmode\pm\else\textpm\fi{}T)|\ensuremath{\rightarrow}\ensuremath{\infty}$ as $T\ensuremath{\rightarrow}\ensuremath{\infty}$, the probability is independent of the decay rate. These results are based on analytic solutions of the time-dependent Schr\"odinger equations for two cases: (a) the energy levels depend linearly on time, and (b) the energy levels are bounded and of the form ${\ensuremath{\varepsilon}}_{1,2}(t)=\ifmmode\pm\else\textpm\fi{}\ensuremath{\varepsilon}tanh(t/\mathcal{T})$. Second, we study LZ transitions affected by dephasing by formulating the Landau-Zener problem with noise in terms of a Schr\"odinger-Langevin stochastic coupled set of differential equations. The LZ survival probability then becomes a random variable whose probability distribution is shown to behave very differently for long and short dephasing times. We also discuss the combined effects of decay and dephasing on the LZ probability.
An analytic theory of the adiabatic passage (AP) regime of interaction of short frequency chirped laser pulses with a two-level quantum system (QS) being initially in the superpositional quantum state is presented. We show that the initial value of the non-diagonal elements of the density matrix can influence the dynamics of the population transfer during the action of the laser pulse, but do not influence essentially the final populations of the states of the QS obtained at the end of the interaction near the AP regime. A novel solution to the Bloch equations is obtained in the form of a converging power series generalizing the ordinary AP solution.
We describe the effect of pure dephasing upon the time-dependent dynamics of two-state quantum systems in the framework of a Lindblad equation for the time evolution of the density matrix. A uniform approximate formula is derived, which modifies the corresponding lossless transition probability by an exponential factor containing the dephasing rate and the interaction parameters. This formula is asymptotically exact in both the diabatic and adiabatic limits; comparison with numerical results shows that it is highly accurate also in the intermediate range. Several two-state models are considered in more detail, including the Landau-Zener, Rosen-Zener, Allen-Eberly, and Demkov-Kunike models, along with several other models, such as a Gaussian model and a Landau-Zener model with a pulsed coupling.