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Physical Review A

Uniform analytic description of dephasing effects in two-state transitions

X. Lacour, S. Guérin, Леонід Петрович Яценко, Nikolay V. Vitanov, Hans Rudolf Jauslin

Published Mar 27, 2007• 32 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.75.033417

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Physical Review A•Jun 1, 1996•256 citations
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.
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.
We present an analytically exactly soluble two-state model, in which a hyperbolic-secant-shaped pulsed interaction has a phase jump of $\ensuremath{\phi}$ at the time of its maximum. The detuning has a constant part and a hyperbolic-tangent chirp term. For $\ensuremath{\phi}=0$, this model reduces to the Demkov-Kunike model, which in turn contains as particular cases three other well-known models: the Rosen-Zener, Allen-Eberly, and Bambini-Berman models. A nonzero $\ensuremath{\phi}$ induces dramatic changes in the transition probability, ranging from complete population inversion to complete population return. The analytic results are particularly instructive in the adiabatic limit and demonstrate that complete population inversion can always occur for a suitable choice of $\ensuremath{\phi}$. The jump phase $\ensuremath{\phi}$ can therefore be used as a control parameter for the two-state transition probability.
We present the generalization of the Landau-Zener model for a constant coupling of a finite duration. The exact evolution matrix is expressed in terms of sums of by-products of parabolic cylinder functions estimated at the turn-on time and at the turn-off time of the coupling. Various approximations in terms of simpler functions are derived and applied to several physically distinct cases. They allow us to study the dependence of the transition probability on the interaction parameters: coupling strength, coupling duration, and detuning slope. Furthermore, the analytic approximations reveal the effects of the finite coupling duration as well as those caused by adding a constant detuning shift, absence of a level crossing, turn-on time or turn-off time near the crossing (``half crossing''), turn-on time and turn-off time near the crossing (``nonsubstantial crossing''). The results are used to obtain analytic approximations to the time evolution in the original Landau-Zener model. Furthermore, following related studies on other models, we define the Landau-Zener class of models that, along with the finite Landau-Zener model presented in this work, contains an infinite number of members that give the same transition probability. Comparison of this class to the Allen-Eberly class shows that the two classes contain members with the same coupling but different detuning chirps as well as members with the same chirp but different couplings. The former case reveals chirp effects while the latter demonstrates shape effects. \textcopyright{} 1996 The American Physical Society.