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Physical review. B, Condensed matter

Dephasing of interference in Landau-Zener transitions

Efrat Shimshoni, Ady Stern

Published Apr 15, 1993• 43 citations
Publisher

Abstract

We study the effect of dephasing on the Landau-Zener transition. Our approach is based on the interpretation of the transition as an interference process. We derive a phase-uncertainty factor for a general coupling to an environment, and demonstrate the enhancement of the asymptotic transition probability due to this phase uncertainty. We discuss the dependence of this enhancement on the properties of a specific environment at hand, the form of the coupling term, and the coupling strength. Given a certain type of coupling to the environment, we find the characteristic dephasing time (${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\varphi}}}$) corresponding to the crossover from weak to strong dephasing. We argue that this crossover ${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\varphi}}}$ can be regarded as a probe of the relevant time scale of phase oscillations, which are suppressed by the specific dephasing mechanism.

DOI
10.1103/physrevb.47.9523

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We study the effect of dephasing on the Landau-Zener transition. Our approach is based on the interpretation of the transition as an interference process. We derive a phase-uncertainty factor for a general coupling to an environment, and demonstrate the enhancement of the asymptotic transition probability due to this phase uncertainty. We discuss the dependence of this enhancement on the properties of a specific environment at hand, the form of the coupling term, and the coupling strength. Given a certain type of coupling to the environment, we find the characteristic dephasing time (\ensuremathτ\ensuremathφ{\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\varphi}}}\ensuremathτ\ensuremathφ​) corresponding to the crossover from weak to strong dephasing. We argue that this crossover \ensuremathτ\ensuremathφ{\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\varphi}}}\ensuremathτ\ensuremathφ​ can be regarded as a probe of the relevant time scale of phase oscillations, which are suppressed by the specific dephasing mechanism.
The quantum dynamics of the Landau-Zener transition in a dissipative environment is studied and analytical results for the transition probability are given in terms of temperature, coupling strength, and the Landau-Zener time. For short Landau-Zener time there is no effect of dissipation. In the oppotise limit we distinguish various temperature regimes: The adiabatic limit is shown to be restricted to low temperatures and no effect of dissipation is present at zero temperature. At intermediate temperatures thermal transitions dominate for weak coupling and high temperatures correspond to the strong-coupling limit.
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.
Landau-Zener tunneling, which describes the transition in a two-level system during a sweep through an anti-crossing, is a model applicable to a wide range of physical phenomena. Realistic quantum systems are affected by dissipation due to coupling to their environments. An important aspect of understanding such open quantum systems is the relative energy scales of the system itself and the system-environment coupling, which distinguishes the weak- and strong-coupling regimes. Using a tunable superconducting flux qubit, we observe the crossover from weak to strong coupling to the environment in Landau-Zener tunneling. Our results confirm previous theoretical studies of dissipative Landau-Zener tunneling in the weak and strong coupling limits. We devise a spin bath model that effectively captures the crossover regime. This work is relevant for understanding the role of dissipation in quantum annealing, where the system is expected to go through a cascade of Landau-Zener transitions before reaching the target state.
By considering a quantum critical Lipkin-Meshkov-Glick model we analyze a new type of Landau-Zener transitions where the population transfer is mediated by interaction rather than from a direct diabatic coupling. For this scenario, at a mean-field level the dynamics is greatly influenced by quantum interferences. In particular, regardless of how slow the Landau-Zener sweep is, for certain parameters almost no population transfer occurs, which is in stark contrast to the regular Landau-Zener model. For moderate system sizes, this counterintuitive mean-field behaviour is not duplicated in the quantum case. This can be attributed quantum fluctuations and the fact that multi-level Landau-Zener-Stückelberg interferences have a `dephasing' effect on the above mentioned phenomenon. We also find a discrepancy between the quantum and mean-field models in terms of how the transfer probabilities scale with the sweep velocity.
When the drive, which causes the level crossing in a qubit, is slow, the probability PLZ{P}_{LZ}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{P}_{LZ}PLZ​. At the same time, the survival probability, $1\ensuremath{-}{P}_{LZ}$, which is exponentially small for a slow drive, can be completely dominated by noise-induced correction. Our main message is that the effect of weak classical noise can be captured analytically by treating it as a perturbation in the Schr\"odinger equation. This allows us to study the dependence of the noise-induced correction to PLZ{P}_{LZ}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 the noise is, the lower the number of quanta for which absorption is allowed energetically is. We consider two conventional realizations of noise: Gaussian noise and telegraph noise.
We consider the Landau-Zener problem for a multilevel quantum system that is coupled to an external environment. In particular, we consider a number of cases of three-level systems coupled to a harmonic oscillator that represents the external environment. We find that, similar to the case of the Landau-Zener problem with a two-level system, when the quantum system and the environment are both initially in their ground states the probability that the system remains in the same quantum state is not affected by the coupling to the environment. The final occupation probabilities of the other states are well described by a common general principle: the coupling to the environment turns each Landau-Zener transition process in the closed system into a sequence of smaller transitions in the combined Hilbert space of the system and environment, and this sequence of transitions lasts a total duration that increases with increasing system-environment coupling strength. These results provide an intuitive understanding of Landau-Zener transitions in open multilevel quantum systems.