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arXiv:1610.04806 · Physical review. B./Physical review. B

Landau-Zener transition driven by slow noise

Zhu-Xi Luo, M. É. Raǐkh

Published Feb 13, 2017• 11 citations
arXivHTMLPDF

Abstract

The effect of a slow noise in nondiagonal matrix element $J(t)$ that describes the diabatic level coupling on the probability of the Landau-Zener transition is studied. For slow noise, the correlation time ${\ensuremath{\tau}}_{c}$ of $J(t)$ is much longer than the characteristic time of the transition. Existing theory for this case suggests that the average transition probability is the result of averaging of the conventional Landau-Zener probability, calculated for a given constant $J$, over the distribution of $J$. We calculate a finite-${\ensuremath{\tau}}_{c}$ correction for this classical result. Our main finding is that this correction is dominated by sparse realizations of noise for which $J(t)$ passes through zero within a narrow time interval near the level crossing. Two models of noise, random telegraph noise and Gaussian noise, are considered. Naturally, in both models the average probability of transition decreases upon decreasing ${\ensuremath{\tau}}_{c}$. For Gaussian noise we identify two domains of this falloff with specific dependencies of average transition probability on ${\ensuremath{\tau}}_{c}$.

DOI
10.1103/physrevb.95.064305

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The effect of a slow noise in nondiagonal matrix element J(t)J(t)J(t) that describes the diabatic level coupling on the probability of the Landau-Zener transition is studied. For slow noise, the correlation time \ensuremathτc{\ensuremath{\tau}}_{c}\ensuremathτc​ of J(t)J(t)J(t) is much longer than the characteristic time of the transition. Existing theory for this case suggests that the average transition probability is the result of averaging of the conventional Landau-Zener probability, calculated for a given constant JJJ, over the distribution of JJJ. We calculate a finite-\ensuremathτc{\ensuremath{\tau}}_{c}\ensuremathτc​ correction for this classical result. Our main finding is that this correction is dominated by sparse realizations of noise for which J(t)J(t)J(t) passes through zero within a narrow time interval near the level crossing. Two models of noise, random telegraph noise and Gaussian noise, are considered. Naturally, in both models the average probability of transition decreases upon decreasing \ensuremathτc{\ensuremath{\tau}}_{c}\ensuremathτc​. For Gaussian noise we identify two domains of this falloff with specific dependencies of average transition probability on \ensuremathτc{\ensuremath{\tau}}_{c}\ensuremathτc​.
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 study Landau-Zener-like dynamics of a qubit influenced by transverse random telegraph noise. The telegraph noise is characterized by its coupling strength $\ensuremath{\upsilon}$ and switching rate $\ensuremath{\gamma}$. The qubit energy levels are driven nonlinearly in time, $\ensuremath{\propto}\mathrm{sgn}(t){\ensuremath{\mid}t\ensuremath{\mid}}^{\ensuremath{\nu}}$, and we derive the transition probability in the limit of sufficiently fast noise, for arbitrary exponent $\ensuremath{\nu}$. The level occupation after the transition depends strongly on $\ensuremath{\nu}$, and there exists a critical ${\ensuremath{\nu}}_{c}$ with qualitative difference between $\ensuremath{\nu}<{\ensuremath{\nu}}_{c}$ and $\ensuremath{\nu}>{\ensuremath{\nu}}_{c}$. When $\ensuremath{\nu}<{\ensuremath{\nu}}_{c}$, the final state is always fully incoherent with equal population of both quantum levels, even for arbitrarily weak noise. For $\ensuremath{\nu}>{\ensuremath{\nu}}_{c}$, the system keeps some coherence depending on the strength of the noise, and in the limit of weak noise, no transition takes place. For fast noise ${\ensuremath{\nu}}_{c}=1∕2$, while for slow noise ${\ensuremath{\nu}}_{c}<1∕2$ and it depends on $\ensuremath{\gamma}$. We also discuss phase coherence, which is relevant when the qubit has a nonzero minimum energy gap. The qualitative dependency on $\ensuremath{\nu}$ is the same for the phase coherence and level occupation. The state after the transition does, in general, depend on $\ensuremath{\gamma}$. For fixed $\ensuremath{\upsilon}$, increasing $\ensuremath{\gamma}$ decreases the final state coherence when $\ensuremath{\nu}<1$ and increases the final state coherence when $\ensuremath{\nu}>1$. Only the conventional linear driving is independent of $\ensuremath{\gamma}$.
We investigate the Landau-Zener transition in two- and three-level systems subject to a classical Gaussian noise. Two complementary limits of the noise being fast and slow compared to characteristic Landau-Zener tunnel times are discussed. The analytical solution of a density matrix (Bloch) equation is given for a long-time asymptotic of transition probability. It is demonstrated that the transition probability induced or assisted by the fast noise can be obtained through a procedure of Bloch's equation averaging with further reducing it to a master equation. In contrast to the case of fast noise, the transition probability for LZ transition induced, or assisted, by the slow classical noise can be obtained by averaging the solution of Bloch's equation over the noise realization. As a result, the transition probability is described by the activation Arrhenius law. The approximate solution of the Bloch's equation at finite times is written in terms of Fresnel's integrals and interpreted in terms of interference pattern. We discuss consequences of a local isomorphism between SU(2) and SO(3) groups and connections between Schr\"odinger and Bloch descriptions of spin dynamics. Based on this isomorphism, we establish the relations between S=1/2S=1/2S=1/2 and 1 transition probabilities influenced by the noise. A possibility to use the slow noise as a probe for tunnel time is discussed.
We analyze the influence of colored classical Gaussian noise on Landau-Zener transitions during a two-level crossing in a time-dependent regular external field. Transition probabilities and coherence factors become random due to the noise. We calculate their two-time correlation functions, which describe the response of this two-level system to a weak external pulse signal. The spectrum and intensity of the magnetic response are derived. Although the noise enters the equation of motion for the Bloch vector in a multiplicative way, nonperturbative analytic results are obtained by a resummation of diagrams in the limit of a short-noise correlation time. Our results also cover regimes where fluctuations are of the same order of magnitude as averages.
A simple stochastic model is proposed describing the nonadiabatic transitions in level crossing with energy fluctuation. The model is an extension of Zener's model having a diagonal energy term fluctuating as a Markoffian Gaussian process. The transition rate P ∞ defined in terms of the diabatic basis is calculated exactly by the formal perturbation expansion with respect to the off-diagonal coupling in the two limiting cases: In the slow fluctuation limit, P ∞ coincides with the Landau-Zener formula, P∞=PLZ≡1−exp⁡(−2πJ2/ℏ∣v∣)P_{\infty}=P_{\text{LZ}} \equiv 1 - \exp (-2 \pi J^{2}/\hbar |v|)P∞​=PLZ​≡1−exp(−2πJ2/ℏ∣v∣), where J is the off-diagonal coupling constant and v is the velocity of the change of the mean energy difference. In the strong damping limit, which is a limiting case of large fluctuation amplitude in the rapid fluctuation limit, P ∞ is given by the formula, P∞=PSD≡{1−exp⁡(−4πJ2/ℏ∣v∣)}/2P_{\infty}=P_{\text{SD}} \equiv \{1 - \exp (-4 \pi J^{2}/\hbar |v|)\}/2P∞​=PSD​≡{1−exp(−4πJ2/ℏ∣v∣)}/2.
We show by direct calculation starting from a microscopic model that the two-state system with time-dependent energy levels in the presence of fast quantum noise obeys the master equation. The solution of master equation is found analytically and analyzed in a broad range of parameters. The fast transverse noise affects the transition probability during much longer time (the accumulation time) than the longitudinal one. The action of the fast longitudinal noise is restricted by the shorter Landau-Zener time, the same as in the regular Landau-Zener process. The large ratio of time scales allows solving the Landau-Zener problem with longitudinal noise only, and then solving the same problem with the transverse noise only and matching the two solutions. The correlation of the longitudinal and transverse noise renormalizes the Landau-Zener transition matrix element and can strongly enhance the survival probability, whereas the transverse noise always reduces it. If the noise is fast, its intensity at which the multiquantum processes become essential corresponds to a deeply adiabatic regime. We briefly discuss possible applications of the general theory to the molecular magnets.