Landau-Zener transition driven by slow noise

The effect of a slow noise in nondiagonal matrix element $J\left(t\right)$ that describes the diabatic level coupling on the probability of the Landau-Zener transition is studied. For slow noise, the correlation time ${\tau }_c$ of $J\left(t\right)$ 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-${\tau }_c$ correction for this classical result. Our main finding is that this correction is dominated by sparse realizations of noise for which $J\left(t\right)$ 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 ${\tau }_c$. For Gaussian noise we identify two domains of this falloff with specific dependencies of average transition probability on ${\tau }_c$.

Figure 1

Schematic of the Landau-Zener transition driven by slow telegraph noise. For a typical noise realization the matrix element switches from $J_0$ to $-J_0$ at time $t_0\sim {\tau }_c\gg {\tau }_{\mathrm{LZ}}$. For such realizations the probability $Q_{\mathrm{LZ}}$ to stay on the same diabatic level is exponentially small. However, for sparse realizations with $t_0\lesssim {\tau }_{\mathrm{LZ}}$, the value $Q_{\mathrm{LZ}}$ is close to 1. Thus, it is these sparse realizations that are responsible for the finite-${\tau }_c$ correction.

Figure 2

The dependence of the probability to stay on the initial diabatic level at $t\to \infty$ is shown schematically versus the moment of switching $t_0$. The probability $Q_{\mathrm{LZ}}\left(t_0\right)$ is described by the Lorentzian Eq. (17) for $t_0\sim {\tau }_{\mathrm{LZ}}$ and approaches asymptotically to $exp(-2\pi |\nu \left|\right)$ for $t_0$ much bigger than the time ${\tau }_{\mathrm{LZ}}$ of the Landau-Zener transition.

Figure 3

If the matrix element $J\left(t\right)$ turns to zero at some $t=t_0$ close to the crossing of the diabatic levels (blue curve), the calculation of the transition probability reduces to the Landau-Zener problem with renormalized velocity and coupling both depending on $t_0$. The moment $t_0$ also defines the shift of minima of adiabatic levels from $t=0$. This shift is smaller than $t_0$, see Eq. (22).

Figure 4

Probability to stay on the same diabatic level after the transition is plotted schematically versus the dimensionless correlation time $v^{1/2}{\tau }_c$. In the domain of a “true” slow noise $v^{1/2}{\tau }_c\gg J_c/v^{1/2}$, the probability $Q_{\mathrm{LZ}}$ deviates from the asymptotic value of ${\left(\frac{v}{\pi J_c^2}\right)}^{1/2}ln\left(\frac{J_c}{v^{1/2}}\right)$ only slightly. In the intermediate domain ${(J_c/v^{1/2})}^{1/3}\ll v^{1/2}{\tau }_c\ll J_c/v^{1/2}$ the crossover from the slow to the fast noise takes place.

Figure 5

In the domain of correlation times $J_c^{1/3}/v^{2/3}\ll {\tau }_c\ll J_c/v$ a crossover between the slow-noise and the fast-noise regimes takes place. In this domain, time-dependent adiabatic levels acquire local minima due to the randomness of $J\left(t\right)$. The duration of the Landau-Zener transition (shown with red) in the vicinity of each minimum is much shorter than ${\tau }_c$. The probability to remain on the same adiabatic level after the transition, given by Eq. (36), is close to 1 so that only the transition in the vicinity of $t=t_0$ is responsible for $Q_{\mathrm{LZ}}$.