Suppression of the Landau-Zener transition probability by weak classical noise

When the drive, which causes the level crossing in a qubit, is slow, the probability $P_{LZ}$ 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 $P_{LZ}$. At the same time, the survival probability, $1-P_{LZ}$, 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 Schrödinger equation. This allows us to study the dependence of the noise-induced correction to $P_{LZ}$ 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.

Figure 1

Schematic illustration of the Landau-Zener transition in the presence of weak transverse noise. The noise causes random fluctuations of the gap, $2J$. When the gap is much bigger than $v^{1/2}$, where $v$ is the sweep velocity, the LZ transition is almost fully adiabatic, so that the “survival” probability $Q_{LZ}$ to stay on the initial diabatic level is exponentially small. Then, even weak noise yields a dominant contribution to $Q_{LZ}$.

Figure 2

The integral $I_2\left(\omega \right)$ [in the units $4i{\left(2\pi \right)}^{1/2}/J^3]$ is plotted from Eq. (25) versus the dimensionless variable, $u=\left(1-\omega /2J\right)$, for two values of the dimensionless parameter $\nu$: $\nu =3$ (orange) and $\nu =3.5$ (blue). For negative $u,I\left(\omega \right)$ oscillates and reproduces the semiclassical result (22) after the first maximum. For positive $u$ it falls off exponentially. Although the $u>0$ tail is slim, it is responsible for the survival probability when the noise is slow.

Figure 3

Survival probability $Q_{LZ}$ is plotted from Eqs. (27, 28, 29) versus the dimensionless noise correlation time (a) for the telegraph noise and (b) for the Gaussian noise. The contribution from the absorption of the high-frequency noise quanta ($\omega >2J$) are shown with red dashed lines (${\mathcal{F}}_1$), while the contributions from the absorption with $\omega <2J$ are shown with purple dashed lines (${\mathcal{F}}_2$). The insets illustrate the evolution of the low-frequency contribution with increasing $\nu =J^2/v$. The smaller the gap is, the stronger the absorption of the subgap quanta is. The characteristic correlation time being $\tau \sim 1/J$, which is much shorter than the time $J/v$ of the LZ transition time, indicates that it is fast noise that suppresses the adiabaticity.

Figure 4

Landau-Zener probability in the presence of slow noise is plotted from Eq. (37) as a function of the dimensionless noise magnitude for different values of the parameter $\nu$, which quantifies the bare survival probability $Q_{LZ}=exp(-2\pi |\nu \left|\right)$. For $2\pi \left|\nu \right|<1/2$ the curves grow monotonically, which suggests that noise enhances the adiabaticity. For $2\pi \left|\nu \right|>1/2$, i.e., when the transition is adiabatic in the absence of noise, the curves exhibit minima, suggesting that weak noise suppresses the adiabaticity, while strong noise, with a magnitude exceeding the gap, enhances it.