Universality in dissipative Landau-Zener transitions

We introduce a random-variable approach to investigate the dynamics of a dissipative two-state system. Based on an exact functional integral description, our method reformulates the problem as that of the time evolution of a quantum state vector subject to a Hamiltonian containing random noise fields. This numerically exact, nonperturbative formalism is particularly well suited in the context of time-dependent Hamiltonians, at both zero and finite temperature. As an important example, we consider the renowned Landau-Zener problem in the presence of an Ohmic environment with a large cutoff frequency at finite temperature. We investigate the “scaling” limit of the problem at intermediate times, where the decay of the upper-spin-state population is universal. Such a dissipative situation may be implemented using a cold-atom bosonic setup.

Figure 1

(a) $P(t)$ as a function of $t$ for various values of $\alpha$, $\Delta =1$, ${\omega }_c=100$, $\epsilon =0$, and $T=0$. We checked that for a given $\alpha$ curves corresponding to different ${\omega }_c/\Delta \gg 1$ scale on top of each other in units of the renormalized tunneling rate ${\Delta }_r=\Delta (\frac{\Delta }{{\omega }_c}){}^{\alpha /(1-\alpha )}$. Quality factor $\Omega /\gamma$ of damped oscillations agrees with prediction $\Omega /\gamma =\cot \frac{\pi \alpha }{2(1-\alpha )}$ from Refs. [25, 26, 29]. Results are obtained with $m_{max}=3000$, $N=5\times {10}^4$. (b) $P(t)$ for different temperatures $T$ (here, $\hslash =k_B=1$), dissipation strength $\alpha =0.1$, and other parameters as in (a).

Figure 2

(a) $p(t)$ for a fast sweep with $v/{\Delta }^2=10$, ${\omega }_c/\Delta =200$, $\alpha =\{0.05,0.1\}$, and $T/\Delta =\{0,1,5\}$ (here, $\hslash =k_B=1$). We choose $m_{max}=4000$, $N=4\times {10}^6$. (b) Slow sweep with $v/{\Delta }^2=0.5$. Other parameters as in (a). (c) Fit of universal decay of $p[\epsilon (t)]$ using Eq. (12) with $\alpha =0.05$ and single fit parameter $C=0.59$.