Landau-Zener transitions in a two-level system coupled to a finite-temperature harmonic oscillator

We consider the Landau-Zener problem for a two-level system (or qubit) when this system interacts with one harmonic oscillator mode that is initially set to a finite-temperature thermal equilibrium state. The oscillator could represent an external mode that is strongly coupled to the qubit, e.g., an ionic oscillation mode in a molecule, or it could represent a prototypical uncontrolled environment. We analyze the qubit's occupation probabilities at the final time in a number of regimes, varying the qubit and oscillator frequencies, their coupling strength, and the temperature. In particular, we find a surprising nonmonotonic dependence on the coupling strength and temperature.

Figure 1

Energy level diagram of a coupled qubit-oscillator system with the qubit bias conditions varied according to the LZ protocol.

Figure 2

Top: Qubit's final excited-state probability $P$ as a function of temperature $k_{B}T$ and coupling strength $g$, both measured relative to the qubit's minimum gap $\Delta$. Middle: $P$ as a function of $k_{B}T/\Delta$ for four values of $g/\Delta$: 0.1 [solid (red) line], 0.3 [dashed (green) line], 1 [dotted (blue) line], and 2 [dash-dotted (magenta) line]. Bottom: $P$ as a function of $g/\Delta$ for three values of $k_{B}T/\Delta$: 1 [solid (red) line], 3 [dashed (green) line], and 5 [dotted (blue) line]. In all panels, the harmonic oscillator frequency is $\hslash \omega /\Delta =0.2$. The sweep rate was chosen such that $P_{\mathrm{LZ}}=0.1$, and this value is the baseline for all of the results plotted.

Figure 3

Same as Fig. 2, but for $\hslash \omega /\Delta =1$.

Figure 4

Same as Fig. 2, but for $\hslash \omega /\Delta =5$.

Figure 5

Final excited-state probability $P$ as a function of the number of excitation quanta $n$ present in the initial state of the oscillator. Here we take $\hslash \omega /\Delta =0.2$. Different lines correspond to different values of the coupling strength: $g/\Delta =0.1$ [solid (red) line], 0.5 [dashed (green) line], 1 [dotted (blue) line], and 2 [dash-dotted (magenta) line].

Figure 6

Qubit's final excited-state probability $P$ obtained from the semiclassical calculation as a function of temperature $k_{B}T$ and coupling strength $g$, both measured relative to $\Delta$. Different panels correspond to different values of the harmonic oscillator frequency: $\hslash \omega /\Delta =0.2$ (top), 1 (middle), and 5 (bottom).