Dissipative Landau-Zener quantum dynamics with transversal and longitudinal noise

We determine the Landau-Zener transition probability in a dissipative environment including both longitudinal as well as transversal quantum-mechanical noise originating from a single noise source. For this, we use the numerically exact quasiadiabatic path integral, as well as the approximative nonequilibrium Bloch equations. We find that transversal quantum noise in general influences the Landau-Zener probability much more strongly than longitudinal quantum noise does at a given temperature and system-bath coupling strength. In other words, transversal noise contributions become important even when the coupling strength of transversal noise is smaller than that of longitudinal noise. We furthermore reveal that transversal noise renormalizes the tunnel coupling independent of temperature. Finally, we show that the effect of mixed longitudinal and transversal noise originating from a single bath cannot be obtained from an incoherent sum of purely longitudinal and purely transversal noise.

Figure 1

Landau-Zener probability $P$ vs sweep velocity $v$ for various temperatures as indicated for ${\omega }_c=10\Delta$. Panels (a) and (c) show $P$ for pure longitudinal system-bath coupling (“LC”) with $\theta =0$, and panels (b) and (c) show $P$ for pure transversal system-bath coupling (“TC”) with $\theta =\pi /2$, respectively. The dissipative coupling strength is $\gamma =2\times {10}^{-4}$ for panels (a) and (b) and $\gamma =2\times {10}^{-3}$ for panels (c) and (d).

Figure 2

Momentary energy levels (black full lines) of the two-level system (energies and $\gamma$ in units of $\Delta )$ and the momentary upward transition rate ${\gamma }_u\left(t\right)$ of Eq. (7) for longitudinal system-bath coupling (blue dashed line) with $\theta =0$ and for transversal system-bath coupling (red dash-dotted line) with $\theta =\pi /2$ vs time.

Figure 3

Landau-Zener probability vs sweep velocity for various low-to-intermediate temperatures for transversal system-bath coupling with strength $\gamma =0.02$ and ${\omega }_c=10$. The black full line represents the pure Landau-Zener probability without dissipation whereas the dotted red line marks the Landau-Zener probability with an effective renormalized tunnel coupling.

Figure 4

Landau-Zener probability $P$ vs sweep velocity $v$ for various mixing angles $\theta$ for $T=25\Delta ,{\omega }_c=10\Delta$, and $\gamma =5\times {10}^{-4}$.

Figure 5

Landau-Zener probability $P$ vs sweep velocity $v$ for mixing angles of opposite sign, $\theta =\pm \pi /30$ (upper figure) and $\theta =\pm \pi /10$ (lower figure) for $T=4\Delta ,{\omega }_c=10\Delta$, and $\gamma =0.01$.

Figure 6

Momentary energy levels (black full lines) of the two-level system (energies and $\gamma$ in units of $\Delta )$ and the momentary upward transition rate ${\gamma }_u\left(t\right)$ of Eq. (7) for $\theta =-\pi /4$ (blue dashed line) and $\theta =\pi /4$ (red dash-dotted line) vs time.