Secular master equation for adiabatically driven time-dependent systems

Relying only on first principles, we derive a master equation of Lindblad form generally applicable for adiabatically driven time-dependent systems. Our analysis shows that the much-debated secular approximation can be valid for slowly time-dependent Hamiltonians when performed in an appropriate basis. We apply our approach to the well-known Landau-Zener problem, where we find that adiabaticity is improved by coupling to a low-temperature environment.

Figure 1

The red line (with the dots representing equidistant time steps) is the path of the Bloch vector of the instantaneous ground state of the LZ problem. The black curve is the third-order superadiabatic ground state and the blue curve is the actual time evolution of an initial ground state. The time evolution operator causes oscillations around the superadiabatic ground state. A low-temperature environment causes the system to relax onto the black curve, rather than the red one, therefore stabilizing the adiabatic theorem.

Figure 2

The transition probability $P_{g\to e}$ as a function of ${\Delta }^2/v$ without coupling (black), and with dephasive coupling $\gamma \left(0\right)/\Delta =0.003$ (red), 0.01 (green), 0.03 (blue), and 0.1 (orange) [from bottom to top in (b)]. In (a) we used Eqs. (7) to (9) and the fourth-order superadiabatic states for $|{\varphi }_{\alpha }^t\rangle$, while in (b) we used the instantaneous eigenstates. Various lines fall on top of each other in (a), meaning that the transition probability is independent of pure dephasive coupling.

Figure 3

Same as Fig. 2, but with an Ohmic environment. Panel (a) is for low temperature $T\ll \Delta$ with ${\gamma }_0/\Delta =0$ (black), 0.01 (red), and 0.03 (green) from top to bottom. In (b) we used $T=0.5$ and ${\gamma }_0/\Delta =0$ (black), 0.01 (red), and 0.1 (green) from bottom to top.