Detuning-induced robustness of a three-state Landau-Zener model against dissipation

A three-state system subjected to a time-dependent Hamiltonian whose bare energies undergo one or more crossings, depending on the relevant parameters, is considered, also taking into account the role of dissipation in the adiabatic following of the Hamiltonian eigenstates. Depending on whether or not the bare energies are equidistant, the relevant population transfer turns out to be very sensitive to the environmental interaction or relatively robust. The physical mechanisms on the basis of this behavior are discussed in detail.

Figure 1

Level-crossing schemes for different values of $\mathrm{\Delta }$: (a) the three bare levels for $\mathrm{\Delta }=0$, (b) the corresponding dressed energies when ${\mathrm{\Omega }}_{12}={\mathrm{\Omega }}_{23}$ and ${\mathrm{\Omega }}_{13}=0$, and the bare energies for (c) $\mathrm{\Delta }<0$ and (d) $\mathrm{\Delta }>0$.

Figure 2

Final population of state $|3\rangle$ (obtained through a numerically exact resolution of the relevant Schrödinger equation) when the system starts with $|1\rangle$ as a function of $\mathrm{\Delta }$ (in units of ${\mathrm{\Omega }}_{23}$), and relevant minimum energy gap between the two highest instantaneous eigenvalues of the Hamiltonian (in units of ${\mathrm{\Omega }}_{23}$) for the cases (a) and (b) ${\mathrm{\Omega }}_{13}/{\mathrm{\Omega }}_{23}=0$ and (c) and (d) ${\mathrm{\Omega }}_{13}/{\mathrm{\Omega }}_{23}=0.5$. The other parameters are $\kappa /{\mathrm{\Omega }}_{23}^2=0.1$ and $\kappa T/{\mathrm{\Omega }}_{23}=100$ for all the curves. In all figures, the four curves correspond to ${\mathrm{\Omega }}_{12}/{\mathrm{\Omega }}_{23}=1$ (solid black line), ${\mathrm{\Omega }}_{12}/{\mathrm{\Omega }}_{23}=0.5$ (bold long-dashed blue line), ${\mathrm{\Omega }}_{12}/{\mathrm{\Omega }}_{23}=0.1$ (dashed red line), and ${\mathrm{\Omega }}_{12}/{\mathrm{\Omega }}_{23}=0$ (dotted green line).

Figure 3

Final population of state $|3\rangle$ (obtained through a numerically exact resolution of the relevant Schrödinger equation) when the system starts in $|1\rangle$ as a function of ${\mathrm{\Omega }}_{13}$ and $\mathrm{\Delta }$ (both in units of ${\mathrm{\Omega }}_{23}$) for (a) $\kappa /{\mathrm{\Omega }}_{23}^2=0.1$ and (b) $\kappa /{\mathrm{\Omega }}_{23}^2=1$; in both cases ${\mathrm{\Omega }}_{12}={\mathrm{\Omega }}_{23}$ and $\kappa T/{\mathrm{\Omega }}_{23}=100$. (c) Relevant minimum energy gap between the two highest instantaneous eigenvalues of the Hamiltonian as a function of ${\mathrm{\Omega }}_{13}$ and $\mathrm{\Delta }$ (all in units of ${\mathrm{\Omega }}_{23}$).

Figure 4

Final population of state $|3\rangle$ (obtained through a numerically exact resolution of the relevant Schrödinger equation) when the system starts in $|1\rangle$ as a function of ${\mathrm{\Omega }}_{13}$ and $\mathrm{\Delta }$ (both in units of ${\mathrm{\Omega }}_{23}$). The relevant parameters are ${\mathrm{\Omega }}_{12}/{\mathrm{\Omega }}_{23}=1$, $\kappa /{\mathrm{\Omega }}_{23}^2=0.1$, and $\kappa T/{\mathrm{\Omega }}_{23}=100$. Concerning the decay rate, the following values have been considered: (a) $\mathrm{\Gamma }/{\mathrm{\Omega }}_{23}=0.001$, (b) $\mathrm{\Gamma }/{\mathrm{\Omega }}_{23}=0.005$, and (c) $\mathrm{\Gamma }/{\mathrm{\Omega }}_{23}=0.025$.

Figure 5

Final population of state $|3\rangle$ (obtained through a numerically exact resolution of the relevant Schrödinger equation) as a function of $\mathrm{\Delta }$ (in units of ${\mathrm{\Omega }}_{23}$) and $\mathrm{\Gamma }$ (in units of ${\mathrm{\Omega }}_{23}$ and in logarithmic scale), when the system starts being in the state $|1\rangle$. The relevant parameters are ${\mathrm{\Omega }}_{12}/{\mathrm{\Omega }}_{23}=1$, $\kappa /{\mathrm{\Omega }}_{23}^2=0.1$, $\kappa T/{\mathrm{\Omega }}_{23}=100$, and (a) ${\mathrm{\Omega }}_{13}=0$, (b) ${\mathrm{\Omega }}_{13}/{\mathrm{\Omega }}_{23}=0.2$, and (c) ${\mathrm{\Omega }}_{13}/{\mathrm{\Omega }}_{23}=0.5$.