Three-state Landau-Zener model in the presence of dissipation

A population transfer based on adiabatic evolutions in a three-state system undergoing an avoided crossing is considered. The efficiency of the process is analyzed in connection with the relevant parameters, bringing to light an important role of the phases of the coupling constants. The role of dissipation is also taken into account, focusing on external decays that can be described by effective non-Hermitian Hamiltonians. Though the population transfer turns out to be quite sensitive to the decay processes, for very large decay rates the occurrence of a Zeno phenomenon allows for restoring a very high efficiency.

Figure 1

Eigenvalues (in units of $\mathrm{\Omega }$) of the Hamiltonian as functions of time (in units of $\mathrm{\Omega }/\kappa$) for (a) $\omega =0$, (b) $\omega /\mathrm{\Omega }=0.5$, (c) $\omega /\mathrm{\Omega }=1$, and (d) $\omega /\mathrm{\Omega }=1.2$. The relevant parameters are $\kappa /{\mathrm{\Omega }}^2=0.1, \mathrm{\Omega }{t}_0=500$, and $\phi =\varphi =0$.

Figure 2

Final population of state $|3\rangle$ when the system starts in state $|1\rangle$ as a function of $\omega /\mathrm{\Omega }$ (in the range [0,2]) and $\kappa /{\mathrm{\Omega }}^2$ (in the range [0.05,5]), for three different values of the phase $\phi$: (a) $\phi =0$, (b) $\phi =\pi /2$, and (c) $\phi =\pi$. In all three cases $\varphi =0$ and $\mathrm{\Omega }{t}_0=500$.

Figure 3

Final population of state $|3\rangle$ when the system starts in state $|1\rangle$ as a function of the decay rate $\mathrm{\Gamma }$ (in units of $\mathrm{\Omega }$ and in logarithmic scale) for three different values of $\omega , \kappa$, and $t_0$: (a) $\omega =0, \kappa /{\mathrm{\Omega }}^2=0.1$, and $\mathrm{\Omega }{t}_0=500$; (b) $\omega =\mathrm{\Omega }, \kappa /{\mathrm{\Omega }}^2=0.1$, and $\mathrm{\Omega }{t}_0=500$; and (c) $\omega =\mathrm{\Omega }, \kappa /{\mathrm{\Omega }}^2=1$, and $\mathrm{\Omega }{t}_0=50$. In all three figures, the black solid line refers to the case ${\mathrm{\Gamma }}_2={\mathrm{\Gamma }}_3=0$ and ${\mathrm{\Gamma }}_1=\mathrm{\Gamma }$, the blue short-dashed line to ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_3=0$ and ${\mathrm{\Gamma }}_2=\mathrm{\Gamma }$, and the red long-dashed line to ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_2=0$ and ${\mathrm{\Gamma }}_3=\mathrm{\Gamma }$. In all cases we have considered $\varphi =\phi =0$.

Figure 4

Final population of state $|3\rangle$ when the system starts in state $|1\rangle$ as a function of ${\log }_{10}({\mathrm{\Gamma }}_2/\mathrm{\Omega })$ (in the range $[-5,5]$) and $\kappa /{\mathrm{\Omega }}^2$ (in the range [0.05,1]), for three different values of the phase $\omega$: (a) $\omega =0$, (b) $\omega =\mathrm{\Omega }/2$, and (c) $\omega =\mathrm{\Omega }$. In all three cases ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_3=0, \phi =\varphi =0$, and $\mathrm{\Omega }{t}_0=500$.

Figure 5

Final population of state $|3\rangle$ when the system starts in state $|1\rangle$ as a function of ${\log }_{10}({\mathrm{\Gamma }}_2/\mathrm{\Omega })$ (in the range $[-5,5]$) and $\omega /{\mathrm{\Omega }}^2$ (in the range [0,2]), for three different values of the phase $\phi$: (a) $\phi =0$, (b) $\phi =\pi /2$, and (c) $\phi =\pi$. In all three cases ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_3=0, \varphi =0, \kappa /{\mathrm{\Omega }}^2=0.1$, and $\mathrm{\Omega }{t}_0=500$.

Figure 6

Final population of state $|3\rangle$ when the system starts in state $|1\rangle$ as a function of ${\log }_{10}({\mathrm{\Gamma }}_2/\mathrm{\Omega })$ (in the range $[-5,5]$) and $\omega /{\mathrm{\Omega }}^2$ (in the range [0,2]), for three different values of the phase $\phi$: (a) $\phi =0$, (b) $\phi =\pi /2$, and (c) $\phi =\pi$. In all three cases ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_3=0, \varphi =0, \kappa /{\mathrm{\Omega }}^2=1$, and $\mathrm{\Omega }{t}_0=50$.