Pseudo-Hermitian Landau-Zener-Stückelberg-Majorana model

We derive the analytical solution of the model of a two-state system interacting with an external coherent field, in which the Hamiltonian is pseudo-Hermitian. We describe in detail the non-Hermitian generalization of the famed Landau-Zener-Stückelberg-Majorana model, but similar generalizations can be derived in a very simple fashion for the other analytically soluble two-state models. The analytical solutions possess a non-Hermitian dynamical invariant, which replaces the probability conservation condition in the Hermitian case. Implementations in waveguide optics and nonlinear frequency conversion are suggested.

Figure 1

Transition probabilities $P_{1\to 1}$ and $P_{1\to 2}$ as a function of scaled time for different values of the parameter $k$ and ${\mathrm{\Omega }}_0=\beta$.

Figure 2

Transition probabilities $P_{1\to 1}$ and $P_{1\to 2}$ as a function of Rabi frequency ${\mathrm{\Omega }}_0$ for different values of the parameter $k$ and $t_f=-t_i=20/\beta$. The dashed lines represent the corresponding asymptotic probabilities.

Figure 3

Transition probabilities $P_{1\to 1}$ and $P_{1\to 2}$ as a function of Rabi frequency ${\mathrm{\Omega }}_0$ for different values of the parameter $k$ and $t_i=0;t_f=20/\beta$. The dashed lines represent the corresponding asymptotic probabilities.