Dynamical invariants for pseudo-Hermitian Hamiltonians

We derive the dynamical invariants for a general $N$-state quantum system described by a pseudo-Hermitian Hamiltonian. Explicit expressions are presented for two- and three-state systems, which are exemplified by explicit analytic solutions for constant couplings. In the two-state case, we derive non-Hermitian analogs of the Bloch vector and the Bloch equation, customary for Hermitian quantum systems. We suggest possible physical implementations of the dynamical invariants in waveguide optics and frequency conversion.

Figure 1

Time evolution of the transition probabilities in the pseudo-Hermitian Rabi model in a two-state system, for (a) ${\mathrm{\Omega }}_2/{\mathrm{\Omega }}_1=0.7, \mathrm{\Delta }={\mathrm{\Omega }}_1$; (b) ${\mathrm{\Omega }}_2/{\mathrm{\Omega }}_1=0.7, \mathrm{\Delta }=0$; (c) ${\mathrm{\Omega }}_2/{\mathrm{\Omega }}_1=1.3, \mathrm{\Delta }={\mathrm{\Omega }}_1$; (d) ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2, \mathrm{\Delta }={\mathrm{\Omega }}_1$. For ${\mathrm{\Omega }}_2>{\mathrm{\Omega }}_1$ [frame (c)], we have ${\rho }_{11}\left(t\right)+{\rho }_{22}\left(t\right)\geqq 1$.

Figure 2

Evolution of populations for pseudo-Hermitian three-level system. (a) ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2$ and ${\mathrm{\Omega }}_3={\mathrm{\Omega }}_4=1.3{\mathrm{\Omega }}_1$. (b) ${\mathrm{\Omega }}_2={\mathrm{\Omega }}_1, {\mathrm{\Omega }}_3={\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_4=0.7{\mathrm{\Omega }}_1$. (c) All couplings are different, $j=1,2,3,4$: ${\mathrm{\Omega }}_2=0.5{\mathrm{\Omega }}_1, {\mathrm{\Omega }}_3=1.3{\mathrm{\Omega }}_1$, and ${\mathrm{\Omega }}_4=1.6{\mathrm{\Omega }}_1$.