Exact Floquet solutions in a Parity-Time-Symmetric Rabi Model

TL;DR

This work addresses finding exact Floquet solutions for a non-Hermitian, PT-symmetric semiclassical Rabi model with a time-periodic drive $\\nu(t)=\\nu_0+\\nu_1\\cos(\\omega t)$. The authors reveal a hidden $sl(2)$ algebraic structure, recasting the second-order equation for the transformed amplitude as an element of the universal enveloping algebra of $sl(2)$ acting on a finite-dimensional polynomial space. Imposing the resonance condition $\\nu_0=(n+1)\\omega$ yields a determinantal constraint det$(\\tilde{H}_n)=0$ that fixes a relation between $\\nu_1$ and $\\gamma$, producing explicit quasi-exact solutions for $n=0\\dots 5$ and revealing degeneracies at $\\nu_1=0$. These results connect the PT-symmetric Rabi problem to the double confluent Heun equation and suggest extensions to broader PT-symmetric quasi-exactly solvable models with potential applications in quantum information and photonics.

Abstract

It is shown that a semiclassical Rabi model with parity-time (PT) symmetry has a hidden $sl(2)$ symmetry and hence possesses quasi-exact solutions. These are located precisely at the exceptional points of the spectrum, the boundaries of the PT-symmetric phase. The corresponding constraints on the model parameters can be interpreted as a resonance relationship between the constant and periodic driving terms.

Figures (3)

• Figure 1: The $n$-photon constraints for $n=0,1,2,3,4$ and $5$.
• Figure 2: The crossing points for doublets $(n_1,n_2)=(2,4)$ on the left and $(n_1,n_2)=(3,5)$ on the right.
• Figure 3: The $n$-photon constraint for $n=20$ . Doubly degenerate solutions happen for $\nu_1=0$ and $\frac{\gamma ^2}{\omega ^2}=\{80,152,216,272,320,360,392,416,432,440\}$ corresponding to the pairs $j=1,\dots,n$ and $j'=(n+1-j)$ in \ref{['doublydegeneratesolutions']}; the "almost degenerate" point is clearly visible (here) at $(\nu_1,\frac{\gamma ^2}{\omega ^2})=(0,80)$ corresponding to the pair $(j,j')=(1,20)$.