Exceptional points and dynamics of an asymmetric non-Hermitian two-level system

We investigated a damped two-level system interacting with a circularly polarized light as described by an asymmetric non-Hermitian Hamiltonian. This is a simple enough system to be studied analytically while complicated enough to exhibit a rich variety of behaviors. This system exhibits a ring of exceptional points in the parameter space of the real and imaginary dipole couplings where within the ring the energy eigenvalue of the system does not change. This leads to unstable regions inside the exceptional ring, which is shown using a linear stability analysis. These unstable regions are unique to gain-loss systems and have the surprising property that no matter how small the gain/loss ratio, the gain always prevails at long times. We also report on eigenvalue switching, phase rigidity and dynamics of the system around the exceptional points. We highlight that some of these properties are different from those in the widely studied case of symmetric non-Hermitian Hamiltonians.

Figure 1

Schematic of a two-level system interacting with left circularly polarized light (${\mathrm{\Omega }}_r-i{\mathrm{\Omega }}_i$). Decays out of the system are captured by ${\gamma }_{1,2}$.

Figure 2

Real (left) and imaginary (right) parts of one of the eigenvalues of $H_{nh}$ plotted in $({\mathrm{\Omega }}_r,{\mathrm{\Omega }}_i)$ parameter space with $\mathrm{\Delta }=0$ THz and ${\gamma }_1=0.3$ THz and ${\gamma }_2=-0.3$ THz. We can clearly see the exceptional ring at $\left|\mathrm{\Omega }\right|=({\gamma }_2-{\gamma }_1)/2$.

Figure 3

The phase rigidity for $H_{nh}^{\prime }$ (top) and for $H_{nh}$ (bottom). Parameters are ${\mathrm{\Omega }}_i=1$ THz, ${\gamma }_2=4.4721$ THz, and ${\gamma }_1=0$ THz. The exceptional points are at ${\mathrm{\Omega }}_r=\pm 2$ THz. The phase rigidity in red and blue is calculated using each eigenvector using Eq. (9). The phase rigidity in black is calculated using a biorthogonal product definition of phase rigidity [Eq. (8)]. The figure is explained in the text below.

Figure 4

Populations and polarization dynamics obtained from the solution of the Bloch equations for the initial conditions ($n_1=0.7$, $n_2=0.3$, $P_R=0.4583$, $P_I=0$). The parameters are $\mathrm{\Delta }=0\mathrm{THz}$, ${\gamma }_1=0.025\mathrm{THz}$, ${\gamma }_2=0.1\mathrm{THz}$, ${\mathrm{\Omega }}_r=0.08\mathrm{THz}$, and ${\mathrm{\Omega }}_i=0.25\mathrm{THz}$. Blue and red curves are for $H_{nh}^{\prime }$ while green and black are for $H_{nh}$. Population in ground state (blue, black) and excited state (red, green). Polarization: real (blue, black) and imaginary (red, green).

Figure 5

Population of the ground state with time for a gain-loss system. The parameters are as follows: Initial condition $[C_1=1$, $C_2=0]$, ${\mathrm{\Omega }}_i=0.001\mathrm{THz}$, ${\gamma }_1=0.1\mathrm{THz}$, and ${\gamma }_2=-0.01\mathrm{THz}$. The instability ring is at ${\mathrm{\Omega }}_r=0.0316\mathrm{THz}$. Top: Far inside the boundary of the stability ring (${\mathrm{\Omega }}_r=0.01\mathrm{THz}$) and far outside the ring boundary (${\mathrm{\Omega }}_r=0.05\mathrm{THz}$). Bottom: Close inside the boundary of the ring (${\mathrm{\Omega }}_r=0.031\mathrm{THz}$) and just outside the ring (${\mathrm{\Omega }}_r=0.032\mathrm{THz}$). Note the difference in time scales in upper and lower panels.