Exploring the topology induced by non-Markovian Liouvillian exceptional points

Abstract

Non-Hermitian (NH) systems can display exotic topological phenomena without Hermitian counterparts, enabled by exceptional points (EPs). So far, investigations of NH topology have been restricted to EPs of the NH Hamiltonian, which governs the system dynamics conditional upon no quantum jumps occurring. The Liouvillian superoperator, which combines the effects of quantum jumps with NH Hamiltonian dynamics, possesses EPs (LEPs) that are significantly different from those of the corresponding NH Hamiltonian. We here study the topological features of the LEPs in the system consisting of a qubit coupled to a non-Markovian reservoir. We find that two distinct winding numbers can be simultaneously produced by executing a single closed path encircling the twofold LEP2, formed by two coinciding LEP2s, each involving a pair of coalescing eigenvectors of the extended Liouvillian superoperator. We experimentally demonstrate this purely non-Markovian phenomenon with a circuit, where a superconducting qubit is coupled to a decaying resonator which acts as a reservoir with memory effects. The results push the exploration of exceptional topology from the Markovian to non-Markovian regime.

Figures (3)

• Figure 1: (color online). Sketch of the theoretical model. A qubit with the frequency $\omega_{q}$ interacts with a non-Markovian reservoir containing a continuum of electromagnetic field modes with the spectral width $\kappa$ and central frequency $\omega_{0}$. Such a reservoir is equivalent to a leaky cavity storing an effective photonic mode, referred to as a pseudomode (PM), with the frequency $\omega_{0}$ and decaying rate $\kappa$. The coupling strength $\Omega$ and detuning $\Delta =\omega_{q}-\omega_{0}$ between the qubit and the PM serve as the control parameters of the extended Liouvillian superoperator, which governs the dynamics of the qubit-PM system.
• Figure 2: (color online). Experimental measurement of the Liouvillian spectrum. (a) Parameter-space loop. We here set $\Omega =\kappa /2+r\cos k$ and $\Delta=r\sin k$, so that the LEPs (star), located at $\left\{ r=\kappa/4,k=\pi \right\}$, are enclosed in the loop when $r>\kappa /4$. (b)-(i) Measured Liouvillian eigenvalues, $\lambda_{1}$ to $\lambda_{8}$, versus $k$. $\lambda_{j}$ for each value of $k$ is extracted by tracking the dynamical evolution of the qubit-PM system starting with the initial state $(\left\vert g\right\rangle +i\left\vert e\right\rangle )\left\vert 0\right\rangle /\sqrt{2}$, and exponentially fitting the amplitude associated with the eigenvector ${\bf V}_{j}$, $A_{j}(t)=A_{j}(0)e^{\lambda_{j}t}$. All the data are measured for $r\approx0.327\kappa$.
• Figure 3: (color online). Characterization of the hybrid topological invariant. Trajectories of the measured Liouvillian eigenvalues: $\lambda_{1}$ and $\lambda_{3}$ (a); $\lambda_{2}$ and $\lambda_{4}$ (b); are plotted on the complex $\mathop{\rm Re}(\lambda)$-$\mathop{\rm Im}(\lambda)$ plane (in unit of $\kappa$). The control parameters are varied along the circle shown in Fig. \ref{['fig3']}. The solid lines denote the trajectories of the calculated eigenvalues.