Manifolds of exceptional points and effective Zeno limit of an open two-qubit system

Abstract

We analytically investigate the Liouvillian exceptional point manifolds (LEPMs) of a two-qubit open system, where one qubit is coupled to a dissipative polarization bath. Exploiting a Z_2 symmetry, we block-diagonalize the Liouvillian and show that one symmetry block yields two planar LEPMs while the other one exhibits a more intricate, multi-sheet topology. The intersection curves of these manifolds provide a phase diagram for effective Zeno transitions at small dissipation. These results are consistent with a perturbative extrapolation from the strong Zeno regime. Interestingly, we find that the fastest relaxation to the non-equilibrium steady state occurs on LEPMs associated with the transition to the effective Zeno regime.

Figures (10)

• Figure 1: An open two-qubit system is schematized as two $XYZ$ Heisenberg spins $1/2$ interacting via anisotropic exchange energies of strength $J\equiv(J_x,J_y,J_z)$. The system is coupled to a polarization dissipative bath (green box with a large arrow inside) only through one of the two spins. The red and blue dashed lines represent the Heisenberg exchange interaction between the two spins and the coupling $\Gamma$ between the first spin and the bath, respectively. Small spheres indicate the sites on which the spins are located.
• Figure 2: Real (top panel) and imaginary (bottom panel) parts of the eigenvalues $\lambda$ of the Liouvillian operator (\ref{['liouv']}) versus $\Gamma$. Red and blue lines refer to eigenvalues of the $\Sigma_+$ and $\Sigma_-$ blocks of $\mathcal{L}_\mathrm{vec}$, respectively. LEPs coincide with the eigenvalue branching points and correspond to broken/resumed symmetries (change of eigenvalue degeneracies). Dashed lines denote perturbative extrapolations of the eigenvalues from the strong, $\Gamma \gg \Gamma_{cr}$, Zeno regime, to the low dissipation Zeno regime, $\Gamma \approx \Gamma_{cr}$, with $\Gamma_{cr}$ given in (\ref{['GaCrit']}). Note the logarithmic horizontal scale and the rescaling of $\mathop{\rm Re} \lambda$ by $\Gamma$ operated in the top panel. Parameters are fixed as: $\gamma =0.6$, $\Delta=0.4$, corresponding to point b in the bottom panel of Fig. \ref{['Fig-PhaseDiagram']}.
• Figure 3: Two-dimensional LEP manifolds of the two-qubit open system (\ref{['LME']}) in the three-dimensional parameter space $\{\gamma, \Delta, \Gamma\}$. Only the LEP manifolds originating from the $\Delta$-dependent block $\Sigma_-$ of the Liouvillian are shown here.
• Figure 4: Top panel. Behavior of $\Gamma_{cr}(\gamma,\Delta)$, light-blue surface with a mesh, for $\Delta \ge 0.05$. Within the region delimited by the red continuous line this surface coincides with the straight planes $\Gamma_{cr}=8$ (displayed in gray) and $\Gamma_{cr}=8\gamma$ (displayed in green). Bottom panel. Two-dimensional diagram of $\Gamma_{cr}(\gamma,\Delta)$. The detailed branching point structure of the Liouvillian eigenvalues at the point labelled b is shown in Fig. \ref{['FigPar1']}. The same info at points a, c, and d is provided in SM. In SM we also show the sections of $\Gamma_{cr}(\gamma,\Delta)$ at the planes $\Delta=0.1,0.3,$ and $0.7$ whose projections are indicated here by tiny solid lines.
• Figure 5: Distance $d(t)\equiv {\|\rho(t)-\rho_{\text{NESS}}(\gamma_f)\|}_2$ versus time $t$ for rapid quenches in the parameter $\gamma$, from $\gamma_{i}=0.4$ to $\gamma_{f}=0.8$, on different constant-$\Gamma$ planes as indicated in the legend, for arbitrary $\Delta$. The left panel shows data for planes above the $\Gamma=8$ LEPM, while the right panel data for planes below it.