Quantum Mpemba effect in parity-time symmetric systems

TL;DR

The work shows that quantum Mpemba effect (QMPE) can arise in intrinsic non-Hermitian, PT-symmetric qubit systems coupled to a bosonic bath. By combining numerical simulations with a long-time analytic framework, it identifies parameter regimes where QMPE occurs as an odd number of intersections between trajectories of dynamical quantifiers, and demonstrates that QMPE is not governed by Hamiltonian or Liouvillian exceptional points. The study confirms QMPE across multiple quantifiers (trace distance, Frobenius distance, and quantum relative entropy) and proves its robustness to dephasing and extension to multi-qubit PT-symmetric systems, highlighting potential experimental realizations with current platforms. Overall, this work broadens the scope of QMPE to intrinsic non-Hermitian physics and reveals nuanced interplay with non-Hermitian features beyond exceptional points, offering practical pathways for observation and applications.

Abstract

The quantum Mpemba effect (QMPE), an anomalous relaxation phenomenon, has been demonstrated in both closed and open Hermitian quantum systems. While some studies have linked the QMPE to Liouvillian exceptional points--non-Hermitian features emerged at the Liouvillian level--in open Hermitian quantum systems, it remains largely unexplored whether the QMPE can occur in intrinsic non-Hermitian systems, where non-Hermiticity is inherent at the Hamiltonian level. Here, we demonstrate unequivocally the occurrence of QMPE in experimentally realizable parity-time-symmetric qubit systems immersed in a bosonic bath. Using established quantifiers for QMPE, we show numerically that the QMPE persists across parameter regimes both near and far from Hamiltonian and Liouvillian exceptional points, but disappears entirely when Hermitian Hamiltonian is restored. Interestingly, neither Hamiltonian nor Liouvillian exceptional points demarcate the boundaries of the QMPE regime. To complement numerical results, we develop an analytical description based on a long-time approximation of the relaxation dynamics of quantifiers. This approach allows us to decipher the number of intersections between two dynamical trajectories of quantifier starting from two initial conditions in the validity regime of the long-time approximation, thereby providing additional information that delineate the parameter regimes supporting the genuine QMPE. We further demonstrate the robustness of QMPE against increasing the number of qubits and dephasing effect. Our findings not only broaden the scope of the QMPE but also suggest its intricate interplay with non-Hermitian features beyond exceptional points.

Figures (10)

• Figure 1: The dynamical trajectories of the trace distance $D(t)$ [Eq. (\ref{['trace_distance']})] for (a) a Hermitian qubit system with Hamiltonian $\hat{H}=\hat{\sigma}_x$ showing no intersections, and (b) its PT-symmetric counterpart with Hamiltonian $\hat{H}=\hat{\sigma}_x+ia\hat{\sigma}_z$ ($a=1.2$) exhibiting a single intersection point indicating the QMPE. For both models: We obtain $D(t)$ by solving Eq. \ref{['rho(t)_Lindblad']} under two different initial conditions $\hat{\rho}^{\rm I}(0)=\frac{1}{2}(\hat{\sigma}_z+\hat{\mathrm{I}})$ (red curves) and $\hat{\rho}^{\rm{II}}(0)=\frac{1}{2}\hat{\mathrm{I}}$ (green curves), where "$\hat{\mathrm{I}}$" is the identity matrix. $\gamma_1=0.6$ and $\gamma_2=1$.
• Figure 2: Real (solid curves) and imaginary (dashed curves) parts of the eigenvalues $\{\mu_j\}$ of $\mathcal{L}_0$ for $\gamma_1=0.5$. The orange, red, blue, and green curves correspond to the eigenvalue branches $\mu_1$, $\mu_2$, $\mu_3$, and $\mu_4$, respectively. The eigenvalues remain non-degenerate throughout the parameter range except at the Liouvillian exceptional point (LEP, vertical dashed line), where two eigenvalues coalesce.
• Figure 3: (a) Contour plot of the number of intersection points (non-negative integers) counted up to $t=20$ between two dynamical trajectories for $D^{\mathrm{I,II}}(t)$, calculated from Eq. (\ref{['eq:rho_t']}) under the initial conditions $\hat{\rho}^{\rm I}(0)=(\hat{\sigma}_z+\hat{\rm{I}})/2$ and $\hat{\rho}^{\rm{II}}(0)=\hat{\rm{I}}/2$, where $\hat{\rm{I}}$ is the identity matrix. Red dashed lines: analytical boundaries from Eq. (\ref{['X=1']}) (see more details in Fig. \ref{['fig:x_trace_left']} and main text). Black dashed line: LEP. The orange star ($a=1,\gamma_1=0.4$) and red star ($a=1.3,\gamma_1=0.4$) mark parameters used in (b) and (c), respectively. (b) and (c) Detailed dynamics of $D^{\rm{I,II}}(t)$ for the parameters marked in (a). Insets: Difference $\Delta D(t)=D^{\rm I}(t)-D^{\rm{II}}(t)$ calculated from Eq. (\ref{['eq:rho_t']}) (solid line) and Eq. (\ref{['rho_M']}) (dash-dotted line); red circles indicate exact intersection points. The label $T=2.3$ in the inset of (c) denotes the theoretical period $\pi/|\mathrm{Im}(\mu_{3,4})|$. For all panels, $\gamma_2=1$.
• Figure 4: Contour plot of magnitudes of theoretical solutions: (a) $|[X(\tau)]_{+}|$ and (b) $|[X(\tau)]_{-}|$ given in Eq. (\ref{['X_solution']}) in the regime to the left of Liouvillian exceptional point (LEP, black solid lines). Green dashed lines are theoretical boundaries for the white shaded regime that validates the constraint in Eq. (\ref{['X=1']}) and are used as red dashed lines in Fig. \ref{['fig:long_trace_left']} (a). Gray shaded regime to the right of the LEP is not considered. $\gamma_2=1$ for all plots. We adopt the same initial conditions as Fig. \ref{['fig:long_trace_left']}.
• Figure 5: Contour plot of the number of intersection points (non-negative integers) counted up to $t=20$ between two dynamical trajectories for $D^{\mathrm{I,II}}(t)$, calculated from Eq. (\ref{['eq:rho_t']}) under the initial conditions $\hat{\rho}^{\rm I}(0)=\frac{1}{2}\hat{\mathrm{I}}-0.3\hat{\sigma}_z-0.2\hat{\sigma}_y$ and $\hat{\rho}^{\rm{II}}(0)=\frac{1}{2}\hat{\mathrm{I}}$. The red, orange and green dashed lines are theoretical boundaries predicted by Eq. (\ref{['X_01']}) (see more details in Fig. \ref{['fig:x_trace_right']} and main text). The black dashed line denotes the position of LEP. $\gamma_2=0.5$.