Local and Global Master Equations through the Lens of Non-Hermitian Physics

Abstract

We investigate the relation between non-Hermitian Hamiltonian and Lindblad dynamics in nonequilibrium open quantum systems. Non-Hermitian models can extend phase diagrams and enable sensing advantages, but such effects often rely on postselection, raising questions about their relevance for unconditional dynamics. Using a minimal two-qubit setup mediating a heat current, we compare local and global Markovian master equations with their non-Hermitian counterparts. We observe that exceptional points emerge only in the local master equation and in the corresponding non-Hermitian Hamiltonian at sufficiently strong nonequilibrium. We further consider hybrid configurations, where one bath is treated with a Lindblad description and the other with a non-Hermitian approach, interpolating between the two extremes. Our results contribute understanding the role of quantum jumps and exceptional points in nonequilibrium open quantum systems and identify a simple, experimentally accessible architecture, realizable, for instance, in circuit-QED platforms, for their exploration.

Figures (11)

• Figure 1: Schematic representation of the model studied in this work. Two detuned qubits interacting with each other and coupled to independent hot and cold bosonic baths.
• Figure 2: ($a$) Evolution of the trace of the non-normalized state $\Omega_{\rm nH}$ as a function of dimensionless time for two values of the coupling $g$, plotted on a logarithmic scale, with a zoomed-in view shown in linear scale. ($b$) Trace distance in Eq. \ref{['eq:trdist']} and upper bound in Eq. \ref{['eq:upperbound']} as functions of dimensionless time, for the same two values of the coupling $g$, on a logarithmic scale. The inset provides a zoomed-in view with shaded regions highlighting the range between the upper bound and the trace distance (i.e., a lower bound), on a linear scale. The normalized non-Hermitian Hamiltonian evolved state $\rho_{\rm nH}(t)$ in the local (loc) approach (see Eq. \ref{['eq:nhl']}) is compared with the global (glob) approach (see Eq. \ref{['eq:nhg']}). The parameters used in our simulations are: $\epsilon_c = \epsilon_h$, $\alpha_c = 0.02\epsilon_h$, $\alpha_h = 0.005\epsilon_h$, $T_c = 0.1\epsilon_h$, $T_h = \epsilon_h$, $\omega_c = 10\epsilon_h$. We tune the coupling parameter $g \in \{0.22,0.62\} \epsilon_h$ (see legend).
• Figure 3: Comparison between the dynamics generated by Lindblad master equations and non-Hermitian Hamiltonians. We plot the trace distance in Eq. \ref{['eq:trdist']} as a function of dimensionless time for four values of the coupling $g$. The Lindblad evolved state $\rho_L(t)$ is compared with the normalized non-Hermitian Hamiltonian state $\rho_{\rm nH}(t)$ for the local approach, as described by Eqs. \ref{['eq:dissipl']} and \ref{['eq:nhl']} (panel ($a$)), and for the global approach, as described by Eqs. \ref{['eq:dissipg']} and \ref{['eq:nhg']} (panel ($b$)). The parameters used in our simulations are: $\epsilon_c = \epsilon_h$, $\alpha_c = 0.2\epsilon_h$, $\alpha_h = 0.05\epsilon_h$, $T_c = 0.1\epsilon_h$, $T_h = \epsilon_h$, $\omega_c = 10\epsilon_h$. We tune the coupling parameter $g$ over the range $g \in [0.03,0.93] \epsilon_h$ (see legend).
• Figure 4: ($a$) Non-Hermitian Hamiltonian von Neumann entropy and ($b$) irreversible entropy production in Lindblad dynamics as functions of dimensionless time for two values of the hot bath temperature $T_h$. Results are shown for both local (loc) and global (glob) approaches. The parameters used in our simulations are: $\epsilon_c = \epsilon_h$, $\alpha_c = 0.2\epsilon_h$, $\alpha_h = 0.05\epsilon_h$, $T_c = 0.1\epsilon_h$, $\omega_c = 10\epsilon_h$, $g = 0.8\epsilon_h$. We tune the temperature of the hot bath $T_h \in \{0.59,1.50\}\epsilon_h$ (see legend).
• Figure 5: Exceptional point in the non-Hermitian Hamiltonian derived from the local master equation. We report the coalescence of eigenvalues and eigenvectors 2 and 3. This is shown by examining the real part in panel ($a$) and the imaginary part in panel ($b$) of the eigenvalues as functions of $g$ (in units of $\epsilon_h$). Non-orthogonality of eigenvectors 2 and 3 is computed by evaluating $1 - |\langle v_i | v_j \rangle|^2$ for each pair of right eigenvectors $i,j$ as a function of $g$ (in units of $\epsilon_h$) and shown in panel ($c$). Divergence of the condition number of the eigenvector matrix $\kappa(V)$ is shown in panel ($d$) as a function of the coupling $g$ (in units of $\epsilon_h$). The parameters used in our simulations are: $\epsilon_c = \epsilon_h$, $\alpha_c = 0.2\epsilon_h$, $\alpha_h = 0.05\epsilon_h$, $T_c = 0.1\epsilon_h$, $T_h = \epsilon_h$, $\omega_c = 10\epsilon_h$. We tune the coupling $g$ over the range $g \in [0.0,0.6] \epsilon_h$ and find $\mathrm{EP}$ at $g \approx 0.12\epsilon_h$.