Limits of the non-Hermitian description of decay models

Abstract

We present a general proof that non-Hermitian dynamics and Lindblad dynamics with only decay terms are equivalent in the highest particle subspace. We then propose an unbiased method to determine if a system's dynamics in the highest-particle subspace is non-Hermitian. We exemplify this for a simple two-site decay system connected to two baths, and find that the exact solution is well approximated by non-Hermitian dynamics only in the weak-coupling and in the singular-coupling limits, where a Lindbladian description was already known to be accurate. The fact that an accurate non-Hermitian description is so limited, even for such a simple system, raises doubts about how valid such descriptions are for more complicated systems away from these asymptotic limits. Finally, we prove that for models with a nondegenerate system Hamiltonian, exceptional points cannot occur in the weak-coupling limit. This result is relevant for the design of experiments that aim to identify such exceptional points.

Figures (4)

• Figure 1: ($a$) Sketch of the system+bath model $H=H_A+H_B+H_C$ described in eqs. \ref{['HA']}, \ref{['HB']} and \ref{['HC']}. ($b$)-($e$) $R_{\theta,\phi}$ of Eq. \ref{['R']} as a function of $\theta$ and $\phi$ for $C_1=C$, $C_2=3C$, $B_1=B$, $B_2=2B$ and $\frac{C^2}{B}=0.5$, and ($b$) $B=2$, $C=1$, ($c$) $B=4$, $C=\sqrt{2}$, ($d$) $B=8$, $C=2$ and ($e$) $B=16$, $C=2\sqrt{2}$. As the singular limit is approached, the two minima gradually approach zero, indicating the appearance of two non-mixing states and validating non-Hermitian dynamics in this limit.
• Figure 2: Mixing parameter $\mathcal{R}$ of Eq. \ref{['RR']} when $C_1=C$, $C_2=3C$, $B_1=B$, $B_2=2B$. Upper panel: colormap of $\mathcal{R}$ as a function of $\frac{C}{A}$ and $\frac{B}{A}$. The thick blue line is Eq. \ref{['bound']} and the region of interest (where the baths are indeed valid baths) is below it. Here, possible non-Hermitian dynamics is signalled by $\mathcal{R} \rightarrow 0$ (dark regions) and is approached asymptotically only in the weak coupling region (bottom region) and in the singular coupling region (right side). Lower panel: $\mathcal{R}$ as a function of $\frac{C}{A}$ for $\frac{B}{A} \approx 2.05$ and $\frac{B}{A} \approx 3.01$, showing a finite $\mathcal{R}$ everywhere along those cuts, except in the weak-coupling limit.
• Figure 3: Comparison between $n_{nH}(t)$ (black line) and $n_b(t)$ (dashed lines) as a function of $t$. The accuracy of the non-Hermitian approximation improves as the singular limit is approached. Here $A=1$.
• Figure 4: ($a$) Single-site bath model. ($b$) Two-site bath model considered in the main article.