Dissipation-Induced Steady States in Topological Superconductors: Mechanisms and Design Principles

TL;DR

The paper addresses the existence and controllability of degenerate nonequilibrium steady states in dissipative topological superconductors hosting Majorana modes, using the GKSL framework and third quantization. It derives a fundamental relation for the number of zero kinetic modes, N0 = 2N_M − rk B, where B encodes the hybridization between MM wavefunctions and linear dissipative fields, and shows how dissipation can both destroy and stabilize Majorana-like modes. The authors develop practical recipes to engineer weak ZKM and demonstrate their framework on a generalized Kitaev chain in the BDI class, including analytical and numerical results and a bath-mediated Majorana transfer protocol. The work provides design principles for stabilizing dissipation-induced steady states in open topological superconductors, with potential implications for robust quantum information storage and manipulation in nonunitary environments.

Abstract

The search for conditions supporting degenerate steady states in nonequilibrium topological superconductors is important for advancing dissipative quantum engineering, a field that has attracted significant research attention over the past decade. In this study, we address this problem by investigating topological superconductors hosting unpaired Majorana modes under the influence of environmental dissipative fields. Within the Gorini-Kossakowski-Sudarshan-Lindblad framework and the third quantization formalism, we establish a correspondence between equilibrium Majorana zero modes and non-equilibrium kinetic zero modes. We further derive a simple algebraic relation between the numbers of these excitations expressed in terms of hybridization between the single-particle wavefunctions and linear dissipative fields. Based on these findings, we propose a practical recipes how to stabilize degenerate steady states in topological superconductors through controlled dissipation engineering. To demonstrate their applicability, we implement our general framework in the BDI-class Kitaev chain with long-range hopping and pairing terms -- a system known to host a robust edge-localized Majorana modes.

Figures (5)

• Figure 1: Schematic illustration of different terms in the generalized Kitaev chain \ref{['Ham_Kit_M']} for $N_r=2$: (a) general situation of nonzero $t_r$ and $\Delta_r$ for $r=1,2$, (b) the symmetric point with $\mu = t_1=\Delta_1=0$ и $t_2=\Delta_2=t$. In the later case there are zero energy localized excitations corresponding to the Majorana operators $w_1$,$w_3$, $w_{2N-2}$, and $w_{2N}$.
• Figure 2: The effect of dissipative fields on the eigen functions and eigen values of $X$. Panels (a), (d), (g), (j): The spatial distribution of the dissipative fields $\underline{l}^r$ (red curve) and $\underline{l}^i$ (blue curve) along the chain. Panel (b): The spatial distribution of the MM wave function in the isolated system. Panel (c): The eigen values $\beta$ of the matrix $X$ for the isolated system. Panels (e), (h), (k): The right eigen vectors of the matrix $X$ corresponding to the eigen values with $\mathop{\mathrm{Im}}\nolimits\beta=0$ and minimal magnitudes $\mathop{\mathrm{Re}}\nolimits\beta$. Panels (f), (i), (l): The eigen values $\beta$ of the matrix $X$. The first row corresponds to the isolated system while the next rows correspond to the dissipative fields chosen according to criteria (a), (b), and (c) in Sec. \ref{['Sec:Recipes']}, respectively.
• Figure 3: Spectral evolution as a function of the dissipation intensity $\gamma$, with color encoding $\gamma$-values according to the colorbar. (a) The case $\hat{L}\neq\hat{L}^+$, $\mathop{\mathrm{rk}}\nolimits\mathcal{B}=2$, $N_0=2$. (b) The case $\hat{L}=\hat{L}^+$, $\mathop{\mathrm{rk}}\nolimits\mathcal{B}=1$, $N_0=3$. The dissipation matrix $\gamma\,\underline{l}\,\underline{l}^\dagger$ governs the $\gamma$-dependence, while the other parameters match those in Fig. \ref{['2']}. Blue-green and red-purple colorbars correspond to spectral branches selected to demonstrate a role of exceptional points with a horizontal-to-vertical change of motion (and vice versa) under the increase of dissipation (the parameter $\gamma$).
• Figure 4: (a) Spatial profile of the gain field $\underline{\nu}$ related with the loss one $\underline{\mu}$ via $\underline{\mu} = e^{i\phi}\underline{\nu}$ (Eq. \ref{['L_w']}). Solid and dashed lines show the real ($\underline{\nu}^{r}$) and imaginary ($\underline{\nu}^{i}$) components, respectively. (b) Phase dependence of kinetic mode energies $|\,\beta\,|$ on $\phi$. Red dots mark a weak zero kinetic mode present at all values of $\phi$. (c) Color map showing the $\phi$-dependent spatial distribution of the normalized ZKM weights across Majorana sublattices, see Eq. \ref{['eq:vecU:111']}. Color scale indicates magnitude of $|{U}_{j}|(\phi)$. At $\phi=0$ ($\phi=\pi$), the ZKM localizes near the right (left) chain edge. Intermediate $\phi$ values yield localization at the both edges due to bath-mediated Majorana hybridization. Other parameters match Fig. \ref{['2']}.
• Figure 5: Dependence of subgap excitation energies on chain length $N$ for the isolated (hollow circles) and open (squares and dots) Kitaev chains (Sec. \ref{['sec5.1']}) with parameters $\mu=0.5$, $t_{r} = \delta_{r,N_{M}}$, $\Delta_r = 0.8\,\delta_{r,N_{M}}$ (all energies are in units of $|t|$). (a,b) Homogeneous systems with $N_M=2$ and $N_M=3$, respectively. (c) Disordered system with $N_M=2$, where on-site energies $\epsilon_j$ (Eq. \ref{['Ham_Kit_F']}) are drawn from a Gaussian distribution (mean $\bar{\epsilon}=0$ and dispersion $\sigma_\epsilon=\mu/2$). The energies $\beta$ for the open system are averaged over 500 random realizations of the dissipation fields $\underline{\mu}, \underline{\nu} \in \mathbb{C}^{N}$.

Theorems & Definitions (1)

• Theorem 1