Controlling Quantum Transport in a Superconducting Device via Dissipative Baths

TL;DR

This work develops a microscopic quantum-field framework for open superconducting devices coupled to fermionic baths, using GKSL dynamics on the Keldysh contour to treat dissipation in quadratic Liouvillians. It derives a generalized Meir-Wingreen formula and Onsager relations that account for bath-induced self-energies, and introduces loss currents and quantum kinetic equations for the open system. The approach is applied to Kitaev-chain–like models to show how dissipation suppresses Majorana zero-bias peaks and how NESS degeneracy maps onto local transport signatures, including conditions under which conductance quantization is preserved. The results highlight bath engineering as a potential tool for probing Majorana physics and for designing non-equilibrium Majorana-based devices with controlled dissipation.

Abstract

Within the quantum field-theoretical approach describing the evolution of a quadratic Liouvillian in the basis of Keldysh contour coherent states, we investigate the spectral and transport properties of a dissipative superconducting system coupled to normal Fermi reservoirs. We derive a generalization of the Meir-Wingreen formula and Onsager matrix for a superconducting system subject to an arbitrary number of fermionic baths. Following Kirchhoff's rule, we obtain an expression describing the dissipation induced loss current and formulate modified quantum kinetic equations. For wide-band contacts locally coupled to individual sites, we find that each contact reduces the degeneracy multiplicity of the non-equilibrium steady state by one. These results are numerically verified through several cases of the extended Kitaev model at symmetric points with a single contact. Furthermore, in the linear response regime at low temperatures, we demonstrate that (non-)degenerate non-equilibrium steady states correspond to (non-)quantized conductance peaks. Revisiting a paradigmatic problem of resonant transport in the Majorana mode of the Kitaev model we demonstrate that the dissipation accounts for the zero-bias peak suppression and its asymmetry.

Figures (3)

• Figure 1: Sketch of the model. A $p$-wave superconducting device interacts with arbitrary number of fermionic reservoirs (or leads) and baths. Transport problem is analyzed in general situation when each lead and bath are coupled with all chain sites. The interaction parameters at the $n$-th site are $t_{jkn}$ and $\mu_{vn},\nu_{vn}$ for the $j$-th lead and $v$-th bath, respectively.
• Figure 2: The effect of a single weak bath with real amplitudes $\underline{\mu}_{1}, \underline{\nu}_{1}$ on the low-energy excitations and local tunnel transport in the standard Kitaev chain model. The top (bottom) row corresponds to the case of a non-Hermitian (Hermitian) bath. (a,d) Spatial distributions of the probability densities of the states in the closed system $|\phi_{1,2}|^2$ (dashed curves) and in the open system $|\psi_{1,2}|^2$, corresponding to energies with zero real part (solid curves). (b(c),e(f)) Voltage dependence of the conductance of lead attached to the left (right) end of the wire, $G_{1L}(V)$ ($G_{1R}(V)$). In panels (b,c,e,f), the dotted curve, $G_{1}^{0}(V)$, corresponds to the situation of the single-lead transport in the absence of dissipation ($\underline{\mu}_{1}=\underline{\nu}_{1}=0$). The red dashed curve shows the contribution from the local Andreev reflection to the conductance $G_{1L,1R}^{L}$, while the blue dash-dotted line represents the contribution from the bath $G_{1L,1R}^{B}$. Inset of panel (a): spatial distribution of the dissipative fields $\underline{\mu}_{1},~\underline{\nu}_{1}$. Inset of panel (f): zero-bias dip of $G_{1R}^{B}$ when the bath is Hermitian. We use the following parameters: $N=10$, $\mu=0$, $t=1$, $\Delta=-0.95$, $\Gamma_{1}=0.1$.
• Figure 3: The effect of the NESS degeneracy on the local transport in the extended Kitaev model in the linear response and low-temperature regimes. The first column shows the dependence of the number of non-decaying zero-energy excitations ($N_0$) on the magnitude of the topological invariant of the isolated Kitaev chain ($N_{M}$), evaluated near the symmetric points, and on the number of baths ($N_B$) in the presence of the lead. The second column displays the corresponding dependence of the conductance $G$ of the lead connected to the leftmost site of the chain. (a,b) All bath Lindblad operators are non-Hermitian. (c,d) Only the Lindblad operator of the first bath is Hermitian. Parameters: $N=20$, $\mu=0.001$, $t=1$, $\Delta=-0.999$, $\Gamma_{1}=0.1$.