Distinguishing Majorana bound states from accidental zero-energy modes with a microwave cavity

Abstract

Transport measurements of hybrid nanowires often rely on the observation of a zero-bias conductance peak as a hallmark of Majorana bound states (MBSs). However, such signatures can also be produced by trivial zero-energy Andreev bound states (ABSs) or by quasi-Majorana bound states (QMBSs), complicating their unambiguous identification. Here we propose microwave absorption visibility, extracted from parity-dependent cavity-nanowire susceptibility measurements, as a complementary probe of MBSs nonlocality. We study a Rashba spin-orbit nanowire consisting of a proximitized superconducting segment and an uncovered quantum-dot region, capacitively coupled to a single-mode microwave cavity. We show that true MBSs yield finite visibility only when both MBSs are simultaneously coupled to the cavity, reflecting their intrinsic nonlocality. In contrast, ABSs and QMBSs exhibit visibility extrema even when the cavity couples only locally to part of the nanowire. We further demonstrate that this distinction persists in the presence of Gaussian disorder, which may otherwise generate trivial subgap states. Motivated by recent experiments, we also analyze ``poor man's" Majoranas in double-quantum-dot setups, where analytical results confirm the same nonlocal visibility criterion. Finally, we discuss a cavity-driven scheme for initializing the electronic system in a given parity state. Our results establish cavity-based visibility as a robust and versatile probe of MBSs, providing a clear route to distinguish them from trivial zero-energy states in hybrid superconducting platforms.

Figures (18)

• Figure 1: Schematic of a Rashba nanowire (gray) consisting of $L = L_d + L_s$ lattice sites, partially covered by an $s$-wave superconductor (SC, green) and subjected to a magnetic field $B$ oriented along the positive $x$ axis. The uncovered segment comprising the first $L_d$ sites defines the quantum dot (QD) or normal region. The remaining $L_s$ sites (green) form the proximized superconducting region, where pairing is induced by an adjacent $s$-wave SC (not shown). The nanowire is capacitively coupled (yellow) to a one-dimensional microwave cavity (blue), with the coupling extending up to a specific site $j_c$ along the wire. By tuning system parameters—such as the chemical potential and the Zeeman splitting induced by the magnetic field $B$—the system can be driven into a topological SC phase, where zero-energy MBSs $\gamma_{L,R}$ appear localized at the ends of the topological SC (shown in red). The presence of these electronic excitations alters the cavity’s frequency $\omega_c$ and decay rate $\kappa$, which can be probed through dispersive readout methods.
• Figure 2: Microwave absorption of a pristine topological wire with MBSs. (a) The absorption visibility vs. the fraction of the wire that is coupled to cavity $j_c/L_s$ for several transitions from the MBS to excited states, ($n = 1, 2, 3, 4$, and $\omega_{0n}=\epsilon_n-\epsilon_0$). The visibility becomes nonzero only when the cavity overlaps with both MBSs. (b) The left (right) MBSs wavefunction probability amplitude $|\psi_{L(R)}(j)|^2$ along the wire, showing exponential localization at the left (right) edge. (c) The susceptibility for the odd (even) parity $|\chi"_{o(e)}(\omega)|$, for a nanowire that is fully coupled to the cavity ($j_c=L_s$) as a function of probe frequency $\omega$. The difference in the peak amplitude helps to distinguish the two parities. Here $\Delta_{\rm min}$ is the (minimal) SC gap for a wire with $L_s=500$ sites, while the values of all model parameters are listed in Table \ref{['parametertable']}.
• Figure 3: Microwave absorption of a topological wire with an adjacent QD (vertical green dashed line marks the QD–SC interface). [(a)--(c)] Imaginary part of the susceptibility $|\chi_{o(e)}"(\omega,j_c)|$ for the odd (even) parity as a function of the frequency $\omega$ when the wire is fully covered by the cavity. The insets show the probability left (right) amplitudes $|\psi_{L(R)}(j)|^2$ corresponding to the zero-energy states as function of the position $j$ in the wire for MBSs, ABSs and QMBSs, respectively. [(d)--(f)] Visibility $\nu(\omega_{0n},j_{c})$ as function of the last site that couples to the cavity, $j_c$, for transitions between the zero-energy state and the excited states $n=1,2,3,4$ for MBSs, ABSs and QMBSs cases, respectively. Unlike the ABSs and QMBSs, the MBSs visibility vanishes unless the cavity couples to both SC edges, a signature for their nonlocality. Figures [(g)--(i)] show visibility $\nu(\omega, j_c)$ as a function of the probe frequency $\omega$ for different fractions $j_c/L = 0.14, 0.71, 1$. (g) MBSs visibility, which becomes nonzero only when the cavity couples to both Majorana end modes at resonant $\omega$. (h) ABSs localized near the interface yield nonzero visibility even when the coupling to the cavity only occurs near the left edge (i) The QMBSs exhibit nonzero visibility once the cavity covers both of them, which happens for $j_c/L>0.64$ for the depicted setup. The parameters utilized are presented in Table \ref{['parametertable']} for each case.
• Figure 4: Spectrum of the electronic system as a function of the Zeeman energy, normalized by the proximity-induced gap $\Delta$. (a) Topological superconducting wire coupled to a QD displaying MBSs at its endpoints. In (b), the system is in a trivial regime characterized by accidental zero-energy ABSs localised at the QD-SC interface in some region of Zeeman energy. (c) System supports QMBSs arising from an inhomogeneous chemical potential. Each schematic includes red dots marking the positions where susceptibility and visibility analyses are conducted.
• Figure 5: Plot of the chemical-potential profile in Eq. \ref{['ChemcialPotentialForArxivQMBSProfile']}, based on the parameters in Table \ref{['parametertable']}, indicating the topological superconducting (TS) and nontopological superconducting (NTS) region. In this configuration, the TS region converts to an NTS region at $j_\xi / L \approx 0.64$, yielding a minimum energy of $\epsilon_0 / \Delta_{\min} = 9.52 \times 10^{-6}$. Although this corresponds to one realization of a QMBS (e.g., with $j_\xi / L$ located at the QD-SC interface), Appendix \ref{['NewQMBSAppendix']} presents an alternative setup where the TS region is shorter than the NTS one, which increases $\epsilon_0 / \Delta_{\min}$ due to stronger wavefunction overlap.