Excess photon-assisted noise of Majorana and Andreev bound states

TL;DR

This work tackles the challenge of distinguishing Majorana bound states from trivial zero-energy Andreev bound states in topological-superconductor platforms by analyzing photon-assisted noise under a dc+ac bias. It develops an effective tunneling model and a scattering-matrix framework to compute the excess photon-assisted noise $S^{exc}$, revealing that $S^{exc}$ vanishes at nonzero integers of $eV_{dc}/\Omega$ for MBSs/QMBSs but remains negative for zero-energy ABSs. The authors corroborate these predictions with numerical calculations and Bogoliubov–de Gennes simulations in semiconductor–superconductor hybrid nanowires, including setups with quantum dots and smooth potentials. The results provide a practical, local probe-based criterion to rule out trivial zero-energy ABSs and assess platform quality, complementing nonlocal transport protocols for identifying topological superconductivity.

Abstract

Photon-assisted tunneling arises under an ac bias, with the drive frequency setting the photon energy. The excess photon-assisted noise is defined as the difference between the shot noise under a combined dc and ac bias and that under a dc bias alone. We investigate this quantity in tunneling into Majorana or Andreev bound states, which are of great interest in the search for topological superconductors. Under a harmonic bias $V(t)=V_\mathrm{dc}[1-\cos(Ωt)]$, the excess photon-assisted noise exhibits distinct behaviors: for Majorana or quasi-Majorana bound states, it undergoes multiple sign reversals as $V_\mathrm{dc}$ increases and vanishes at nonzero integer values of $eV_\mathrm{dc}/Ω$ (with $e$ the elementary charge), whereas for zero-energy Andreev bound states--particularly those producing nearly quantized zero-bias conductance peaks--it remains strictly negative over the entire $V_\mathrm{dc}$ range.

Figures (10)

• Figure 1: (a) Schematic of a metallic probe tunnel-coupled to an ABS, with a bias containing both dc and ac components applied to the probe. The ac bias component enables photon-assisted tunnelings, with the drive frequency setting the photon energy. The ABS with energy $\varepsilon_B$ is represented as a pair of hybridized Majoranas, $\gamma_1$ and $\gamma_2$. $\lambda_{1\sigma}$ and $\lambda_{2\sigma}$ denote the spin-dependent tunneling amplitudes between the probe and $\gamma_{1,2}$. (b) Characteristic spatial profiles of the wave functions of ABSs, MBSs, and QMBSs in semiconductor–superconductor hybrid nanowire systems in the Majorana basis. These indicate that when $\lambda_{2\sigma}\ne 0$, the probe detects an ABS, whereas when $\lambda_{2\sigma}= 0$, it detects an MBS or QMBS. Whether $\gamma_1$ represents an MBS or a QMBS depends on the position of its partner $\gamma_2$, which, however, is inaccessible in the single-probe setup shown in (a).
• Figure 2: (a) Zero-temperature, zero-bias conductance $G(0)$ (in units of $2e^2/h$) and (b) $\Delta Y$, formulated by Eq. \ref{['deltay2']}, as functions of the parameters $r$ and $\theta$ (see Sec. \ref{['secIIA']} for their physical meanings). Here $r=0$ corresponds to MBSs/QMBSs, while $r>0$ corresponds to zero-energy ABSs.
• Figure 3: [(a)--(c)] $K_{1,m^\prime}$ and [(d)--(f)] $K_{2,m^\prime}$, defined in Eqs. \ref{['P1']} and \ref{['P2']}, respectively, as a function of the reduced energy $E/\Omega$ for various $m^\prime$ and $\Gamma/\Omega$ values as indicated. Calculations are performed with $r=0$.
• Figure 4: Current noise as a function of the reduced dc bias $eV_{\mathrm{dc}}/\Omega$ for MBSs/QMBSs. Here, $S^{\mathrm{dc}}$ and $S^{\mathrm{dc+ac}}$ denote the current noise with the ac bias component switched off and on, respectively, while $S^{\mathrm{exc}}$ represents the excess noise defined as $S^{\mathrm{dc+ac}}-S^{\mathrm{dc}}$. The insets in (b) show magnified views around $eV_{\mathrm{dc}}/\Omega = 1$ and $2$. The parameters are: (a) $k_{B}T=0.001$ and $\Gamma=0.02$; (b) $k_{B}T=0.001$; and (c) $\Gamma=0.02$. Other parameters are specified in each panel.
• Figure 5: Conductance and current noise for a zero-energy ABS with $r=0.4$ and $\theta=0.25\pi$. (a) The ABS exhibits a quantized ZBCP arising from the additive contributions of two eigenchannels with transmissions $\tau_+$ and $\tau_-$. (b) Current noise as a function of the reduced dc bias $eV_\textrm{dc}/\Omega$ at $k_BT=0.001$ and $\Gamma=0.02$, where $S^\textrm{dc}$ and $S^\textrm{dc+ac}$ denote the noise with the ac bias component switched off and on, respectively. (c) Excess noise for $\Gamma=0.02$ at different temperatures as indicated.