Hanbury Brown and Twiss noise correlations in a topological superconductor beam splitter

We study Hanbury Brown and Twiss current cross-correlations in a three-terminal junction where a central topological superconductor (TS) nanowire, bearing Majorana bound states at its ends, is connected to two normal leads. Relying on a nonperturbative Green function formalism, our calculations allow us to provide analytical expressions for the currents and their correlations at subgap voltages, while also giving exact numerical results valid for arbitrary external bias. We show that when the normal leads are biased at voltages $V_1$ and $V_2$ smaller than the gap, the sign of the current cross-correlations is given by $-\text{sgn}\left(V_1{V}_2\right)$. In particular, this leads to positive cross-correlations for opposite voltages, a behavior in stark contrast with the one of a standard superconductor, which provides direct evidence of the presence of the Majorana zero mode at the edge of the TS. We further extend our results, varying the length of the TS (leading to an overlap of the Majorana bound states) as well as its chemical potential (driving it away from half-filling), generalizing the boundary TS Green function to those cases. In the case of opposite bias voltages, $\text{sgn}\left(V_1{V}_2\right)=-1$, driving the TS wire through the topological transition leads to a sign change of the current cross-correlations, providing yet another signature of the physics of the Majorana bound state.

Figure 1

Schematic view of the setup: a grounded TS nanowire is tunnel coupled (with hopping amplitudes ${\lambda }_1,{\lambda }_2$) to two normal-conducting (${\mathrm{N}}_1$, ${\mathrm{N}}_2$) leads which are biased at voltages $V_1$ and $V_2$, respectively.

Figure 2

Zero-temperature local $J_{11}\left(\omega \right)$ (left panel) and non-local $J_{12}\left(\omega \right)$ (right panel) differential conductances (in units of $2e^2/h$) vs $\omega /\mathrm{\Delta }$ for ${\lambda }_1={\lambda }_2$ and several values of the total transmission probability $\tau$.

Figure 3

Current $I_1$ (in units of $e\mathrm{\Delta }/h)$ vs voltage $V=V_1$ in the case of equal (red dashed curve) and opposite (blue full curve) bias voltages $V_{1,2}$ at zero temperature, ${\lambda }_1={\lambda }_2$, and several values of $\tau$. In all figures, voltages are given in units of $\mathrm{\Delta }$.

Figure 4

HBT cross-correlations $S_{12}$ (in units of $e^2/h$) vs $V=V_1$ in the case of equal (red dashed curve) and opposite (blue full curve) voltages $V_{1,2}$ for ${\lambda }_1={\lambda }_2$ and several transparencies $\tau$ (as noted on each panel), at zero temperature. The bottom right panel combines all curves to show the overall scale. $S_{12}$ is negative (positive) for equal (opposite) bias voltages, and the values of $S_{12}$ are simply opposite in sign in the two cases for $\left|V\right|\ll \mathrm{\Gamma }$.

Figure 5

Schematic picture of the current partitioning between the TS and two normal leads (${\mathrm{N}}_{1,2}$) at low voltage $\left|eV\right|\ll \mathrm{\Gamma }$. Electrons (holes) are shown as full (empty) circles. Left panel: the case of equal voltages, where a noiseless stream of electrons from the Majorana bound state is partitioned between the two normal leads, with perfect anti-correlations of the two electron streams. Right panel: the case of opposite voltages where lead ${\mathrm{N}}_2$ is biased at potential $-V$, so that electrons are emitted into ${\mathrm{N}}_1$ while holes are emitted into ${\mathrm{N}}_2$. The two fermion streams are perfectly anticorrelated, which leads to positive cross-correlation noise. The arrows indicate directions of quasiparticle motion.

Figure 6

Cross-correlations $S_{12}$ vs $V=V_1$ in the case of opposite bias voltages ($V_1=-V_2$) for a symmetric junction (${\lambda }_1={\lambda }_2$) with total transmission $\tau =0.5$ and several values of the TS wire length $L$. Each curve is labeled by the corresponding value of $L$ in units of ${\xi }_0=\hslash v_F/\mathrm{\Delta }$. For $L\gg 1,S_{12}$ is identical to the infinite TS case, while for $L\sim 1$ the cross-correlations $S_{12}$ become negative over the range $\left|V\right|/\mathrm{\Delta }\lesssim {({\xi }_0/L)}^2$

Figure 7

Current correlations $S_{12}$ vs $V=V_1$ in the setup with an infinite TS wire for several values of its chemical potential $\mu$ (each curve is labeled by the corresponding value of $\mu /\mathrm{\Delta }$). The top (bottom) panel shows the case of opposite (equal) voltages. We consider a symmetric junction (${\lambda }_1={\lambda }_2$) with a total transparency $\tau =0.5$, and a hopping strength (TS wire bandwidth) $t_0=10\mathrm{\Delta }$. The black curve (labeled “ref”) on each plot corresponds to the wide-band limit [using Eq. (4)] and serves as a reference for comparison. There is a clear transition when $\mu$ reaches the bandwidth $t_0$: the peak behavior around $V=0$ disappears when $\mu >t_0$, signaling the absence of a Majorana bound state for $\mu >t_0$.

Figure 8

Panels (a) and (b) : schematic illustration of the behavior of the system for the choice of $\mathrm{\Lambda }<1$ and $\mathrm{\Lambda }>1$, in the case of a strong reflection ($r$ close to 1). While $\mathrm{\Lambda }={\mathrm{\Lambda }}_0<1$ and $\mathrm{\Lambda }=1/{\mathrm{\Lambda }}_0>1$ correspond to the same transparency between the TS and the normal leads, the transparency of the direct channel between the two normal leads is totally different in the two cases. Panel (c) : plots of the current $I_1\left(V\right)$ in the $V_1=-V_2=V$ configuration, for $\mathrm{\Lambda }=4$ and $\mathrm{\Lambda }=1/4$.