Quantum Coulomb drag signatures of Majorana bound states

Abstract

Majorana bound states (MBSs), with their non-Abelian statistics and topological protection, are key candidates for fault-tolerant quantum computation. However, their unambiguous identification in solid-state systems remains a fundamental challenge. Here, we present a theoretical study demonstrating that drag transport in a capacitively coupled double quantum dot system offers a robust and nonlocal probe of weakly coupled MBSs. Using the master equation approach, we investigate both steady-state and transient dynamics and uncover a distinctive signature of MBSs, i.e., the emergence of pronounced split peaks in the drag transconductance, directly linked to inter-MBS coupling. We further show that the dynamics of quantum coherence exhibit an inverse correlation with the emergence and enhancement of MBS-induced split peaks in the drag transconductance as the inter-MBS coupling increases. A comparative analysis with Andreev bound states (ABSs) reveals key differences, that is, MBS-induced transconductance peaks are symmetric and robust, while ABS features are asymmetric and sensitive to perturbations. These findings establish clear experimental criteria for distinguishing MBSs and provide a practical framework for probing Majorana physics through nonlocal transport.

Figures (4)

• Figure 1: Steady-state and transient transport signatures of MBSs. (a) Schematic illustration of the experimental setup for observing MBS-induced drag transport. A bias voltage applied across QD1 generates a drive current from source ($S$) to drain ($D$). Capacitive coupling between QD1 and QD2 induces a drag current through the unbiased QD2, which is connected to a normal metallic lead ($N$) and a proximitized Rashba nanowire hosting two weakly coupled MBSs, $\eta_L$ and $\eta_R$, at its ends. The tunneling amplitudes between QD2 and the MBSs are denoted by $t_{M,L}$ and $t_{M,R}$. (b) Illustration of a representative sequence of tunneling events that drive the transition from the state $\left|10\right\rangle$ to $\left|11\right\rangle$, breaking the symmetry between forward and backward transport through QD2. (c) Drive current through QD1 as a function of gate voltages $V_s$ and $V_d$ under a fixed bias voltage $V_b = 1.04U_{12}$. (d) Differential conductance $dI_D/dV_b$ at fixed $V_d = 0.5U_{12}$, showing a pronounced conductance peak at $(eV_b, eV_s) = (0, 2.5U_{12})$. (e) Drag current through QD2 as a function of $V_s$ and $V_d$ at $V_b = 1.04U_{12}$, revealing two symmetric pairs of MBS-induced current peaks. The upper and lower split peaks in one such pair near $eV_s = 1.5U_{12}$, located at $(eV_d, eV_s) = (0.52, 1.51)U_{12}$ and $(0.52, 1.49)U_{12}$, are labeled as $I^{+}_{\text{peak}}$ and $I^{-}_{\text{peak}}$, respectively. (f) Dependence of the peak current ratio $I^{+}_{\text{peak}} / I^{-}_{\text{peak}}$ on the inter-MBS coupling strength $g$, extracted from data in (e) with additional values of $g$ [see Figs. S3(a)–S3(l) in Supplementary Sec. II B]. As $g \to 0$, the peak ratio approaches unity, indicating no observable splitting. The rapid suppression of $I^{+}{\text{peak}}$ with increasing $g$ leads to a sharp decline in the ratio. (g) Drag transconductance $|dI_N/dV_b|$ at fixed $eV_d = 0.5U_{12}$ showing four pairs of MBS-induced split peaks centered around $V_s = 1.5U_{12}$ and $V_s = -0.5U_{12}$. In the upper-right pair, the peaks located at $(eV_b, eV_s) = (1.06, 1.54)U_{12}$ and $(1.04, 1.46)U_{12}$ are denoted as $G_{\text{peak}}^+$ and $G_{\text{peak}}^-$, respectively. (h) The conductance peak ratio $G_{\text{peak}}^+ / G_{\text{peak}}^-$ as a function of $g$, obtained from data in (g) with extended values of $g$ [see Figs. S4(a)–S4(g) in Supplementary Sec. II C]. (i, k) Time evolution of the differential conductance of the drive current $|dI_D/dV_b|$ and the drag transconductance $|dI_N/dV_b|$ (in units of $e^2$), respectively, for $(eV_d, eV_s) = (0.5, 2.5)U_{12}$ and $(0.5, 1.46)U_{12}$. (j, l) Voltage slices of (i) and (k) taken at $eV_b = 0$ and $eV_b = 1.04U_{12}$, respectively. The long-time limits of (i) and (k) correspond to the steady-state points in (d) and (g) at $eV_s = 2.5U_{12}$ and $1.46U_{12}$, respectively. $\mu_M$ schematically represents the characteristic energy of the MBSs that serve as an effective bath for QD2. Other parameters are chosen as follows (in units of $U_{12}$): $\mu_S^{(0)} = \mu_D^{(0)} = \mu_N^{(0)} = 0$, $\Gamma_S = \Gamma_D = \Gamma_N = \Gamma_B = \Gamma_A = 0.01$, $k_B T = 0.05$, $g = 0.02$, and $\sigma = 0.01$. Drive and drag currents are given in units of $e\Gamma$, while differential conductances are expressed in units of $e^2$. The cDQD system is initialized in the state $\left| {\psi \left( 0 \right)} \right\rangle = \left| {00} \right\rangle$.
• Figure 2: The time-evolution characteristics of the drag conductance. Panels (a)–(h) depict the temporal evolution of differential drag conductance across the ${V_b}$–${V_s}$ parameter plane ($t$ in units of $1/U_{12}$). Other parameters the same as those in Fig. \ref{['fig1']} except for $eV_d=0.5 U_{12}$. The initial state for the time evolution is chosen as $\left|\psi\left(0 \right) \right\rangle = \left|00\right\rangle$.
• Figure 3: Correlation between quantum coherence and conductance characteristics under different inter-MBS coupling strengths. Panels (a)–(c) show the evolution of the relative quantum coherence $c_{\mathrm{rel}}$ as a function of gate voltage $V_s$ and time $t$, with (a) $g = 0.02U_{12}$, (b) $g = 0.06U_{12}$, and (c) $g = 0.10U_{12}$. Correspondingly, panels (d)–(f) present the evolution of the total off-diagonal coherence $\mathcal{C}$ under the same coupling strengths. Panels (g)–(i) display the steady-state differential conductance $|dI_N/dV_b|$, highlighting its behavior near $V_s \approx 1.5U_{12}/e$. Other parameters are the same as those in Fig. \ref{['fig1']} except for ${V_d} =0.5U_{12}/e$. The initial state for the time evolution is chosen as $\left| {\psi \left( 0 \right)} \right\rangle = \frac{1}{\sqrt{2}}(|b\rangle + |c\rangle)$.
• Figure 4: Distinguishing MBSs from ABSs. Drag-current differential conductance versus ${V_b}$ and ${V_s}$ for (a) weakly coupled MBSs, (b) an ABS, and (c) an isolated MBS, taken along $V_s - V_d = 1$ [see the orange dashed line in Fig. \ref{['fig1']}(e)]. (d) Conductance profiles along the white dashed lines in (a)–(c) ($V_s = -V_b/2 + 1$) are compared for different states: two weakly coupled MBSs (green), an isolated MBS (blue), and a near-zero-energy ABS (orange). The inter-MBS coupling and ABS energy are set to $g = 0.02$ and other parameters are identical to those in Fig. \ref{['fig1']}.