Dynamic current susceptibility as a probe of Majorana bound states in nanowire-based Josephson junctions

We theoretically study a Josephson junction based on a semiconducting nanowire subject to a time-dependent flux bias. We establish a general density-matrix approach for the dynamical response of the Majorana junction and calculate the resulting flux-dependent susceptibility using both microscopic and effective low-energy descriptions for the nanowire. We find that the diagonal component of the susceptibility, associated with the dynamics of the Majorana state populations, dominates over the standard Kubo contribution for a wide range of experimentally relevant parameters. The diagonal term, explored, in this Rapid Communication, in the context of Majorana physics, allows probing accurately the presence of Majorana bound states in the junction.

Figure 1

Scheme of the setup: A nanowire (in blue) proximitized with a bulk $s$-wave superconductor (in gray) and subject to both a dc flux ${\mathrm{\Phi }}_{\mathrm{dc}}$ and an ac flux $\delta \mathrm{\Phi }\left(t\right)$. The superconducting phase across the junction and the external flux threading the ring are related by the condition $\varphi =2e\mathrm{\Phi }/\hslash$ with $\mathrm{\Phi }={\mathrm{\Phi }}_{\mathrm{dc}}+\delta \mathrm{\Phi }$. The bulk $s$-wave superconductor (in gray) is interrupted under the weak link (in green, thinner line). The red circles indicate four MBSs ${\gamma }_1-{\gamma }_4$, whereas $l_{12}$ and $l_{34}$ are the sizes of the left and right topological regions, respectively. The two inner MBSs ${\gamma }_2$ and ${\gamma }_3$ overlap through the weak link whereas the outer MBSs ${\gamma }_1$ and ${\gamma }_4$ can in principle overlap through the superconducting ring.

Figure 2

Top: The imaginary part of the susceptibility ${\chi }^{\prime \prime }(\varphi ,\omega )$ as a function of phase $\varphi$ in the topological regime for various values of the Zeeman field and assuming all possible transitions. The black (full), red (dashed), blue (dotted), brown (dot-dashed), and green (small dots) curves correspond to $V_Z=1.2,1.3,1.4,1.5$, and $1.6(\times {\mathrm{\Delta }}_w)$, respectively. [The inset: The black (full), red (dashed), and blue (dotted) curves correspond to $V_Z=1.7,2$, and $2.3(\times {\mathrm{\Delta }}_w)$, respectively.] The topological transition takes place at $V_Z={\mathrm{\Delta }}_w$. We expressed all energies in terms of the hopping $t$ with ${\mathrm{\Delta }}_w=0.05,\alpha =0.08,\omega =1.6\times {10}^{-4},T=0.8\omega ,\gamma ={10}^{-8}$. Bottom: Many-body spectrum and the allowed transitions for $V_Z=1.2{\mathrm{\Delta }}_w$ (all the other parameters are as above). The red (full) [blue (dashed)] levels correspond to the $\tau =1(\tau =-1)$ parity state. The left (right) vertical arrows depict the parity-conserving (flipping) transitions.

Figure 3

The imaginary part of the total susceptibility $\chi {}^{\prime \prime }(\varphi ,\omega )$ from the effective model as a function of $\varphi$ with and without the parity constraints for several values of the resonator frequency $\omega$ (relative to the frequency in Fig. 2). The black (full), blue (dashed), and red (gray) curves correspond to the unconstrained (Fermi-Dirac) distributions and the constrained distributions with parities $\tau =-1$ and $\tau =1$, respectively. The parameters of the effective model are extracted from the full Hamiltonian at $V_Z=1.5$, whereas the rest of the parameters are the same as in Fig. 2. A crossover from ${\chi }_D^{\prime \prime }$ to ${\chi }_{\mathrm{ND}}^{\prime \prime }$ dominated dissipation is clearly visible in the range of frequencies depicted in the plot for the unconstrained parity case, whereas the dissipation is dominated by ${\chi }_{\tau ,\mathrm{ND}}^{\prime \prime }$ in the constrained case.