Josephson radiation patterns in underdamped topological Josephson junctions

Josephson radiation is a useful signature for detecting Majorana zero modes in topological superconductors. We study the Josephson radiation of the underdamped topological Josephson junction within the quantum resistively and capacitively shunted junction model. We show that the quantum dynamics of the Majorana two-level system induce oscillatory patterns in the Josephson emission spectra. With the Floquet method, we obtain analytical results for these oscillatory patterns and find that they are well described by Bessel functions. We perform numerical simulations to verify the analytical results and demonstrate that these Bessel radiation patterns exist for a wide range of junction parameters.

Figure 1

(a) Schematic of a Josephson junction with Majorana zero modes ${\gamma }_L,{\gamma }_L^{\prime },{\gamma }_R,\mathrm{and}{\gamma }_R^{\prime }$. Upon current injection above the critical current, the voltage across the junction oscillates, and the junction radiates electromagnetic waves. (b) The “tilted-washboard” potential representation of the quantum resistively and capacitively shunted junction model where the dynamics of the Josephson phase are equivalent to the semiclassical motion of a spin-one-half particle. The potential function for the spin-up state (blue line) and the spin-down state (the red line) have avoided crossing points around $\theta =(2n+1)\pi$. The Landau-Zener transitions would happen at these avoided-crossing points. (c) The semiconductors picture for the Josephson radiation in the voltage-biased scenario. The coherent Cooper pair tunneling results in a radiation with frequency $f_{\mathrm{J}}=2\mathrm{eV}/\mathrm{h}$, whereas the coherent single-particle tunneling through the Majorana zero modes contributes radiation with frequency $f_{\mathrm{M}}=\mathrm{eV}/\mathrm{h}$.

Figure 2

(a) Radiation spectrum for typical underdamped junctions simulated with the quantum resistively and capacitively shunted junction model. Junction parameters are taken as $C=5\times {10}^5{e}^2/2E_{\mathrm{M}}, R=5\times {10}^{-3}\hslash /e^2, I_{c2}=e{E}_{\mathrm{M}}/\hslash , I_{c1}=8e{E}_{\mathrm{M}}/\hslash , \delta =0.05E_{\mathrm{M}}$. The Stewart-McCumber number is ${\beta }_c=100$, making the junction well into the underdamped region. (b) The lowest-order oscillation curves of the analytical results shown in Eq. (10). The black dotted line shows the half Josephson frequency $f=f_{\mathrm{J}}/2$.

Figure 3

Radiation spectrum for the underdamped junction simulated with the quantum resistively and capacitively shunted junction model with typical parameters. Parameters are taken as (a) $C=2.5\times {10}^5{e}^2/2E_{\mathrm{M}}, R=5\times {10}^{-3}\hslash /e^2, I_{c2}=e{E}_{\mathrm{M}}/\hslash , I_{c1}=8e{E}_{\mathrm{M}}/\hslash , \delta =0.1E_{\mathrm{M}}$; (b) $\delta =0.2E_{\mathrm{M}}$ with other parameters the same; (c) $C=1.25\times {10}^5{e}^2/2E_{\mathrm{M}}$ with other parameters the same; (d) $C=6\times {10}^4{e}^2/2E_{\mathrm{M}}, R=0.02\hslash /e^2, I_{c1}=2e{E}_{\mathrm{M}}/\hslash$ with other parameters the same.

Figure 4

(a) Numerical results of the radiation spectrum when the coupling energy Majorana zero modes at one side of the junction is large. Junction parameters are taken the same as in Fig. 2 except for $\delta /E_{\mathrm{M}}=0.35$. (b) Analytical results of the oscillation patterns given by Eq. (11) (blue dotted curve), and the straight lines connecting the intersection points given by Eq. (12) (black solid line); the Josephson frequencies of $f=n{f}_{\mathrm{J}}/2$ are marked with the red lines.