Hybrid light-matter excitations and spontaneous time-reversal symmetry breaking in two-dimensional Josephson Junctions

Abstract

In the context of hybrid superconductor-semiconductor systems, Josephson junctions based on two-dimensional materials, such as graphene, offer promising opportunities because of their scalability and gate-tunable electronic properties. In this work, we investigate the inductive coupling between a quantum LC resonator and a superconducting loop embedding a short, ballistic, planar Josephson junction, with the graphene-based case as a representative example. Within a mean-field formalism, we analyze how the properties of the global system depend on the light-matter interaction coupling, the Fermi level of the two-dimensional material, and temperature. Our findings reveal that the current-phase relation can show features indicative of spontaneous time-reversal symmetry breaking. Furthermore, starting from the mean-field theory, we determine the low-energy spectrum of collective hybridized light-matter excitations.

Figures (11)

• Figure 1: \ref{['fig:GJJ_LC_circuit_schematic']} The resonant circuit is represented as a lumped-element LC resonator (red) with a capacitance $C_{\rm r}$ and an inductance $L_{\rm r}$. The LC circuit interacts inductively, through a mutual inductance $M$, with a loop containing a single short GJJ (blue). The superconducting phase difference across the GJJ is $\varphi$. \ref{['fig:GJJ_schematic']} A GJJ made by a monolayer graphene (grey) deposited on a substrate (green) and covered by two superconducting leads (blue). The uncovered grey region represents the graphene stripe in the normal phase. In this picture, $L$ represents the junction channel length along the $x$-direction (longitudinal) and $W$ is the width of the device along the $y$-direction (transverse).
• Figure 2: CPR and reciprocal inductance of the GJJ for coupling constant $g = 0.1$ (red solid line) and $g = 0$ (blue dashed line), shown respectively in \ref{['fig:I_vs_phi_g_comparison']} and \ref{['fig:L_vs_phi_g_comparison']}. The quantities are expressed in units of $\mathcal{N}\Delta_0/\phi_0$ and $\mathcal{N}\Delta_0/\phi_0^{2}$. In the inset of \ref{['fig:I_vs_phi_g_comparison']}, two finite supercurrents of opposite sign appear at $\varphi=\pi$ when the coupling constant $g$ is switched on. In the inset of \ref{['fig:L_vs_phi_g_comparison']}, the coupling produces only a small renormalization near $\varphi=\pi$. All results are obtained at zero temperature. Other parameters are $\hbar\omega_{\rm r}=0.6\Delta_0$ and $\mu_0 = 10\hbar v_{\rm F}/L$.
• Figure 3: Modulus of the photonic mean-field $\bar{\alpha}$ minimizing the energy-density functional as a function of the coupling constant $g$ at zero temperature. The red line shows the value of $\bar{\alpha}$ obtained from the full numerical solution of the self-consistent approach, while the blue dashed line corresponds to Eq. \ref{['eq:baralpha']}, with $\kappa=0$. Because of the logarithmic singularity of the energy-density functional in the variable $g\alpha$ as $g\alpha\to0$, the numerical search is terminated at $g \lesssim 0.03$, where $|\bar{\alpha}|\sim10^{-25}$, beyond which resolution limitations prevent a stable solution. Other parameters are $\hbar\omega_{\mathrm r} = 0.6\Delta_0$ and $\mu_0 = 10\hbar v_{\mathrm F}/L$.
• Figure 4: Supercurrent $I$ at $\varphi = \pi^{-}$, in units of $\mathcal{N}\Delta_0/\phi_{0}$, as a function of the Fermi level $\mu_{0}$, expressed in units of $\hbar v_{\mathrm{F}}/L$, for three temperatures: $k_{\mathrm{B}}T/\Delta_0 = 0$ (blue solid line), $0.002$ (green solid line), and $0.01$ (red solid line). Vertical dashed lines mark values of $\mu_{0} L/(\hbar v_{\mathrm{F}})$ equal to $n \pi$ (where $n$ is an integer). The remaining parameters are $\hbar\omega_{\mathrm{r}} = 0.6 \Delta_0$ and $g = 0.1$.
• Figure 5: Phase diagram referring to the time-reversal breaking (TRB) instability. Gray shaded region represents the values of the coupling constant $g$ and the temperature $T$ (in units of $\Delta_0/k_{\rm B}$) where the global system, composed of GJJ and LC quantum harmonic oscillator, is in TRB phase, where a finite supercurrent emerges at $\varphi=\pi$, while the white region corresponds to the phase where no supercurrent is present. The boundary between the two regions denotes the critical temperature $T_{\mathrm c}$. The comparison, in logarithmic scale, between the numerical solution and the analytical estimate in Eq. \ref{['eq:kbT_vs_g_analytical']} (blue dashed line) shows that the approximated expression captures the qualitative behavior of the spontaneous instability. Here, the other parameters are fixed at $\hbar\omega_{\mathrm r}=0.6\Delta_0$ and $\mu_0 = 10\hbar v_{\mathrm F}/L$.