Cavity superconductor-polaritons

Following the recent success of realizing exciton-polariton condensates in cavities, we examine the hybridization of cavity photons with the closest analog of excitons within a superconductor, states called Bardasis-Schrieffer modes. Although these modes do not typically couple linearly to light, one can engineer a coupling with an externally imposed supercurrent, leading to the formation of hybridized Bardasis-Schrieffer-polariton states, which we obtain both as poles of the bosonic Green's function and through the derivation of an effective Hamiltonian picture for the model. These new excitations have nontrivial overlap with both the original photon states and $d$-wave superconducting fluctuations. We conjecture that a phase-coherent density of these objects could produce a finite $d$-wave component of the superconducting order parameter—an $s\pm id$ superconducting state.

Figure 1

The dispersion of the Bardasis-Schrieffer-polariton modes (dot-dashed), calculated both numerically and with a simplified analytic method [24]—the two give visually identical results. An external supercurrent causes the BS mode and cavity photons to hybridize, and the polariton states have significant overlap with each. The “dark” photon mode (dashed) remains decoupled. The splitting of otherwise degenerate photon modes is a result of a supercurrent-induced self-energy contribution. Temperature and supercurrent angle are chosen to maximize hybridization (see Fig. 2). Inset: schematic of the system: a two-dimensional superconductor with an applied supercurrent $I_S$ at the center of a planar cavity.

Figure 2

The hybridization matrix element $g$ in the effective Hamiltonian as a function of temperature, superfluid velocity, and ${\theta }_S$, the angle between the direction of the supercurrent and the axis defined implicitly by the $d$-wave form factor $f_d\left({\varphi }_k\right)$, all scaled by their respective maxima. Left: $g\left(T\right)$ is maximized for a temperature $T_{\text{max}}\approx 0.42T_c$. Center: $g\left(v_S\right)$ is sharply peaked for large superfluid velocity around $v_S\approx 0.96\mathrm{\Delta }(v_S=0)/k_F$. [Note, ${\mathrm{\Delta }}_0\equiv \mathrm{\Delta }(v_S=0)$.] Right: $g\left({\theta }_S\right)$ is maximal for ${\theta }_S=m\pi /2,m\in \mathbb{Z}$, and vanishes when the supercurrent runs along a node of $f_d,{\theta }_S=(2m+1)\pi /4$. Inset: the orientation of the supercurrent with respect to the $d$-wave form factor. The color of the lobes gives the relative sign of $f_d$ for different angles, and the dashed lines are the nodes where $f_d=0$. The plots use $T=T_{\text{max}},v_S=0.9\mathrm{\Delta }(v_S=0)/k_F$, and ${\theta }_S=0$ where applicable, and fixed detuning ${\omega }_0=0.96{\mathrm{\Omega }}_{\text{BS}}$.

Figure 3

The Bardasis-Schrieffer component of the eigenvectors of the effective Hamiltonian, Eq. (15) [24]. The upper (solid) and lower (dot-dashed) polaritons have significant photon and Bardasis-Schrieffer character, indicating strong hybridization between the systems. One can also clearly see the “dark” photon mode (dashed) which does not hybridize with the superconductor's collective mode.