Cavity-control of the Ginzburg-Landau stiffness in superconductors

Abstract

Confining light around solids via cavities enhances the coupling between the electromagnetic fluctuations and the matter. We predict that in superconductors this cavity-enhanced coupling enables the control of the order-parameter stiffness, which governs key length scales such as the coherence length of Cooper pairs and the magnetic penetration depth. We explain this as a renormalization of the Cooper-pair kinetic mass caused by photon-mediated repulsive interactions between the electrons building the pair. This effect is generic for Bardeen-Cooper-Schriffer superconductors and is most pronounced in low-$T_c$ materials. The strength of this effect can be tuned via the length of the cavity and we estimate it to be sizable for cavities in the infrared range.

Figures (2)

• Figure 1: (a) Sketch of a superconductor (green) inside a Fabry-Pérot cavity (grey). "Deformation" of the order parameter by the operator $\hat{\bf D}_\text{c{}l}=-i\nabla-2e{\bf A}_\text{c{}l}$ contributes to the GL free energy $\delta F_\text{GL}$ via the GL stiffness parameter $\tilde{a}_2$, which depends on cavity size. (b) Gapping of the photon field spectrum due to cavity confinement and the Meissner effect (for $\tilde{\lambda}_\text{L} \lesssim d_\text{S}$) inside the superconductor. (c) $\tilde{a}_2$ versus electron-photon interaction energy $\Sigma_0$ with bare value $a_2(0) = \nu_{2D}\xi^2(0)7 \zeta(3)/8$. (d) Renormalized magnetic penetration depth $\tilde{\lambda}_\text{L}$ as a function of cavity length $L_z$ for four typical materials with zero-temperature values $\lambda_\text{L}(T=0)$. Black dashed line shows self-consistent Meissner renormalization effect for Al at $T/T_c=0.8$.
• Figure 2: (a) Dyson equation for the Green function $\tilde{G}_0$ dressed by the effective electron-photon interaction vertex $V_{{\bf k, k'}}({\bf q})$ via the self-energy $\Sigma({\bf k})$. (b) Inverse propagator of the gap field ${[ \mathcal{D}_\Delta({\bf p}) ]^{-1} = \Pi(0)|_{T_c} - \tilde{\Pi}({\bf p})}$ with the bare BCS interaction $\lambda^{-1}=\Pi(0)|_{T_c}$ and the vertex corrections $\Gamma({\bf k},{\bf p})$.