Cavity Quantum Eliashberg Enhancement of Superconductivity

Driving a conventional superconductor with an appropriately tuned classical electromagnetic field can lead to an enhancement of superconductivity via a redistribution of the quasiparticles into a more favorable nonequilibrium distribution—a phenomenon known as the Eliashberg effect. Here, we theoretically consider coupling a two-dimensional superconducting film to the quantized electromagnetic modes of a microwave resonator cavity. As in the classical Eliashberg case, we use a kinetic equation to study the effect of the fluctuating, dynamical electromagnetic field on the Bogoliubov quasiparticles. We find that when the photon and quasiparticle systems are out of thermal equilibrium, a redistribution of quasiparticles into a more favorable nonequilibrium steady state occurs, thereby enhancing superconductivity in the sample. We predict that by tailoring the cavity environment (e.g., the photon occupation and spectral functions), enhancement can be observed in a variety of parameter regimes, offering a large degree of tunability.

Figure 1

(a) Relative enhancement of the gap function as a function of cavity frequency ${\omega }_0$ for a particular value of the overall scaling constant $\pi \alpha XD{\tau }_{\mathrm{in}}/c^2$ (we take $X=133$ and $\pi \alpha D{\tau }_{\mathrm{in}}{T}_c^2/c^2=9.17\times {10}^{-5}$ with $T_c$ set to unity). Curves are colored and labeled according to the ratio $T_{\text{cav}}/T_{\mathrm{qp}}$, comparing the photon and quasiparticle temperatures. The enhancement is seen set in after the cavity frequency surpasses the pair-breaking energy $2{\mathrm{\Delta }}_0$. (b) Schematic picture of the system used for calculation. The lowest cavity resonator mode with cutoff frequency ${\omega }_0$ is shown, as is the 2D superconducting (SC) layer. (c) Depiction of the various processes which contribute to the quasiparticle collision integral, plotted against the equilibrium $n(E)$. The blue arrows depict the down-scattering terms captured by $f(\mathrm{\Omega },E)$, the red arrows depict the up-scattering terms captured by $f(-\mathrm{\Omega },E)$, and the green arrows represent the pair processes captured by $f(-\mathrm{\Omega },-E)$.

Figure 2

Change in quasiparticle distribution function due to cavity photons. The two curves are at the same temperature ($T_{\text{cav}}/T_{\mathrm{qp}}=0.5$) but different cavity frequencies ${\omega }_0/{\mathrm{\Delta }}_0$. For low cavity frequency (orange), the gap $\mathrm{\Delta }$ is diminished due to an accumulation of cooler quasiparticles near the gap edge, due to a down-scattering of particles. For higher cavity frequency (blue), the recombination processes are more dominant and lead to a net reduction in quasiparticles, enhancing the gap $\mathrm{\Delta }$. The kink features labeled $A$ and $C$ reflect the onset of the term $f(\mathrm{\Omega },E)$ in Eq. (14), which is nonzero only for $E>{\omega }_0+{\mathrm{\Delta }}_0$. At higher cavity frequencies (${\omega }_0>2{\mathrm{\Delta }}_0$) an additional kink feature (located at $B$) emerges at $E={\omega }_0-{\mathrm{\Delta }}_0$. For $E<{\omega }_0-{\mathrm{\Delta }}_0$, the term $f(-\mathrm{\Omega },E)$ (which represents the pair processes) contributes over the entire integration region of $\mathrm{\Omega }>{\omega }_0$, while for $E>{\omega }_0-{\mathrm{\Delta }}_0$ the integral only captures some of the frequencies where this term contributes.

Figure 3

Gap enhancement $\delta {\mathrm{\Delta }}_0$ for a single-mode cavity, for both cold and hot photons. The $y$ axis is determined by the overall scale $4\pi \alpha D{\tau }_{\mathrm{in}}{T}_c^2/((\pi \sqrt{3}{)}^3{c}^2)X$; with the same values chosen for $X$ and ${\tau }_{\mathrm{in}}$, ${\tau }_{\mathrm{el}}$, $v_F/c$ as in Fig. 1. Curves are colored and labeled according to the ratio $T_{\text{cav}}/T_{\mathrm{qp}}$, comparing the photon and quasiparticle temperatures. Here, the cavity width is held fixed at $1/2{\tau }_{\text{cav}}=10{\omega }_0$.