Long-Range Photon Fluctuations Enhance Photon-Mediated Electron Pairing and Superconductivity

Recently, the possibility of inducing superconductivity for electrons in two-dimensional materials has been proposed via cavity-mediated pairing. The cavity-mediated electron-electron interactions are long range, which has two main effects: firstly, within the standard BCS-type pairing mediated by adiabatic photons, the superconducting critical temperature depends polynomially on the coupling strength, instead of the exponential dependence characterizing the phonon-mediated pairing; secondly, as we show here, the effect of photon fluctuations is significantly enhanced. These mediate novel non-BCS-type pairing processes, via nonadiabatic photons, which are not sensitive to the electron occupation but rather to the electron dispersion and lifetime at the Fermi surface. Therefore, while the leading temperature dependence of BCS pairing comes from the smoothening of the Fermi-Dirac distribution, the temperature dependence of the fluctuation-induced pairing comes from the electron lifetime. For realistic parameters, also including cavity loss, this results in a critical temperature which can be more than 1 order of magnitude larger than the BCS prediction. Moreover, a finite average number of photons (as can be achieved by incoherently pumping the cavity) adds to the fluctuations and leads to a further enhancement of the critical temperature.

Figure 1

(a) The real-time Keldysh formulation of the Bethe-Salpeter equation for the vertex function $\mathrm{\Gamma }$ in the pairing channel. It contains the bare vertex ${\mathrm{\Gamma }}_0$ (first diagram), the standard BCS terms ${\mathrm{\Gamma }}_{\mathrm{BCS}}$ (second and third diagrams), and the “fluctuation term," ${\mathrm{\Gamma }}_{\text{fluct}}$ (fourth diagram). The latter leads to a non-BCS-type of pairing. (b) The electronic energy structure can be divided between on shell electrons in the vicinity of the FS: ${\omega }_1\sim {\epsilon }_{k_F\pm q_0}$ and off shell electrons excited to photonic frequencies: ${\omega }_1\sim {\delta }_c$ [see Eq. (3)]. (c) Standard BCS pairing: the scattering via off shell adiabatic photons leaves the electrons on shell. (d) Fluctuation-induced pairing: on shell nonadiabatic photons at frequencies $\pm {\delta }_c$ induce pairing between on shell electrons through an intermediate transition to off shell electronic states.

Figure 2

Critical temperature $T_c$ vs dimensionless coupling strength $\tilde{g}$ in log plot. (a) For thermal photons, the blue solid line with circles corresponds to prediction (5), based on solely the BCS pairing [see Fig. 1]. The solid magenta line corresponds to the prediction which also includes the fluctuation-induced pairing [see Fig. 1], as well as a finite Fermi-liquid-type electron lifetime [see Eq. (7)]. The green dotted line shows the prediction for long-lived electrons from Eq. (6), while the red dashed line corresponds to the prediction of Eq. (8) for short-lived electrons. (b) For driven dissipative photons, solid line with circles show the BCS prediction which is almost unaffected by photon loss and incoherent pump. Solid lines with triangles show the fluctuation-assisted enhancement in the purely lossy case $\gamma =\kappa$ which is further amplified by incoherent pumping $\gamma >\kappa$ shown by solid line with squares. The inset (linear plot) shows $T_c$ increases almost linearly with $\gamma$. We choose ${\delta }_c=0.19\mathrm{THz}$, ${\epsilon }_{k_F+q_0}/{\delta }_c=0.007$, $\kappa /{\delta }_c=0.01$, and for the inset $\tilde{g}=0.004$.