Microwave Dressed States and Vacuum Fluctuations in a Superconducting Condensate

Abstract

Microwave dressed states are found to emerge within the superconducting condensate when coupled to a quantized electromagnetic field due to photon-Cooper pair entanglement. The renormalized energy separation between these states exceeds the prediction of BCS theory, with the enhancement depending on the number of photons and also arising from electromagnetic vacuum fluctuations. Our work introduces an equilibrium quantum model of microwave-enhanced superconductivity, expanding the theoretical description beyond Eliashberg's non-equilibrium theory. We further demonstrate that the superconducting condensate exerts a back-action on the electromagnetic field, suppressing electric field fluctuations, including those from the vacuum state. This result is consistent with Glauber and Lewenstein's field quantization in dielectric media.

Figures (2)

• Figure 1: Schematic illustration of photon-Cooper pair coupling in a superconducting condensate. Left: Bare (uncoupled) states consisting of the BCS ground state ($\psi_o$) with $n+1$ photons, and a pair-excited state ($\psi_e$) with $n$ photons. The excitation corresponds to the promotion of a Cooper pair into its constituent electron and hole pairs at opposite momentum states, $+k_e$ and $-k_e$, resulting in a localized charge separation that interacts with the quantized electromagnetic field. Right: Dressed (coupled) states formed by coherent superpositions of the bare states, exhibiting an energy splitting enhanced by photon-Cooper pair entanglement, as described by Equation (\ref{['eqn:vacRabiSplit']}). Here, $\xi_{k_e}$ is the kinetic energy of the excited Cooper pair relative to the Fermi level, $\Delta$ is the superconducting gap, $v_{k_e}^2$ is the Cooper pair occupation probability at momentum $k_e$, and $V$ is the volume in which the condensate is confined.
• Figure 2: Single-photon Cooper pair excitation energy ($\hbar \Omega$) for a 1 cm$^3$ aluminum superconductor, calculated from Equation (\ref{['eqn:vacRabiSplit']}). The energy is shown as a function of photon number $n$, for a field frequency chosen near the pair-breaking energy of the superconducting gap ($\omega_\lambda=2\Delta/\hbar$). The result is plotted near the Fermi level ($k_F$) and normalized to the gap energy ($\Delta$), with the large-volume limit ($V\rightarrow\infty$) used to define the BCS baseline.