Cavity quantum-electrodynamical polaritonically enhanced electron-phonon coupling and its influence on superconductivity

Abstract

So far, laser control of solids has been mainly discussed in the context of strong classical nonlinear light-matter coupling in a pump-probe framework. Here, we propose a quantum-electrodynamical setting to address the coupling of a low-dimensional quantum material to quantized electromagnetic fields in quantum cavities. Using a protoypical model system describing FeSe/SrTiO₃ with electron-phonon long-range forward scattering, we study how the formation of phonon polaritons at the two-dimensional interface of the material modifies effective couplings and superconducting properties in a Migdal-Eliashberg simulation. We find that through highly polarizable dipolar phonons, large cavity-enhanced electron-phonon couplings are possible, but superconductivity is not enhanced for the forward-scattering pairing mechanism due to the interplay between coupling enhancement and mode softening. Our results demonstrate that quantum cavities enable the engineering of fundamental couplings in solids, paving the way for unprecedented control of material properties.

Fig. 1. Setup of 2D material in optical cavity, phonon polariton frequency dispersions, and momentum-dependent electron-phonon coupling vertices for the polariton branches.

(A) We consider a setup with a 2D material on a dielectric substrate inside a small optical cavity with mirrors as shown. (B) Schematic phonon, photon, and upper and lower polariton dispersions versus 2D in-plane momentum q. The coupling of the phononic dipole current to the photonic vector potential leads to a splitting determined by the plasma frequency ω_P. In the cavity, ω_P is controlled by the cavity volume. (C) Momentum-dependent squared electron-boson vertex g²(q). For forward scattering, the squared bare electron-phonon vertex ${\mathit{g}}^2(\mathit{q})={\mathit{g}}_0^2\text{exp}(-2\mathit{q}/{\mathit{q}}_0)$ is peaked near q = 0. In the polaritonic case (ω_P > 0), the upper polariton branch inherits some of the electron-phonon coupling at small q.

Fig. 2. Temperature-dependent electron-phonon coupling for different coupling ranges and plasma frequencies.

(A) Dimensionless electron-phonon coupling strength extracted from the normal self-energy at ${\overrightarrow{\mathit{k}}}_{\mathrm{F}}$ at the smallest Matsubara frequency, $\mathrm{\lambda }\equiv \mathit{Z}({\overrightarrow{\mathit{k}}}_{\mathrm{F}},\mathit{i}\mathrm{\pi }/\mathrm{\beta })-1$, as a function of temperature for a value of the coupling range in momentum space q₀/k_F = 0.105 representative of FeSe/SrTiO₃, and different phononic plasma frequencies ω_P as indicated. The case ω_P = 0 represents the system without cavity. For increasing ω_P, λ increases. Below the superconducting transition, which also shifts with ω_P (see Fig. 3), λ decreases consistently for all values of ω_P. (B) Temperature-dependent λ for smaller q₀/k_F = 0.053 and different ω_P. As for the superconducting order parameter, the effects of the cavity coupling that is parameterized by ω_P are more pronounced. (C) For even smaller q₀/k_F = 0.021, we obtain a strongly enhanced λ accompanied by the shift in the superconducting transition that shows up as a cusp in λ(T), which reaches a maximum at T_c.

Fig. 3. Temperature-dependent superconducting gap for different coupling ranges and plasma frequencies.

(A) The superconducting gap at ${\overrightarrow{\mathit{k}}}_{\mathrm{F}}$ at the smallest Matsubara frequency, $\mathrm{\Delta }\equiv \mathrm{\varphi }({\overrightarrow{\mathit{k}}}_{\mathrm{F}},\mathit{i}\mathrm{\pi }/\mathrm{\beta })/\mathit{Z}({\overrightarrow{\mathit{k}}}_{\mathrm{F}},\mathit{i}\mathrm{\pi }/\mathrm{\beta })$, as a function of temperature for a value of the coupling range in momentum space q₀/k_F = 0.105 representative of FeSe/SrTiO₃, and different phononic plasma frequencies ω_P (measured in electron volts for the FeSe example) as indicated. The case ω_P = 0 represents the system without cavity. For the decreasing cavity volume, ω_P increases, causing a decrease in Δ and the superconducting critical temperature T_c. (B) Temperature-dependent gap for smaller q₀/k_F = 0.053 and different ω_P. The light-suppressed superconductivity is more pronounced. (C) For even smaller q₀/k_F = 0.021, strongly reduced Δ values are observed with increasing ω_P.