Resolution of Gauge Ambiguities in Molecular Cavity Quantum Electrodynamics

This work provides the fundamental theoretical framework for molecular cavity quantum electrodynamics by resolving the gauge ambiguities between the Coulomb gauge and the dipole gauge Hamiltonians under the electronic state truncation. We conjecture that such ambiguity arises because not all operators are consistently constrained in the same truncated electronic subspace for both gauges. We resolve this ambiguity by constructing a unitary transformation operator that properly constrains all light-matter interaction terms in the same subspace. We further derive an equivalent and yet convenient expression for the Coulomb gauge Hamiltonian under the truncated subspace. We finally provide the analytical and numerical results of a model molecular system coupled to the cavity to demonstrate the validity of our theory.

Figure 1

Shin-Metiu model (transferring proton and electron between two fixed ions) coupled to an optical cavity. (a) Diabatic potentials ${\mathcal{V}}_{\phi \varphi }(R)$ (upper panel) and dipole $\tilde{\mu }$ (lower panel), with the inset describes characters of the diabatic states. (b) Polariton potential energy surface ${\mathcal{E}}_k$ for the molecule-cavity hybrid system at $A_0=0.2$ and $\hslash {\omega }_c=3\mathrm{eV}$, from ${\widehat{\mathcal{H}}}_{\mathrm{pl}}^D$ (solid), ${\widehat{\mathcal{H}}}_{\mathrm{pl}}^C$ (dotted), and ${\widehat{\mathcal{H}}}_{\mathrm{pl}}^{C^{\prime }}$ (dashed). The polariton eigenenergies as a function of $A_0$ are depicted at (c) $R=0$ and (d) $R=-1.74\mathrm{a}.\mathrm{u}.$.