Cavity Quantum Electrodynamics at Arbitrary Light-Matter Coupling Strengths

Quantum light-matter systems at strong coupling are notoriously challenging to analyze due to the need to include states with many excitations in every coupled mode. We propose a nonperturbative approach to analyze light-matter correlations at all interaction strengths. The key element of our approach is a unitary transformation that achieves asymptotic decoupling of light and matter degrees of freedom in the limit where light-matter interaction becomes the dominant energy scale. In the transformed frame, truncation of the matter or photon Hilbert space is increasingly well justified at larger coupling, enabling one to systematically derive low-energy effective models, such as tight-binding Hamiltonians. We demonstrate the versatility of our approach by applying it to concrete models relevant to electrons in crystal potential and electric dipoles interacting with a cavity mode. A generalization to the case of spatially varying electromagnetic modes is also discussed.

Figure 1

(Top) Effective parameter ${\xi }_g$ characterizing interaction strength in the asymptotically decoupled frame against the bare light-matter coupling $g$. (Bottom) Exact spectrum obtained by diagonalizing Eq. (4), or equivalently Eq. (5), for an electron in periodic potential. In the extremely strong coupling (ESC) regime, it exhibits equally spaced flat bands narrowing as $\propto 1/g$, corresponding to localized electrons with the large renormalized mass (inset). Numerical values are shown in the unit ${\omega }_c=\hslash =m=1$ throughout this Letter. The potential depth and lattice constant are $v=5$ and $d=4$, respectively.

Figure 2

(a)–(d) Exact electron-polariton bands obtained by diagonalizing Eq. (5). Gray dashed curves indicate dispersions at $g=0$. (e) Comparisons between the exact results and the tight-binding models at $g=0.1$, 1, 2, 10 from top to bottom. Black dashed (red dotted) curves indicate the tight-binding results in the asymptotically decoupled frame (Coulomb gauge). We set $v=5$ and $d=4$ in (a)–(e).

Figure 3

Low-energy spectra for (a) deep and (b) shallow double-well potentials with photon-number cutoff $n_c=100$. Blue solid curves (red dotted curves) show the results in the AD frame (PZW frame). We choose the parameters (a) $\lambda =50$, $\mu =95$ and (b) $\lambda =3$, $\mu =3.85$ such that the transition frequency of the two lowest matter levels is resonant with ${\omega }_c$ in each case.