Strong quantum interaction between excitons bound by cavity photon exchange

TL;DR

This paper addresses the problem of achieving strong interactions between polaritons formed by coupling a cavity mode to bound-to-continuum intersubband transitions in doped quantum wells. It develops a theoretical framework that yields a bound polariton state via exact diagonalization of a quadratic light–matter Hamiltonian and then computes the resulting polariton–polariton interaction, revealing two concurrent contributions: a Coulomb term and a Pauli-blocking saturation term. The key finding is that reducing the bound-state binding energy (via cavity detuning) expands the polariton Bohr radius, causing the Coulomb-driven interaction strength $g_{PP}$ to grow dramatically, reminiscent of Rydberg physics. This suggests giant quantum optical nonlinearities in the mid- to far-infrared, a spectral region with rich fundamental and practical potential for infrared quantum photonics and strongly correlated states.

Abstract

We theoretically predict the interaction between polaritonic excitations arising from the coupling of a cavity photon mode with bound to continuum intersubband transitions in a doped quantum well. The resulting exciton bound by photon exchange, recently demonstrated experimentally, exhibits a binding energy that can be continuously tuned by varying the cavity frequency. We show that polariton-polariton interactions, originating from both Coulomb interactions and Pauli blocking, can be dramatically enhanced by reducing the exciton binding energy, thereby increasing the effective Bohr radius along the growth direction. This regime is reminiscent of Rydberg atoms, where weak binding leads to strong quantum interactions. Our predictions indicate that this physics can give rise to giant quantum optical nonlinearities in the mid and far infrared, a spectral region that remains largely unexplored in quantum optics and offers exciting opportunities for both fundamental studies and applications.

Figures (4)

• Figure 1: Schematic representation of the single-particle electronic spectrum of a quantum well with one quantum-confined conduction subband separated from the continuum by an ionization energy gap $\epsilon_g$. (a) Square modulus of the envelope functions of the quantum-well electron eigenstates along the $z$ direction, in the absence of a cavity mode. The quantum-well potential and energy levels are indicated by the black dashed line. (b) Cavity-dressed system. The coupling between the bound-to-continuum transitions and the cavity mode gives rise to a polariton bound state, whose electron-component probability density is shown as a red solid line. The degree of localization of this electron component depends sensitively on the polariton binding energy, defined as the difference between the polariton excitation energy $\hbar\omega_\text{pol}$ and the gap energy $\epsilon_g$. This binding energy can be tuned by adjusting the cavity frequency, as discussed in the main text.
• Figure 2: Spectral properties of the bound polariton mode. (a) Contour plot of the photonic spectral function $A(\epsilon,\hbar\omega_c)$ as a function of the cavity frequency $\omega_c$ and the energy $\epsilon$. To account for cavity losses, a finite imaginary part $\gamma_{\text{c}}$ was added to the self-energy of the Green’s function used to compute the spectral function. (b) Matter and photonic fractions of the bound polariton as a function of the cavity frequency. The results correspond to a quantum well of width $L_{\text{QW}} = 3$ nm, ionization energy $\epsilon_{\text{ion}} = 138$ meV, effective electron mass $m_e^* = 0.06m_e$, electron density $n = 5\times10^{12}$ cm${^{-2}}$, dielectric constant $\varepsilon = 13\varepsilon_0$, and cavity broadening $\gamma_{\text{c}} = 5$ meV.
• Figure 3: Squared wavefunction of the bound polariton state for different values of the polariton binding energy, controlled by the detuning of the cavity mode frequency. Other parameters are the same as in Fig. \ref{['fig:Spectral-function']}, which also shows the dependence of $\epsilon_B$ on the cavity detuning.
• Figure 4: (a) Dependence of three relevant length scales (electron hole Bohr radius $a_B$, Coulomb scattering length $\alpha_B$, and localization length $\ell_{\epsilon_B}$) on the polariton binding energy $\epsilon_B$. The numerical result (green solid line) is in agreement with our analytical prediction (black dashed line) in the $\epsilon_B \to 0$ limit. (b) Polariton-polariton interaction $g_{PP}$ as a function of the polariton binding energy, using the experimental parameters from Ref. Cortese2021. The corresponding interaction energy is $U = g_{PP}/S$, where $S$ is the transverse area.