"Cavity Polaritons" Chapter 1 "Dispersion of Cavity Polaritons"

Hello! I am Sakata, an M1 student at the Laboratory of Quantum Theory of Light and Matter (Hikari-ron Lab)!
At the Hikari-ron Lab, we conduct Journal Clubs using various literature on quantum optics with the goal of understanding polaritons, which are quasiparticles formed by quantizing electromagnetic waves in matter.
This time, I will introduce Chapter 1, "Dispersion of Cavity Polaritons," from "Cavity Polaritons" by Alexey Kavokin et al.

Chapter 1

In Chapter 1, we consider the structure of a resonator (microcavity) in which a quantum well is sandwiched between Bragg mirrors, as shown in the figure below. We derive the optical responses (reflection coefficients, transmission coefficients, etc.) for both the quantum well and the Bragg mirrors, and use the transfer matrix method to combine them to derive the dispersion equation.

Quantum Well

A quantum well (QW) is a very thin (nanometer-scale) semiconductor layer in which the direction of electron movement is confined. Electrons and holes are trapped within this thin layer, forming excitons bound by Coulomb force. Excitons respond very strongly to light near their resonance frequency.

In this section, starting from Maxwell's equations, we calculate the reflection and transmission coefficients when light near the resonance frequency is incident on a quantum well.

At this time, to describe the optical properties of the excitons, we use the following non-local dielectric response theory instead of a simple local model.

$$
\begin{align*}
4\pi P_{\text{exc}}(z) &= \int_{-\infty}^{\infty} \chi(z, z')E(z')dz' \\\
\chi(z, z') &= \tilde{\chi}(\omega)\Phi(z)\Phi(z') \\\
\tilde{\chi}(\omega) &= \frac{Q}{\omega_0 - \omega - i\gamma}, \quad Q = \epsilon_B \omega_{LT} \pi a_B^3
\end{align*}
$$

This is a more realistic model in which the polarization changes not only at the point where the exciton is generated, but throughout the entire range where its wavefunction extends.

Bragg Mirror

A Bragg mirror is a periodic structure in which two types of materials with different refractive indices are stacked alternately with a thickness of 1/4 of the wavelength of light.

Due to this periodic structure, a stop band is formed where only light of a specific wavelength is reflected with very high reflectivity.

In a microcavity, it functions as a "mirror" to efficiently confine light.

Microcavity and Polaritons

To describe the behavior of this entire system, the transfer matrix method is used. First, we define the transfer matrix $${T_{QW}}$$ for crossing the quantum well. Next, by multiplying this $${T_{QW}}$$ with the matrix representing the propagation of light within the cavity, we construct the transfer matrix $${T_c}$$ for the entire cavity layer.

Next, we find the resonance frequency, that is, the frequency of the eigenmode at which light can be confined and continue to oscillate within the cavity even without external input.

What is important here is that when light makes one round trip inside the cavity, it must satisfy the interference condition to exist in a self-regenerative manner, that is, the condition where amplitude and phase perfectly match (constructive interference). This condition is expressed as a dispersion equation as a boundary condition that includes all the effects of light propagation, transmission/reflection at the quantum well, and reflection by the Bragg mirror.

The dispersion equation is simplified as the equation for two coupled harmonic oscillators.
$${\left( \tilde{\omega}_0 - \omega - i\gamma \right)\left( \omega_c - \omega - i\gamma_c \right) = V^2}$$

• Strong Coupling Regime:

  • When the coupling strength V is greater than the decay rates of both the photon and the exciton, the photon and exciton can no longer exist as separate particles. A new quasiparticle, a mixture of both, called a "polariton", is formed. In experiments, this is observed as an "anti-crossing", where the energies of the two polaritons split significantly at the point where the original photon and exciton energies match. This splitting width is called "Rabi splitting".

• Weak Coupling Regime:

  • When the coupling is weak, polaritons are not formed, and the quantum well simply promotes the absorption and emission of light. This is the regime utilized in devices such as Vertical-Cavity Surface-Emitting Lasers (VCSELs).

• Splitting due to polarization

  • When light is incident at an angle, its properties also change depending on the polarization. It is stated that the coupling strength and energy differ slightly between s-polarization and p-polarization, which serves as a clue for investigating the finer properties of polaritons.

That is all for this time. Thank you for reading until the end!

From Chapter 2 onwards, we will also cover cases with multiple quantum wells and cases with multiple resonators arranged in a row. Look forward to the next installment!!

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