quantum electrodynamic cavity QED polariton semiconductor
# Quantum Electrodynamic Cavity QED Formalism, Exciton-Polariton Physics, and Strong Light-Matter Coupling in Microcavity Structures
## 1. Introduction: The Jaynes-Cummings Model and Strong Coupling Regime
The Jaynes-Cummings model describes the quantum interaction between a two-level atomic system and a single quantized electromagnetic mode confined in a cavity. This framework is fundamental to understanding cavity QED and provides a platform for exploring quantum effects at the intersection of photonics and solid-state physics.
In semiconductors, the relevant two-level system is typically an exciton (electron-hole pair) coupled to the cavity photon mode. When the coupling strength exceeds the loss rates of both the cavity and the exciton, the system enters the strong coupling regime—characterized by Rabi oscillations and an energy splitting proportional to the coupling strength.
The Jaynes-Cummings model reveals:
- Vacuum Rabi splitting: Energy separation between dressed states in the absence of external excitation
- Purcell effect: Enhancement or suppression of spontaneous emission
- Rabi oscillations: Coherent oscillation of population between atomic and photonic states
- Photon blockade: Nonlinear response enabling single-photon quantum gates
## 2. Jaynes-Cummings Hamiltonian and Its Derivation
The Jaynes-Cummings Hamiltonian in the rotating wave approximation:
$$H = \hbar\omega_c a^\dagger a + \frac{\hbar\omega_e}{2} \sigma_z + \hbar g(a^\dagger \sigma_- + a \sigma_+)$$
where:
- $a^\dagger, a$ are creation/annihilation operators for cavity photons
- $\omega_c$ is the cavity resonance frequency
- $\sigma_+, \sigma_-, \sigma_z$ are Pauli matrices for the two-level system
- $\omega_e$ is the atomic transition frequency
- $g$ is the light-matter coupling strength
### Coupling Strength
The coupling constant relates to the Rabi frequency:
$$g = \frac{d \mathcal{E}_0}{2\hbar}$$
where:
- $d = \langle e|x|g
angle$ is the electric dipole moment
- $\mathcal{E}_0 = \sqrt{\hbar\omega_c / 2\epsilon_0 V}$ is the vacuum electric field in the cavity
- $V$ is the cavity mode volume
For tight confinement (small $V$) or strong dipole moments, $g$ can exceed dissipation rates, enabling strong coupling.
## 3. Dressed States and Vacuum Rabi Splitting
Diagonalizing the Jaynes-Cummings Hamiltonian yields dressed states that are entangled superpositions of photonic and atomic excitations. In the case of resonant coupling ($\omega_c = \omega_e = \omega_0$), the eigenstate energies are:
$$E_\pm = \hbar\omega_0 + \hbar g\sqrt{n+1}$$
where $n$ is the photon number. For the single-photon case ($n=1$), the vacuum Rabi splitting separates the ground state from the first excited doublet:
$$\hbar\Omega_R = 2\hbar g$$
This splitting is independent of cavity loss and represents a fundamental quantum effect: even in the ground state (zero photons), the coupling creates an energy splitting due to quantum vacuum fluctuations.
### Rabi Oscillations
Starting in the excited atomic state with zero photons, the system coherently oscillates between:
- $|e, 0
angle$: Atom excited, cavity empty
- $|g, 1
angle$: Atom ground, cavity has one photon
The oscillation frequency is $\Omega_R = 2g$ (in absence of detuning and damping).
## 4. Purcell Effect: Cavity-Modified Spontaneous Emission
The Purcell factor quantifies the modification of spontaneous emission rate by the cavity:
$$F_P = \frac{3}{4\pi^2}\left(\frac{\lambda}{n} ight)^3 \frac{Q}{V}$$
where:
- $\lambda$ is the wavelength in vacuum
- $n$ is the refractive index of the cavity medium
- $Q$ is the cavity quality factor (ratio of stored to dissipated energy per cycle)
- $V$ is the cavity mode volume
A large Purcell factor ($F_P > 1$) indicates Purcell enhancement: the cavity suppresses all electromagnetic modes except the cavity mode, accelerating emission into that mode. This is crucial for:
- Single-photon sources: Forcing emitters to radiate exclusively into the cavity mode
- Quantum optics: Creating non-classical light with sub-Poissonian statistics
Conversely, $F_P < 1$ indicates Purcell suppression, useful for storing quantum information in long-lived atomic states.
## 5. Exciton-Polariton Dispersion and Band Structure
In a photonic crystal cavity, the cavity mode interacts with exciton resonances, creating hybrid exciton-polariton quasiparticles. The dispersion relation is found by solving:
$$\begin{pmatrix} \omega - \omega_c & -g \\ -g & \omega - \omega_e \end{pmatrix} \begin{pmatrix} a \\ \sigma_- \end{pmatrix} = 0$$
This yields:
$$\omega_\pm(k) = \frac{1}{2}\left[(\omega_c + \omega_e) \pm \sqrt{(\omega_c - \omega_e)^2 + 4g^2} ight]$$
The lower polariton branch (LPB) at $\omega_-(k)$ is predominantly photonic at large $k$ (free-space photon dispersion) but becomes exciton-like near the energy minimum. The upper polariton branch (UPB) shows the opposite character.
### Polariton Effective Mass
Near the band minimum, the polariton effective mass:
$$m_{LP}^* = \frac{\hbar k^2}{2\omega_-(k)}$$
can be much smaller than the free exciton mass, enabling polariton condensation at higher temperatures than exciton condensates.
## 6. Strong vs. Weak Coupling Regimes
The transition from weak to strong coupling is characterized by the cooperativity parameter:
$$C = \frac{g^2}{\gamma \kappa}$$
where $\gamma$ is the atomic linewidth and $\kappa$ is the cavity decay rate.
- Weak coupling ($C \ll 1$): Cavity acts as a passive filter; Purcell factor controls spontaneous emission rate
- Strong coupling ($C > 1$): Rabi oscillations coherently exchange energy between atom and cavity before dissipation; vacuum Rabi splitting observable
- Ultra-strong coupling ($C \gg 1$): Coupling approaches the transition frequency; counter-rotating terms in the Hamiltonian become significant; polariton nonlinearity dominant
## 7. Photonic Crystal Cavity Design and Mode Confinement
A photonic crystal (PhC) cavity achieves strong confinement through:
- Photonic bandgap: Periodic dielectric structure prevents photon propagation in certain frequencies
- Defect mode: Point or line defect in the periodic structure creates localized cavity mode
- Quality factor: $Q = \omega_0 / \kappa$ can exceed $10^4$ for high-Q cavities
The cavity mode volume $V$ is minimized by tight transverse confinement:
$$V \approx \left(\frac{\lambda}{2n} ight)^3$$
For state-of-the-art PhC cavities in GaAs at 1.5 μm wavelength: $V \sim 0.1-1 \, (\lambda / 2n)^3$, achieving $g / \gamma > 10$ (strong coupling).
## 8. Python Implementation: Jaynes-Cummings Spectrum and Rabi Oscillations
import numpy as np
import matplotlib.pyplot as plt
from scipy.linalg import eigh
from scipy.integrate import odeint
def jaynes_cummings_hamiltonian(n_cutoff, omega_c, omega_e, g, detuning=0):
"""
Construct Jaynes-Cummings Hamiltonian matrix.
H = ℏω_c a†a + ℏω_e σ_z/2 + ℏg(a† σ_- + a σ_+)
Parameters:
-----------
n_cutoff : int
Maximum photon number in Hilbert space truncation
omega_c : float (eV)
Cavity resonance frequency
omega_e : float (eV)
Atomic transition frequency
g : float (eV)
Light-matter coupling strength
detuning : float (eV)
Detuning: δ = ω_c - ω_e
Returns:
--------
H : array (2*(n_cutoff+1), 2*(n_cutoff+1))
Hamiltonian matrix
"""
dim = 2 * (n_cutoff + 1)
H = np.zeros((dim, dim), dtype=complex)
# Cavity photon energy: ℏω_c |n⟩⟨n|
for n in range(n_cutoff + 1):
H[2*n, 2*n] += omega_c * (n + 0.5) # Photon term
H[2*n + 1, 2*n + 1] += omega_c * (n + 0.5)
# Atomic energy: ℏω_e σ_z/2
omega_a = omega_e - detuning # Reference to cavity
for n in range(n_cutoff + 1):
H[2*n + 1, 2*n + 1] += omega_a
# Coupling: g(a† σ_- + a σ_+)
# a† |g, n⟩ = |e, n+1⟩, a |e, n⟩ = |g, n-1⟩
for n in range(n_cutoff):
H[2*(n+1), 2*n + 1] += g * np.sqrt(n + 1) # a† σ_-
H[2*n + 1, 2*(n+1)] += g * np.sqrt(n + 1) # a σ_+ (adjoint)
return H
def rabi_oscillation_lindblad(t, psi0, omega_c, omega_e, g, gamma, kappa):
"""
Solve Lindblad master equation for Rabi oscillations.
dρ/dt = -i[H, ρ] + γ L_σ_- ρ + κ L_a ρ
Parameters:
-----------
t : array (fs)
Time array
psi0 : array
Initial state vector (will be converted to density matrix)
omega_c, omega_e, g : float
Cavity and atomic parameters
gamma : float (eV)
Atomic decay rate
kappa : float (eV)
Cavity decay rate
Returns:
--------
occupation_e : array
Excited state population vs. time
occupation_photon : array
Photon number vs. time
"""
n_cutoff = 3
dim = 2 * (n_cutoff + 1)
H = jaynes_cummings_hamiltonian(n_cutoff, omega_c, omega_e, g)
# Lindblad operators
# σ_- = |g⟩⟨e|
L_sigma_minus = np.zeros((dim, dim), dtype=complex)
for n in range(n_cutoff + 1):
L_sigma_minus[2*n, 2*n + 1] = 1.0 # |g,n⟩⟨e,n|
# a = Σ√n |n-1⟩⟨n|
L_a = np.zeros((dim, dim), dtype=complex)
for n in range(1, n_cutoff + 1):
L_a[2*n - 2, 2*n] = np.sqrt(n) # |n-1⟩⟨n|
def lindblad_rhs(rho_vec, t):
# Reshape vector to density matrix
rho = rho_vec.reshape((dim, dim))
# Commutator: -i[H, ρ]
drho_H = -1j * (H @ rho - rho @ H)
# Decay: γ (2 L_σ_- ρ L_σ_-† - {L_σ_-† L_σ_-, ρ})
drho_decay = gamma * (2 * L_sigma_minus @ rho @ L_sigma_minus.conj().T -
(L_sigma_minus.conj().T @ L_sigma_minus @ rho +
rho @ L_sigma_minus.conj().T @ L_sigma_minus))
# Cavity loss: κ (2 L_a ρ L_a† - {L_a† L_a, ρ})
drho_kappa = kappa * (2 * L_a @ rho @ L_a.conj().T -
(L_a.conj().T @ L_a @ rho +
rho @ L_a.conj().T @ L_a))
drho = drho_H + drho_decay + drho_kappa
return drho.flatten()
# Initial condition: |e, 0⟩
rho0 = np.zeros((dim, dim), dtype=complex)
rho0[1, 1] = 1.0 # Excited state with no photons
# Solve
t_femtosec = t / 0.02419 # Convert eV·t to fs
solution = odeint(lindblad_rhs, rho0.flatten(), t_femtosec)
# Extract populations
occupation_e = np.zeros(len(t))
occupation_photon = np.zeros(len(t))
for i, rho_vec in enumerate(solution):
rho = rho_vec.reshape((dim, dim))
# Excited state: sum of all |e, n⟩⟨e, n|
occupation_e[i] = np.sum([rho[2*n + 1, 2*n + 1].real for n in range(n_cutoff + 1)])
# Photon number: ⟨a†a⟩
occupation_photon[i] = 0
for n in range(1, n_cutoff + 1):
occupation_photon[i] += n * rho[2*n, 2*n].real
return occupation_e, occupation_photon
# Parameters (realistic GaAs PhC cavity)
omega_cavity = 1.50 # eV (0.827 μm wavelength)
omega_exciton = 1.504 # eV (detuned from cavity)
coupling = 0.003 # eV (strong coupling regime)
decay_rate = 0.0001 # eV (narrow exciton linewidth)
cavity_decay = 0.0002 # eV (high-Q cavity)
# Eigenstate calculation
H_jc = jaynes_cummings_hamiltonian(5, omega_cavity, omega_exciton, coupling, detuning=0)
eigenvalues, eigenvectors = eigh(H_jc)
# Time evolution
t = np.linspace(0, 500, 1000) # 500 fs
P_e, N_photon = rabi_oscillation_lindblad(t * 0.02419, omega_cavity, omega_exciton,
coupling, decay_rate, cavity_decay)
# Plotting
fig, axes = plt.subplots(2, 2, figsize=(13, 10))
# Panel A: Energy eigenvalue diagram (vacuum Rabi splitting)
n_states_plot = 8
axes[0, 0].plot(np.arange(n_states_plot), eigenvalues[:n_states_plot]*1e3, 'bo-',
markersize=8, linewidth=2)
axes[0, 0].set_xlabel('State Index', fontsize=11)
axes[0, 0].set_ylabel('Energy (meV)', fontsize=11)
axes[0, 0].set_title('Jaynes-Cummings Energy Spectrum', fontsize=12, fontweight='bold')
axes[0, 0].grid(True, alpha=0.3)
axes[0, 0].axhline(y=omega_cavity*1000, color='r', linestyle='--', alpha=0.5, label='ω_c')
axes[0, 0].axhline(y=omega_exciton*1000, color='g', linestyle='--', alpha=0.5, label='ω_e')
axes[0, 0].legend()
# Panel B: Vacuum Rabi splitting
vacuum_splitting = eigenvalues[2] - eigenvalues[1] # Gap in ground state manifold
axes[0, 1].bar(['Vacuum Rabi
Splitting'], [vacuum_splitting*1e3], color='purple', alpha=0.7, edgecolor='black')
axes[0, 1].set_ylabel('Energy (meV)', fontsize=11)
axes[0, 1].set_title(f'Rabi Splitting: {vacuum_splitting*1e3:.3f} meV', fontsize=12, fontweight='bold')
axes[0, 1].text(0, vacuum_splitting*1e3/2, f'2g = {2*coupling*1e3:.2f} meV',
ha='center', fontsize=10, fontweight='bold')
# Panel C: Rabi oscillations
axes[1, 0].plot(t, P_e, 'b-', linewidth=2, label='Excited State Pop.')
axes[1, 0].plot(t, N_photon, 'r--', linewidth=2, label='Photon Number')
axes[1, 0].set_xlabel('Time (fs)', fontsize=11)
axes[1, 0].set_ylabel('Population', fontsize=11)
axes[1, 0].set_title('Rabi Oscillations with Damping', fontsize=12, fontweight='bold')
axes[1, 0].legend(fontsize=10)
axes[1, 0].grid(True, alpha=0.3)
# Panel D: Phase space (Wigner function-like)
detuning_array = np.linspace(-0.01, 0.01, 50)
Purcell_factors = []
for delta in detuning_array:
H_det = jaynes_cummings_hamiltonian(3, omega_cavity, omega_exciton + delta, coupling)
eigs_det, _ = eigh(H_det)
Q = omega_cavity / cavity_decay
V_cavity = (1.5e-6)**3 # Mode volume ~(λ/2n)³
F_P = (3 / (4*np.pi**2)) * (1.5e-6 / 3.0)**3 * Q / V_cavity
Purcell_factors.append(F_P)
axes[1, 1].plot(detuning_array*1e3, Purcell_factors, 'g-o', linewidth=2, markersize=5)
axes[1, 1].set_xlabel('Detuning (meV)', fontsize=11)
axes[1, 1].set_ylabel('Purcell Factor F_P', fontsize=11)
axes[1, 1].set_title('Purcell Enhancement vs. Detuning', fontsize=12, fontweight='bold')
axes[1, 1].grid(True, alpha=0.3)
plt.tight_layout()
plt.savefig('cavity_qed_dynamics.png', dpi=150, bbox_inches='tight')
plt.show()
print("=== Jaynes-Cummings Cavity QED Analysis Complete ===")
print(f"Cavity frequency: {omega_cavity:.3f} eV")
print(f"Exciton frequency: {omega_exciton:.3f} eV")
print(f"Coupling strength: {coupling*1e3:.2f} meV")
print(f"Vacuum Rabi splitting (2g): {2*coupling*1e3:.2f} meV")
print(f"Cooperativity C = g²/(γκ): {(coupling**2) / (decay_rate * cavity_decay):.1f}")## 9. Polariton Condensation and Non-Equilibrium Phase Transitions
At high excitation densities, exciton-polariton condensation occurs when the lower polariton branch population exceeds a critical density. Unlike BEC, this is a driven-dissipative phase transition:
$$\frac{d}{dt}\langle a_- angle = -i(\omega_- - i\gamma_-/2)\langle a_- angle + F(t)$$
where $F(t)$ is a driving field and $\gamma_-$ is the polariton decay rate.
The condensate exhibits:
- Long-range spatial coherence: Macroscopic wavefunction
- Temporal coherence: Narrow linewidth despite finite decay
- Nonlinear interactions: $n^2$ term suppresses stimulated scattering → coherent light emission
## 10. Nonlinear Optical Effects and Photon Blockade
At high photon numbers, the Jaynes-Cummings system exhibits photon blockade: the presence of one photon in the cavity effectively prevents absorption of a second photon through AC Stark shift:
$$\Delta E = 2g$$
This creates non-Poissonian photon statistics—the key requirement for single-photon sources and quantum computing.
The second-order correlation function:
$$g^{(2)}( au) = \frac{\langle a^\dagger a^\dagger a a angle}{\langle a^\dagger a angle^2}$$
For a coherent state: $g^{(2)}(0) = 1$ (Poissonian)
For photon blockade: $g^{(2)}(0) \ll 1$ (sub-Poissonian, single photon)
## 11. Applications to Semiconductor Lasers and Quantum Light Sources
Microcavity lasers exploit strong coupling to achieve:
- Reduced threshold current: Stimulated scattering dominates spontaneous emission
- Narrow linewidth: Cavity filtering
- Enhanced modulation bandwidth: Faster switching due to strong coupling
Quantum dot-cavity systems generate deterministic single photons via:
$$|\psi
angle = |e, 0
angle \xrightarrow{\Omega_R} |g, 1
angle \xrightarrow{ ext{cavity decay}} | ext{1 photon}
angle$$
## 12. Coupling to External Modes and Leaky Cavities
Real cavities have output coupling to free-space modes. The coupling rate $\kappa_{ ext{out}}$ determines the escape efficiency:
$$\beta = \frac{\kappa_{ ext{out}}}{\kappa_{ ext{in}} + \kappa_{ ext{out}}}$$
For $\beta = 1$ (all decay to external modes), the cavity becomes an open system; for $\beta < 1$, some energy is lost to other modes (parasitic absorption, scattering).
## 13. Recent Advances in PhC and Micropillar Cavities
Modern structures achieve:
- Ultra-high Q: $Q > 10^6$ in buried structures
- Small mode volumes: $V < 0.01(\lambda/2n)^3$ in silicon
- Integrated circuits: On-chip quantum optics with multiple cavities
These enable applications from quantum computing (superconducting qubits coupled to cavities) to quantum sensing (enhanced phase sensitivity).
## 14. Limitations and Decoherence Mechanisms
Main loss channels:
- Cavity absorption: Residual absorption in dielectric reduces Q
- Disorder scattering: Roughness and point defects couple cavity photons to radiation modes
- Phonon coupling: Lattice vibrations dephase exciton coherence ($/T_2$ time ~ 100 fs at room temperature)
- Spectral diffusion: Fluctuating electric fields from charged defects shift exciton energy
## 15. Future Directions: Many-Body QED and Topological Polaritons
Emerging frontiers:
- Dicke superradiance: Collective coupling of many emitters → enhanced coupling strength
- Topological edge polaritons: Protected transport against disorder in PhC lattices
- Time-crystals: Periodically-driven polariton systems with persistent oscillations
- Quantum simulation: Engineering effective Hamiltonians for condensed-matter physics using photons