Cavity Quantum Electrodynamics explores how light-matter interactions change in confined spaces. The Purcell effect, a key phenomenon, shows how cavities can boost or suppress spontaneous emission rates compared to free space.

Understanding the Purcell effect is crucial for controlling light emission in quantum systems. By tweaking cavity design and emitter placement, we can manipulate photon properties, opening doors for quantum tech and faster optical communication.

The Purcell Effect in Cavity QED

Spontaneous Emission Enhancement and Suppression

• The Purcell effect describes the enhancement or suppression of spontaneous emission rates of an emitter placed inside an optical cavity compared to its emission in free space
• The effect arises from the modification of the local density of optical states (LDOS) experienced by the emitter due to the presence of the cavity
• The spontaneous emission rate is proportional to the LDOS at the emitter's position and transition frequency

Cavity Parameters Influencing the Purcell Effect

• Cavities with high quality factors (Q) and small mode volumes (V) can significantly enhance the LDOS, leading to increased spontaneous emission rates
  • High Q cavities have longer photon lifetimes, allowing for stronger emitter-cavity interaction
  • Small V concentrates the cavity field, increasing the electric field intensity at the emitter's position
• The Purcell effect enables the control of spontaneous emission directionality, spectral properties, and lifetime of the emitter
  • Directional emission can be achieved by designing cavities with specific mode profiles (Fabry-Pérot, whispering gallery modes)
  • Spectral properties can be tailored by matching the cavity resonance to the emitter's transition frequency
  • Lifetime reduction occurs when the Purcell factor is greater than 1, enabling faster emission rates

Purcell Factor Derivation

Purcell Factor Formula

• The Purcell factor (Fp) quantifies the enhancement of spontaneous emission rate in a cavity compared to free space
• $Fp = (3/4π²) × (λ³/V) × (Q/n³)$, where λ is the emission wavelength, V is the cavity mode volume, Q is the cavity quality factor, and n is the refractive index of the cavity medium
• The derivation of the Purcell factor involves calculating the LDOS in the cavity and comparing it to the LDOS in free space

Cavity Parameters in the Purcell Factor

• The cavity mode volume (V) represents the spatial extent of the cavity mode and is inversely proportional to the electric field intensity at the emitter's position
  • Smaller V leads to higher field intensities and stronger emitter-cavity coupling
• The cavity quality factor (Q) describes the lifetime of photons in the cavity and is related to the cavity's ability to store electromagnetic energy
  • Higher Q implies longer photon lifetimes and narrower cavity resonance linewidths
• The emitter's dipole moment and its alignment with the cavity mode's electric field also influence the Purcell factor
  • Maximum Purcell enhancement occurs when the emitter is positioned at the antinode of the cavity mode and its dipole moment is aligned with the electric field

Controlling Spontaneous Emission

Cavity Designs for Spontaneous Emission Control

• Optical cavities can be designed to control the spontaneous emission properties of emitters, such as atoms, molecules, or quantum dots
• Fabry-Pérot cavities, whispering gallery mode resonators, and photonic crystal cavities are commonly used for spontaneous emission control
  • Fabry-Pérot cavities consist of two parallel mirrors forming a standing wave pattern
  • Whispering gallery mode resonators confine light by total internal reflection in circular or spherical geometries
  • Photonic crystal cavities use periodic dielectric structures to create localized modes
• The cavity's resonance frequency can be tuned to match the emitter's transition frequency to achieve maximum Purcell enhancement

Applications of Spontaneous Emission Control

• Cavity-emitter coupling can be optimized by precisely positioning the emitter within the cavity mode volume
• Spontaneous emission control has applications in quantum information processing, such as the generation of single photons on demand and the realization of quantum networks
  • Single-photon sources are crucial for secure quantum key distribution and quantum computing algorithms
• Cavity-enhanced single-photon sources exhibit high purity, indistinguishability, and efficiency, which are crucial for quantum cryptography and quantum computing
• Spontaneous emission control can also be used to modify the radiative lifetime of emitters, enabling the realization of fast and efficient light-emitting devices
  • Shorter lifetimes allow for higher modulation speeds in optical communication systems

Cavity Losses and Detuning

Impact of Cavity Losses

• Cavity losses, such as mirror transmission, absorption, and scattering, reduce the cavity quality factor (Q) and limit the achievable Purcell enhancement
• The presence of losses broadens the cavity resonance linewidth, reducing the spectral overlap between the emitter and the cavity mode
  • Broader linewidths decrease the sensitivity to emitter-cavity detuning
• Cavity losses can be mitigated by using high-reflectivity mirrors, low-loss materials, and active frequency stabilization techniques
  • Distributed Bragg reflectors (DBRs) and dielectric mirrors provide high reflectivity over a broad wavelength range
  • Low-loss dielectrics (silicon, silicon nitride) minimize absorption and scattering losses

Effect of Detuning

• Detuning between the emitter's transition frequency and the cavity resonance frequency diminishes the Purcell effect
• The Purcell factor decreases as the detuning increases, following a Lorentzian profile with a linewidth determined by the cavity Q
  • Maximum enhancement occurs at zero detuning when the emitter and cavity are resonant
• Off-resonant coupling between the emitter and the cavity mode can lead to modified emission spectra, such as the Mollow triplet or the Autler-Townes splitting
  • Mollow triplet appears as three emission peaks due to strong emitter-cavity coupling and large detuning
  • Autler-Townes splitting occurs when the cavity acts as a strong driving field, creating dressed states
• The interplay between cavity losses, detuning, and emitter-cavity coupling strength determines the overall efficiency and fidelity of spontaneous emission control in cavity QED systems