Purcell Effect and Cavity-Modified Emission

The Purcell effect is the quiet workhorse of quantum photonics. In semiconductor single-photon sources, an InAs quantum dot is grown inside a micropillar so a large Purcell factor does three jobs at once: it shortens the radiative lifetime, funnels nearly all the emission into one collectable mode (driving the coupling factor $\beta\to1$), and outruns the dephasing that would make successive photons distinguishable. Devices built this way — pioneered in Pascale Senellart's group at C2N and elsewhere — now emit on-demand single photons with indistinguishability above 99%, feeding boson-sampling machines and photonic processors. The same $\beta\to1$ physics yields microcavity lasers whose thresholds nearly vanish. The strong-coupling end lives most vividly in circuit quantum electrodynamics, where a superconducting transmon qubit plays the atom and a microwave resonator the cavity, reaching cooperativities in the thousands on a chip. There the Purcell effect is double-edged: engineers deliberately add a "Purcell filter" — a second, detuned resonator — to suppress the qubit's decay into the readout line, using inhibited emission to extend $T_1$, while exploiting strong coupling for fast readout and two-qubit gates. From silicon-vacancy centres in diamond nanocavities to plasmonic antennas that Purcell-enhance molecular fluorescence, the lesson of the 1946 abstract is now instrumentation: an emitter's lifetime is a design parameter.

Picture spontaneous emission as an atom trying to leave a room through whatever doorways the vacuum provides — one doorway per field mode at the transition frequency. In free space the doorways are uniform, and the atom ambles out at the Einstein rate $\Gamma_0=A_{21}$. A cavity rebuilds the room: near a resonance it throws open one enormous door — a band of modes crammed into the narrow linewidth — and the atom rushes out; tune the atom away and the doorways are bricked up, so with no modes to radiate into the excited state is trapped and the lifetime stretches. Two knobs set how dramatic this is. The mode volume $V$ fixes how strong the vacuum field is — the zero-point amplitude of one cavity photon scales as $\sqrt{\hbar\omega/2\varepsilon_0 V}$, so a smaller mode means a stronger field. The quality factor $Q$ fixes how sharply the density of states spikes: it counts how many round trips a photon survives, the same as how narrow the resonance is, $\kappa=\omega_c/Q$. Purcell's factor $F_P\propto Q/V$ is exactly the product of "strong field per mode" and "many modes per bandwidth."

In the strong-coupling limit the picture inverts. The emitted photon does not escape — it is re-absorbed by the same atom before it can leak out, then re-emitted, then re-absorbed. Atom and single cavity photon fuse into one object, a pair of dressed states, and the excitation oscillates between them at the vacuum Rabi frequency $2g$; in the spectrum the single line splits into two. The two panels below contrast this reversible Rabi flopping with irreversible Purcell decay.

The whole subject is one number and its consequences. The Purcell factor is the peak on-resonance enhancement of the spontaneous emission rate, and it depends only on the ratio of quality factor to mode volume. It is a maximum, realized for an emitter on resonance, at a field antinode, dipole aligned to the cavity field.

where: $F_P$ — peak enhancement $\Gamma_{\mathrm{cav}}/\Gamma_0$; $\lambda$ — free-space transition wavelength; $n$ — index of the cavity medium; $Q=\omega_c/\kappa$ — cavity quality factor; $V$ — mode volume. Physically, $\lambda^3/V$ measures how tightly the field is squeezed and $Q$ how narrow the resonance is, so a wavelength-scale cavity with $Q$ in the thousands buys enhancements of hundreds. The reference against which all this is measured is the free-space rate — the same atom with no cavity — set by the transition frequency and dipole moment, and identically the Einstein $A_{21}$ coefficient.

where: $\Gamma_0$ — free-space decay rate; $\omega_a$ — transition angular frequency; $\mathbf{d}$ — transition dipole $\langle e|\hat{\mathbf{d}}|g\rangle$; $\varepsilon_0$ — vacuum permittivity. The steep $\omega_a^3$ dependence is why optical transitions decay in nanoseconds and microwave ones can live for seconds. The microscopic coupling to a single cavity mode is $g$, the transition dipole dotted into the vacuum field of one photon — the zero-point field inherited from field quantization, confined to volume $V$.

where: $g$ — single-photon coupling (half the vacuum Rabi frequency); $\mathbf{E}_{\mathrm{vac}}$ — zero-point field of one cavity photon at the antinode; $\omega_c$ — cavity frequency; $V$ — mode volume. Smaller $V$ means a larger vacuum field and larger $g$ — the same $V$ as in $F_P$. When the cavity leaks faster than $g$ can recycle the photon ($g\ll\kappa$) the emission stays exponential but runs at the Purcell-enhanced rate, which on resonance takes a compact form — $4g^2/\kappa$ from microscopic quantities, $F_P\Gamma_0$ from the cavity figures of merit.

where: $\Gamma_{\mathrm{cav}}$ — on-resonance enhanced rate; $\kappa=\omega_c/Q$ — cavity energy decay rate (full linewidth). Both forms describe the same on-resonance decay. Detuning turns the enhancement down along the cavity's own Lorentzian and adds back whatever the emitter radiates into the uncaptured (leaky) modes — so one expression carries both enhancement (at $\Delta=0$) and suppression (large $\Delta$).

where: $\Delta=\omega_a-\omega_c$ — detuning; $\kappa$ — cavity linewidth; $\Gamma_{\mathrm{leaky}}\approx\Gamma_0$ — residual emission into non-cavity modes. One linewidth of detuning already halves the enhancement. In strong coupling the coherent exchange outruns the losses and the excited-state probability oscillates rather than decays, at the vacuum Rabi frequency — which equals the normal-mode splitting seen in the spectrum.

where: $\Omega_R$ — vacuum Rabi frequency and normal-mode splitting; $P_e(t)$ — excited-state probability for one quantum initially in the atom. The atom is fully de-excited at $gt=\pi/2$, re-excited at $gt=\pi$: energy shuttling between atom and vacuum field. The single figure of merit ranking a cavity is the cooperativity: coherent coupling squared over the product of the loss rates.

where: $C$ — single-atom cooperativity; $\gamma=\Gamma_0$ — atomic decay into non-cavity modes; $\kappa$ — cavity linewidth (FWHM, energy decay). $C\lesssim1$ is the weak-coupling Purcell regime, $C\gg1$ is strong coupling. (Texts that define $\kappa$ as the amplitude decay rate — half the energy rate — write the familiar $F_P=2C$, $1+2C$; identical physics.)

The behaviour is fixed by where the emitter sits relative to the resonance and by the hierarchy of $g$ against the losses $\kappa$ (cavity) and $\gamma$ (atom).

1 · A quantum-dot single-photon source (the real thing)

An InAs quantum dot emitting at $\lambda_0=920$ nm sits in a GaAs micropillar ($n=3.5$) with $Q=10^4$ and mode volume $V=5(\lambda/n)^3$. Its free-space radiative lifetime is $\tau_0=1.0$ ns. Find the ideal on-resonance Purcell factor, the mode volume in physical units, and the cavity-shortened lifetime.

Step 1. Only the ratio $Q/V$ matters, and $V$ is given in units of $(\lambda/n)^3$, so those cancel: $F_P=\dfrac{3}{4\pi^2}\cdot\dfrac{10^4}{5}=0.07599\times2000=\mathbf{152}$. Halving $V$ or doubling $Q$ would each double this.

Step 2. For concreteness, $\lambda/n=920/3.5=262.9$ nm, so $(\lambda/n)^3=1.82\times10^{-2}\ \mu\mathrm{m}^3$ and $V=5(\lambda/n)^3=\mathbf{0.091\ \mu m^3}$ — a mode squeezed to a tenth of a cubic micron, which is what makes the large $Q/V$ possible.

Step 3. On resonance, at the antinode, dipole aligned, the lifetime shortens by $F_P$: $\tau_{\mathrm{cav}}=\tau_0/F_P=1000\ \mathrm{ps}/152=\mathbf{6.6\ ps}$. The ideal Purcell factor is $F_P\approx152$, shrinking the $1.0$ ns lifetime to about $6.6$ ps. Outrunning the dot's dephasing this way makes the photons both bright and indistinguishable — though a real, detuned, off-antinode device measures less.

2 · Weak or strong coupling? Cooperativity and vacuum Rabi splitting

A quantum dot couples to a photonic-crystal cavity at $\lambda=920$ nm with $g/2\pi=20$ GHz, cavity linewidth $\kappa/2\pi=10$ GHz, and emitter linewidth $\gamma/2\pi=0.2$ GHz. Classify the regime, compute the cooperativity, and give the vacuum Rabi splitting and its period.

Step 1. The coupling $g/2\pi=20$ GHz exceeds both the cavity loss $10$ GHz and the emitter loss $0.2$ GHz, so $g>\kappa,\gamma$ — the strong-coupling regime, where coherent exchange beats the leaks and excitation oscillates rather than decays.

Step 2. Cooperativity is a ratio of rates, so the $/2\pi$ values may be used throughout: $C=\dfrac{g^2}{\kappa\gamma}=\dfrac{20^2}{10\times0.2}=\dfrac{400}{2}=\mathbf{200}$, deep in strong coupling ($C\gg1$).

Step 3. The spectrum splits at $\Omega_R=2g$, so the splitting is $2g/2\pi=\mathbf{40\ GHz}$. In time the population flops as $\cos^2(gt)$: it fully transfers to the cavity at $t=\pi/(2g)=\mathbf{12.5\ ps}$ and returns after a full period $1/(2g/2\pi)=\mathbf{25\ ps}$. Strong coupling with $C=200$: a $40$ GHz doublet and a $25$ ps reversible oscillation — the atom hands its quantum to the vacuum field and takes it back before either can leak away. (Had this been weak coupling, detuning by three linewidths would cut the enhancement to $\tfrac{(\kappa/2)^2}{(\kappa/2)^2+\Delta^2}\approx3\%$ of its peak — $F_P$ is a resonant maximum, not a fixed property.)

The Purcell effect is one of physics' great one-liners: a two-sentence conference abstract, buried among nuclear-magnetic-resonance results, that took thirty-five years to be tested and now underwrites a whole industry of quantum light sources.

Texts. Loudon, The Quantum Theory of Light, 3rd ed., Ch. 1 and 6 (spontaneous emission, cavity effects, vacuum Rabi) · Foot, Atomic Physics, Ch. 7 (Einstein coefficients) · Saleh and Teich, Fundamentals of Photonics, 2nd ed., Ch. 13–14 (photons in cavities) · Haroche and Raimond, Exploring the Quantum (cavity QED, Rydberg atoms) · Hecht, Optics, 5th ed. (optical resonators) · Demtröder, Laser Spectroscopy (linewidths and cavity effects).

Provenance. The Purcell factor and its density-of-states derivation come from Purcell, Phys. Rev. 69, 681 (1946), developed in Loudon Ch. 6; the free-space rate $\Gamma_0=A_{21}$ is standard (Foot Ch. 7). The Jaynes–Cummings vacuum-Rabi solution and the $2g$ splitting are in Loudon Ch. 6 and Haroche and Raimond; the cavity density of states and $\beta$-factor laser physics follow Saleh and Teich Ch. 13–14. A verifier should recompute $F_P=\frac{3}{4\pi^2}(\lambda/n)^3 Q/V$ from the DOS ratio and confirm it equals $4g^2/(\kappa\gamma)$ built from the vacuum field; check that $\Gamma_0$ reduces to the Einstein $A$ coefficient; and verify $P_e=\cos^2(gt)$ with eigenvalue splitting $2g$.

Prerequisites. Quantization of the Electromagnetic Field · Vacuum State and Zero-Point Energy · Fock States and the Photon Number Basis. Leads to. Single-Photon Sources · Photon Antibunching · Coherence Functions and the Degree of Coherence.

Ask the tutor: ask me to work through a Purcell-factor or cooperativity estimate for your own cavity — pillar, photonic crystal, or superconducting resonator — and I will show whether it lands in the weak or strong-coupling regime.