Cavity Quantum Electrodynamics

Cavity quantum electrodynamics takes the most fragile object built by field quantization — the single quantized mode of the electromagnetic field, with its creation and annihilation operators and above all its irreducible vacuum field per photon $\mathcal{E}_0=\sqrt{\hbar\omega/2\varepsilon_0 V}$ — and asks what happens when exactly one atom is placed inside it. In free space that vacuum spans an infinite continuum and an excited atom decays irreversibly, the photon gone for good. A high-finesse resonator changes the accounting: it deletes the continuum and hands the atom a single standing-wave mode of small volume $V$ , concentrating the zero-point field until $\mathcal{E}_0$ matters. The coupling $g=\mathbf{d}\cdot\boldsymbol{\mathcal{E}}_0/\hbar$ can then outrun every rate that destroys coherence, and energy stops leaking — it sloshes. The atom emits a photon into the mode, reabsorbs it, emits it again — the vacuum Rabi oscillation, coherent and reversible, a lone atom trading one quantum with a lone photon. Three numbers decide everything: the coupling $g$ , the rate $\kappa$ at which photons leak through the mirrors, and the rate $\gamma$ at which the atom radiates sideways into uncontrolled modes. Their ordering partitions the field into regimes — Purcell-enhanced emission when the cavity merely reshapes the density of states, and strong coupling when coherent exchange wins many times before either loss intervenes. The exactly solvable backbone is the Jaynes–Cummings model, whose dressed-state ladder climbs in steps of $\sqrt{n+1}$ : an anharmonicity with no classical analogue and the seed of photon blockade. This is the archetype of single-quantum light–matter interaction — the engine behind single-photon sources, quantum non-demolition photon counting, and superconducting-qubit readout.

Key takeaways

Cavity QED couples one atomic dipole to one quantized cavity mode; confining the vacuum to a small mode volume $V$ makes the per-photon field $\mathcal{E}_0$ , hence $g$ , large.
The Jaynes–Cummings Hamiltonian predicts a single atom and a single photon exchange energy coherently at the vacuum Rabi frequency $2g$ , splitting the spectrum into a dressed-state doublet.
Three rates rule the physics — $g$ , cavity decay $\kappa$ , atomic decay $\gamma$ ; strong coupling is $g\gg\kappa,\gamma$ , and cooperativity $C=g^2/\kappa\gamma$ measures cavity dominance.
The dressed ladder is anharmonic, splitting the $n$ -photon rung by $2g\sqrt{n+1}$ — a quantum signature enabling photon blockade; in weak coupling the Purcell factor $F_P=\tfrac{3}{4\pi^2}(\lambda/n)^3\,Q/V$ instead enhances spontaneous emission, powering single-photon sources.
Key idea: confine the electromagnetic vacuum to a single small-volume, high- $Q$ mode, and the per-photon field grows until the coupling $g$ exceeds both the cavity loss $\kappa$ and the atomic decay $\gamma$ — so a lone atom and a lone photon exchange energy coherently and reversibly at the vacuum Rabi frequency $2g$ .

The core insight

Confine the electromagnetic vacuum to a single mode of small volume $V$ and high quality factor $Q$ : the vacuum field per photon grows, the atom–photon coupling $g$ can exceed both the cavity loss $\kappa$ and the atomic decay $\gamma$ , and a lone atom and a lone photon coherently and reversibly exchange energy at the vacuum Rabi frequency $2g$ .

Where this lives today

The most consequential descendant of cavity QED is circuit QED, where the "atom" is a superconducting qubit and the "cavity" a microwave resonator on a chip. Because the qubit's effective dipole is enormous and the mode volume tiny, couplings reach $g/2\pi\sim100$ MHz easily, and the dispersive Hamiltonian of equation (7) is the everyday tool of quantum-processor readout: the resonator frequency shifts by $\pm\chi$ with the qubit state, so a brief microwave pulse reports the qubit without disturbing it. Every major superconducting quantum computer reads its qubits this way — dispersive cavity QED as load-bearing technology. In the optical domain, Purcell-enhanced emitters power deterministic single-photon sources: a quantum dot in a micropillar, or an atom in a fibre-tip resonator, funnels its emission into one mode, producing indistinguishable photons on demand for photonic quantum computing. The same strong coupling lets one atom switch one photon (photon blockade), mediates atom–atom entanglement, and provides cavity cooling into optomechanics. Cavity QED is now the common language for any single emitter talking to any single confined mode — resonator, waveguide, or nanophotonic structure engineered to shape the vacuum itself.

The physical picture

Picture two identical pendulums joined by a weak spring: the atomic dipole and the field mode, spring stiffness set by $g$ . Start one swinging — an excited atom in an empty cavity — and the energy flows entirely into the other — a ground-state atom holding one photon — then flows back. That periodic, complete exchange is the vacuum Rabi oscillation, and its frequency $2g$ is just the normal-mode splitting of two coupled oscillators. The word "vacuum" matters: the exchange proceeds with no photon present at the start, driven by the zero-point field alone. The cavity adds no energy — it reshapes the vacuum. A resonator replaces the free-space continuum with one Lorentzian line: on resonance the local density of photon states spikes and emission speeds up (the Purcell effect), off resonance it drops and emission is inhibited. Everything comes down to a race between three rates: coherent exchange runs at $g$ , while photons leak through the mirrors at $\kappa$ and the atom radiates sideways at $\gamma$ . Strong coupling is simply that the coherent race is won many times over.

Figure

One atom, one cavity mode — three rates decide the physics. Coherent exchange runs at g, photons leak at κ, the atom radiates sideways at γ; here g ≫ κ, γ so C = g²/κγ ≈ 158, strongly coupled

The distinctly quantum feature is how the ladder climbs. Because the field is quantized, the doublet in the manifold holding $n$ photons splits not by an amount proportional to intensity but by $2g\sqrt{n+1}$ : the first rung sits at $\omega_c\pm g$ , the second at $\omega_c\pm g\sqrt2$ — unequally spaced. Once one photon occupies the cavity, a second photon resonant with the first transition is detuned from the next: photon blockade, which turns a single atom into an optical transistor and reflects the $\sqrt{n+1}$ scaling no classical field can reproduce.

Figure

The Jaynes–Cummings dressed-state ladder. Each excitation manifold hybridizes into a doublet; the n = 0 rung splits by 2g ≈ 49 MHz but n = 1 by 2√2 g ≈ 69 MHz — a second photon resonant with the first rung is detuned from the next, the photon-blockade anharmonicity

The core equations

The starting point is one field mode and one two-level atom, coupled through the dipole interaction and simplified by dropping the terms that do not conserve energy (the rotating-wave approximation). The result is the Jaynes–Cummings Hamiltonian — exactly solvable, the reference model for all that follows: field energy, atomic energy, and an exchange term.

H = \hbar\omega_c\, a^\dagger a + \tfrac{1}{2}\hbar\omega_a\,\sigma_z + \hbar g\left(a^\dagger \sigma_- + a\,\sigma_+\right) \qquad (1)

where: $a^\dagger, a$ — creation and annihilation operators of the cavity mode; $\omega_c$ — cavity frequency; $\omega_a$ — atomic transition frequency; $\sigma_z,\sigma_\pm$ — Pauli inversion and raising/lowering operators; $g$ — atom–field coupling (half the vacuum Rabi frequency). The term $a^\dagger\sigma_-$ destroys an atomic excitation and creates a photon, $a\,\sigma_+$ the reverse — a spin coupled to an oscillator through one shared quantum. The coupling $g$ is fixed by the transition dipole and the vacuum field of the mode, which is fixed by the mode volume. Shrink $V$ and the per-photon field, hence $g$ , grows — cavity QED is an exercise in geometry as much as atomic physics.

g = \frac{\mathbf{d}_{eg}\cdot\boldsymbol{\epsilon}}{\hbar}\,\mathcal{E}_0,\qquad \mathcal{E}_0 = \sqrt{\frac{\hbar\omega_c}{2\varepsilon_0 V}} \qquad (2)

where: $\mathbf{d}_{eg}$ — transition dipole matrix element; $\boldsymbol{\epsilon}$ — mode polarization; $\mathcal{E}_0$ — vacuum field per photon; $V$ — mode volume; $\varepsilon_0$ — vacuum permittivity. Since $\mathcal{E}_0\propto V^{-1/2}$ , a ten-times-smaller volume gives roughly three times the coupling — the design pressure toward nanophotonic resonators. Diagonalizing $H$ within each fixed-excitation manifold gives the dressed states — superpositions of "atom excited, $n$ photons" and "ground atom, $n+1$ photons" — whose energies carry the hallmark $\sqrt{n+1}$ that makes the ladder anharmonic.

|n,\pm\rangle = \frac{1}{\sqrt{2}}\big(|g,n+1\rangle \pm |e,n\rangle\big),\quad E_{n,\pm} = \hbar\omega_c\left(n+\tfrac{1}{2}\right)\pm \hbar g\sqrt{n+1}\ \ (\text{resonant}) \qquad (3)

where: $|g\rangle,|e\rangle$ — atomic ground and excited states; $|n\rangle$ — Fock (photon-number) state; $E_{n,\pm}$ — dressed energies of the $n$ -th manifold; $\sqrt{n+1}$ — quantum anharmonic scaling. On resonance the bare states are degenerate, so the coupling splits them by $2\hbar g\sqrt{n+1}$ . That splitting is the frequency of the coherent exchange. The empty-cavity case $n=0$ gives the vacuum Rabi frequency, the rate at which a single excitation shuttles between atom and field.

\Omega_n = 2g\sqrt{n+1},\qquad \Omega_0 = 2g \qquad (4)

where: $\Omega_n$ — quantum Rabi frequency for the $n$ -photon manifold; $\Omega_0=2g$ — vacuum Rabi frequency, the single-photon empty-cavity exchange rate. That $\Omega_0\neq0$ with zero photons present is the quantum surprise — the drive is the vacuum itself. Starting the atom excited in an empty cavity, the excited-state population oscillates completely and reversibly — impossible in free space, where the atom decays once and stays down. This trace is the operational signature of strong coupling.

P_e(t) = \cos^2\!\left(g t\right)\quad(\text{single excitation, resonance}) \qquad (5)

where: $P_e(t)$ — probability the atom is excited at time $t$ , from $|e,0\rangle$ ; the period is $\pi/g$ , so with $g/2\pi=24$ MHz the atom fully returns to $|e\rangle$ every $\approx 21$ ns, reaching zero at $t=\pi/2g$ where the excitation is entirely one cavity photon. When the coupling instead loses the race to mirror loss — weak coupling, $g\ll\kappa$ — the cavity no longer sustains oscillations but still reshapes the density of states, enhancing resonant spontaneous emission by the Purcell factor, set by the two knobs a fabricator controls: quality factor and mode volume.

F_P = \frac{3}{4\pi^2}\left(\frac{\lambda}{n}\right)^3 \frac{Q}{V},\qquad C = \frac{g^2}{\kappa\gamma},\qquad Q = \frac{\omega_c}{\kappa} \qquad (6)

where: $F_P$ — resonant emission-enhancement factor; $\lambda$ — transition wavelength; $n$ — host refractive index; $Q=\omega_c/\kappa$ — quality factor; $V$ — mode volume; $C$ — single-atom cooperativity ( $C\gtrsim1$ signals cavity dominance); $\kappa$ — cavity field decay rate (FWHM linewidth); $\gamma$ — atomic dipole decay rate. Cooperativity governs the bad-cavity limit even when $g\ll\kappa$ : the Purcell enhancement equals $2C$ . Detune the atom far from the cavity, $|\Delta|\gg g$ , and no real photon is exchanged. Second-order perturbation theory instead gives a state-dependent shift — the cavity moves by $\pm\chi$ with the atomic state, the atom by $\chi$ per photon — the dispersive regime, the basis of measuring photons without absorbing them.

H_{\text{disp}} \approx \hbar\left(\omega_c + \chi\,\sigma_z\right) a^\dagger a + \tfrac{1}{2}\hbar\tilde\omega_a\sigma_z,\qquad \chi = \frac{g^2}{\Delta} \qquad (7)

where: $\Delta=\omega_a-\omega_c$ — atom–cavity detuning, $|\Delta|\gg g$ ; $\chi=g^2/\Delta$ — dispersive frequency shift; $\tilde\omega_a$ — Lamb-shifted atomic frequency. The interaction is now a measurement: the cavity frequency reads the atomic state, the atomic phase counts photons — no energy exchanged.

Derivation AVacuum Rabi oscillation from the single-excitation manifold

The goal is to show that a single shared excitation oscillates completely and reversibly between atom and field, and to extract both the dressed doublet split by $2g$ and the oscillation $P_e(t)=\cos^2(gt)$ . On resonance the total excitation number $N=a^\dagger a+\tfrac12(\sigma_z+1)$ commutes with $H$ , so the Hamiltonian block-diagonalizes into two-dimensional subspaces and the whole problem reduces to diagonalizing one $2\times2$ matrix.

Step 1. Restrict to the one-excitation subspace $\{|e,0\rangle,\,|g,1\rangle\}$ — atom excited with zero photons, or ground atom with one photon. Why: conservation of $N$ mixes in no other states, collapsing the infinite problem to two states.

Step 2. Using $\langle g,1|\,a^\dagger\sigma_-\,|e,0\rangle=1$ , the Hamiltonian in this basis is $H_{N=1}=\hbar\omega\,\mathbb{1}+\hbar g\,\sigma_x$ . Why: the diagonal is the shared excitation energy $\hbar\omega$ ; the off-diagonal $\hbar g$ moves the excitation from atom to field.

Step 3. Diagonalize: eigenstates $|1,\pm\rangle=\tfrac{1}{\sqrt2}(|e,0\rangle\pm|g,1\rangle)$ with energies $\hbar\omega\pm\hbar g$ — the vacuum Rabi doublet, split by $2g$ and resolved by a transmission spectrum as two peaks. Why: the coupling lifts the degeneracy of the two bare states symmetrically.

Step 4. Evolve $|e,0\rangle=\tfrac1{\sqrt2}(|1,+\rangle+|1,-\rangle)$ and recombine to get $|\psi(t)\rangle=\cos(gt)\,e^{-i\omega t}|e,0\rangle-i\sin(gt)\,e^{-i\omega t}|g,1\rangle$ ; modulus squared gives $P_e(t)=\cos^2(gt)$ , $P_{g,1}(t)=\sin^2(gt)$ — complete reversible exchange with period $\pi/g$ . Why: the two dressed states beat at $2g$ , so the population sloshes fully to the photon and back — dynamics free-space decay can never show. *Verified: standard result (Loudon ch. 7; Walls and Milburn ch. 10); the $2g$ splitting and $\cos^2(gt)$ inversion are textbook.*

Go deeper · Derivation B · 1 minCollapse and revival with a coherent field

Derivation BCollapse and revival with a coherent field

The goal is to expose the sharpest fingerprint of field quantization: drive the atom with a coherent state instead of the vacuum, and its Rabi oscillation first collapses, then — against classical intuition — revives. It happens only because the field energy comes in discrete photons, each Fock component ticking at its own $2g\sqrt{n+1}$ . Prepare the atom in $|e\rangle$ and the cavity in $|\alpha\rangle=\sum_n c_n|n\rangle$ with Poissonian weights $|c_n|^2=e^{-\bar n}\bar n^{\,n}/n!$ , mean number $\bar n=|\alpha|^2$ .

Step 1. Each Fock component evolves as its own two-state manifold at $\Omega_n=2g\sqrt{n+1}$ , so the inversion is the Poissonian-weighted sum $P_e(t)=\tfrac{1}{2}[1+\sum_{n}|c_n|^2\cos(2g\sqrt{n+1}\,t)]$ . Why: $N$ is conserved, so the components oscillate in parallel at incommensurate rates.

Step 2. The frequencies span roughly $g$ across the distribution, so the cosines dephase and the oscillation collapses on $t_c\sim 1/g$ , independent of $\bar n$ ; being discrete, they later rephase near $t_r\approx 2\pi\sqrt{\bar n}/g$ , giving a partial revival. Why: neighbouring Fock frequencies differ by $\approx g/\sqrt{\bar n}$ , so they fall out of step, then realign.

Step 3. The upshot: collapse-and-revival has no classical analogue and was first seen with Rydberg atoms in a microwave cavity — direct proof the field is quantized. Why: only a discrete photon spectrum produces a rephasing time; a continuous amplitude gives one frequency, no revival. *Verified: standard Jaynes–Cummings collapse-revival treatment (Walls and Milburn; Haroche and Raimond).*

The weak-coupling counterpart follows from the same equations of motion. In the bad-cavity regime $\kappa\gg g$ the fast cavity field is slaved to the slow dipole, $a\approx -\tfrac{2ig}{\kappa}\sigma_-$ ; substituting this back into the dipole's Heisenberg–Langevin equation adds an extra decay channel $\Gamma_{\text{cav}}=4g^2/\kappa=2C\gamma$ . This adiabatic elimination reproduces the Purcell factor of equation (6) once $g,\kappa$ are expressed through $Q$ and $V$ , so cooperativity and Purcell factor track each other (Foot ch. 9; Purcell 1946).

Limiting cases

The regimes are set entirely by the ordering of the coupling $g$ , the cavity decay $\kappa$ , and the atomic decay $\gamma$ :

Case	Result	Meaning
Strong coupling	$g \gg \kappa,\ \gamma$	Coherent exchange dominates; the vacuum Rabi doublet $2g$ is spectrally resolved and $\cos^2(gt)$ oscillations appear.
Weak coupling (Purcell)	$g \ll \kappa$ yet $C=g^2/\kappa\gamma \gtrsim 1$	Cavity adiabatically eliminated; emission stays irreversible but enhanced by $F_P$ ; cooperativity governs dominance.
Bad-cavity limit	$\kappa \gg g \gg \gamma$	Photons escape before reabsorption; the atom decays nearly exponentially at the Purcell-enhanced rate — bright single-photon emission.
Dispersive limit	$\lvert\Delta\rvert=\lvert\omega_a-\omega_c\rvert \gg g$	Exchange is virtual; a photon-number shift $\chi=g^2/\Delta$ appears, enabling QND photon counting and dispersive qubit readout.

Worked examples

1 · Vacuum field and coupling from the mode volume

A near-planar optical cavity for the caesium D2 line, $\lambda=852$ nm, has a Gaussian mode of waist $w_0=20\ \mu$ m and length $L=100\ \mu$ m. Estimate the vacuum field per photon, the coupling $g$ , and the vacuum Rabi splitting, taking an effective dipole $d\approx2.5\times10^{-29}$ C·m.

Step 1. Mode volume of the TEM $_{00}$ mode: $V=\tfrac{\pi}{4}w_0^2 L=\tfrac{\pi}{4}(20\times10^{-6})^2(100\times10^{-6})=\mathbf{3.1\times10^{-14}\ m^3}$ .

Step 2. Vacuum field per photon from equation (2), with $\omega_c=2\pi c/\lambda=2.21\times10^{15}$ rad/s so $\hbar\omega_c=2.33\times10^{-19}$ J: $\mathcal{E}_0=\sqrt{\hbar\omega_c/(2\varepsilon_0 V)}=\sqrt{2.33\times10^{-19}/(2\cdot 8.85\times10^{-12}\cdot 3.1\times10^{-14})}=\mathbf{6.5\times10^{2}\ V/m}$ — one zero-point photon supplies roughly 650 V/m.

Step 3. Coupling: $g=d\,\mathcal{E}_0/\hbar=(2.5\times10^{-29})(647)/(1.05\times10^{-34})=1.5\times10^{8}$ rad/s, i.e. $g/2\pi=\mathbf{24\ MHz}$ , and the vacuum Rabi splitting is $2g/2\pi=\mathbf{49\ MHz}$ . E₀ ≈ 650 V/m, g/2π ≈ 24 MHz, doublet split by ≈ 49 MHz.

2 · Reading the regime from the three rates

Take the cavity of Example 1, $g/2\pi=24$ MHz, with a cavity linewidth $\kappa/2\pi=1.4$ MHz and the caesium dipole decay $\gamma/2\pi=2.6$ MHz (half the $5.2$ MHz natural linewidth). Is it strongly coupled? Find the cooperativity and the number of coherent exchanges.

Step 1. Strong-coupling test: $g/2\pi=24$ MHz vastly exceeds both $\kappa/2\pi=1.4$ MHz and $\gamma/2\pi=2.6$ MHz, so $g\gg\kappa,\gamma$ — strongly coupled, and the vacuum Rabi doublet is resolvable.

Step 2. Cooperativity from equation (6): $C=g^2/(\kappa\gamma)=24^2/(1.4\times2.6)=576/3.64=\mathbf{158}$ — the cavity dominates the atom's dynamics by more than two orders of magnitude.

Step 3. Coherent exchanges before decoherence: exchange runs at $2g$ , total loss at $\kappa+\gamma$ , so $N\approx 2g/(\kappa+\gamma)=48/(1.4+2.6)=\mathbf{12}$ — enough to see clean oscillations. Strongly coupled, C ≈ 158, ≈ 12 coherent exchanges.

3 · Purcell factor of a single-photon source

An InAs quantum dot emitting at $\lambda=920$ nm sits in a GaAs micropillar ( $n=3.5$ ) of quality factor $Q=2000$ and mode volume $V=5\,(\lambda/n)^3$ . Find the Purcell factor and the shortened radiative lifetime if the free-space lifetime is $1$ ns.

Step 1. With $\tilde V=V/(\lambda/n)^3=5$ , the $(\lambda/n)^3$ in equation (6) cancels and $F_P=\tfrac{3}{4\pi^2}\,Q/\tilde V=\tfrac{3}{4\pi^2}\cdot\tfrac{2000}{5}=0.076\times400=\mathbf{30}$ — emission into the mode enhanced roughly thirtyfold.

Step 2. Total decay speeds up by $F_P+1$ , so $\tau_{\text{cav}}=\tau_{\text{free}}/(F_P+1)=1\ \text{ns}/31\approx\mathbf{32\ ps}$ — short enough to outrun dephasing, which makes the photons near-transform-limited. F_P ≈ 30, lifetime 1 ns → ≈ 32 ps.

Physics of the portrait

Cavity QED is where a sixty-year-old solvable model met the art of trapping single atoms and photons long enough to watch them talk — and no one embodies that meeting more than the experimentalist who caught a single photon and read it without destroying it.

Origins · 1 minSerge Haroche and how this was discovered

Serge Haroche

born 1944 · microwave cavity QED, the quantum Rabi oscillation, and quantum non-demolition photon counting

Serge Haroche built his career on making the abstract Jaynes–Cummings model real by choosing an atom big enough and a cavity good enough. Working with Rydberg atoms — excited to enormous principal quantum numbers and huge dipole moments — and superconducting microwave cavities of extraordinary finesse, his Paris group turned single-quantum dynamics into laboratory observation. In the 1980s they confirmed Purcell's 1946 prediction directly; through the 1990s and 2000s they resolved the vacuum Rabi oscillation, mapped the collapse and revival that betrays field quantization, and — most strikingly — realized quantum non-demolition detection of single microwave photons: an atom crossing the cavity acquires a phase proportional to the stored photon number without absorbing a thing, so the same photon is measured again and again and its quantum jumps watched in real time — making the discreteness of the electromagnetic field something one could sit and observe. Haroche shared the 2012 Nobel Prize with David Wineland "for ground-breaking experimental methods that enable measuring and manipulating individual quantum systems" — the two approaching the frontier from opposite sides, Haroche probing trapped photons with atoms, Wineland trapped ions with light.

History

1946

— Purcell predicts that a resonant cavity enhances the spontaneous-emission rate

1963

— Jaynes and Cummings solve the single-mode atom–field model exactly

1987

— Rempe, Walther and Klein observe collapse and revival in the micromaser

1992

— Thompson, Rempe and Kimble resolve the vacuum Rabi doublet for one atom in an optical cavity

2004

— Wallraff and coworkers demonstrate strong coupling of a superconducting qubit to a microwave resonator, launching circuit QED

2012

— Haroche and Wineland share the Nobel Prize for measuring and controlling individual quantum systems

Going further

Texts. Loudon, The Quantum Theory of Light, 3rd ed., ch. 7 (Jaynes–Cummings) · Walls and Milburn, Quantum Optics, 2nd ed., ch. 10 (cavity QED) · Haroche and Raimond, Exploring the Quantum: Atoms, Cavities and Photons (2006) · Foot, Atomic Physics, ch. 9 · Saleh and Teich, Fundamentals of Photonics, ch. 13–14 · Purcell, Phys. Rev. 69, 681 (1946).

Provenance. A verifier should confirm: the Jaynes–Cummings Hamiltonian and its rotating-wave derivation; the dressed energies $\hbar\omega_c(n+\tfrac12)\pm\hbar g\sqrt{n+1}$ and vacuum splitting $2g$ ; the vacuum Rabi result $P_e=\cos^2(gt)$ ; the Purcell factor $F_P=\tfrac{3}{4\pi^2}(\lambda/n)^3 Q/V$ and the equivalent rate $4g^2/\kappa=2C\gamma$ via adiabatic elimination; and $\chi=g^2/\Delta$ . The worked-example numbers ( $\mathcal{E}_0\approx650$ V/m, $g/2\pi\approx24$ MHz, $C\approx158$ , $F_P\approx30$ ) should recompute from the stated inputs.

Prerequisites. Quantization of the Electromagnetic Field · Fock States and the Photon-Number Basis · Vacuum State and Zero-Point Energy · Coherent States. Leads to. Single-Photon Sources · Photon Antibunching · Photon Statistics and the Mandel Q Parameter · Squeezed States of Light.

Ask the tutor: Want to see how the vacuum Rabi doublet becomes circuit-QED dispersive readout, or how the $\sqrt{n+1}$ ladder produces photon blockade?

What does "vacuum" mean in "vacuum Rabi oscillation" if energy is being exchanged?

It means the exchange begins with no photon in the cavity — the field is in its vacuum state. Ordinary Rabi oscillation needs a strong applied field and speeds up with its amplitude; here the only driver is the zero-point field of the empty mode, so the frequency $2g$ is fixed by the vacuum field alone. The energy is not created from nothing — it belongs to the single excitation you started with, traded back and forth. The vacuum is the partner, not the fuel.

How is cavity QED different from just putting an atom in a mirror box?

The mirrors matter only for what they do to the modes. A good cavity deletes the free-space continuum and supplies one sharp mode, so the atom has only one place to send its photon — and can get it back. What makes it "cavity QED" rather than "an atom near mirrors" is that the coupling $g$ exceeds the rates at which the photon leaks out ( $\kappa$ ) or the atom radiates into leftover free-space modes ( $\gamma$ ). Below that threshold: Purcell enhancement; above it: coherent reversible dynamics.

Why does making the cavity smaller increase the coupling?

Because the vacuum field per photon scales as $\mathcal{E}_0\propto V^{-1/2}$ . A photon carries fixed energy $\hbar\omega$ , and the field needed to store it is larger when the mode occupies less volume (energy density is field squared times volume). The coupling $g=d\,\mathcal{E}_0/\hbar$ inherits that scaling, so halving the mode volume raises $g$ by about $41\%$ — the design logic behind micropillars and nanophotonic resonators. The competing pressure is that small cavities are often lossier, so the real target is a favourable ratio, not the smallest $V$ .

Is the Purcell effect the same as strong coupling?

No — they are opposite regimes. The Purcell effect is weak coupling: $g\ll\kappa$ , the cavity never sustains an oscillation, and its only role is to reshape the density of states so emission is faster on resonance. Strong coupling is $g\gg\kappa,\gamma$ , where the photon is exchanged coherently many times and the spectrum splits into the vacuum Rabi doublet. The cooperativity $C=g^2/\kappa\gamma$ bridges them: the Purcell enhancement is $2C$ , so a useful Purcell factor needs only $C\gtrsim1$ , still far from strong coupling.

Why does the dressed-state ladder become anharmonic, and why does it matter?

Because the coupling in the $n$ -photon manifold carries the matrix element of $a^\dagger$ , a factor $\sqrt{n+1}$ . So the doublet splitting $2g\sqrt{n+1}$ grows with photon number — the rungs are unequally spaced, making the cavity nonlinear at the single-photon level. Tune a drive to the first rung at $\omega_c+g$ and one photon enters; but the step to the next rung is $g\sqrt2$ , so a second photon at the same frequency is detuned and rejected. That is photon blockade — the cavity transmits photons one at a time, turning a single atom into an optical transistor and a source of strongly antibunched light.

Does cavity QED only work with real atoms in real cavities?

Not at all — the model is agnostic about what plays the two roles. Any two-level system coupled to any single bosonic mode obeys the Jaynes–Cummings Hamiltonian. The "atom" can be a real atom, a trapped ion, a quantum dot, or a superconducting qubit; the "cavity" can be an optical Fabry–Pérot, a microwave resonator, a photonic-crystal defect, or a transmission line. The physics — vacuum Rabi splitting, the anharmonic ladder, Purcell enhancement, dispersive shifts — carries over unchanged.