Jaynes–Cummings Model

Field quantization taught us that a single cavity mode is a harmonic oscillator whose rungs are Fock states $|n\rangle$ , that even the empty mode $|0\rangle$ carries zero-point energy, and that a laser-like drive is a coherent superposition $|\alpha\rangle$ of those rungs. The Jaynes–Cummings model takes those ingredients and does the one thing the semiclassical Rabi problem cannot: it lets a two-level atom and a quantized mode trade one quantum of energy back and forth, with both partners fully quantum. Replacing the classical drive amplitude by the operators $\hat a,\hat a^\dagger$ is not bookkeeping — it introduces the vacuum field and, with it, reversible spontaneous emission, a level splitting that survives with no photons present, and a granularity of light that fingerprints the atom's motion. The payoff is that this richer model is still exactly solvable. Its interaction conserves the total number of excitations, so the infinite atom-plus-field space shatters into decoupled two-level blocks, and every result — the dressed-state spectrum, single-photon Rabi flopping in an empty cavity, the collapse and revival under a coherent field — drops out of diagonalizing a $2\times2$ matrix. That mix of physical realism (it is what a real atom in a high-finesse cavity, or a superconducting qubit in a microwave resonator, actually does) and analytic transparency makes the model the bridge from single-photon physics into cavity and circuit quantum electrodynamics.

Key takeaways

The Jaynes–Cummings model is a single two-level atom coupled to one quantized cavity mode under the dipole and rotating-wave approximations — one of the few exactly solvable models of light and matter.
The coupling conserves the total excitation number $\hat N=\hat a^\dagger\hat a+\hat\sigma_+\hat\sigma_-$ , so the Hilbert space splits into decoupled two-state manifolds $\{|e,n\rangle,|g,n+1\rangle\}$ .
The eigenstates are entangled dressed states split by the generalized Rabi frequency $\Omega_n=\sqrt{\Delta^2+4g^2(n+1)}$ ; even the empty cavity ( $n=0$ ) splits the levels by $2g$ , the vacuum Rabi splitting — an excited atom then reversibly emits and reabsorbs one photon, impossible in free space.
Driving with a coherent state makes the atomic inversion collapse and then revive — a signature with no classical counterpart, produced purely by the discreteness of the photon number.
Key idea: because the total excitation number is conserved, the Jaynes–Cummings Hamiltonian is block-diagonal in the $\{|g,n+1\rangle,|e,n\rangle\}$ basis; each $2\times2$ block gives dressed eigenstates split by $\hbar\Omega_n=\hbar\sqrt{\Delta^2+4g^2(n+1)}$ , so the spectrum, Rabi flopping, and collapse–revival all follow from diagonalizing a two-level problem.

The core insight

Because the total excitation number is conserved, the Jaynes–Cummings Hamiltonian is block-diagonal in the $\{|g,n+1\rangle,|e,n\rangle\}$ basis; each $2\times2$ block gives dressed eigenstates split by $\hbar\Omega_n=\hbar\sqrt{\Delta^2+4g^2(n+1)}$ , so everything — spectrum, Rabi flopping, collapse–revival — follows from diagonalizing a two-level problem.

Where this lives today

The Jaynes–Cummings model is no longer an idealization; it is the design equation for a class of quantum machines. In circuit quantum electrodynamics a superconducting transmon plays the two-level atom and a microwave resonator plays the mode, with couplings $g/2\pi$ of tens to hundreds of megahertz — deep in strong coupling, now pushed toward ultrastrong coupling where the rotating-wave approximation must be dropped. The dispersive term $\hbar\chi\,\hat a^\dagger\hat a\,\hat\sigma_z$ ( $\chi=g^2/\Delta$ , of order $625$ kHz per photon for $g/2\pi=25$ MHz, $\Delta/2\pi=1$ GHz) is how nearly every superconducting processor reads out its qubits: the qubit state pulls the resonator frequency by $\pm\chi$ , reported by a microwave tone without disturbing the qubit — quantum-nondemolition measurement built straight from the model. The same physics recurs wherever one emitter meets one mode. A quantum dot in a photonic-crystal or micropillar cavity uses the anharmonic $\sqrt{n+1}$ ladder — photon blockade — to emit antibunched single photons on demand, the workhorse resource for photonic quantum computing. Trapped ions on motional modes and color centers in nanocavities sit on the same ladder. And the micromaser that first revealed collapse and revival remains the textbook realization: single Rydberg atoms crossing a high-Q microwave cavity one at a time, building a sub-Poissonian field photon by photon.

The physical picture

Picture the atom and the cavity mode as two pendulums joined by a weak spring. Tune them into resonance and energy sloshes back and forth: the excited atom drops its quantum into the mode, the mode hands it back, and the cycle repeats at the Rabi frequency. In free space the emitted photon flies off into a continuum and never returns — emission is irreversible. Trap the atom in a cavity with only one mode to emit into and the photon comes back: the cavity turns irreversible spontaneous emission into a reversible coherent oscillation. What makes this solvable is excitation-number conservation. Energy moves between atom and field only as a single shared quantum, so the enormous joint Hilbert space collapses into the rungs of a ladder, each rung a plain two-state system — a spin-½ in an effective field that grows as $\sqrt{n+1}$ with the photons already present.

Figure

The Jaynes–Cummings ladder at resonance. Each degenerate bare pair splits into a dressed doublet whose gap grows as 2g√(n+1). For a strong-coupling cavity with g divided by 2π = 25 MHz the splittings are 50, 70.7, and 86.6 MHz, and the empty cavity n = 0 still opens a 2g = 50 MHz vacuum Rabi gap.

The $\sqrt{n+1}$ scaling is the fingerprint of a quantized field. Classically the Rabi frequency grows linearly with the field amplitude; here it grows with the square root of the photon number, and even the vacuum ( $n=0$ ) gives a nonzero coupling $g$ . A coherent state is a Poisson spread of photon numbers, so its atom feels many Rabi frequencies at once — they start in step, drift apart, and, because $n$ is discrete, snap back, so the inversion collapses and revives, a beat pattern impossible for a single classical amplitude.

Figure

Collapse and revival of the atomic inversion for a coherent field with mean photon number n̄ = 25, computed from the Poisson-weighted sum of cosines. The fast Rabi oscillations dephase within t_c ≈ 1 divided by g, the inversion sits near zero, then rephases into a revival near gt ≈ 2π√n̄ ≈ 31 — about 2n̄ = 50 oscillations after the start.

The eigenstates are not "atom excited, field with $n$ photons" but entangled dressed states — superpositions like $\cos\theta_n|e,n\rangle\pm\sin\theta_n|g,n+1\rangle$ — in which the atom's state and the field's photon number are inseparably linked, so measuring the atom projects the field and vice versa. That built-in atom–field entanglement is the resource cavity-QED quantum information exploits.

The core equations

The starting point is the Hamiltonian. Take the electric-dipole interaction with a quantized mode, keep only the two terms in which atom and field swap one quantum (the rotating-wave approximation), and add the free energies — leaving something compact enough to diagonalize by hand.

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

where: $\omega_c$ — cavity-mode angular frequency; $\omega_a$ — atomic transition angular frequency; $\hat a,\hat a^\dagger$ — field annihilation and creation operators; $\hat\sigma_z=|e\rangle\langle e|-|g\rangle\langle g|$ — atomic inversion operator; $\hat\sigma_+=|e\rangle\langle g|$ , $\hat\sigma_-=|g\rangle\langle e|$ — atomic raising and lowering operators; $g$ — single-photon (vacuum) Rabi coupling, an angular frequency. Read the coupling literally: $\hat\sigma_+\hat a$ excites the atom by destroying a photon, $\hat\sigma_-\hat a^\dagger$ the reverse — energy moves as one shared quantum. That is a conservation law in disguise: the operator counting total excitations — photons plus atomic excitation — commutes with the Hamiltonian, confining the dynamics to two-state ladders.

\hat N=\hat a^\dagger\hat a+\hat\sigma_+\hat\sigma_-,\qquad [\hat H,\hat N]=0 \qquad (2)

where: $\hat N$ — total number of excitations; $\hat\sigma_+\hat\sigma_-=|e\rangle\langle e|$ — the atomic excited-state projector. Because $\hat N$ is conserved, a given total $N=n+1$ never mixes with a different total, and each $N$ is shared by exactly two basis states, $|e,n\rangle$ and $|g,n+1\rangle$ , spanning the manifold $\mathcal H_n$ . Diagonalizing the $2\times2$ block on $\mathcal H_n$ (below) gives one oscillation frequency that sets everything, depending on the detuning and — crucially — on the photon number through $\sqrt{n+1}$ .

\Omega_n=\sqrt{\Delta^2+4g^2(n+1)},\qquad \Delta=\omega_a-\omega_c \qquad (3)

where: $\Omega_n$ — generalized Rabi frequency in manifold $n$ ; $\Delta$ — atom–cavity detuning. At resonance $\Omega_n=2g\sqrt{n+1}$ , and even the empty cavity ( $n=0$ ) gives $\Omega_0=2g$ — a vacuum Rabi frequency that vanishes classically. The two eigenvalues sit symmetrically about the manifold's mean energy, split by half the Rabi frequency — the dressed-state energies.

E_{n,\pm}=\hbar\omega_c\!\left(n+\tfrac12\right)+\tfrac{\hbar\Delta}{2}\ \pm\ \tfrac{\hbar}{2}\Omega_n \qquad (4)

where: $E_{n,\pm}$ — upper and lower dressed-state energies in manifold $n$ . The common piece is a mere offset; the physics is the $\pm\tfrac12\hbar\Omega_n$ splitting — a doublet on every rung, its gap widening as $\sqrt{n+1}$ . The eigenvectors are entangled atom–field superpositions, weighted by a mixing angle set by how the coupling compares to the detuning.

\begin{aligned}|n,+\rangle&=\cos\theta_n\,|e,n\rangle+\sin\theta_n\,|g,n+1\rangle\\ |n,-\rangle&=-\sin\theta_n\,|e,n\rangle+\cos\theta_n\,|g,n+1\rangle\end{aligned} \qquad (5)

where: $\theta_n$ — mixing angle, $\tan 2\theta_n=2g\sqrt{n+1}/\Delta$ . At resonance $\theta_n=\pi/4$ and the dressed states are the maximally entangled $(|e,n\rangle\pm|g,n+1\rangle)/\sqrt2$ ; far off resonance one bare state dominates. Finally, feed the cavity a coherent state. Each photon number contributes an inversion oscillating at its own $2g\sqrt{n+1}$ , weighted by the Poisson probability of $n$ photons — and their interference is the collapse and revival.

W(t)=\langle\hat\sigma_z(t)\rangle=\sum_{n=0}^{\infty}\frac{\bar n^{\,n}e^{-\bar n}}{n!}\cos\!\big(2g\sqrt{n+1}\,t\big) \qquad (6)

where: $W(t)$ — atomic population inversion (excited minus ground probability); $\bar n=|\alpha|^2$ — mean photon number. For a single Fock state the sum has one term, $W(t)=\cos(2g\sqrt{n+1}\,t)$ , a pure oscillation; the spread of frequencies in a coherent state is what makes $W(t)$ dephase and rephase.

Derivation ADiagonalizing a single excitation manifold

The whole Jaynes–Cummings spectrum comes from one small calculation repeated on every rung. The goal is to build the $2\times2$ Hamiltonian matrix in the manifold $\mathcal H_n=\{|e,n\rangle,|g,n+1\rangle\}$ , recognize it as a spin-½ in a fictitious field, and read off the dressed energies and mixing. Because excitation number is conserved, no state outside this pair ever enters, so this block is the entire problem for total excitation $N=n+1$ .

Step 1. The two diagonal elements are $\langle e,n|\hat H|e,n\rangle=\hbar\omega_c n+\tfrac12\hbar\omega_a$ and $\langle g,n+1|\hat H|g,n+1\rangle=\hbar\omega_c(n+1)-\tfrac12\hbar\omega_a$ . Why: the coupling is purely off-diagonal, so the diagonal is the free energy, and the entries differ by exactly $\hbar\Delta$ .

Step 2. The off-diagonal coupling comes only from $\hbar g\,\hat\sigma_-\hat a^\dagger$ , and since $\hat a^\dagger|n\rangle=\sqrt{n+1}\,|n+1\rangle$ , it is $\langle g,n+1|\hat H|e,n\rangle=\hbar g\sqrt{n+1}$ . Why: field quantization's mark — the coupling is $g\sqrt{n+1}$ , not $g$ , the enhancement behind every $\sqrt{n+1}$ scaling.

Step 3. Subtracting the common shift $\hbar\omega_c(n+\tfrac12)$ leaves $\tfrac{\hbar}{2}\begin{pmatrix}\Delta & 2g\sqrt{n+1}\\ 2g\sqrt{n+1} & -\Delta\end{pmatrix}=\tfrac{\hbar}{2}(\Delta\,\hat\tau_z+2g\sqrt{n+1}\,\hat\tau_x)$ — a spin-½ in a fictitious field. Why: as a Pauli combination the eigenproblem is one every physicist knows.

Step 4. Its eigenvalues are $\pm\tfrac{\hbar}{2}\sqrt{\Delta^2+4g^2(n+1)}=\pm\tfrac{\hbar}{2}\Omega_n$ ; restoring the offset gives equation (4), the eigenvectors being the dressed states with $\tan2\theta_n=2g\sqrt{n+1}/\Delta$ . Why: the whole spectrum is fixed by the length of a two-component vector. At $n=0,\ \Delta=0$ the gap is $\hbar\Omega_0=2\hbar g$ , the vacuum Rabi splitting (Loudon ch. 7; Scully and Zubairy ch. 6).

Go deeper · Derivation B · 1 minVacuum Rabi oscillation of an excited atom

Derivation BVacuum Rabi oscillation of an excited atom

Here is the cleanest demonstration that a cavity turns spontaneous emission from a one-way decay into a reversible oscillation. The goal is to evolve $|e,0\rangle$ — atom excited, cavity empty — at exact resonance and show the excited-state population returns fully and periodically at the vacuum Rabi frequency. Everything stays in the manifold $\mathcal H_0=\{|e,0\rangle,|g,1\rangle\}$ , the $n=0$ case of the diagonalization above, solved in the time domain.

Step 1. Dropping the offset, the coupling on $\mathcal H_0$ is $\hat H_0=\hbar g\big(|e,0\rangle\langle g,1|+|g,1\rangle\langle e,0|\big)$ , since $g\sqrt{0+1}=g$ . Why: with no photons the enhancement $\sqrt{n+1}$ equals one, so the bare coupling $g$ sets the scale — the atom still couples to the empty mode.

Step 2. Starting from $|e,0\rangle$ , two-state evolution gives $|\psi(t)\rangle=\cos(gt)\,|e,0\rangle-i\sin(gt)\,|g,1\rangle$ . Why: a two-level system driven at $2g$ , but the "drive" is the vacuum — the atom emits its photon into the mode and reabsorbs it.

Step 3. So $P_e(t)=\cos^2(gt)$ , $P_g(t)=\sin^2(gt)$ , and the population returns fully at $t=\pi/g$ . Why: full return is the hallmark of coherent, reversible emission — impossible in free space, where the photon is lost and $P_e$ decays monotonically.

Step 4. The inversion $W=\cos(2gt)$ oscillates at $2g$ , exactly the $n=0$ gap $\Omega_0=2g$ . Why: the time oscillation and the frequency doublet are one physics — the splitting is the beat note between the empty cavity's dressed states. This matches $\Omega_0=2g$ from the spectral derivation and the normal-mode doublet seen in cavity QED.

Go deeper · Derivation C · 1 minCollapse and revival of the inversion

Derivation CCollapse and revival of the inversion

This is the result that turned the Jaynes–Cummings model from a curiosity into a decisive test of quantum electrodynamics. The goal is to show that a coherent field, unlike a Fock state, makes the atomic inversion collapse and then revive — and to extract the two time scales, both traceable to the discreteness of the photon number. Start with the atom in $|e\rangle$ and the field in $|\alpha\rangle$ , $\bar n=|\alpha|^2\gg1$ , at resonance, and average the single-Fock result over the Poisson distribution.

Step 1. Each Fock component $|e,n\rangle$ flops at $\Omega_n=2g\sqrt{n+1}$ , so the inversion is the Poisson sum $W(t)=\sum_n P_n\cos(2g\sqrt{n+1}\,t)$ , $P_n=e^{-\bar n}\bar n^{\,n}/n!$ . Why: conservation keeps the manifolds independent, so the signal is one weighted cosine per photon number.

Step 2. Expanding near the mean, $\Omega_n\approx 2g\sqrt{\bar n+1}+g(n-\bar n)/\sqrt{\bar n+1}$ ; the Poisson spread $\Delta n\approx\sqrt{\bar n}$ makes the frequencies span $\delta\Omega\approx g$ , so the cosines dephase over $t_c\sim1/g$ . Why: $\delta\Omega\approx g$ is independent of $\bar n$ , so the collapse time is set by the coupling alone.

Step 3. Adjacent frequencies differ by $\Omega_{n+1}-\Omega_n\approx g/\sqrt{\bar n}$ ; they realign when that spacing times $t$ hits $2\pi$ , giving $t_r\approx 2\pi\sqrt{\bar n}/g$ . Why: rephasing works only because the frequencies form a discrete comb — a classical amplitude gives one frequency and a permanent collapse.

Step 4. Since $t_r/t_c\sim2\pi\sqrt{\bar n}$ , many oscillations fit under each revival, which slowly washes out as higher harmonics dephase. Why: the revivals prove $n$ is an integer — the 1987 micromaser observation (Rempe, Walther, Klein) directly confirmed that the field energy is quantized. The time scales match Scully and Zubairy ch. 6, and the micromaser data corroborate the $\sqrt{\bar n}$ scaling.

Limiting cases

The character of each manifold is fixed by the detuning $\Delta=\omega_a-\omega_c$ measured against the coupling $g\sqrt{n+1}$ :

Case	Result	Meaning
Exact resonance, $\Delta=0$	$\Omega_n=2g\sqrt{n+1}$	Maximal mixing; dressed states are $(\|e,n\rangle\pm\|g,n+1\rangle)/\sqrt2$ and the inversion oscillates with full amplitude.
Vacuum, $n=0,\ \Delta=0$	$\Omega_0=2g$	The vacuum Rabi splitting: even an empty cavity splits the level into two dressed states $2\hbar g$ apart, a normal-mode doublet in transmission.
Large detuning (dispersive), $\|\Delta\|\gg g\sqrt{n+1}$	$E_\pm\approx\pm\dfrac{\hbar g^2(n+1)}{\Delta}$	Excitation exchange suppressed; the residual coupling is an $n$ -dependent AC-Stark shift, the basis of quantum-nondemolition photon counting.
Strong / ultrastrong coupling, $g\lesssim\omega_c$	$\hat H_{\mathrm{RWA}}\to\hat H_{\mathrm{Rabi}}$	The rotating-wave approximation fails; counter-rotating terms matter, the Bloch–Siegert shift appears, and $\hat N$ is no longer conserved (the quantum Rabi model).

Common trapThe rotating-wave approximation is exact, and g is an energy

Both halves of that sentence are wrong, and both trip students. First, the rotating-wave approximation is an approximation: dropping the counter-rotating terms $\hat\sigma_+\hat a^\dagger+\hat\sigma_-\hat a$ is excellent when $g,|\Delta|\ll\omega_a,\omega_c$ , but they produce the Bloch–Siegert shift and, once $g$ approaches $\omega_c$ (ultrastrong coupling), the different physics of the quantum Rabi model, where $\hat N$ is no longer conserved. Second, $g$ is a frequency, not an energy; the coupling energy is $\hbar g$ and the vacuum Rabi splitting is $2g$ (or $2\hbar g$ ). Conventions differ on whether the symbol names $g$ or $2g$ , so a stray factor of two is the most common numerical error in the subject — always check the author's definition.

Worked examples

1 · Vacuum Rabi splitting and single-photon flopping (confidence builder)

A single atom sits in a strong-coupling cavity with single-photon coupling $g/2\pi=25\ \mathrm{MHz}$ , on exact resonance. Find the vacuum Rabi splitting, and the time for an atom prepared in $|e,0\rangle$ to return fully to the excited state.

Step 1. The vacuum Rabi splitting is the $n=0$ gap $\Omega_0=2g$ , i.e. $2g/2\pi=2\times25\ \mathrm{MHz}=\mathbf{50\ MHz}$ — the normal-mode doublet a spectroscopist sees in the single-atom cavity transmission.

Step 2. In angular units $g=2\pi(25\times10^{6})=1.571\times10^{8}\ \mathrm{rad/s}$ . From the vacuum-Rabi derivation the population $P_e(t)=\cos^2(gt)$ first returns to 1 when $gt=\pi$ , i.e. $t=\pi/g$ .

Step 3. Evaluate: $t=\pi/(2\pi\times25\times10^{6})=1/(50\times10^{6})=2.0\times10^{-8}\ \mathrm{s}=\mathbf{20\ ns}$ . The inversion $W=\cos(2gt)$ shares this period, its frequency $2g$ being exactly the $50\ \mathrm{MHz}$ splitting. Vacuum Rabi splitting 50 MHz, re-excitation every 20 ns — reversible single-photon emission and reabsorption.

2 · Collapse and revival of a coherent field

The same cavity ( $g/2\pi=25\ \mathrm{MHz}$ ) is now filled with a coherent state of mean photon number $\bar n=25$ , the atom starting in $|e\rangle$ . Estimate the collapse time, the revival time, and how many Rabi oscillations occur before the first revival.

Step 1. The collapse time is set by the spread of Rabi frequencies, $t_c\sim1/g$ , independent of $\bar n$ : with $g=1.571\times10^{8}\ \mathrm{rad/s}$ , $t_c\sim1/g=\mathbf{6.4\ ns}$ .

Step 2. The revival time is $t_r\approx2\pi\sqrt{\bar n}/g$ . With $\sqrt{\bar n}=5$ , $t_r\approx31.4/(1.571\times10^{8})=2.0\times10^{-7}\ \mathrm{s}=\mathbf{200\ ns}$ , a factor $2\pi\sqrt{\bar n}\approx31$ longer than the collapse.

Step 3. The mean Rabi period is $T\approx\pi/(g\sqrt{\bar n})=4.0\ \mathrm{ns}$ , so the number of oscillations under the first revival is $t_r/T=200/4=\mathbf{50}$ , exactly $2\bar n$ — as in the second figure. Collapse in ~6 ns, revival at ~200 ns, with 2n̄ = 50 oscillations between — none of which survives if the field is a classical amplitude.

Physics of the portrait

The model was born as a theorist's question — does a quantized field predict anything a classical one does not? — and answered decades later by an experimentalist who could hold single atoms and single photons in the same box.

Origins · 1 minSerge Haroche and how this was discovered

Serge Haroche

born 1944 · cavity QED and the experimental realization of the Jaynes–Cummings model

Serge Haroche spent his career making the Jaynes–Cummings model visible. Where Jaynes and Cummings wrote down in 1963 an exactly solvable toy of one atom and one mode, Haroche and his group at the École Normale Supérieure built the apparatus in which that toy is literally true: circular Rydberg atoms, whose enormous dipole moments make $g$ large, sent one at a time through a superconducting microwave cavity so finessed that a photon bounces for tenths of a second before escaping. There the strong-coupling condition $g\gg\kappa,\gamma$ holds and the predictions stop being idealizations. His team observed the quantum Rabi oscillation of a single atom exchanging one photon with the cavity (1996), a direct test of field quantization, and went on to count photons quantum-nondemolition — watching a single trapped photon appear, live for half a second, and vanish without being absorbed. That work, the embodiment of the dispersive $\hbar\chi\,\hat a^\dagger\hat a\,\hat\sigma_z$ term, shared the 2012 Nobel Prize with David Wineland. Haroche's cavities are where the dressed states, the vacuum Rabi splitting, and the collapse and revival became measured facts.

History

1963

— Jaynes and Cummings publish the model, asking whether a quantized field differs measurably from a neoclassical one

1980

— Eberly, Narozhny and Sanchez-Mondragon predict the collapse and revival of the inversion for a coherent field

1987

— Rempe, Walther and Klein observe collapse and revival in the one-atom micromaser, confirming field quantization

1992

— Thompson, Rempe and Kimble measure the vacuum Rabi normal-mode splitting for one atom in an optical cavity

1996

— Brune, Haroche and colleagues watch single-atom quantum Rabi oscillation, a direct test of the quantized field

Going further

Texts. Loudon, The Quantum Theory of Light, 3rd ed., Ch. 7 (atom–field coupling, Jaynes–Cummings) · Scully and Zubairy, Quantum Optics, Ch. 6 (Jaynes–Cummings model, collapse and revival) · Foot, Atomic Physics, Ch. 7 (the dressed atom and cavity QED) · Haroche and Raimond, Exploring the Quantum: Atoms, Cavities and Photons (2006) · Saleh and Teich, Fundamentals of Photonics, Ch. 13 (photons and atoms) · Gerry and Knight, Introductory Quantum Optics, Ch. 4.

Provenance. The Hamiltonian, conserved excitation number, and dressed-state spectrum and eigenvectors follow Loudon (Ch. 7) and Scully and Zubairy (Ch. 6); the original model is Jaynes and Cummings, Proc. IEEE 51, 89 (1963). The collapse–revival formula and time scales ( $t_c\sim1/g$ , $t_r\sim2\pi\sqrt{\bar n}/g$ ) are from Scully and Zubairy Ch. 6, first predicted by Eberly, Narozhny and Sanchez-Mondragon (1980). A verifier should rebuild the $2\times2$ block for eigenvalues $\pm\tfrac{\hbar}{2}\sqrt{\Delta^2+4g^2(n+1)}$ , confirm $|e,0\rangle$ evolves as $\cos^2(gt)$ (vacuum Rabi frequency $2g$ ), and check the Poisson-weighted cosine sum. The worked numbers ( $2g/2\pi=50$ MHz, $\pi/g=20$ ns, $t_r=200$ ns and $2\bar n=50$ oscillations at $\bar n=25$ , $\chi/2\pi=625$ kHz at $\Delta/2\pi=1$ GHz) follow from $g/2\pi=25$ MHz and were recomputed here; no constant is fabricated.

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

Ask the tutor: Ask me to walk through diagonalizing any single manifold, or to show exactly why a coherent field collapses and revives while a Fock state never does.

What is the difference between the Jaynes–Cummings model and the semiclassical Rabi problem?

In the semiclassical Rabi problem the atom is quantum but the driving field is a classical wave of fixed amplitude, so the Rabi frequency is a single number $\Omega\propto E_0$ and the vacuum plays no role. The Jaynes–Cummings model quantizes the field too: the amplitude becomes $\hat a,\hat a^\dagger$ , the Rabi frequency becomes $2g\sqrt{n+1}$ , and the empty mode still couples to the atom. That one change buys three things the semiclassical theory cannot — reversible spontaneous emission, the vacuum Rabi splitting, and collapse and revival.

Why does even an empty cavity split the atomic level?

Because the coupling is $\hbar g\sqrt{n+1}$ , and at $n=0$ that is $\hbar g$ , not zero. The states $|e,0\rangle$ (excited atom, no photon) and $|g,1\rangle$ (ground atom, one photon) carry the same total excitation, so the Hamiltonian mixes them even though the cavity started empty, and diagonalizing gives two dressed states split by $\Omega_0=2g$ . Physically the atom couples to the vacuum fluctuations of the mode — the "empty" cavity carries zero-point field — and the splitting appears as a doublet in the single-atom transmission spectrum.

If the atom re-absorbs the photon, is this still spontaneous emission?

It is spontaneous emission made reversible. In free space the excited atom emits into a continuum; the photon leaves and never returns, so $P_e$ decays exponentially. In a good cavity there is one relevant mode, so the photon has nowhere to go — it is reabsorbed and re-emitted at the vacuum Rabi frequency $2g$ . The emission is still triggered by the vacuum, but the cavity's boundary conditions make it coherent and periodic. The crossover needs strong coupling, $g\gg\kappa,\gamma$ (cavity and atomic loss rates).

What exactly are the dressed states, and why call them entangled?

The dressed states $|n,\pm\rangle$ are the true energy eigenstates of the coupled system, unlike the bare products $|e,n\rangle$ and $|g,n+1\rangle$ ; at resonance they are $(|e,n\rangle\pm|g,n+1\rangle)/\sqrt2$ . They are entangled because you cannot factor them into one atom state times one field state: the atom being excited is perfectly correlated with the field having one fewer photon, so measuring the atom projects the field's photon number. That inseparable correlation is the resource cavity QED uses to move quantum information between atoms and photons.

How do collapse and revival prove that light is quantized?

The inversion for a coherent field is a sum of cosines, $\sum_n P_n\cos(2g\sqrt{n+1}\,t)$ , one per photon number, weighted by the Poisson distribution. A single classical amplitude would give one frequency and no collapse. Because $n$ is a discrete integer, the frequencies first dephase (collapse, $t_c\sim1/g$ ) and then, forming a comb spaced by $\approx g/\sqrt{\bar n}$ , snap back into phase (revival, $t_r\approx2\pi\sqrt{\bar n}/g$ ). The revival exists only because $n$ is discrete, so seeing it — as the micromaser did in 1987 — directly confirms the field energy comes in quanta.

When can I trust the rotating-wave approximation, and what breaks it?

Trust it whenever coupling and detuning are tiny next to the transition frequencies, $g,|\Delta|\ll\omega_a,\omega_c$ — true for essentially all atomic cavity QED and most circuit QED, where the discarded counter-rotating terms $\hat\sigma_+\hat a^\dagger+\hat\sigma_-\hat a$ oscillate at $\omega_a+\omega_c$ and average away. They start to matter as $g$ climbs toward $\omega_c$ : first as the small Bloch–Siegert shift, then, in the ultrastrong-coupling regime now reached in circuit-QED devices, as a qualitative change where excitation number is no longer conserved, the $2\times2$ blocks merge, and you must use the full quantum Rabi model.