Quantum Light

Here is a scandal this book has been carrying since Chapter 5, and it is time to face it. Take one hydrogen atom, promote its electron to the 2p state, and put it in a perfect vacuum, in perfect darkness, coupled to nothing. Our quantum mechanics is unequivocal about what happens next: nothing. The 2p state is an energy eigenstate — a stationary state — and Chapter 5 proved that stationary states only turn their phase. Every probability, every observable, frozen forever. The atom should sit excited until the end of time.

It falls in 1.6 nanoseconds.

Something must perturb it — Chapter 12's golden rule is happy to compute decay rates, but it needs a perturbation to work with, and we just removed them all. No field, no collisions, no light. Unless… the electromagnetic field cannot actually be removed. Suppose the field is a quantum system in its own right. Then “perfect darkness” is not the absence of the field — it is the field in its ground state, and if the field is anything like every other quantum system in this book, its ground state fidgets. A vacuum with fluctuations in it is not nothing; it is a perturbation that cannot be switched off, tickling every excited atom in the universe. That is the hypothesis. This chapter builds it, checks its statistics against experiments no classical wave can survive, and then cashes it in: we will compute the 1.6 nanoseconds.

How do you quantize a field? The same way we have quantized everything: find the classical degrees of freedom, spot the Hamiltonian, promote to operators. Put the field in a box and expand it in standing-wave modes. Classically, each mode's amplitude obeys $\ddot{q}=-{\omega }^{2}q$ — each mode of the electromagnetic field is a harmonic oscillator, with the electric field playing position and the magnetic field playing momentum. Maxwell already knew that; we just take it seriously:

Everything Chapter 7 earned now pays out at once. The ladder of each oscillator counts something: a photon is a rung — not a little ball flying through space, but one quantum of excitation of a mode, created by a†, destroyed by a, obeying Chapter 15's boson arithmetic because [a, a†] = 1 was boson arithmetic all along. Planck's E = ħω, Chapter 1's clicks, Chapter 15's n+1 appetite — all of them are this one identification. We quantized light in Chapter 7 and didn't know it.

And the payout has a dark side, which is the point. The oscillator's ground state is not “q = 0, p = 0” — the uncertainty principle forbids that corner of phase space. It is a Gaussian of unremovable width, and for the field that means

The vacuum has no mean field but a nonzero mean square field: zero-point fluctuations, one half-quantum per mode, in every mode in the universe. Look at it directly:

Those fluctuations are not a formal garnish. They push measurably: two neutral plates half a micron apart exclude some vacuum modes between them and get pressed together (Casimir, 1948 — measured to percent precision); they jiggle the electron in hydrogen enough to split 2s from 2p₁/₂ by 1057 MHz (the Lamb shift, exercise 7 estimates it); and — our scandal — they are the perturbation that no darkness can remove.

If a mode is an oscillator, then a beam of light is a quantum state of oscillators, and different states of the same intensity can be physically different kinds of light. Three matter most. The number state |n⟩: sharp energy, no phase — the ring in the widget above. The coherent state |α⟩: Chapter 7's minimum-uncertainty blob, following the classical orbit — this is what a laser emits, and its photon count is Poisson-distributed. And thermal light — a lamp, a star — a classical mixture with geometric photon statistics. To tell them apart, Hanbury Brown and Twiss taught us the decisive question: put two detectors on the beam and ask, given a click, how likely is a second click right now?

For any classical wave — any intensity I(t), however wild — the Cauchy–Schwarz inequality forces ⟨I²⟩ ≥ ⟨I⟩², so g² ≥ 1: classical light can bunch but never antibunch. Now look at the number state's answer. The operator ordering in (17.3) did the physics: after one photon is absorbed, only n−1 remain, and the second click knows it. Light made of countable quanta can keep its clicks apart.

This is worth saying carefully, because the photoelectric effect usually gets the credit: energy exchange in lumps of ħω can be mimicked by a classical field driving quantized atoms. Antibunching cannot. When Kimble, Dagenais and Mandel drove single sodium atoms in 1977 and measured g²(0) < 1, that was the first optical phenomenon with no classical-wave impersonation — the photon's existence proof, half a century after its name was coined. Today “g² < ½” is the routine certificate that a quantum-dot or diamond-vacancy source emits true single photons for quantum cryptography.

Now cash in the scandal. Couple an excited atom to the quantized field: the interaction is the dipole energy −d·E, with Ê from (17.1). The atom in |e⟩ with the field in vacuum |0⟩ can go to |g⟩ plus one photon in any mode — the matrix element rides Chapter 15's appetite, ⟨1|a†|0⟩ = √(0+1) = 1: spontaneous emission is stimulated emission, stimulated by the half-quantum that is always there. Chapter 12's golden rule then wants the density of final photon states — ρ(ω) = ω²V/π²c³ per unit angular frequency, counting both polarizations over all directions (per unit energy it carries an extra 1/ħ; exercise 5 derives it) — and everything assembles into Einstein's A coefficient:

Every factor is ours now: ω³ is one power from the vacuum's per-mode energy and two from the crowding of modes at high frequency; |d|² is the size of the charge's lever arm; and the whole thing exists only because ⟨E²⟩ ≠ 0. For hydrogen's 2p → 1s the dipole integral is an exercise with Chapter 10's wavefunctions — ⟨1s|r|2p⟩ = 768/(243√6) a₀ ≈ 1.290 a₀ — and the lab runs the arithmetic in SI units, then checks the answer against the measured lifetime:

The computed value itself is in the scoreboard at the end of the chapter, read straight out of the run — a tenth of a percent is simply the gate it has to clear. Five chapters conspired to get it there: the golden rule from 12, the orbitals from 10, boson arithmetic from 15, the oscillator from 7, and this chapter's field modes — predicting, with no adjustable anything, how long an atom lives in the dark. When a theory agrees with experiment to a part in a thousand across that many of its own moving parts, you are allowed to believe the vacuum hums.

Free-space emission is irreversible — the photon leaves into a continuum of modes and never returns. Put the atom in a cavity so good that one mode dominates, and the story changes completely: atom and mode become a closed two-body quantum system, the hydrogen atom of quantum optics. Keep only the energy-conserving terms (the rotating-wave approximation) and you have the Jaynes–Cummings model:

The new symbols are the atom's lowering and raising operators: ${\hat{\sigma }}_{+}$ takes the atom from ground to excited, $|g⟩\to |e⟩$, and ${\hat{\sigma }}_{-}$ does the reverse — they are the two-level analogs of the field's ${\hat{a}}^{†}$ and $\hat{a}$. So each piece of the coupling pairs one atomic flip with one photon of the opposite sign: ${\hat{\sigma }}_{+}\hat{a}$ absorbs a photon while exciting the atom, and ${\hat{\sigma }}_{-}{\hat{a}}^{†}$ emits one while dropping it back down.

It conserves excitation number, so it splits into 2×2 blocks — {|e,n⟩, |g,n+1⟩} — and each block is Chapter 5's two-level problem all over again. Diagonalize one block and you get the dressed states, atom and field hybridized:

Read the n = 0 rung at resonance: the states |e, 0⟩ and |g, 1⟩ — excited atom with no photon, ground atom with one — split by exactly 2g. An energy doublet built from a photon that isn't there: the vacuum Rabi splitting, first resolved in 1992, now the working currency of superconducting quantum processors (their “qubit-cavity g” is this g). And because the splitting grows as √(n+1), the ladder is anharmonic: a drive tuned to the first rung is off-tune for the second, so one photon in the cavity blocks the next — photon blockade, a light switch operated by single quanta.

The model's most famous prediction needs no cavity loss, no noise, nothing but (17.7). Start the atom excited in a coherent field. Each number component n drives Rabi flopping at its own frequency 2g√(n+1); the spread of n dephases them and the flopping collapses — and then, because the frequencies are indexed by integers, they re-phase and the flopping revives:

The collapse alone a classical noisy field could fake. The revival it cannot — a continuum of frequencies never re-phases. Every revival observed (Rempe, Walther and Klein 1987, with one atom crossing a superconducting cavity; Haroche's group in exquisite detail since) is the field being caught counting on its fingers. And note what the collapse window hides: between collapse and revival the field has evolved into a superposition of two classical-like states — a small Schrödinger cat, made of microwaves, which Haroche watched decohere in real time. Chapter 18 will need exactly that thought.

The lab (Rust-QP/ch17-jaynes) builds the Jaynes–Cummings matrix and referees it against its closed form. There is a trap here, and it is worth dwelling on because it is a trap about refereeing, not about physics. We want to certify the vacuum Rabi splitting: that the n = 0 doublet is separated by exactly 2g. The lazy way is to sweep the computed eigenvalues for the one nearest ω−g and the one nearest ω+g and subtract. Do that and the check reports zero error every single time — and it would report zero with the vacuum block left uncoupled altogether, because 2 and 0 are also eigenvalues of other rungs, and a search told what to look for will find something close to it. A referee whose answer is its own search key is not a referee. So the levels are located by position instead: sort the closed-form spectrum with each level carrying its label, and read the numeric eigenvalue at the index the label lands on.

1// g (σ₊ a + σ₋ a†):  |e,n⟩ ↔ |g,n+1⟩ with amplitude g√(n+1)
2if n < NCUT {
3    let amp = g * ((n + 1) as f64).sqrt();
4    h[(2 * n + 1, 2 * (n + 1))] = amp;
5    h[(2 * (n + 1), 2 * n + 1)] = amp;
6}
7// …
8let eigs = sorted(h.symmetric_eigen().eigenvalues.iter().cloned().collect());
9// Closed-form spectrum, each level CARRYING ITS LABEL (block n, branch ±)
10// through the sort. The labels cost nothing and buy everything: they let
11// the checks below read a numeric eigenvalue at a KNOWN POSITION in the
12// sorted list instead of hunting the list for the value they expect.
13let mut exact_lab: Vec<(f64, Option<(usize, i8)>)> = vec![(0.0, None)]; // |g,0⟩
14for n in 0..NCUT {
15    let mid = omega * (n + 1) as f64 + delta / 2.0;
16    let split = (delta * delta / 4.0 + g * g * (n + 1) as f64).sqrt();
17    exact_lab.push((mid - split, Some((n, -1))));
18    exact_lab.push((mid + split, Some((n, 1))));
19}
20exact_lab.sort_by(|a, b| a.0.partial_cmp(&b.0).unwrap()); // stable: ties keep order
21let exact: Vec<f64> = exact_lab.iter().map(|&(e, _)| e).collect();
22for i in 0..exact.len() {
23    spec_worst[case] = spec_worst[case].max((eigs[i] - exact[i]).abs());
24}
25if case == 0 {
26    // where the closed form puts a given dressed level in the sorted list
27    let idx = |n: usize, br: i8| -> usize {
28        exact_lab.iter().position(|&(_, l)| l == Some((n, br))).unwrap()
29    };
30    // VACUUM RABI SPLITTING, by index. Both levels are read out of the
31    // numeric spectrum at the positions the closed-form ORDERING assigns
32    // to E∓(0); nothing about the answer 2g is used to find them, so a
33    // mis-wired coupling moves the eigenvalues and this breaks.
34    //
35    // The trap that makes indexing mandatory: at g = ω the dressed
36    // ladders INTERLEAVE across n, and two of these levels are exactly
37    // degenerate with rungs from other blocks — E−(0) = E(|g,0⟩) = 0 and
38    // E+(0) = E−(3) = 2. Degenerate partners share an eigenvalue, so a
39    // position is still unambiguous; a nearest-VALUE search would instead
40    // snap onto whatever it was told to look for and pass on anything.
41    let split = eigs[idx(0, 1)] - eigs[idx(0, -1)];
42    vac_rabi_err = (split - 2.0 * g).abs();
43// …

The dynamics referee is the book's favorite pattern, two disjoint code paths — a blind eigendecomposition of the full matrix against the exact per-block sum, which shares no line of code with it:

1for it in 0..n_t {
2    let t = 1.4 * t_rev * it as f64 / (n_t - 1) as f64;
3    // ψ(t) = V e^{-iEt} c0
4    let mut pe = 0.0;
5    for n in 0..=NCUT {
6        // amplitude on |e,n⟩ = row 2n+1
7        let mut re = 0.0;
8        let mut im = 0.0;
9        for k in 0..dim {
10            let phase = -vals[k] * t;
11            let a = vecs[(2 * n + 1, k)] * c0[k];
12            re += a * phase.cos();
13            im += a * phase.sin();
14        }
15        pe += re * re + im * im;
16    }
17    // path B: exact per-block solution
18    let mut pe_ex = 0.0;
19    for n in 0..=NCUT {
20        let r = g * ((n + 1) as f64).sqrt() * t;
21        pe_ex += weights[n] * r.cos().powi(2);
22    }
23    two_path_worst = two_path_worst.max((pe - pe_ex).abs());
24    pe_t.push(t);
25    pe_numeric.push(pe);
26    pe_analytic.push(pe_ex);
27}

The same discipline applies to (17.3). Computing $g^{(2)}(0)$ from the number distribution means evaluating ${\sum }_{n}n(n-1)p_n$ — but that formula is a consequence of the ladder amplitudes, so checking it against 2/3 and 1 leaves the amplitudes themselves untested. The lab therefore computes the two pure states' correlations a second way, by building $\hat{a}$ as a matrix and applying it twice:

1// A SECOND, disjoint route to g² for the two pure states: build a as a
2// literal matrix, a|n⟩ = √n |n−1⟩, and apply it. Then
3//     ⟨a†a⟩ = ‖aψ‖²,   ⟨a†a†aa⟩ = ‖a a ψ‖²,
4// which is (17.3) read left to right, with no n(n−1) formula anywhere. The
5// moment sum above and this one can only agree if the ladder amplitudes are
6// right — the √n / √(n+1) slip breaks it, and it is the same slip that would
7// wreck the Jaynes–Cummings coupling.
8let a_op = M::from_fn(NCUT + 1, NCUT + 1, |i, j| {
9    if j == i + 1 { (j as f64).sqrt() } else { 0.0 }
10});
11let g2_ladder = |p: &[f64]| -> f64 {
12    let psi = DVector::from_iterator(NCUT + 1, p.iter().map(|w| w.sqrt()));
13    let one = &a_op * psi; // a|ψ⟩
14    let two = &a_op * &one; // a a|ψ⟩
15    two.norm_squared() / (one.norm_squared() * one.norm_squared())
16};
17let g2_ladder_worst = (g2_ladder(&fock_p) - g2_fock)
18    .abs()
19    .max((g2_ladder(&coh_p) - g2_coh).abs());

And the headline: the lifetime of 2p, computed in SI units from the golden rule, the vacuum's n+1, and a Simpson integral that must land on the closed-form dipole moment first:

1// radial integral I = ∫ R10 r R21 r² dr, R10 = 2e^{-r}, R21 = r e^{-r/2}/(2√6)
2let simpson = |f: &dyn Fn(f64) -> f64, a: f64, b: f64, n: usize| -> f64 {
3    let h = (b - a) / n as f64;
4    let mut s = f(a) + f(b);
5    for i in 1..n {
6        s += f(a + i as f64 * h) * if i % 2 == 1 { 4.0 } else { 2.0 };
7    }
8    s * h / 3.0
9};
10let integrand = |r: f64| 2.0 * (-r).exp() * r * (r * (-r / 2.0).exp() / (2.0 * 6.0f64.sqrt())) * r * r;
11let i_num = simpson(&integrand, 0.0, 60.0, 40_000);
12let i_exact = 768.0 / (243.0 * 6.0f64.sqrt()); // 4!·(2/3)^5/√6 = 1.29027 a₀
13let dipole_err = (i_num - i_exact).abs();
14// …
15// SI: A = ω³ e² (I²/3) a₀² / (3π ε₀ ħ c³)
16let a0 = 5.291_772_109_03e-11;
17let e = 1.602_176_634e-19;
18let eps0 = 8.854_187_812_8e-12;
19let hbar = 1.054_571_817e-34;
20let c = 299_792_458.0f64;
21let ry_ev = 13.605_693_122_994;
22let de_j = 0.75 * ry_ev * e; // 2p → 1s photon energy
23let w = de_j / hbar;
24let d2 = e * e * (i_num * i_num / 3.0) * a0 * a0;
25let a_rate = w.powi(3) * d2 / (3.0 * PI * eps0 * hbar * c.powi(3));
26let tau_ns = 1e9 / a_rate;
27let tau_measured = 1.596;
28let tau_dev = (tau_ns - tau_measured).abs() / tau_measured;

Twelve referees, each printed with the tolerance it is gated at, and the run is not allowed to claim success until all twelve pass. Every tolerance below sits a small factor above the error the run actually achieves — which is the only setting that makes a tolerance mean anything. Loosen one by four orders and it stops being a test:

1// Tolerances are set AT the accuracy this run achieves — a few times the
2// measured error, never orders above it. A tolerance with four spare orders
3// in it is not a referee, it is a decoration.
4let ceil = |name, value, tol| RefereeRow { name, value, tol, pass: value < tol };
5let floor = |name, value, tol| RefereeRow { name, value, tol, pass: value > tol };
6let referees = vec![
7    ceil("spectrum_resonant", referee.spectrum_resonant, 2e-14),
8    ceil("spectrum_detuned", referee.spectrum_detuned, 2e-14),
9    ceil("vacuum_rabi_2g", referee.vacuum_rabi_err, 2e-14),
10    ceil("evolution_two_paths", referee.evolution_two_paths, 2e-14),
11    ceil("collapse_ceiling", referee.collapse_ceiling, 2e-4),
12    floor("revival_peak (floor)", referee.revival_peak, 0.78),
13    ceil("g2_fock_err", referee.g2_fock_err, 1e-15),
14    ceil("g2_coherent_err", referee.g2_coherent_err, 1e-15),
15    ceil("g2_thermal_err", referee.g2_thermal_err, 1e-11),
16    ceil("g2_ladder_worst", referee.g2_ladder_worst, 1e-14),
17    ceil("dipole_integral_err", referee.dipole_integral_err, 1e-13),
18    ceil("tau_vs_measured", referee.tau_vs_measured, 1e-3),
19];
20// …
21let all_passed = data.referees.iter().all(|r| r.pass);
22assert!(all_passed, "at least one ch17 referee FAILED — see the table above");
23println!("ALL REFEREES PASSED.");

1. (F) Carry the quantization through honestly for one cavity mode: from the classical energy (ε₀/2)∫(E² + c²B²)dV, define the mode amplitude q and its conjugate p, check they satisfy the oscillator equations, and recover (17.1) with the quoted normalization √(ħω/2ε₀V).
2. (F) Put numbers on the hum: for a cavity of volume 1 μm³ and optical ω (600 nm), evaluate the vacuum field √⟨E²⟩ in volts per meter. Compare with the field 1 mW of laser light focused to 1 μm² carries. (The vacuum is not weak where the boxes are small — that is why cavity QED works.)
3. (F) Derive all three g² values in (17.3) from the number distributions (sharp, Poisson, geometric), and prove the classical bound: for any probability distribution of intensity, ⟨I²⟩ ≥ ⟨I⟩² forces g² ≥ 1.
4. (C) In the collapse–revival explorer, measure t_rev at n̄ = 4, 9, 16, 25, 36 and verify the √n̄ law. Then confirm the collapse time does not move — and explain both from the spread of Rabi frequencies g(√(n+1) − √n) ≈ g/2√n̄ across the Poisson width Δn = √n̄.
5. (F) Assemble (17.4): golden rule with the free-space mode density ρ(ω) = ω²V/π²c³ (derive it: count k-states in a shell, two polarizations), the dipole matrix element averaged over emission directions, and the vacuum matrix element ⟨1|a†|0⟩ = 1. Then evaluate ⟨1s|r|2p⟩ by hand and confirm 768/(243√6) a₀.
6. (F, hard) Weisskopf–Wigner: go beyond the golden rule by making the ansatz that the amplitude of |e, 0⟩ decays as e^(−Γt/2) and solving the coupled amplitude equations for the continuum. Show the decay is exponential with the same Γ, that the emitted photon's spectrum is a Lorentzian of width Γ (the natural linewidth), and connect Γ·τ = 1 to the energy–time uncertainty relation.
7. (F, hard) Welton's Lamb-shift estimate. Vacuum fluctuations jiggle the electron by δr, smearing the Coulomb potential: ΔV ≈ (1/6)⟨δr²⟩∇²V, and ∇²(1/r) ∝ δ³(r) — only s-states feel it. Estimate ⟨δr²⟩ by driving a free electron with the vacuum field mode by mode (cut off at the electron Compton frequency and at the atomic binding frequency), and show the 2s shift lands within an order of magnitude of the measured 1057 MHz. The vacuum, measured with a radar set in 1947.
8. (P, hard) Go cat-hunting in the lab. At half the revival time, theory says the field is a superposition of two coherent states |αe^(iφ)⟩ and |αe^(−iφ)⟩ — a Schrödinger cat. Extend ch17-jaynes to save the field's reduced state at t = t_rev/2 (trace out the atom from ψ), compute its Husimi Q(α) on a grid, and referee two signatures: two well-separated blobs in Q, and an atom–field entanglement entropy near ln 2. Then compute the same at t = t_rev and watch the cat reassemble into one blob — the revival, seen from the field's side.