Photon Blockade

The blockade has migrated from single trapped atoms to solid-state and superconducting platforms, where it does engineering work. Semiconductor quantum dots in micropillar and photonic-crystal cavities behave as artificial atoms with large dipole moments and reach strong coupling on a chip; driven near blockade, the best emit single photons with $g^{(2)}(0)$ below $0.01$ and near-unity indistinguishability — the front-runner for the deterministic single-photon sources photonic quantum computers need in bulk. In circuit QED, a superconducting qubit coupled to a microwave resonator reaches $g/2\pi$ of tens to hundreds of MHz with far smaller loss, making blockade robust and a primitive for microwave single-photon sources and photon detectors on superconducting processors. A different modern route drops the cavity entirely. In Rydberg-mediated schemes, photons stored as slow-light polaritons in an ultracold gas inherit the long-range interaction of Rydberg atoms, so one stored photon blockades a whole micron-scale volume against a second — the Rydberg blockade imported into optics. These experiments have produced strongly interacting, antibunched photons and even photon–photon bound states, extending "one photon blocks the next" from a single mode to a propagating field.

Picture a staircase whose steps are labelled by photon number. For a bare, linear cavity the steps are evenly spaced, so a resonant drive climbs every rung and the field grows into a coherent state. Coupling the cavity to a single atom warps the staircase: because the vacuum Rabi splitting of the $n$-th manifold scales as $\sqrt{n}$, the second step is a different height than the first. A drive tuned to the first step then misses the second, and the cavity refuses to hold two photons. The first figure makes this explicit.

The blockade is dynamical: the first photon must be present to shift the resonance, so the light gates the light. And it survives only in strong coupling — the anharmonic shift is of order $g$, but the rungs are resolvable only if $g$ beats the blur of loss $\kappa$ and decay $\gamma$; in weak coupling the ladder smears into one line and blockade fails. The second figure shows the resolved doublet in the cavity transmission of a strongly coupled atom.

The whole subject is carried by one model: a single two-level emitter exchanging quanta with one cavity mode. In the rotating-wave approximation this is the Jaynes–Cummings Hamiltonian, from which every result follows.

where: $\hat a,\hat a^\dagger$ — cavity photon operators; $\hat\sigma_z,\hat\sigma_\pm$ — atomic inversion and raising/lowering operators; $\omega_c,\omega_a$ — cavity and atomic frequencies; $g$ — single-photon (vacuum Rabi) coupling. The third term swaps a photon for an atomic excitation and is the only nonlinear term. Because $\hat H_{\mathrm{JC}}$ conserves the total excitation number it splits into independent $2\times2$ blocks, one per manifold, each diagonalizing into a doublet of dressed states (polaritons) whose energies are the backbone of the blockade.

where: $n\ge 1$ — total excitation number; $E_{n,\pm}$ — upper/lower dressed states of the $n$-th manifold; $\delta$ — atom–cavity detuning. On resonance ($\delta=0$) the manifold sits at $\hbar\omega_c n \pm \hbar g\sqrt{n}$, so the splitting $2\hbar g\sqrt{n}$ grows as $\sqrt{n}$ — the anharmonicity. The lowest manifold ($n=1$) is what a linear transmission scan probes, and its splitting is the fingerprint of strong coupling.

where: $\pm g$ — displacement of the two lowest polaritons from the bare cavity frequency $\omega_c$. A single empty-cavity resonance becomes a doublet split by $2g$; resolving it (Figure 2) is what strong coupling means. The blockade lives in the mismatch between the first and second rungs of the lower branch: comparing the one- and two-photon transition energies gives the anharmonicity.

where: $A$ — detuning between the one- and two-photon transitions on the lower branch; $E_0=0$ — ground-state energy. Blockade requires $A$ to exceed the linewidths $\hbar\kappa,\hbar\gamma$; then the one-photon drive leaves the two-photon rung dark. Whether blockade is working is read off the emitted light, not the levels — the normalized zero-delay second-order coherence is the standard metric, counting photon pairs relative to a coherent reference.

where: $g^{(2)}(0)$ — normalized zero-delay intensity correlation; $\hat n=\hat a^\dagger\hat a$. Coherent light gives $1$, thermal light $2$ (bunched), blockaded light $<1$ (antibunched), ideally $\to0$. For a weakly driven cavity the steady state is mostly vacuum with small one- and two-photon admixtures, and $g^{(2)}(0)$ reduces to a ratio of amplitudes.

where: $c_n$ — amplitude of the $n$-photon component, with $|c_0|\approx1\gg|c_1|\gg|c_2|$. Blockade drives $c_2\to0$ while $c_1$ stays finite, so the ratio collapses; a coherent state has $c_2=c_1^2/\sqrt{2}$, reproducing $g^{(2)}(0)=1$. The same statistics collapse into one signed number, the Mandel parameter, certifying nonclassicality from the number variance.

where: $Q$ — Mandel parameter, negative for sub-Poissonian (nonclassical) light; $\langle(\Delta\hat n)^2\rangle$ — photon-number variance. $Q<0$ means a distribution narrower than Poissonian, which no classical field can produce; blockade always gives $Q<0$. Blockade only exists under drive and dissipation, so the honest description is a driven-dissipative master equation: a coherent drive $\hat H_{\rm drive}=\hbar\eta(\hat a^\dagger e^{-i\omega_L t}+\hat a\,e^{+i\omega_L t})$ pumps the cavity while two loss channels remove excitation.

where: $\mathcal{D}[\hat c]\hat\rho = \hat c\hat\rho\hat c^\dagger-\tfrac12\{\hat c^\dagger\hat c,\hat\rho\}$ — Lindblad dissipator; $\kappa$ — cavity decay rate; $\gamma$ — atomic spontaneous-emission rate; $\eta,\omega_L$ — drive amplitude and frequency ($\omega_L=\omega_c-g$ for blockade). Solving for the steady state and evaluating equation (5) is exactly what a numerical study of blockade computes.

Blockade needs an anharmonicity that beats the linewidths; the useful regimes fall out of the equations above.

1 · A real strong-coupling cavity: does it blockade? (confidence builder)

A single caesium atom in a high-finesse cavity has $(g,\kappa,\gamma)/2\pi=(34,\,4.1,\,2.6)$ MHz. Find the vacuum Rabi splitting, the blockade anharmonicity, and the cooperativity, and decide whether blockade appears.

Step 1. The vacuum Rabi doublet (equation 3) is split by $2g$, so $2g/2\pi = 2\times34 = \mathbf{68\ MHz}$ — the peak-to-peak separation in Figure 2. It dwarfs $\kappa/2\pi=4.1$ MHz, so the doublet is cleanly resolved: the sign of strong coupling.

Step 2. The blockade offset is the anharmonicity $A=\hbar g(2-\sqrt2)$ (equation 4), i.e. $A/h=0.5858\times34=\mathbf{19.9\ MHz}$. This is how far the drive at $\omega_c-g$ misses the two-photon transition, and it exceeds both linewidths ($\kappa/2\pi=4.1$, $\gamma/2\pi=2.6$ MHz).

Step 3. The single-atom cooperativity is $C=g^2/(2\kappa\gamma)=34^2/(2\times4.1\times2.6)=\mathbf{54}$. Since $C\gg1$ and $g$ beats $\kappa,\gamma$ individually, the system is deep in strong coupling.

Step 4. The critical photon number $n_0=\gamma^2/(2g^2)=2.6^2/(2\times34^2)=\mathbf{2.9\times10^{-3}}$ is far below one, so a single photon already saturates the nonlinearity. A 68 MHz doublet, a 19.9 MHz anharmonicity beating few-MHz linewidths, and $C\approx54$ with $n_0\ll1$: this cavity blockades, and driving the lower polariton gives antibunched output $g^{(2)}(0)\ll1$.

2 · From suppressed pairs to $g^{(2)}(0)$ and Mandel $Q$

A weakly driven blockade cavity has mean photon number $\langle n\rangle=0.01$. Without the nonlinearity it would be coherent ($g^{(2)}(0)=1$); the blockade suppresses the two-photon amplitude to a fraction $f=0.1$ of its coherent value. Find $g^{(2)}(0)$ and the Mandel $Q$.

Step 1. For weak drive $\langle n\rangle\approx|c_1|^2=0.01$. A coherent state has $c_2=c_1^2/\sqrt2$, giving the reference $g^{(2)}(0)=2|c_2|^2/|c_1|^4=1$ — the baseline we are beating.

Step 2. Blockade rescales $c_2\to f\,c_2^{\rm coh}$, and since $g^{(2)}(0)\propto|c_2|^2$ (equation 6) it scales as $f^2$: $g^{(2)}(0)=f^2\times1=0.1^2=\mathbf{0.01}$. Suppressing the amplitude tenfold cuts the pair probability, and $g^{(2)}(0)$, a hundredfold.

Step 3. The Mandel parameter (equation 7) is $Q=\langle n\rangle(g^{(2)}(0)-1)=0.01\times(0.01-1)=\mathbf{-9.9\times10^{-3}}$. The negative sign certifies sub-Poissonian light; the small magnitude reflects the tiny mean number, since $Q\ge-\langle n\rangle$.

Step 4. A hundredfold suppression of pairs is a strong turnstile; the same drive on a linear cavity gives $Q=0$. $g^{(2)}(0)=0.01$ and $Q=-9.9\times10^{-3}$ — deep antibunching, with $Q<0$ the nonclassicality certificate.

Photon blockade is where the abstract Jaynes–Cummings ladder became laboratory reality — and no one did more to force that transition than the physicist who first resolved the vacuum Rabi splitting for a single atom, then watched one atom turn a laser into single photons.

Texts. Loudon, The Quantum Theory of Light, 3rd ed., Ch. 6–7 (Jaynes–Cummings, resonance fluorescence, correlations) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., Ch. 13–14 (photon statistics, cavity QED) · Foot, Atomic Physics, Ch. 7–9 (two-level atom, dressed states) · Birnbaum et al., Nature 436, 87 (2005) — photon blockade with one atom in a cavity · Kimble, Physica Scripta T76, 127 (1998) — strong coupling and vacuum Rabi splitting.

Provenance. The dressed-state energies and vacuum Rabi doublet are standard Jaynes–Cummings results (Loudon Ch. 6–7; Foot Ch. 7). A verifier should recompute the $2\times2$ block diagonalization, confirm $g(2-\sqrt2)=0.5858\,g$ and $\sqrt2-1=0.4142\,g$, and check $g^{(2)}(0)=2|c_2|^2/|c_1|^4$ and $Q=\langle n\rangle(g^{(2)}(0)-1)$ (Saleh & Teich Ch. 13). The parameters $(g,\kappa,\gamma)/2\pi=(34,4.1,2.6)$ MHz in Worked Example 1 are the single-atom values of Birnbaum et al., Nature 436, 87 (2005); from them $2g/2\pi=68$ MHz, anharmonicity $19.9$ MHz, $C\approx54$, and $n_0\approx2.9\times10^{-3}$ follow directly. The interference condition matches the Liew–Savona / Bamba treatments; dissipators follow the standard Lindblad form.

Prerequisites. Fock States and the Photon Number Basis · Coherent States · Photon Statistics and the Mandel Q Parameter · Coherence Functions and the Degree of Coherence · Quantization of the Electromagnetic Field. Leads to. Photon Antibunching · Single-Photon Sources · Hanbury Brown–Twiss Experiment · Squeezed States of Light.

Ask the tutor: Ask me to walk through how the $\sqrt{n}$ splitting of the dressed states turns into the $0.586\,g$ offset that blocks the second photon, or to compare conventional and unconventional blockade for a cavity you specify.