Circuit Quantum Electrodynamics

Circuit QED is not a curiosity; it is the operating system of superconducting quantum computers. Every transmon in Google's and IBM's processors is read out dispersively — a weak microwave tone probes a readout resonator, the qubit-conditioned phase shift is amplified first by a near-quantum-limited Josephson parametric or travelling-wave amplifier, and single-shot fidelities above $99\%$ are achieved in tens to a few hundred nanoseconds. Two-qubit gates ride the same machinery: the cross-resonance gate drives one transmon at a neighbour's frequency through their shared bus, and tunable couplers modulate an effective $J$ to switch entangling interactions on and off. Beyond gate-model computing, long-lived three-dimensional cavities store logical qubits directly in superpositions of photon-number states — the "cat" and GKP bosonic codes pursued at Yale and by AWS — where a single ancilla transmon, coupled dispersively, both writes and error-corrects the code. The field also runs quantum-optics experiments impossible with real atoms: photon-number-resolved spectroscopy that splits a qubit line into a comb of Fock-state peaks, Wigner tomography of engineered cavity states, and genuine ultrastrong coupling where the ground state itself is squeezed and holds virtual photons. The live frontier is scaling without losing coherence: taming measurement-induced state transitions, suppressing readout crosstalk, and engineering Purcell filters and dedicated readout lines so that fast measurement and long qubit lifetime — the two things equation (7) puts in tension — can finally coexist across thousands of qubits.

Picture a swing and a child who can push it with a fixed shove. The swing is the resonator; the child is the qubit. When their natural frequencies match, energy sloshes back and forth — the child pumps the swing, the swing lifts the child, over and over. That is a vacuum Rabi oscillation: one quantum of excitation, emitted as a photon and reabsorbed as a qubit flip, exchanged at rate $g$. Strong coupling simply means the sloshing repeats many times before friction — the cavity leaking photons at rate $\kappa$, or the qubit decaying at rate $\gamma$ — damps it out. In a modern device $g$ can beat $\kappa$ by a factor of a hundred and $\gamma$ by ten thousand, so hundreds of clean exchanges fit inside one coherence time. The first figure fixes the two ingredients and the two regimes that the single detuning $\Delta=\omega_q-\omega_r$ selects between. On resonance the qubit and cavity trade energy directly; far off resonance they cannot trade energy at all, but each still shifts the other's frequency — the child's weight changes the swing's period without any push landing.

The reason cQED reaches couplings that optical cavity QED cannot comes down to two multiplicative advantages. The superconducting "atom" is macroscopic, so its charge matrix element — the dipole that the field grips — is enormous, and the resonator confines the field to a one-dimensional mode with a per-photon vacuum voltage $V_{\rm rms}=\sqrt{\hbar\omega_r/2C_r}$ that is large because $C_r$ is tiny. Multiply a big dipole by a big vacuum field and $g$ vaults over the loss rates. The second figure shows the payoff most directly: the dispersive readout, where the cavity's own resonance frequency reports the qubit's state.

Underneath both figures is the same fact, worth holding onto: the dressed-state ladder. Each rung of the joint qubit–photon spectrum is split into a doublet by the coupling, and the rungs are spaced by $2g\sqrt{n}$ — uneven in $\sqrt{n}$. That uneven spacing is the fingerprint of genuinely quantized light: you can resolve the one-, two-, and three-photon transitions at distinct frequencies, something no classical field could ever show.

The workhorse is the Jaynes–Cummings Hamiltonian, the model of a two-level system exchanging single quanta with one harmonic mode. It is what survives when you write down the capacitive coupling of qubit charge to cavity voltage and keep only the energy-conserving terms.

where: $\omega_q$ — qubit transition frequency between $|g\rangle$ and $|e\rangle$; $\omega_r$ — resonator mode frequency; $a,a^\dagger$ — photon annihilation and creation operators; $\sigma_z,\sigma_\pm$ — qubit Pauli-$z$ and raising/lowering operators with $\sigma_+=|e\rangle\langle g|$; $g$ — vacuum Rabi (single-photon) coupling. Reading it: the first two terms are the free qubit and the free photons; the third swaps one qubit excitation for one photon and conserves the total excitation number $N=a^\dagger a+\sigma_+\sigma_-$. The coupling $g$ is not a free parameter — it is fixed by the circuit layout, the product of the cavity's per-photon voltage and the transmon's charge matrix element funneled through the coupling capacitor.

where: $C_g$ — qubit–resonator coupling capacitance; $C_\Sigma$ — total qubit capacitance; $C_r$ — resonator capacitance; $E_J,E_C$ — Josephson and charging energies of the transmon; $e$ — electron charge. Reading it: the voltage divider $C_g/C_\Sigma$ sets how much of the cavity voltage the qubit sees, $\sqrt{\hbar\omega_r/C_r}$ is the vacuum voltage, and $(E_J/8E_C)^{1/4}$ is the slowly growing transmon dipole; typical values land at $g/2\pi\sim50$–$300$ MHz. Diagonalizing (1) inside each excitation manifold gives the dressed states, whose energies are the levels probed in spectroscopy.

where: $n=0,1,2,\dots$ — photon-number index of the manifold; $\Delta=\omega_q-\omega_r$ — detuning; $E_{\pm,n}$ — upper/lower dressed level. Reading it: the square-root is an avoided crossing; the two bare states repel by an amount set by both the coupling $g$ and the detuning $\Delta$. On resonance ($\Delta=0$) the square root collapses to a clean, quantized splitting.

where: $g$ — single-photon coupling; $n$ — excitation number of the manifold. Reading it: the lowest doublet ($n=1$) splits by $2g$, the vacuum Rabi splitting seen as two peaks in cavity transmission separated by $g/\pi$ in ordinary frequency; higher rungs split by $2g\sqrt{2}$, $2g\sqrt{3}$, the anharmonic signature of Fock states. Detune far past the coupling and the physics inverts: the qubit and cavity stop trading energy and instead shift each other's frequency. A Schrieffer–Wolff transformation turns (1) into the dispersive Hamiltonian.

where: $\chi$ — dispersive (state-dependent) frequency shift; the term $\chi\sigma_z\,a^\dagger a$ — the qubit-conditioned cavity pull used for readout; $\tfrac12\chi\sigma_z$ — the Lamb shift of the qubit. Reading it: the cavity resonance moves to $\omega_r\pm\chi$ according to the qubit state, and because $[\sigma_z,H_{\rm disp}]=0$, measuring the cavity does not flip the qubit — the measurement is quantum-nondemolition. For a real transmon the third level $|2\rangle$ participates, and the two-level $\chi$ must be corrected — sometimes drastically, even in sign.

where: $\alpha=\omega_{12}-\omega_{01}\approx-E_C/\hbar<0$ — transmon anharmonicity; $\Delta$ — detuning of the $0\!\to\!1$ transition from the cavity. Reading it: the ideal formula $g^2/\Delta$ is recovered only when $|\alpha|\to\infty$; for a finite negative $\alpha$ the true shift is smaller. Two more numbers bound the design. The cavity gives the qubit an unwanted decay channel (the Purcell effect), and the dispersive picture holds only below a critical photon number.

where: $\kappa$ — cavity energy decay rate; $\gamma_P$ — cavity-induced qubit relaxation; $\bar n_{\rm crit}$ — intracavity photon number at which the dispersive approximation fails. Reading it: pushing $\Delta$ larger suppresses Purcell decay but also weakens $\chi$ and raises $\bar n_{\rm crit}$, so readout speed, readout power, and qubit lifetime are all traded against the same detuning.

The behaviour is set by where the detuning $\Delta=\omega_q-\omega_r$ sits relative to $g$, and by the hierarchy of $g$, $\kappa$, and $\gamma$.

1 · Designing a dispersive readout (fully worked)

A transmon has $g/2\pi=100$ MHz, qubit frequency $\omega_q/2\pi=5.0$ GHz, resonator $\omega_r/2\pi=7.0$ GHz, anharmonicity $\alpha/2\pi=-300$ MHz, and cavity linewidth $\kappa/2\pi=1$ MHz. Find the detuning, the two-level and multilevel dispersive shifts, the critical photon number, and the Purcell-limited lifetime.

Step 1. Detuning: $\Delta/2\pi=\omega_q/2\pi-\omega_r/2\pi=5.0-7.0=\mathbf{-2000\ MHz}$. The qubit is below the cavity, so $\Delta<0$ and the shift $\chi$ will be negative.

Step 2. Two-level shift: $\chi/2\pi=(g/2\pi)^2/(\Delta/2\pi)=(100)^2/(-2000)=\mathbf{-5\ MHz}$. Naïvely this predicts the two cavity peaks separated by $2|\chi|/2\pi=10$ MHz, matching the figure above.

Step 3. Multilevel correction: $\chi=g^2\alpha/[\Delta(\Delta+\alpha)]$. With $\Delta+\alpha=-2000+(-300)=-2300$ MHz, $\chi/2\pi=(100)^2(-300)/[(-2000)(-2300)]=-3.0\times10^{6}/4.6\times10^{6}=\mathbf{-0.65\ MHz}$. The true pull is nearly eight times smaller than the two-level guess — using $g^2/\Delta$ here would badly overestimate the readout contrast.

Step 4. Critical photon number: $\bar n_{\rm crit}=\Delta^2/4g^2=(2000)^2/[4(100)^2]=4.0\times10^6/4.0\times10^4=\mathbf{100}$. Readout must stay well below $\sim100$ intracavity photons to remain dispersive and QND.

Step 5. Purcell decay: $\gamma_P=\kappa g^2/\Delta^2$, so $\gamma_P/2\pi=(1)(100/2000)^2=1\times0.0025=2.5$ kHz, giving $T_1^{\rm Purcell}=1/\gamma_P=1/(2\pi\times2500\ {\rm s^{-1}})=\mathbf{64\ \mu s}$. This is the ceiling the cavity alone places on the qubit lifetime at this detuning; a Purcell filter is needed to push $T_1$ into the hundreds of microseconds. Δ = −2 GHz, χ_2level = −5 MHz, χ_transmon = −0.65 MHz, n_crit = 100, T₁ᴾᵘʳᶜᵉˡˡ ≈ 64 μs.

2 · Strong coupling and vacuum Rabi

The same device sits on resonance with $g/2\pi=100$ MHz, cavity linewidth $\kappa/2\pi=1$ MHz, and qubit linewidth $\gamma/2\pi=0.01$ MHz (a $16\ \mu$s $T_1$). Is it strongly coupled, and what does its spectrum look like?

Step 1. Strong-coupling test: $g/\kappa=100/1=\mathbf{100}$ and $g/\gamma=100/0.01=\mathbf{10^4}$, so $g\gg\kappa,\gamma$ by wide margins — the system is deep in strong coupling and a single photon is exchanged thousands of times before it is lost.

Step 2. Cooperativity: $C=g^2/(\kappa\gamma)=(100)^2/[(1)(0.01)]=10^4/10^{-2}=\mathbf{10^6}$. A cooperativity of a million means the coherent coupling utterly dominates the combined dissipation.

Step 3. Vacuum Rabi splitting: the lowest doublet is split by $2g/2\pi=\mathbf{200\ MHz}$, seen as two transmission peaks $200$ MHz apart. The next rung splits by $2g\sqrt2/2\pi=\mathbf{283\ MHz}$, and the $\sqrt2$ (rather than $2\times$) is the direct spectroscopic proof that the field is quantized. g/κ = 100, C = 10⁶, vacuum Rabi splitting 2g/2π = 200 MHz, second rung 2g√2/2π = 283 MHz.

Circuit QED is a young field with a clear origin: the moment someone realized that a superconducting qubit and an on-chip resonator could reenact, with room to spare, the strong-coupling physics that atomic physicists had chased for decades.

Texts. Blais, Grimsmo, Girvin and Wallraff, Circuit quantum electrodynamics, Rev. Mod. Phys. 93, 025005 (2021) — the definitive review · Koch et al., Charge-insensitive qubit design derived from the Cooper pair box, Phys. Rev. A 76, 042319 (2007) — the transmon · Wallraff et al., Strong coupling of a single photon to a superconducting qubit, Nature 431, 162 (2004) — first on-chip strong coupling · Nielsen and Chuang, Quantum Computation and Quantum Information, §7 · Preskill, Lecture Notes on Quantum Computation, Ch. 8 · Haroche and Raimond, Exploring the Quantum: Atoms, Cavities, and Photons. The key relations used throughout this article are: the dressed energies $E_{\pm,n}=\hbar\omega_r(n+\tfrac12)\pm\tfrac{\hbar}{2}\sqrt{4g^2(n+1)+\Delta^2}$ and vacuum Rabi splitting $2g\sqrt n$; the dispersive shift $\chi=g^2/\Delta$ (two-level, by Schrieffer–Wolff) and the transmon multilevel form $\chi=g^2\alpha/[\Delta(\Delta+\alpha)]$; the Purcell rate $\kappa g^2/\Delta^2$ and critical photon number $\bar n_{\rm crit}=\Delta^2/4g^2$; and the cavity-mediated exchange $J=\tfrac12 g_1g_2(1/\Delta_1+1/\Delta_2)$. Worked-example numbers are self-consistent within the quoted device parameters; the ranges ($g/2\pi\sim$ tens–hundreds of MHz, $\omega_r/2\pi\sim$ few GHz, single-shot fidelity $>99\%$) are representative of current devices, not tied to any single experiment.

Prerequisites. Superconducting Qubits · The Transmon Qubit · The Josephson Junction · DiVincenzo Criteria. Leads to. Dispersive Readout · Trapped-Ion Qubits · Sideband Cooling.

Ask the tutor: Ask me to walk through the Schrieffer–Wolff derivation of $\chi$ step by step, or to size up a readout — detuning, photon budget, and Purcell lifetime — for a transmon you have in mind.