Circuit Quantum Electrodynamics (Circuit QED)

Figure

Description

Circuit quantum electrodynamics (circuit QED) is the on-chip analogue of cavity QED: a superconducting artificial atom couples coherently to a quantized mode of a microwave resonator. In the original Blais et al. proposal, a quasi-one-dimensional coplanar-waveguide (CPW) resonator confined its zero-point energy to an effective volume of about $10^{-5}$ cubic wavelengths, producing ${\mathcal{E}}_{\mathrm{rms}}\sim 0.2V/m$. Together with the Cooper-pair-box dipole moment ($d\sim 2\times 10^{4}e{a}_0$), this led to the proposal-scale estimate $g/\pi \sim 100\mathrm{MHz}$—not $g/2\pi \sim 100\mathrm{MHz}$.

The defining regime is strong coupling, $g\gg \kappa ,\gamma$, where coherent qubit–photon exchange outpaces cavity loss $\kappa$ and qubit decoherence $\gamma$. Wallraff et al. first demonstrated it in 2004 with $g/2\pi \approx 5.8\mathrm{MHz}$, $\kappa /2\pi =0.8\mathrm{MHz}$, and $\gamma /2\pi =0.7\mathrm{MHz}$, observing a vacuum-Rabi splitting of $2g/2\pi \approx 11.6\mathrm{MHz}$.

Most processor applications operate instead in the dispersive regime, $|\Delta |=|{\omega }_q-{\omega }_r|\gg g$. The qubit shifts the resonator frequency by a state-dependent amount, enabling QND readout, while virtual photons mediate interactions between separated qubits. Detuning suppresses cavity-mediated relaxation to the Purcell rate ${\Gamma }_P\simeq \kappa (g/\Delta {)}^2$; engineered Purcell filters can suppress that channel further. Circuit QED now underlies superconducting-qubit readout, resonator buses, bosonic encodings, microwave quantum optics, and hybrid quantum interfaces.

Hamiltonian

The two-level quantum-Rabi model is

$H_{\mathrm{Rabi}}=\hslash {\omega }_r{a}^{†}a+\frac{\hslash {\omega }_q}{2}{\sigma }_z+\hslash g(a+a^{†}){\sigma }_x.$

Near resonance and for $g\ll {\omega }_r,{\omega }_q$, the rotating-wave approximation gives the Jaynes–Cummings Hamiltonian

$H_{\mathrm{JC}}=\hslash {\omega }_r{a}^{†}a+\frac{\hslash {\omega }_q}{2}{\sigma }_z+\hslash g(a^{†}{\sigma }_{-}+a{\sigma }_{+}).$

Here ${\omega }_r$ is the resonator frequency, ${\omega }_q$ is the qubit transition frequency, $g$ is the single-photon coupling rate, and $a^{(†)}$ are photon annihilation (creation) operators. At exact resonance, the $n\ge 1$ excitation manifold contains

$|n,\pm ⟩=\frac{|e,n-1⟩\pm |g,n⟩}{\sqrt{2}},$

split by $2g\sqrt{n}$. This ordering is the central vacuum-Rabi signature.

For the original Cooper-pair-box geometry at its charge sweet spot,

$g=\frac{\beta e}{\hslash }\sqrt{\frac{\hslash {\omega }_r}{cL}}$

where $\beta =C_g/C_{\Sigma }$, $c$ is capacitance per unit length, and $L$ is resonator length.

With $\Delta \equiv {\omega }_q-{\omega }_r$ and $|\Delta |\gg g$, a two-level dispersive approximation is

$H_{\mathrm{disp}}\approx \hslash ({\omega }_r+\chi {\sigma }_z)a^{†}a+\frac{\hslash ({\omega }_q+\chi )}{2}{\sigma }_z,\chi =\frac{g^2}{\Delta }.$

For a weakly anharmonic transmon, higher levels modify the shift to approximately $\chi \simeq g^2\alpha /[\Delta (\Delta +\alpha )]$, where $\alpha ={\omega }_{12}-{\omega }_{01}<0$. The measured resonator separation is $2|\chi |$ in the ideal two-state readout picture.

Motivation

Compared with earlier lumped-element proposals, lithographic transmission-line resonators offered reproducible modes, large zero-point voltages, spatially separated antinodes, and controlled input/output coupling. Those ingredients made strong coupling, dispersive measurement, and cavity-mediated state transfer practical. Modern implementations add dedicated Purcell filters because a readout resonator is both a measurement channel and, without filtering, a radiative decay channel.

Experimental Status

First strong coupling — Wallraff et al. (2004):

• Coupled a Cooper pair box to a 6.04-GHz CPW resonator with $g/2\pi \approx 5.8\mathrm{MHz}$.
• Observed a vacuum-Rabi splitting of about 11.6 MHz, larger than both $\kappa /2\pi =0.8\mathrm{MHz}$ and $\gamma /2\pi =0.7\mathrm{MHz}$.

Cavity quantum bus — Majer et al. (2007):

• Demonstrated coherent state transfer between two spatially separated superconducting qubits through virtual photons in a shared transmission-line cavity.

Long-lived cavity modes and bosonic encodings — Teoh et al. (2023):

• Used two superconducting cavities with photon lifetimes above 1 ms as the modes of a dual-rail qubit, with transmon-assisted control and erasure detection.

Fast multiplexed dispersive readout — Spring et al. (2025):

• Combined readout and filter resonators with intrinsic notch filtering, obtaining 99.77% average assignment fidelity across four qubits in a 56-ns integration window; the best channel exceeded 99.9%.

Millimeter-wave circuit QED — Dixit et al. (2026):

• Coupled a 3.1-GHz transmon to a 34.7-GHz cavity at $g/2\pi =1.3\mathrm{GHz}$ and achieved greater than 99% readout fidelity without a quantum-limited first-stage amplifier, demonstrating a highly detuned extension of the architecture.

Key Metrics

Metric	Value	Notes	Fidelity reference
First strong-coupling demonstration	$g/2\pi \approx 5.8$ MHz; $2g/2\pi \approx 11.6$ MHz	6.04-GHz CPW resonator; $\kappa /2\pi =0.8$ MHz, $\gamma /2\pi =0.7$ MHz	Wallraff et al. 2004
Long-lived cavity photon	$T_1>1$ ms	Two 3D stub cavities used for dual-rail encoding	Teoh et al. 2023
Multiplexed assignment fidelity	99.77% average; best $>99.9\%$	Four qubits, 56-ns integration, resonator linewidths up to 42 MHz	Spring et al. 2025
Millimeter-wave readout	$>99\%$ fidelity	34.7-GHz cavity, 3.1-GHz transmon, $g/2\pi =1.3$ GHz	Dixit et al. 2026

References

Original proposal

• A. Blais et al., “Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation,” Phys. Rev. A 69, 062320 (2004) — arXiv:cond-mat/0402216

First experimental demonstration

• A. Wallraff et al., “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004) — arXiv:cond-mat/0407325

Review and milestones

• A. Blais et al., “Circuit quantum electrodynamics,” Rev. Mod. Phys. 93, 025005 (2021) — arXiv:2005.12667
• J. Majer et al., “Coupling superconducting qubits via a cavity bus,” Nature 449, 443–447 (2007) — arXiv:0709.2135
• J. D. Teoh et al., “Dual-rail encoding with superconducting cavities,” PNAS 120, e2221736120 (2023) — arXiv:2212.12077
• P. A. Spring et al., “Fast Multiplexed Superconducting-Qubit Readout with Intrinsic Purcell Filtering Using a Multiconductor Transmission Line,” PRX Quantum 6, 020345 (2025) — arXiv:2409.04967
• A. V. Dixit et al., “Millimeter wave readout of a superconducting qubit,” Phys. Rev. Lett., accepted 3 September 2026 — arXiv:2603.13837

Linked Papers

• blais-2004-circuit-qed
• blais-2021-circuit-electrodynamics
• dixit-2026-millimeter-wave-readout
• majer-2007-cavity-bus
• spring-2025-fast-multiplexed-readout
• strickland-2024-gatemon-losses
• teoh-2023-dual-rail-cavity
• wallraff-2004-strong-coupling-single
• walter-2017-rapid-readout
• rathnakaran-2026-squeezing-fueled-quantum-otto-engine

Evergreen context

• jaynes-cummings-in-circuits — the atom-cavity Hamiltonian in its superconducting form
• dispersive-readout-mechanism — why off-resonant coupling yields QND measurement
• resonator-as-quantum-bus — how the cavity mediates long-range qubit-qubit coupling
• vacuum-rms-field-scaling — why CPW resonators achieve unusually large zero-point fields
• purcell-protection-via-detuning — how detuning and filters suppress spontaneous emission

Related Entries

• transmon — most common qubit used in circuit QED architectures
• cooper-pair-box-charge-qubit — original qubit in the Blais et al. proposal
• fluxonium — also operates within circuit QED, requires auxiliary readout
• qubit-readout — dispersive readout is the primary application of circuit QED
• bosonic-qubit — long-lived cavity modes host oscillator encodings
• dual-rail-superconducting-qubit — two circuit-QED cavity modes encode an erasure-biased qubit