Circuit Quantum Electrodynamics (Circuit QED)

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Description

Circuit quantum electrodynamics (circuit QED) is the solid-state realization of cavity QED, where a superconducting qubit plays the role of the atom and a coplanar waveguide (CPW) transmission line resonator plays the role of the optical cavity. Proposed by Blais, Huang, Wallraff, Girvin, and Schoelkopf (2004), the key insight is that the zero-point energy of a quasi-1D resonator is concentrated in an extremely small effective volume ($\approx 10^{-5}$ cubic wavelengths), producing rms vacuum fields ${\mathcal{E}}_{\text{rms}}\sim 0.2\text{V/m}$ — about 100× larger than 3D microwave cavities. Combined with the enormous transition dipole moment of the Cooper pair box ($d\sim 2\times 10^4$ atomic units), this yields coupling strengths three orders of magnitude beyond atomic microwave cQED.

The system reaches the strong coupling regime where the vacuum Rabi frequency $g/2\pi \sim 100\text{MHz}$ vastly exceeds both the cavity decay rate $\kappa$ and the qubit decay rate $\gamma$. In the dispersive regime ($|\Delta |=|{\omega }_q-{\omega }_r|\gg g$), the cavity frequency shifts by $\pm \chi =\pm g^2/\Delta$ depending on the qubit state, enabling quantum non-demolition (QND) readout. The resonator simultaneously provides Purcell protection — suppressing spontaneous emission by a factor $(g/\Delta {)}^2$ — and acts as a quantum bus for entangling qubits at different antinodes via virtual photon exchange.

Circuit QED is the foundational measurement and coupling architecture underlying virtually all modern superconducting quantum processors.

Hamiltonian

At the charge degeneracy point, the system is described by the Jaynes-Cummings Hamiltonian:

$H=\hslash {\omega }_r\left(a^{†}a+\frac{1}{2}\right)+\frac{\hslash {\omega }_q}{2}{\sigma }_z+\hslash g(a^{†}{\sigma }_{-}+a{\sigma }_{+})$

where ${\omega }_r$ is the resonator frequency, ${\omega }_q$ is the qubit frequency, $g$ is the vacuum Rabi coupling, and $a^{(†)}$ are the resonator photon operators.

The coupling strength is:

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

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

In the dispersive regime ($|\Delta |\gg g$), the effective Hamiltonian becomes:

$H_{\text{disp}}\approx \hslash ({\omega }_r+\chi {\sigma }_z)a^{†}a+\frac{\hslash ({\omega }_q+\chi )}{2}{\sigma }_z$

where $\chi =g^2/\Delta$ is the dispersive shift. The qubit-state-dependent cavity frequency shift enables QND measurement of the qubit.

Motivation

Previous proposals for coupling superconducting qubits used discrete LC circuits or large Josephson junctions, which suffered from parasitic resonances and 1/f noise sensitivity. The transmission line resonator approach provides: strong inhibition of spontaneous emission (Purcell protection), high-fidelity QND readout via dispersive phase shifts, a natural quantum bus for entangling qubits separated by centimeter distances, and compatibility with standard lithographic fabrication. Circuit QED made dispersive readout and multi-qubit coupling practical, enabling the scaling of superconducting quantum processors.

Experimental Status

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

• Achieved strong coupling between a Cooper pair box and a CPW resonator with $g/2\pi \sim 100\text{MHz}$.
• Observed vacuum Rabi splitting in transmission spectroscopy.

Dispersive readout established:

• QND measurement of qubit state via cavity phase shift became the standard readout method for all transmon-based processors.

Quantum bus demonstrations:

• Two qubits at different antinodes entangled via virtual photon exchange with effective coupling $g_1{g}_2/\Delta$.

3D circuit QED:

• Extension to 3D microwave cavities achieved resonator lifetimes $>1\text{ms}$, enabling long-lived bosonic codes.

Key Metrics

Metric	Value	Notes	Fidelity reference
Vacuum Rabi coupling $g/2\pi$	50–400 MHz	Depending on qubit type and geometry	—
Resonator $T_1$	1–10 μs (2D), >1 ms (3D)	Planar vs 3D cavity	—
Dispersive shift $\chi /2\pi$	0.5–5 MHz	$\chi =g^2/\Delta$	—
Readout speed	200–500 ns	Dispersive measurement	—
Purcell limit $T_1^P$	>1 ms	With Purcell filter	—
Resonator footprint	~5 mm (λ/2)	Coplanar waveguide	—
Operating temperature	10–20 mK	Dilution refrigerator	—
Strong coupling ratio $g/\kappa$	10–1000	$\kappa$ = resonator linewidth	—

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 (2004)

Linked Papers

• blais-2004-circuit-qed

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
• microwave-photonic-qubit — itinerant microwave photons for inter-module state transfer