Circuit Quantum Electrodynamics (Circuit QED)

Circuit QED maps cavity quantum electrodynamics onto superconducting circuits: a Cooper pair box qubit couples to the quantized field of a 1D transmission line resonator, reaching the strong-coupling regime with vacuum Rabi rates $g/\pi \sim 100\text{MHz}$.

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 transmission line resonator plays the role of the optical cavity. 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.

At the charge degeneracy point ($N_g=1/2$), the system Hamiltonian reduces to the Jaynes-Cummings form:

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

with vacuum Rabi frequency:

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

where $\beta =C_g/C_{\Sigma }$ is the capacitive coupling ratio.

Figure

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. The critical photon number $m_0={\gamma }^2/2g^2\le 10^{-6}$ and minimum detectable atom number $N_0=2\gamma \kappa /g^2\le 6\times 10^{-5}$ confirm access to the very strong coupling regime.

Key Findings

• Strong coupling achieved with $g/\pi \sim 100\text{MHz}$, vastly exceeding both cavity decay rate $\kappa$ and qubit decay rate $\gamma$.
• In the dispersive regime ($|\Delta |=|{\omega }_r-\Omega |\gg g$), the cavity frequency shifts by $\pm g^2/\Delta$ depending on qubit state, enabling QND readout.
• Purcell effect suppresses spontaneous emission by factor $(g/\Delta {)}^2$ when qubit is detuned from cavity.
• Resonator acts as quantum bus: two qubits at different antinodes can be entangled via virtual photon exchange with effective coupling $g_1{g}_2/\Delta$.
• Readout SNR analysis predicts measurement fidelity limited by amplifier noise temperature, not backaction.

Linked Papers

• blais-2004-circuit-qed

Related Entries

• transmon
• cooper-pair-box-charge-qubit
• circuit-qed
• dispersive-readout-mechanism
• jaynes-cummings-in-circuits
• vacuum-rms-field-scaling
• purcell-protection-via-detuning
• resonator-as-quantum-bus

Physics

A superconducting qubit (e.g., Cooper pair box) coupled to a coplanar waveguide resonator via the vacuum electric field. The system is described by the Jaynes-Cummings Hamiltonian:

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

where $g$ is the vacuum Rabi coupling. The qubit is placed at the voltage antinode of the fundamental $\lambda /2$ mode to maximize $g$. In the dispersive regime ($|\Delta |=|{\omega }_q-{\omega }_r|\gg g$), the qubit state shifts the resonator frequency by $\pm \chi =\pm g^2/\Delta$, enabling dispersive readout.

Related Qubits

• transmon — most common qubit used in circuit QED
• cooper-pair-box-charge-qubit — original qubit in Blais proposal
• fluxonium — also operates in circuit QED architecture

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	—