Confining a resonator to quasi-1D amplifies vacuum fields by 100× over 3D cavities

Geometry layer of circuit QED

The rms vacuum voltage in a transmission line resonator scales as:

$V_{\text{rms}}^0\sim \sqrt{\frac{\hslash {\omega }_r}{cL}}$

where $c$ is capacitance per unit length and $L$ is resonator length. For a coplanar waveguide at ${\omega }_r/2\pi =10\text{GHz}$ with a $10\mu \text{m}$ gap, this gives $V_{\text{rms}}\sim 2\mu \text{V}$ and ${\mathcal{E}}_{\text{rms}}\sim 0.2\text{V/m}$.

The effective mode volume is $\sim 10^{-5}$ cubic wavelengths — five orders of magnitude smaller than free space. This is what makes circuit QED strong coupling “easy” compared to atomic cavity QED: you don’t need a heroic cavity Q or a Rydberg atom, because field confinement does much of the work.

This same principle explains why superconducting quantum circuits achieve coupling-to-loss ratios ($g/\kappa$, $g/\gamma$) that atomic systems struggle to match.

Routing boundary

This note is the geometry and zero-point-field layer of circuit QED. It explains why the resonator presents a large vacuum voltage before a particular qubit is attached.

• Stay here when the live question is how mode volume, capacitance per unit length, resonator length, or impedance set the zero-point voltage available for coupling.
• Switch to jaynes-cummings-in-circuits when the live question is how that voltage couples through a qubit dipole matrix element to produce $g$, dressed states, or vacuum-Rabi splitting.
• Continue to dispersive-readout-mechanism or resonator-as-quantum-bus once the single-qubit coupling has been reduced to a useful off-resonant measurement or two-qubit interaction.

The abstraction boundary is therefore

$\text{resonator geometry}⟶V_{\mathrm{rms}}^0⟶g⟶\text{effective readout or bus interaction}.$

This note owns the first arrow; jaynes-cummings-in-circuits owns the second.

Circuit-QED abstraction ladder

1. Start here for the vacuum field supplied by the resonator geometry.
2. Read jaynes-cummings-in-circuits for the one-qubit one-mode interaction built from that field.
3. Branch to dispersive-readout-mechanism for measurement or resonator-as-quantum-bus for cavity-mediated qubit-qubit exchange.

Source: blais-2004-circuit-qed Related: circuit-qed, jaynes-cummings-in-circuits, dispersive-readout-mechanism, resonator-as-quantum-bus, high-quality-superconducting-cavities-coupled-to-nonlinear