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

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 the mode volume does all 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.

Source: blais-2004-circuit-qed Related: circuit-qed, high-quality-superconducting-cavities-coupled-to-nonlinear