Cutoff-Free Circuit Quantum Electrodynamics

Any quantum-confined electronic system coupled to the electromagnetic continuum is subject to radiative decay and renormalization of its energy levels. When coupled to a cavity, these quantities can be strongly modified with respect to their values in vacuum. Generally, this modification can be accurately captured by including only the closest resonant mode of the cavity. In the circuit quantum electrodynamics architecture, it is, however, found that the radiative decay rates are strongly influenced by far off-resonant modes. A multimode calculation accounting for the infinite set of cavity modes leads to divergences unless a cutoff is imposed. It has so far not been identified what the source of divergence is. We show here that unless gauge invariance is respected, any attempt at the calculation of circuit QED quantities is bound to diverge. We then present a theoretical approach to the calculation of a finite spontaneous emission rate and the Lamb shift that is free of cutoff.

Figure 1

(a) A transmon qubit coupled to an open superconducting resonator. The black dashed line is a cartoon of the fundamental bare mode of the resonator, while the red solid curve represents the modified resonator mode. (b) The transmission $|T{|}^2$ is shown versus the real frequency for the bare resonator modes (solid black curves). Capacitively coupling the qubit, whose transition frequency ${\omega }_j$ is slightly above the fundamental resonator frequency ${\nu }_1$, gives rise to hybridized modes (dashed red curves). Alternatively, one may study the positions of these resonances in the complex frequency plane, where the bare resonator and qubit poles (black points) are displaced into hybridized resonatorlike and qubitlike resonances (red points). The Purcell decay and the Lamb shift are obtained as the displacement of the qubitlike pole. The bare (hybridized) complex frequencies are the poles (zeros) of the characteristic function $D_j(s)$.

Figure 2

Dependence of (a) coupling strength $g_n$, (b) resonator decay rate ${\kappa }_n$ (See Ref. [12] for derivation), and (c) Purcell decay rate in the dispersive regime $(g_n/{\delta }_n{)}^2{\kappa }_n$ on mode number $n$ for different values of ${\chi }_s=\{0,{10}^{-3},{10}^{-2},{10}^{-1}\}$. Other parameters are set as ${\chi }_R={\chi }_L={10}^{-3}$ and $x_0=0^{+}$.

Figure 3

Comparison of (a), (b) spontaneous decay rate between the linear theory (blue solid) and the dispersive limit result ${\gamma }_P$ (black dashed) as a function of ${\omega }_j$. (c), (d) Lamb shift between the linear theory (blue solid), leading order perturbation (red dotted) and the dispersive limit result ${\mathrm{\Delta }}_L$ (black dashed). (a), (c) ${\chi }_g=0.001$ and (b), (d) ${\chi }_g=0.1$. Both values of ${\chi }_g$ are in strong coupling regime, i.e., $g_1/{\alpha }_j\gg 1$. However, ${\chi }_g=0.1$ ($g_1/{\nu }_1=0.1033$) reaches ultrastrong coupling [35], where multimode effects are non-negligible. The nonlinearity is set as $\epsilon =0.1$, while other parameters are ${\chi }_R={\chi }_L={10}^{-3}$ and ${\chi }_j=0.05$. The vertical dash-dotted black line shows the position of the fundamental frequency of the resonator.