Convergence of the multimode quantum Rabi model of circuit quantum electrodynamics

Circuit quantum electrodynamics (QED) studies the interaction of artificial atoms, open transmission lines, and electromagnetic resonators fabricated from superconducting electronics. While the theory of an artificial atom coupled to one mode of a resonator is well studied, considering multiple modes leads to divergences which are not well understood. Here, we introduce a first-principles model of a multimode resonator coupled to a Josephson junction atom. Studying the model in the absence of any cutoff, in which the coupling rate to mode number $n$ scales as $\sqrt{n}$ for $n$ up to $\infty$, we find that quantities such as the Lamb shift do not diverge due to a natural rescaling of the bare atomic parameters that arises directly from the circuit analysis. Introducing a cutoff in the coupling from a nonzero capacitance of the Josephson junction, we provide a physical interpretation of the decoupling of higher modes in the context of circuit analysis. In addition to explaining the convergence of the quantum Rabi model with no cutoff, our work also provides a useful framework for analyzing the ultrastrong coupling regime of a multimode circuit QED.

Figure 1

(a) Schematic representation of cavity QED: A multilevel atom (in blue) coupled to the electromagnetic cavity modes (dashed black lines). (b) Circuit QED example covered in this work: A Josephson junction anharmonic LC oscillator, or “artificial atom,” coupled to modes of a transmission line. (c) Lumped element equivalent of (b).

Figure 2

Renormalization of the Hamiltonian parameters and dressing of the atom by higher modes. For the plots, we choose ${\omega }_0/2\pi =10$ GHz, $Z_0=50\mathrm{\Omega }$, $C_c=50$ fF, and $E_J/h=20$ GHz. Although this places the AA in the transmon limit, we note that the scaling shown in the figure is exact for all regimes, including the Cooper pair box limit. (a) Coupling of the ground to first excited-state transition of the atom $|g\rangle \to |e\rangle$ to the resonator modes as a function of the mode number $m$, $\hslash g_m^{\left(M\right)}=\sqrt{2m+1}{V}_{\text{zpf,0}}2e{\langle g|}^{\left(M\right)}{\widehat{N}}_J{|e\rangle }^{\left(M\right)}$, for different values of $M$. The $M$ dependence of the coupling is detailed in (b). (b), (c) Renormalization of the coupling strength and the atomic frequency through the change in charging energy $E_C^{\left(M\right)}$ as a function of $M$. For large $M$, the coupling diminishes with $1/M$ and the atomic frequency increases linearly with $M$. (d)–(f) Schematic energy diagrams of the renormalization procedure in the case of the atom and fundamental mode at resonance. (a) $M=1$ and (b) $M=2$. Adding a mode shifts the bare atomic energy upwards and changes the values of the couplings. (f) Dressing the atom with the second mode results in a dressed atomic state with atomic frequency and coupling to the $m=1$ mode close to that of the $M=1$ model.

Figure 3

Calculated spectrum as a function of the number of modes $M$ included in the model. (a) Dots (squares) correspond to a diagonalization of the circuit (non-)renormalized extended-Rabi model. The frequency obtained by black-box quantization of the circuit Fig. 1 (dashed line) provides a point of reference corresponding to the case when all modes are included. (b) Zoom of dashed box in (a). Triangles show the prediction based on a first-order classical approximation of the Lamb shift: ${\chi }_M={\tilde{\omega }}_a^{(M+1)}-{\tilde{\omega }}_a^{\left(M\right)}\simeq -2[{\left(g_M^{\left(0\right)}\right)}^2/{\omega }_M][{\left({\omega }_a^{\left(0\right)}\right)}^2/{\omega }_M^2]$ [23].

Figure 4

High-frequency cutoff for $C_J\ne 0$. The capacitive loading at the left boundary of the resonator shown in the inset transforms this point from a voltage antinode to a voltage node for higher modes. The mode $m_c\simeq 35$ marks this transition. The solid line corresponds to the coupling strength as a function of the number of modes. With $C_J\ne 0$ the coupling strength converges to a nonzero value for large $M$; hence the choice $M=3000$. Dashed lines: asymptotic values of the coupling, with $g_0=g_0^{(M=3000)}$ and $C=C_c{C}_J/(C_c+C_J)$.