Lifetime renormalization of weakly anharmonic superconducting qubits. I. Role of number nonconserving terms

The dynamics of a weakly anharmonic superconducting qubit in a complex electromagnetic environment is generally well described by an effective multimode Kerr Hamiltonian at sufficiently weak excitation. This Hamiltonian can be embedded in a master equation with losses determined by the details of the electromagnetic environment. Recent experiments indicate, however, that when a superconducting circuit is driven with microwave signals populating the system with sufficiently high excitations, the observed relaxation rates appear to be substantially different from expectations based on the electromagnetic environment of the qubit alone. This issue is a limiting factor in the optimization of superconducting qubit readout schemes. We claim here that an effective master equation with drive-power-dependent parameters is an efficient approach to model such quantum dynamics. In this sequence of papers, we derive effective master equations, whose parameters exhibit nonlinear dependence on the excitation level of the circuit as well as the electromagnetic environment of the qubit. We show that the number nonconserving terms in the qubit nonlinearity generally lead to a renormalization of dissipative parameters of the effective master equation, while the number conserving terms give rise to a renormalization of the system frequencies. Here, in Paper I, we consider the excitation-relaxation dynamics of a transmon qubit that is prepared in a certain initial state, but is not driven otherwise. A unitary transformation technique is introduced to study the renormalization of (i) qubit relaxation due to coupling to a generic bath and (ii) Purcell decay. Analytic expressions are provided for the dependence of the nonlinear dissipative terms on the details of the electromagnetic environment of the qubit. The perturbation technique based on unitary transformations developed here is generalized to the continuously driven case in Paper II [A. Petrescu et al., Phys. Rev. B 101, 134510 (2020)].

Figure 1

(a) A weakly anharmonic qubit ($E_{\text{j}}\gg E_{\text{c}}$) linearly (capacitively) coupled to an open cavity. (b) Separation of linear and anharmonic parts of the Josephson potential. (c) Josephson potential and its perturbative spectrum for a weak anharmonicity characterized by $E_{\text{j}}=50E_{\text{c}}$. The solid black curve shows the normalized Josephson potential, while the dashed black and dashed-dotted blue curves are the resulting quadratic and quartic theories along with their corresponding energy levels.

Figure 2

Comparison between different theories mentioned in Table 2 for $E_{\text{j}}=50E_{\text{c}}(\epsilon =0.2)$ and qubit initial condition $|{\mathrm{\Psi }}_{\text{a}}\left(0\right)\rangle =\frac{1}{\sqrt{4}}{\sum }_{m=0}^3|m\rangle$. This initial condition is adopted to have nonzero matrix elements for both single- and three-photon relaxation processes. (a) Phase space during the first period for ${\kappa }_{\text{a}}={\kappa }_{\text{a3}}=0$, (b) phase space, and (c) qubit occupation number for ${\kappa }_{\text{a}}={\kappa }_{\text{a3}}={\omega }_{\text{a}}/25$. The bath spectral function $S_{YY}\left(\omega \right)$ is assumed to be flat for simplicity.

Figure 3

Hybridization as a function of $g$ for ${\overline{\omega }}_{\text{a}}=0.8{\overline{\omega }}_{\text{c}}$ (${\overline{\omega }}_{\text{a}}<{\overline{\omega }}_{\text{c}}$). (a), (b) Show hybridization coefficients obtained from Eqs. (A32a) and (A32b), (c) show the normal-mode frequencies obtained from Eqs. (A30a) and (A30b), and (d) show the order-$\epsilon$ correction to the normal-mode dissipators according to Eqs. (42a) and (42b). The last value of $g$ is chosen in each case such that the lower normal-mode frequency hits $0^{+}$.

Figure 4

Hybridization as a function of $g$ for ${\overline{\omega }}_c=0.8{\overline{\omega }}_{\text{a}}$ (${\overline{\omega }}_{\text{a}}>{\overline{\omega }}_c$). (a), (b) Show hybridization coefficients obtained from Eqs. (A32a) and (A32b), (c) show the normal-mode frequencies obtained from Eqs. (A30a) and (A30b), and (d) show the order-$\epsilon$ correction to the normal-mode dissipators according to Eqs. (42a) and (42b). The last value of $g$ is chosen in each case such that the lower normal-mode frequency hits $0^{+}$.

Figure 5

Comparison of different theories given in Table 2 for $E_{\text{j}}=50E_{\text{c}}(\epsilon =0.2)$, ${\omega }_{\text{a}}=0.8{\overline{\omega }}_c$, and $g\approx 0.27{\overline{\omega }}_c$, resulting in hybridized quantities ${\omega }_{\text{a}}\approx 0.55{\omega }_{\text{c}}$, ${\omega }_{\text{c}}\approx 1.15{\omega }_{\text{c}}$, $u_{\text{aa}}=u_{\text{ac}}\approx 0.69$, and $v_{\text{cc}}=v_{\text{ca}}\approx 0.76$. We assume a flat-bath spectral function $S_{YY}\left({\omega }_{\text{a}}\right)=S_{YY}\left({\omega }_c\right)$, such that $Q_{\text{a}}\equiv {\omega }_{\text{a}}/\left[v_{\text{ca}}^2{S}_{YY}\left({\omega }_{\text{a}}\right)\right]\approx 20.7$ and $Q_{\text{c}}\equiv {\omega }_c/\left[v_{\text{cc}}^2{S}_{YY}\left({\omega }_c\right)\right]\approx 42.7$. Low-$Q$ values are chosen for simplicity and are irrelevant under the flat-bath assumption. The qubit and cavity initial conditions are set as $|{\mathrm{\Psi }}_{\text{a}/\text{c}}\left(0\right)\rangle =(|0_{\text{a}/\text{c}}\rangle +|1_{\text{a}/\text{c}}\rangle )/\sqrt{2}$. (a) Schematic of the harmonic parameters, (b), (c) qubit and cavity occupation numbers. We note that the parameters chosen here correspond to a sizable hybridization $g\approx 0.27{\overline{\omega }}_{\text{c}}$ (sometimes referred to as ultrastrong coupling [36, 37, 38]).