Purcell effect with microwave drive: Suppression of qubit relaxation rate

We analyze the Purcell relaxation rate of a superconducting qubit coupled to a resonator, which is coupled to a transmission line and pumped by an external microwave drive. Considering the typical regime of the qubit measurement, we focus on the case when the qubit frequency is significantly detuned from the resonator frequency. Surprisingly, the Purcell rate decreases when the strength of the microwave drive is increased. This suppression becomes significant in the nonlinear regime. In the presence of the microwave drive, the loss of photons to the transmission line also causes excitation of the qubit; however, the excitation rate is typically much smaller than the relaxation rate. Our analysis also applies to a more general case of a two-level quantum system coupled to a cavity.

Figure 1

(a) Schematic of the considered system with (b) the energy-level diagram. The qubit is off-resonantly ($\Delta ={\omega }_q-{\omega }_r$) coupled to the resonator, with coupling constant $g$. The resonator decays with the rate $\kappa$, which causes the energy relaxation (Purcell relaxation) of the qubit. In a circuit QED qubit measurement setup, a microwave (rf) drive with resonant frequency ${\omega }_{\mathbf{r}}$ and normalized amplitude $\varepsilon$ is applied to the resonator. We show that the qubit relaxation rate decreases with increasing strength of this drive, and that there exists a relatively weak qubit excitation.

Figure 2

The Jaynes-Cummings ladder in the bare-state basis (black solid lines) and the eigenstate basis (red dashed lines). Here transitions (“jumps”) from the right set to the left set of eigenstates correspond to qubit relaxation with rate ${\Gamma }_R$, while transitions from the left to the right set of eigenstates give rise to qubit excitation with rate ${\gamma }_E$.

Figure 3

The Purcell relaxation rate ${\Gamma }_R$ normalized by the no-drive value ${\Gamma }_P$, as a function of the mean photon number $\overline{n}$ induced by the drive for several values of normalized detuning $\Delta /g$: 15, 10, and 5. The dashed red lines show ${\Gamma }_R\left(\overline{n}\right)$ calculated using Eq. (29), the blue solid lines show ${\Gamma }_R$ averaged over the coherent-state distribution using Eq. (30). For the upper curve we also show the truncated expansion [up to ${\lambda }^8$, Eq. (33)] by the green dot-dashed line. The blue dots on the lines indicate $n_{\mathrm{crit}}$.

Figure 4

The Purcell relaxation rate ${\Gamma }_R$ [Eq. (30)] normalized by the no-drive value ${\Gamma }_P$ [Eq. (17)] vs $\overline{n}/n_{\mathrm{crit}}$ for several values of $\Delta /g$: 5, 10, 15, and 20 (blue solid lines). The curves for $\Delta /g\ge 10$ are practically indistinguishable from each other. The dashed red lines show the approximations (34) and (36) for $\overline{n}\ll n_{\mathrm{crit}}$ and $\overline{n}\gg n_{\mathrm{crit}}$, respectively. The dot-dashed black line (almost indistinguishable from the lowest blue line) shows the approximation (35). In the approximations we assume large $\Delta /\mathit{g}$, so that ${\Gamma }_P\approx {\Gamma }_d$.

Figure 5

The qubit excitation rate ${\gamma }_E$ [Eq. (39)] normalized by the no-drive relaxation rate ${\Gamma }_P$ vs $\overline{n}/n_{\mathrm{crit}}$ for several values of $\Delta /\mathit{g}$: 5, 10, 15, and 20 (blue solid lines). The curves for $\Delta /\mathit{g}\ge 15$ are practically indistinguishable from each other. The red dashed lines show the approximations (41) and (42) for $\overline{n}\ll n_{\mathrm{crit}}$ and $\overline{n}\gg n_{\mathrm{crit}}$, respectively. The dot-dashed black line (barely distinguishable) shows the approximation (43). In the approximations we assume large $\Delta /\mathit{g}$.

Figure 6

Four neighboring levels of the Jaynes-Cummings ladder of Fig. 2, redrawn in a slightly different way. The shown energy is relative to the energy of the state $|g,n\rangle$. Assuming $n\approx \overline{n}\gg 1$, the level splitting in both pairs of eigenstates is approximately $|{\Omega }_S|=\sqrt{{\Delta }^2+4{\mathit{g}}^2\overline{n}}$. The level splitting increases with $\overline{n}$. There are four possible transitions marked by down arrows; two of them have frequencies almost coinciding with ${\omega }_r$ (neglecting the dispersive shift), and two of them are at frequencies ${\omega }_r\pm {\Omega }_S$. These transitions can be thought of as the Mollow triplet [30] at nonzero detuning.

Figure 7

Normalized Purcell rate ${\Gamma }_R/{\Gamma }_P$ vs the mean photon number for $\kappa =\mathit{g}=2\pi \times 50$ MHz and several values of detuning: $\Delta /g=5$, 10, 15, and 20. The red dots show the results obtained numerically, blue solid lines are calculated using Eq. (30). The large blue dots indicate the critical photon number $n_{\mathrm{crit}}$.

Figure 8

Numerical results for the normalized Purcell rate ${\Gamma }_R/{\Gamma }_P$ as a function of $\overline{n}$ for $\kappa /\mathit{g}=0.1$ (blue diamonds) and $\kappa /\mathit{g}=1$ (red dots); the detuning parameter is $\Delta /g=5$, 10, 15, and 20 (as in Fig. 7).

Figure 9

The qubit excitation rate ${\gamma }_E$ (normalized by the no-drive relaxation rate ${\Gamma }_P$) as a function of $\overline{n}$ for $\kappa =\mathit{g}$ (we used $\mathit{g}/2\pi =50$ MHz) and several values of detuning, $\Delta /g=5$, 10, 15, and 20. The red dots show the numerical results, the blue solid lines are obtained analytically using Eq. (39). The large blue dots indicate $n_{\mathrm{crit}}$.