Back-action of a driven nonlinear resonator on a superconducting qubit

We study the backaction of a driven nonlinear resonator on a multilevel superconducting qubit. Using unitary transformations on the multilevel Jaynes-Cummings Hamiltonian and quantum optics master equation, we derive an analytical model that goes beyond linear response theory. Within the limits of validity of the model, we obtain quantitative agreement with experimental and numerical data, both in the bifurcation and in the parametric amplification regimes of the nonlinear resonator. We show in particular that the measurement-induced dephasing rate of the qubit can be rather small at high drive power. This is in contrast to measurement with a linear resonator where this rate increases with the drive power. Finally, we show that, for typical parameters of circuit quantum electrodynamics, correctly describing measurement-induced dephasing requires a model going beyond linear response theory, such as the one presented here.

Figure 1

Representation of one possible implementation of the system considered in this paper. This represents a stripline resonator (blue) made nonlinear with an embedded Josephson junction (dark green), capacitively coupled to a transmon qubit between the central conductor and the ground planes. The model described in this paper, however, applies to various other nonlinear resonators and qubits (see text).

Figure 2

Phase space representation of the pointer states ${\alpha }_0$ (full blue circle) and ${\alpha }_1$ (red dashed circle) given by Eq. (3.3). Parameters are $\chi /2\pi =4$MHz, $\kappa /2\pi =10$ MHz, ${\omega }_p={\omega }_r$, and ${\epsilon }_p/2\pi =30$ MHz.

Figure 3

(a) Amplitude of the resonator internal field (arbitrary units) in response to a drive of reduced frequency $\Omega =2({\omega }_r-{\omega }_d)/\kappa$ (${\Omega }_C=\sqrt{3}$) for increasing drive amplitudes ${\epsilon }_d$. The horizontal lines indicate regions for which linear response theory would (full green lines, check marks) and would not (dashed red lines, X marks) be valid for modeling a qubit-resonator system. (b) Stability diagram of the resonator (see text for explanations). As discussed in Sec. 6a, the vertical dashed lines represent the two operating points studied in this paper.

Figure 4

Experimental (cf., Ref. [16]) and analytical qubit excited-state $\left|1\right.〉$ population for the two operating points indicated in the legend of Fig. 3. The qubit is a transmon with bare parameters $({\omega }_{1,0},{\omega }_{2,1},\gamma ,{\gamma }_{\varphi })/2\pi =(5720,5421.6,0.22,0.25)$ MHz. The resonator's bare parameters are $({\omega }_r,K,K^{\prime },\kappa ,{\kappa }_{\mathrm{NL}})/2\pi =(6453.5,-0.625,-0.00125,9.6,0)$ MHz, and the qubit-resonator couplings are $(g_0,g_1)/2\pi =(42.4,58.4)$ MHz. These parameters were chosen to fit those of Ref. [16]. Couplings to higher transitions as well as higher transition frequencies can be computed from the transmon Hamiltonian [6]. The experimental attenuation required to link the experimental power in dB to the theoretical parameter ${\epsilon }_p$ was calibrated in Ref. [16]. Top: experimental spectroscopy results from Ref. [16]. Bottom: analytical stationary solution Eq. (6.1). Left (Right): pump frequency ${\omega }_p/2\pi =6430\left(6450\right)$ MHz, corresponding to $\Omega /{\Omega }_C=3.1\left(0.7\right)$. These points are identified by vertical lines on Fig. 3. The amplitude ${\epsilon }_s/2\pi =3$ MHz was chosen to fit the experimental power broadening of the qubit lines.

Figure 5

Lamb and ac-Stark shifted qubit frequency as a function of the drive strength ${\epsilon }_p$ for the two operating points indicated in the legend of Fig. 3 and used in Ref. [16]. Parameters are the same as those described in the legend of Fig 4. Points are experimentally (black circles) and numerically (orange squares) extracted qubit transition frequency ${\omega }_{1,0}$. Lines are analytically computed ${\omega }_1^{\prime \prime \prime }-{\omega }_0^{\prime \prime \prime }$ with the complete Eq. (5.26a) (full black lines), when setting ${\mathbb{K}}_i^p=0$ (dotted red lines) or when taking ${\omega }_p={\omega }_r$ in ${\mathbb{S}}_i^p$ and ${\mathbb{K}}_i^p$ (dashed green lines).

Figure 6

(a) and (b): Qubit line's half-width at half-maximum as a function of the drive strength ${\epsilon }_p$ for the two operating points indicated in the legend of Fig. 3 and used in Ref. [16]. Points are experimentally (gray circles) and numerically (orange squares) extracted qubit linewidths. Lines are analytical solutions corresponding to ${\gamma }_2$, where the fields ${\alpha }_i$ are computed according to the nonlinear response Eq. (5.19) (full black lines), to linear response Eq. (6.3) (dashed green lines) and by replacing $\kappa |{\alpha }_1-{\alpha }_0{|}^2/2$ with the linear resonator result Eq. (6.4) (dotted red lines). (c) and (d): Phase space representation of the fields ${\alpha }_0$ (black lines, circles) and ${\alpha }_1$ (red lines, squares) as given by the linear (dashed lines, empty symbols) and nonlinear (full lines, full symbols) response theories. All four symbols of a given set (1 or 2) correspond to a given pump amplitude ${\epsilon }_p$.

Figure 7

Maximum dispersive coupling $S_{\max }$ for which the linear response theory is valid with 10$\%$ uncertainty at the region of maximum gain as a function of the reduced detuning $\Omega /{\Omega }_C$ in the parametric amplification regime.