Nonlinear resonant behavior of a dispersive readout circuit for a superconducting flux qubit

A nonlinear resonant circuit comprising a SQUID magnetometer and a shunting capacitor is studied as a readout scheme for a persistent-current qubit. The flux state of the qubit is detected as a change in the Josephson inductance of the SQUID magnetometer, which in turn mediates a shift in the resonant frequency of the readout circuit. The nonlinearity and resulting hysteresis in the resonant behavior are characterized as a function of the power of both the input drive and the associated resonance-peak response. Numerical simulations based on a nonlinear circuit model shows that the observed nonlinearity is dominated by the effect due to an ac flux rather than current bias through the Josephson inductor.

Figure 1

Experimental setup: (a) Circuit schematic of the resonant readout circuit. The designed component values were $L_1=69\mathrm{nH}$, $L_2=0.78\mathrm{nH}$, $C_1=1.4\mathrm{pF}$, and $C_2=100\mathrm{pF}$. The SQUID inductance $L_J$ was approximated to be $0.1\mathrm{nH}$ for the circuit design. $R_s$ and $R_L$ were $50\Omega$ impedances. (b) Optical micrograph of the actual device. (c) Electronic setup at different temperature stages of the ${}^3\mathrm{He}$ cryostat.

Figure 2

(Color online) (a) Modulation of the resonant frequency with external dc flux bias. Qubit steps are observed at $0.18\mathrm{mA}$ and $-0.495\mathrm{mA}$. (b) Modulation of the resonant frequency for various input power. The amount of modulation is reduced in the nonlinear regime $(f_{b,NL}-f_{a,NL})$ compared to the linear regime $(f_{b,L}-f_{a,L})$. The circular markers represent the inflection points where $d^2{f}_o∕d{\Phi }^2=0$.

Figure 3

(Color) Evolution of the magnitude and phase spectra of the readout circuit from the linear to the nonlinear regime with increasing input power. Data are shown for flux biases at ${\Phi }_{\mathit{dc}}=0$, 0.3 ${\Phi }_o$, and 0.5 ${\Phi }_o$. The nonlinear spectrum evolves from having a lower resonant frequency at $\Phi =0$ to having a higher resonant frequency at $\Phi =0.5{\Phi }_o$. The phase spectrum at $-54\mathrm{dBm}$ for $\Phi =0.3{\Phi }_o$ was arbitrarily shifted for display purpose. A self-resonant dip was observed near the resonant frequency of the spectrum, e.g., at $419.5\mathrm{MHz}$ for ${\Phi }_{\mathit{dc}}=0.5{\Phi }_o$, and accounted for the general tilt in the shape of the spectrum. The self-resonance is believed to be due to parasitic capacitive coupling between the input and output ports.

Figure 4

(Color online) Illustration of the resonant spectra in the nonlinear regime [(a) and (b)] and the linear regime [(c) and (d)] for $\Phi =0.5{\Phi }_o$ and $\Phi =0$, respectively. The shaded region of the nonlinear spectrum marks the region over which multiple solutions occur. The solid line traces the actual spectrum observed experimentally with a forward frequency sweep, and the circular marker corresponds to the peak frequency that was being measured. The bending of the nonlinear spectra to opposite sides accounts for the reduced separation of resonant frequencies $(f_{b,NL}-f_{a,NL})$ compared to the linear case $(f_{b,L}-f_{a,L})$, as was observed in Fig. 2b.

Figure 5

(Color online) Bending $\delta f$ as a function of input power from $-74\mathrm{dBm}\text{to}-54\mathrm{dBm}$ in $1\mathrm{dB}$ steps. Measurements were made with a forward frequency sweep. The data are shown for various dc flux biases between 0 and $0.5{\Phi }_o$. The sign of $\delta f$ indicates the polarity of the asymmetric spectrum. The red line corresponds to a linear fit in the low-power regime.

Figure 6

(Color online) Top plots: Hysteretic resonant spectrum for flux biases at ${\Phi }_{\mathit{dc}}=0$, $0.3{\Phi }_o$, and $0.5{\Phi }_o$. The extent of the bistable region is given by $\mid f_t-f_b\mid$. Middle plots: $\delta f_t$ and $\delta f_b$ as a function of input power from $-66\mathrm{dBm}\text{to}-54\mathrm{dBm}$ in $1\mathrm{dB}$ steps. The onset of hysteretic regime occurs at $P_H=-61\mathrm{dBm}$ for ${\Phi }_{\mathit{dc}}=0.5{\Phi }_o$ and $P_H=-59\mathrm{dBm}$ for ${\Phi }_{\mathit{dc}}=0$. Bottom plots: $\delta f_t$ and $\delta f_b$ replotted as a function of resonance-peak power. The dotted line is a linear fit for low power.

Figure 7

(Color online) (a) Circuit schematic of the simulated nonlinear LRC circuit model. The circuit is driven by a current source $I\sin {\omega }_{s}t$. The readout SQUID is modeled by a flux-dependent nonlinear inductor given by Eq. (9). The simulated circuit parameters are $R=500\Omega$, $C=100\mathrm{pF}$, and $I_{\mathit{co}}=2.3\mu A$, which corresponds to $L_{Jo}=0.1\mathrm{nH}$. The $Q$ of the simulated circuit is about 500. (b) A plot of $L_J\left({\Phi }_{\mathit{ext}}\right)$ given by Eq. (9) for different values of $\beta$. The circular markers represent the inflection points where $d^2{L}_J∕d{\Phi }^2$ is zero. The illustration shows that depending on the dc flux bias, the ac modulation of $L_J$ due to ${\Phi }_{\mathit{ac}}$ can result in a lower $({\Phi }_{\mathit{dc}}=0.5{\Phi }_o)$ or higher $({\Phi }_{\mathit{dc}}=0)$ effective inductance.

Figure 8

(Color online) Simulated magnitude and phase spectra of $V_L$ for increasing drive amplitude $I$. The results qualitatively reproduce the experimentally observed behavior in Fig. 3. $V_L$ is plotted in reduced units of ${\Phi }_o∕\sqrt{L_{Jo}C}$ and $I$ in units of $2I_{\mathit{co}}$. The frequency axes are normalized with respect to the resonant frequency of the linear spectrum: $f_o=1.88\mathrm{GHz}$ $({\Phi }_{\mathit{dc}}=0)$, $1.59\mathrm{GHz}$ $\left(0.3{\Phi }_o\right)$, and $1.33\mathrm{GHz}$ $\left(0.5{\Phi }_o\right)$. The phase spectra at the highest drives $I=0.1$ and 0.2 for ${\Phi }_{\mathit{dc}}=0.3{\Phi }_o$ are arbitrarily shifted for display purpose.

Figure 9

(Color online) Top row: simulated hysteretic behavior at various flux bias for drive amplitude $I=0.01$. Middle row: $\delta f_b$ and $\delta f_t$ as a function of the square of the drive amplitude, which is proportional to the input power. The simulations were performed for $I$ between 0.001 and 0.05. Bottom row: $\delta f_b$ and $\delta f_t$ as a function of the square of the voltage response $V_L$, which is proportional to the resonance-peak power. $V_L$ is in reduced units of ${\Phi }_o∕\sqrt{L_{Jo}C}$ and $I$ is in units of $2I_{\mathit{co}}$.