Superconducting quantum interference device with frequency-dependent damping: Readout of flux qubits

Recent experiments on superconducting flux qubits, consisting of a superconducting loop interrupted by Josephson junctions, have demonstrated quantum coherence between two different quantum states. The state of the qubit is measured with a superconducting quantum interference device (SQUID). Such measurements require the SQUID to have high resolution while exerting minimal backaction on the qubit. By designing shunts across the SQUID junctions appropriately, one can improve the measurement resolution without increasing the backaction significantly. Using a path-integral approach to analyze the Caldeira-Leggett model, we calculate the narrowing of the distribution of the switching events from the zero-voltage state of the SQUID for arbitrary shunt admittances, focusing on shunts consisting of a capacitance $C_s$ and resistance $R_s$ in series. To test this model, we fabricated a dc SQUID in which each junction is shunted with a thin-film interdigitated capacitor in series with a resistor, and measured the switching distribution as a function of temperature and applied magnetic flux. After accounting for the damping due to the SQUID leads, we found good agreement between the measured escape rates and the predictions of our model. We analyze the backaction of a shunted symmetric SQUID on a flux qubit. For the given parameters of our SQUID and realistic parameters for a flux qubit, at the degeneracy point we find a relaxation time of $113\mu \mathrm{s}$, which limits the decoherence time to $226\mu \mathrm{s}$. Based on our analysis of the escape process, we determine that a SQUID with purely capacitive shunts should have narrow switching distributions and no dissipation.

Figure 1

Schematic of flux qubit coupled to readout SQUID.

Figure 2

Idealized SQUID switching distribution. As $I_b$ is increased, the probability of the SQUID switching increases from 0 to 1 over a characteristic width $\delta I_s^{\left(\Delta p\right)}$.

Figure 3

Transformation from SQUID with shunt admittances $Y$ to equivalent single Josephson junction.

Figure 4

SQUID potential energy. Contour plot showing the critical point ${\mathbf{x}}_{\mathbf{c}}$, the minimum ${\mathbf{x}}_{\mathbf{0}}$ and the saddle point ${\mathbf{x}}_{\mathbf{01}}$. Open circles show the starting point for search taken from analytic approximations. White broken lines show contours where $\partial U∕\partial ({\gamma }_1-{\gamma }_2)=0$ (dotted) and $\partial U∕\partial ({\gamma }_1+{\gamma }_2)=0$ (dashed). SQUID parameters are: ${\Phi }_S=0.05{\Phi }_0$, $I_0=1.96\mu \mathrm{A}$, $I_b=1.49\mu \mathrm{A}$, and $L_S=500\mathrm{pH}$, corresponding to ${\beta }_L=0.95$.

Figure 5

Micrographs of RC-SQUID. (a) Overview showing flux bias line; leads and pads for applying bias current and detecting voltage are labeled at top. The pulse current is split symmetrically between pads “SQUID A” and “SQUID B” so that fluxes generated by this current do not couple to the qubit. (b) Enlarged view showing Josephson junctions, interdigitated capacitors, and AuCu resistors, indicated by the ellipse. (c) Capacitor detail. (d) Junction detail.

Figure 6

(a) $I_s^{50\%}$ vs ${\Phi }_S$ for an RC-shunted SQUID. Thick line shows data, thin lines show calculated modulation curves for ${\beta }_L=1.05$ (top), ${\beta }_L=0.95$ (middle), and ${\beta }_L=0.86$ (bottom). (b) Switching width $\delta I_s^{\left(\Delta p\right)}p∕I_s^{50\%}$ vs ${\Phi }_S$. (c) Resolution ${\rho }^{\left(\Delta p\right)}$ vs ${\Phi }_S$. Measurements made at $24\mathrm{mK}$.

Figure 7

Measured 90% switching width vs temperature for several different ${\Phi }_S$. Solid lines show the predicted distribution widths for RC-shunted SQUID with $R_s=40\Omega$, $C_j=9.3\mathrm{fF}$, $I_0=1.96\mu \mathrm{A}$, $C_s=20\mathrm{fF}$, and a fit value $R_l=52\Omega$. Dashed line shows calculated width for corresponding unshunted SQUID with $R_l=\infty$ at ${\Phi }_S=0.01{\Phi }_0$.

Figure 8

Suppression of $T^{*}$ for RC-shunted SQUID $2\pi k_B{T}^{*}∕\hslash {\omega }_0$ vs ${\omega }_c∕{\omega }_0$. Four curves are shown for different values of $\kappa \equiv C_s∕C_j$.

Figure 9

$J\left(\omega \right)$ for a flux qubit coupled to an RC-shunted SQUID with parameters given in text. Solid line corresponds to case A and dashed line corresponds to case B.

Figure 10

Dephasing (${\tau }_{\varphi }$, black) and relaxation (${\tau }_R$, gray) times for flux qubit coupled to RC-shunted SQUID. Solid lines correspond to case A and dashed lines correspond to case B.