Transformed dissipation in superconducting quantum circuits

Superconducting quantum circuits must be designed carefully to avoid dissipation from coupling to external control circuitry. We describe a reformulation of dissipation theory based on a current-to-current transformer model. We test this theory with an experimentally determined impedance transformation of $\sim {10}^5$ and find quantitative agreement better than a factor of 2 between this transformation and the reduced lifetime of a phase qubit coupled to a tunable transformer. Higher-order corrections from quantum fluctuations are also calculated with this theory, but found not to limit the qubit lifetime. We also illustrate how this simple connection between current and impedance transformation can be used to rule out dissipation sources in experimental qubit systems.

Figure 1

Impedance transformation by a two-port dissipationless coupler. (a) Qubit is connected through a dissipationless coupler to environment, described by admittance $Y_1\left(\omega \right)$ of the external circuitry. (b) The transfer function of the coupler relates current source $I_1$ across the input port to current $I_2$ across the shorted output. (c) The effective dissipation seen by the qubit at port 2 is transformed by the squared derivative of the transfer function.

Figure 2

Micrograph (a) and schematic (b) of the phase qubit and SQUID. The overlap of the qubit and SQUID loops increases their mutual inductance $M_{sq}$, while their gradiometric layout reduces their sensitivity to external flux. The flux bias coil couples to the qubit to tune its frequency, but has negligible mutual inductance with the SQUID. The shunt resistor $R_s$ reduces quasiparticle generation in the SQUID when it switches (Ref. 15). The qubit sees $R_s$ transformed by coupling through the SQUID. For the tested device, qubit capacitance and critical current are $1\mathrm{pF}$ and $2\mathrm{\mu }\mathrm{A}$. In addition, we have $I_0=2\mathrm{\mu }\mathrm{A}$, $\alpha =1.5$, $L_s=300\mathrm{pH}$, $L_q=720\mathrm{pH}$, $M_{sq}=70\mathrm{pH}$, $M_{fq}=2\mathrm{pH}$, $R_s=30\Omega$ and $C_s=1\mathrm{pF}$.

Figure 3

Theoretical transfer function of the three-junction SQUID. The plot shows the induced qubit current $I_q$ versus SQUID bias current $I_{sq}$, for four values of the junction size ratio $\alpha$, with device parameters as for Fig. 2. At points where $d{I}_q∕d{I}_{sq}=0$, the qubit will be insensitive to noise and dissipation from the SQUID. The design value of $\alpha =1.7$ ensures the existence of an insensitive point that is not too close to the critical current of the SQUID. The physical origin of decoupling is easily understood for $\alpha =2$. At the bias $I_{sq}\approx 0$ the Josephson inductances of the upper and lower branches are equal, producing a symmetric flow of current and no net flux to the qubit.

Figure 4

Measured SQUID transfer function and its effect on qubit lifetime. (a) Qubit current $I_q$ versus SQUID bias $I_{sq}$, measured as described in the text. Outside the range shown, the SQUID switches prematurely, preventing reliable qubit operation. (b) Measured qubit lifetime $T_1$ (dots) along with theoretical curves. The dashed line is the prediction from the transfer function alone, while the solid line adds to this a constant dissipation corresponding to a lifetime of $450\mathrm{ns}$. Best fit is for $R_s=11\Omega$.