Tunable-cavity QED with phase qubits

We describe a tunable-cavity quantum electrodynamics (QED) architecture with an rf SQUID phase qubit inductively coupled to a single-mode, resonant cavity with a tunable frequency that allows for both microwave readout of tunneling and dispersive measurements of the qubit. Dispersive measurement is well characterized by a three-level model, strongly dependent on qubit anharmonicity, qubit-cavity coupling, and detuning. A tunable-cavity frequency provides a way to strongly vary both the qubit-cavity detuning and coupling strength, which can reduce Purcell losses, cavity-induced dephasing of the qubit, and residual bus coupling for a system with multiple qubits. With our qubit-cavity system, we show that dynamic control over the cavity frequency enables one to avoid Purcell losses during coherent qubit evolutions and optimize state readout during qubit measurements. The maximum qubit decay time $T_1=1.5\mu$s is found to be limited by surface dielectric losses from a design geometry similar to planar transmon qubits.

Figure 1

(a) Circuit schematic showing the cavity (red) and qubit (blue) subcircuits, the $Z_o=50\Omega$ microwave feedline (green), their shared inductance $M$ (purple), and their respective gradiometric bias lines $L_{cB}$ (red) and $L_{qB}$ (blue). (b) Fabrication layout for design $B$ and SEM image of angle-evaporated Josephson junction. (c) Table highlighting differences between designs $A$ and $B$. Values common to both designs are $L_s=1.79$ nH, ${\mathcal{L}}_c=L_c+M=0.75$ nH, ${\mathcal{L}}_q=L_q+M=2.5$ nH, $L_x=0.27$ nH, $I_{oc}=I_{oq}\approx 0.32\mu$A, and $M=61$ pH. Both the phase qubit and the cavity were based on rf SQUID designs with ratios ${\mathcal{L}}_c/L_{Jc}=0.7$ $\left({\mathcal{L}}_q/L_{Jq}=2.8\right)$ for the cavity (qubit) (see text).

Figure 2

Coupling rate $2g/2\pi$ (design $A)$ as a function of cavity frequency ${\omega }_c/2\pi$. The solid red (blue) line is the prediction from Eq. (3) (including $L_x$ and $C_J$'s). The dotted (dashed) line is the prediction for capacitive coupling with $C=15$ fF $(C=5$ fF). The solid circles were measured spectroscopically (see text). At the lowest cavity frequency, the solid $☆$ results from a fit to the Purcell data, discussed later in Sec. 3c2. The gray region highlights where the phase qubit (design $A)$ remains stable enough for operation (see text).

Figure 3

(a) Cavity spectroscopy (design $A)$ while sweeping the cavity flux bias with the qubit far detuned, biased at its maximum frequency. The solid line is a fit to the model including the junction capacitance. (b) Zoom-in near the maximum cavity frequency showing a slot mode. (c) Line-cut on-resonance along the dashed line in (b) with a fit to a skewed Lorentzian (solid line).

Figure 4

(a) The cavity response (design $A)$ as the qubit flux bias is swept. Two different data sets (with the qubit reset at ${\varphi }_q=\pm 2)$ have been overlaid to show the double-valued or hysteretic regions. The straight arrows show the direction of operation and the curved arrows indicate tunneling transitions. (b) Cavity transmission for each qubit flux state near ${\varphi }_q=-0.5$, dashed line in (a). (c) Tunneling probability $P_T$ for the $|g\rangle$ state and $|e\rangle$ state versus measure pulse (MP) amplitude. The maximum raw measurement contrast is obtained by maximizing the difference between the $P_T$ curves, $|e\rangle -|g\rangle$.

Figure 5

(a) Qubit spectroscopy (design $A)$. The dashed line is a fit to the model described in the main text. Potential well configurations at various flux biases are sketched with a dot denoting the metastable minimum used for qubit operation. (b) The inset shows that at the deepest well with a minimum anharmonicity $\left({\alpha }_r=-0.2\%\right)$, the qubit is still sufficiently coherent for the $f_{01}$ transition to be spectroscopically distinguishable from the $f_{12}$ (and $f_{02}/2)$ transition, allowing for qubit operations. Tracking these spectroscopic peaks over the entire spectrum provides us with a measure of the qubit's anharmonicity, as shown later in Fig. 9.

Figure 6

(a) Time domain measurements (design $A)$. Rabi oscillations for frequencies near $f_{01}=7.38$ GHz. (b) Line-cut on-resonance along the dashed line in (a). The fit (solid line) yields a Rabi oscillation decay time of $T^{\prime }=409$ ns. (c) Ramsey oscillations versus qubit flux detuning near $f_{01}=7.38$ GHz. (d) Line-cut along the dashed line in (c). The fit (solid line) yields a Ramsey decay time of $T_2^{*}=106$ ns. With $T_1=600$ ns, this implies [56] a phase coherence time $T_2=310$ ns.

Figure 7

(a) Qubit spectroscopy (design $A)$ overlaid with cavity spectroscopy at two frequencies, $f_c=6.58$ and 6.78 GHz. (b) Zoom-in of the split cavity spectrum in (a) when $f_c=6.78$ GHz with corresponding fit lines. (c) Zoom-in of the split cavity spectrum in (a) when $f_c=6.58$ GHz with corresponding fit lines. (d) Cavity spectroscopy (design $B)$ while sweeping the qubit flux with $f_c=7.07$ GHz showing a large normal-mode splitting when the qubit is resonant with the cavity. All solid lines represent the uncoupled qubit and cavity frequencies and the dashed lines show the new coupled normal-mode frequencies. Notice in (d) the additional weak splitting from a slot mode just below the cavity, and in (c) and (d), qubit tunneling events are visible as abrupt changes in the cavity spectrum.

Figure 8

(a) Pulse sequence. (b) Rabi oscillations (design $A)$ for various pulse durations obtained using dispersive measurement at $f_{01}=7.18$ GHz, with ${\Delta }_{01}=+10g$. (c) A single, averaged time trace along the vertical dashed line in (b). (d) Rabi oscillations extracted from the final population at the end of the drive pulse, along the dashed diagonal line in (b). (e) Zoom-in of dashed box in (b) showing Rabi oscillations observed during continuous driving.

Figure 9

(a) Relative qubit anharmonicity ${\alpha }_r$ versus qubit frequency ${\omega }_{01}/2\pi$ (design $A)$. The solid red line is a polynomial fit to the experimental data, used to calculate the three-level model curves in (b)–(d), while the blue line is a theoretical prediction of the relative anharmonicity (including $L_x$, but neglecting $C_J)$ using perturbation theory and no adjustable parameters. (b) Full dispersive shift $2\chi$ versus relative detuning ${\Delta }_{01}/{\omega }_{01}$ for four different cavity frequencies, $f_c=6.78$, 6.68, 6.58, and 6.48 GHz. Symbols represent the data with lines showing the three-level model predictions. The bold dashed line shows the two-level model prediction when $f_c=6.58$ GHz. (c) Full dispersive shift $2\chi$ versus the relative anharmonicity ${\alpha }_r$. (d) Full dispersive shift $2\chi$ versus both the relative anharmonicity ${\alpha }_r$ and relative detuning ${\Delta }_{01}/{\omega }_{01}$.

Figure 10

(a) The Purcell effect for design $A$. (b) The Purcell effect for design $B$. The solid (open) symbols represent data taken dynamically (statically) with tunneling (dispersive) measurement. The horizontal dashed line shows the predicted decay time $T_1$ due solely to coupling of the flux bias line to a 50 $\Omega$ environment. The other dashed lines show $T_1$ from Purcell loss through the single-mode tunable cavity at each frequency. The dotted lines are the predicted $T_1$ due to dielectric loss with $Q_d=82400$. The dash-dot lines represent the limiting $T_1$ due to coupling to the flux bias line and dielectric loss. The solid lines represent the predicted $T_1$ by combining all the decay rates influencing the qubit: the flux bias line, the dielectric loss, and the Purcell effect (also see Table 1).