Universal Stabilization of a Parametrically Coupled Qubit

We autonomously stabilize arbitrary states of a qubit through parametric modulation of the coupling between a fixed frequency qubit and resonator. The coupling modulation is achieved with a tunable coupling design, in which the qubit and the resonator are connected in parallel to a superconducting quantum interference device. This allows for quasistatic tuning of the qubit-cavity coupling strength from 12 MHz to more than 300 MHz. Additionally, the coupling can be dynamically modulated, allowing for single-photon exchange in 6 ns. Qubit coherence times exceeding $20\mu \mathrm{s}$ are maintained over the majority of the range of tuning, limited primarily by the Purcell effect. The parametric stabilization technique realized using the tunable coupler involves engineering the qubit bath through a combination of photon nonconserving sideband interactions realized by flux modulation, and direct qubit Rabi driving. We demonstrate that the qubit can be stabilized to arbitrary states on the Bloch sphere with a worst-case fidelity exceeding 80%.

Figure 1

(a) Optical image and (b) circuit diagram of our device. The lumped-element resonator is formed by a “$C$” shaped capacitor pad and an isolated meander line inductor. The inductor line protrudes to the common node where both the qubit Josephson junction and the coupler SQUID loop are connected. Two voltage ports are placed at the two sides of the resonator’s capacitor pad enabling transmission measurements. The qubit-cavity coupling strength is tuned with the SQUID-loop flux by modulating the current that flows through the flux line. The qubit can be probed via a separate qubit drive line that is weakly coupled to the qubit’s shunting capacitor. Insets show the details of the qubit Josephson junction and dc SQUID loop.

Figure 2

(a) Spectroscopy showing the qubit excited state population as a function of flux through the coupler. The qubit frequency is insensitive over nearly the entire flux range. (Inset) Number splitting of the qubit peak due to photons in the resonator, used to calibrate the static coupling between the qubit and the resonator. (b) Qubit coherence and qubit-cavity coupling strength as a function of the flux through the coupler. The dephasing time ($T_2^{*}$) is comparable to the energy relaxation time ($T_1$) over the entire tuning range. The coherence times drop near $\mathrm{\Phi }=0.5{\mathrm{\Phi }}_0$ as a result of the Purcell effect due to the strong coupling to the readout resonator, as indicated by the black dashed line.

Figure 3

Red-sideband interactions probed by applying a rf flux tone to the tunable coupler to generate sidebands. (a) Spectroscopy of the (normalized) resonator transmission as a function of sideband and resonator probe frequency, showing the stimulated vacuum Rabi spitting. (b) Stimulated vacuum Rabi oscillations between the qubit and resonator, measured as an oscillation of the qubit excited state population. A single photon is loaded into the qubit before the flux pulse.

Figure 4

Resonator spectroscopy showing (normalized) transmission near the blue-sideband resonance condition. Experimental data (a) and master equation simulations (b) show excellent agreement. (c) The energy level diagram corresponding to Eq. (3) provides a map to the spectroscopic features $A$, $B$, $C$, and $D$ at different modulation frequencies, indicated by arrows in (a). $A$: When the modulation frequency is far detuned from the blue-sideband resonance, the qubit stays in its ground state. $B$: The excited state of the qubit is stabilized, causing the cavity to be shifted down by $2\chi$. $C$: The crossing of $|e1⟩$ and $|g0⟩$, manifest as an avoided crossing. The qubit excited state is also maximally stabilized at this frequency due to the resonance of $|e1⟩$ and $|g0⟩$. $D$: Enhanced cavity transmission appears when $|e0⟩\to |g0⟩$ and $|g0⟩\to |g1⟩$ transition energies are equal. The asymmetry of the unshifted cavity peak line centered at the blue-sideband resonance is likely due to interactions between higher levels $|g,n⟩\to |e,n+1⟩$.

Figure 5

Illustration of the universal stabilization scheme for single-qubit states. In the lab frame (a), qubit Rabi drive and blue-sideband modulation are applied with appropriately chosen detuning and strength. In the rotating frame (b), these two drives result in the dressing of the qubit state into arbitrary superpositions $|\tilde{g}⟩$, $|\tilde{e}⟩$, with resonant coupling between $|\tilde{e}0⟩$ and $|\tilde{g}1⟩$. Together with the aid of the fast cavity decay, these finally lead to the stabilization of the $|\tilde{g}0⟩$ state. (c) The stabilization purity $|⟨\overrightarrow{\sigma }⟩|$, plotted against the polar angle $\theta$ of the stabilization axis, both obtained from qubit tomography. Purities exceeding 80% are achieved over the entire Bloch sphere, while purities $>90\%$ and $>99\%$ are reached for stabilizing the $|e⟩$ ($\theta =180°$) and $|g⟩$ ($\theta =0°$) states, respectively. Experimental data qualitatively agree with the analytical calculation from Eq. (5) (red line) and numerical master equation simulation (black dashed line). The stabilization experiment was performed at zero flux, where qubit and cavity frequencies are ${\omega }_q/2\pi =4.343\mathrm{GHz}$ and ${\omega }_r/2\pi =5.439\mathrm{GHz}$, with the linewidths being $\gamma /2\pi \approx 7.6\mathrm{kHz}$, ${\gamma }_{\varphi }/2\pi \approx 3\mathrm{kHz}$, and $\kappa /2\pi \approx 1.6\mathrm{MHz}$. Left inset: stabilization angles predicted by theory closely match the experimental values. Right inset: trajectory of the qubit state in the dynamic process of stabilization, for the specific case of $\theta =135°$ (red triangle) with measured purity of 87%. Starting from $|g⟩$, the qubit state moves in a helical path along the stabilization axis, until it saturates around the rotating frame ground state, $|\tilde{g}⟩$.