Simultaneous sweet-spot locking of gradiometric fluxonium qubits

Abstract

Efforts to scale up superconducting processors that employ flux-qubits face numerous challenges, among which is the crosstalk created by neighboring flux lines, which are necessary to bias the qubits at the zero-field and $Φ_0/2$ sweet spots. A solution to this problem is to use symmetric gradiometric loops, which incorporate a flux locking mechanism that, once a fluxon is trapped during cooldown, holds the device at the sweet spot and limits the need for active biasing. We demonstrate this technique by simultaneously locking multiple gradiometric fluxonium qubits in which an aluminum loop retains the trapped fluxon indefinitely. By compensating the inductive asymmetry between the two loops of the design, we are able to lock the effective flux-bias within $Φ_{eff} = -3 \times 10^{-4} Φ_0$ from the target, corresponding to only 15 % degradation in $T_{2,E}$ when operated in zero external field. The design strategy demonstrated here reduces integration complexity for flux qubits by minimizing cross-talk and potentially eliminating the need for local flux bias.

Figures (12)

• Figure 1: Demonstration of flux locking. (a) False-colored microscope picture of the gradiometric fluxonium featuring a standard Al/$\mathrm{AlO}_x$/Al junction (blue), Al capacitive pads to couple to the resonator (purple) and granular aluminum (grAl) inductors with normal-state resistivity $\rho_n \approx 2e3µΩ \cm$ and thickness $40nm$ (red). Purple parts are stacks of grAl above two layers of Al shunt. The fluxes threading the two loops $\Phi'_\mathrm{ext}$ and $\Phi"_\mathrm{ext}$ give rise to an effective flux bias $\Phi_\mathrm{eff}$ (see \ref{['eq:Phi_eff']}). (b) Reflection measurement setup. All samples, each consisting of a qubit capacitively coupled to a LC resonator, are placed in a single 3D waveguide and measured in reflection by a Josephson Parametric Amplifier (JPA) winkel_nondegenerate_2020. (c) Spectra of 8 gradiometric fluxonium samples measured in the vicinity of $\Phi_\mathrm{eff}=0$, after cooling down either in zero field or (d) in a field ${\Phi_0}/{2} < \Phi'_\mathrm{ext}+\Phi"_\mathrm{ext} < {3\Phi_0}/{2}$, corresponding to $\Phi_0$ trapped in the aluminum ring of the device. Markers depict the frequency of the $0-1$ transition extracted from a two-tone spectroscopy and lines represent a fit to the fluxonium Hamiltonian in \ref{['eq:hamiltonian']}. Vertical lines indicate the position of the minimal frequency and are shown in detail in the bottom inset.
• Figure 2: Tuning of the asymmetry. Inset: Circuit diagram of the gradiometric fluxonium, showing the Josephson junction (blue), the granular aluminum inductances (red) in the middle, and the low-value inductances (purple) of the aluminum ring. The difference in loop widths $w'-w"$ is tuned while keeping the total size constant. (a) Inductive asymmetry $\alpha=(L"-L')/(L"+L')$ vs. loop width asymmetry. $\alpha$ is obtained from the offset of the spectrum when the fluxonium is $\pi$-locked (cf. \ref{['fig:fig1']}), such that $\Phi_\mathrm{offset}/\Phi_0=\alpha/2$. Perfect sweet-spot locking is realized when $\alpha=0$. (b) Effective area $A_\mathrm{eff}$ vs. loop width asymmetry. $A_\mathrm{eff}$ is extracted from fitting the spectra together with fluxonium parameters $E_\mathrm{J}$, $E_C$ and $E_L$. Zero susceptibility to homogeneous magnetic field is achieved when $A_\mathrm{eff}=0$.
• Figure 3: Qubit time-domain characterization. $T_1$ (in blue) and $T_{2,\mathrm{E}}$ (in orange) coherence times for samples $\textcolor{color_a}{\bullet}$ a (top) and $\textcolor{color_b}{\bullet}$ b (bottom) with $m=1$ fluxon trapped vs. effective flux $\Phi_\mathrm{eff}$ around zero field, illustrating the precision of the locking mechanism. Markers show the measured values with uncertainty. The gray vertical line indicates zero field. In each title, $\chi$ is the dispersive shift at the sweet spot and $\kappa$ is the resonator's linewidth.
• Figure 4: Optical microscope pictures of the qubit (top) and the aluminum LC resonator (bottom) for one of the samples. Each sample's resonator is designed with a different number of meanders in order to spread the frequencies between 6 and 8 $GHz$. Each chip contains two of such samples and does not have any ground plane.
• Figure 5: Cryostat and measurement setup. For frequency-domain experiments, we use a Vector Network Analyzer (VNA) and a microwave source as indicated in the top-left corner. The cryostat is set up with all black components, in particular the total attenuation is 60 dB on the readout input line is and 30 dB on the qubit drive line. For time-domain experiment, we use a Quantum Machines' Operator-X or an Intermodulation Products' Presto. The cryostat is set up with the additional attenuators in grey, increasing the total attenuation to 80 dB on the readout input line and 40 dB on the qubit drive line.