Spin Environment of a Superconducting Qubit in High Magnetic Fields

Abstract

Superconducting qubits equipped with quantum non-demolition readout and active feedback can be used as information engines to probe and manipulate microscopic degrees of freedom, whether intentionally designed or naturally occurring in their environment. In the case of spin systems, the required magnetic field bias presents a challenge for superconductors and Josephson junctions. Here we demonstrate a granular aluminum nanojunction fluxonium qubit (gralmonium) with spectrum and coherence resilient to fields beyond one Tesla. Sweeping the field reveals a paramagnetic spin-1/2 ensemble, which is the dominant gralmonium loss mechanism when the electron spin resonance matches the qubit. We also observe a suppression of MHz range fast flux noise in magnetic field, suggesting the freezing of surface spins. Using an active state stabilization sequence, the qubit hyperpolarizes long-lived two-level systems (TLSs) in its environment, previously speculated to be spins. Surprisingly, the coupling to these TLSs is unaffected by magnetic fields, leaving the question of their origin open. The robust operation of gralmoniums in Tesla fields offers new opportunities to explore unresolved questions in spin environment dynamics and facilitates hybrid architectures linking superconducting qubits with spin systems.

Figures (9)

• Figure 1: Gradiometric gralmonium qubit resilient to tesla magnetic field.a False-colored scanning electron microscope (SEM) image of the qubit circuit, galvanically coupled to the readout resonator. The device consists of a 20 thick single layer of grAl. The colored regions (ocher & violet) illustrate the $10\%$ mismatched areas of the two flux loops in the gradiometric design Gusenkova2022Jan, which result in an effective flux bias $\Phi_\text{ext}$ in perpendicular magnetic field $B_\perp$ (cf. \ref{['eq:phiExtTotal']}). Inset: zoom-in on the $\sim \qty{20}{\nano\meter}$ wide grAl nanojunction of the qubit (cf. gralmonium). b Circuit schematic for the gradiometric qubit depicted in a: the nanojunction (red) is shunted by an interdigitated capacitor and two flux loops (ocher & violet) with inductances $L_1 + L_\mathrm{s}$ and $L_3$, respectively. The inductance shared between the loops is $L_2$. The qubit is inductively coupled via $L_\mathrm{s}$ to the readout resonator (inductance $L_\mathrm{r}$, capacitance $C_\mathrm{r}$) for which we measure the single-port reflection coefficient $S_{11}$. c Two-tone (TT) spectroscopy at the half flux sweet spot $\Phi_\textbf{ext} = \Phi_0/2$ in $B_\parallel = \qty{0}{\tesla}$. d Increase of the sweet spot qubit frequency in magnetic field up to 1,2. Inset: TT-spectroscopy in $B_\parallel = \qty{1,2}{\tesla}$. e Qubit spectrum: ground to excited ($f_{\text{ge}}$ in dark blue markers) and ground to second-excited ($f_{\text{gf}}$ in light blue markers) state transitions extracted from TT-spectroscopy. From a fit (black line) to the fluxonium Hamiltonian (\ref{['eq:fluxoniumHamiltonian']}), we obtain $E_\mathrm{J}/h = \qty{32.2}{\giga\hertz}$ (i.e. critical current $I_\mathrm{c}=\qty{64.9}{\nano\ampere}$), $E_\mathrm{c}/h= \qty{14.1}{\giga\hertz}$ ($C=\qty{1.37}{\femto\farad}$) and $E_\mathrm{L}/h = \qty{0.454}{\giga\hertz}$ ($L_\mathrm{q}=\qty{360}{\nano\henry}$). f Suppression of the grAl superconducting gap $\Delta$ in magnetic field. The red and orange markers, corresponding to the qubit nanojunction and inductor superconducting gaps ($\Delta_{E_\text{J}}$, $\Delta_{L_\mathrm{q}}$), are obtained from fitted $E_\text{J}$ and $E_\text{L}$ values (cf. e) at each magnetic field. The capacitance $C$ is fixed to the fit value obtained in $B_\parallel =\qty{0}{\tesla}$. The green markers are obtained from the shift of the readout resonator frequency $f_{\text{r}}(B_\parallel)$. The black lines show fits to the field dependence of the superconducting gap, indicating a $40\%$ higher critical field for the nanojunction.
• Figure 2: Qubit coherence in magnetic field: signatures of environmental spin polarization.a Energy relaxation time $T_1$, Ramsey and echo coherence time, $T_{2 \mathrm{R}}$ and $T_{2 \mathrm{E}}$ respectively, of the gradiometric gralmonium in magnetic field up to 1. b, c Ramsey fringes measured in $B_\parallel = \qty{0}{\tesla}$ and $B_\parallel = \qty{1}{\tesla}$, respectively. A two-frequency fit (black line) indicates a similar beating pattern (dotted envelope) for both magnetic fields. d Energy relaxation $T_1$ up to 120: similarly to observations on resonators Samkharadze2016AprKroll2019JunBorisov2020Sep, the drop in $T_1$ suggests coupling to the electron spin resonance (ESR) of paramagnetic impurities of unknown origin. Inset: The fields $B_\text{ESR} = h f_\mathrm{q} / g \mu_\mathrm{B}$ at which the ESR matches different qubit frequencies in different cooldowns, correspond to the expectation for a spin $s=1/2$ ensemble with gyromagnetic factor $g=2$ (black line). Note that we use the same device for which the qubit frequency changes between cooldowns (cf. Ref. gralmonium). e Dephasing times $T_{\varphi \mathrm{R}}$, $T_{\varphi \mathrm{E}}$ remain unaffected by the ESR. f Flux noise echo dephasing rate $\Gamma_{\varphi \text{E}}^\Phi$ in the vicinity of the sweet spot for three in-plane magnetic fields. Dashed lines show fits to \ref{['eq:GammaE_flux']}. g Flux noise amplitude $\sqrt{A_\Phi}$ in magnetic field with fit to \ref{['eq:APhi_vs_B']}, corresponding to a spin freezing with a spin temperature of $T_\mathrm{S}=\qty{85}{\milli\kelvin}$. In all panels, the errorbars represent the standard deviation obtained from successive measurements.
• Figure 3: Magnetic susceptibility of long-lived two-level-systems (TLSs) in high field.a Sketch of the qubit preparation sequence used in panels (b-d). The repeated ($N=10^4$) active reset of the qubit state in $\ket{\text{g}}$ or $\ket{\text{e}}$ (blue and red traces in all panels, respectively) results in the hyperpolarization of environmental, long-lived TLS Spiecker2023Sep. The last step of the preparation sequence consists in a qubit initialization in $\ket{\text{g}}$ or $\ket{\text{e}}$. We use a 540 rectangular readout pulse and a 32 Gaussian $\pi$-pulse. b Qubit population relaxation after the preparation sequence for different magnetic fields $B_\parallel$. We fit the data (semi-transparent) to the theoretical model Spiecker2023SepSpiecker2024May (opaque). For reference, the black dashed lines show an exponential decay with the qubit energy relaxation rate $\Gamma_1$. In zero field, we reproduce the signatures of TLS hyperpolarization recently observed in other superconducting qubits Spiecker2023SepOdeh2023Dec, i.e. undershoot (blue) and overshoot (red) compared to the single exponential decay. c At the ESR resonance field $B_\text{ESR}$, the hyperpolarization signatures are suppressed due to energy relaxation of the qubit into the paramagnetic ensemble. d The signatures of TLS hyperpolarization on qubit relaxation in magnetic fields exceeding 1T are comparable to zero field, indicating a very low susceptibility of the long-lived TLSs to magnetic field.
• Figure 4: Cylindrical waveguide sample holder within the vector magnet. The 2D vector magnet is thermalized on the 4 stage of the cryostat and separated by a 1 gap from the cylindrical pipe of the sample holder, which is anchored to the 30 stage. The waveguide, with a 3 inner diameter and 0,3 wall thickness, has a cut-off frequency of $\sim\qty{60}{\giga\hertz}$, operating in the sub-wavelength regime. This results in coupling via the evanescent microwave field of a stripped coaxial pin, with an exponential decay in coupling strength relative to the chip-to-pin distance (cf. Rieger2023Julgralmonium). The circuit is positioned on the bottom of a 3 x 10 sapphire chip and fixed with a copper dowel, clamped against the cylindrical copper pipe walls. Apiezon N vacuum grease on the dowel provides additional thermal anchoring. High magnetic fields are applied in the substrate plane via a solenoid coil, while magnetic flux tuning is achieved with a Helmholtz pair aligned perpendicular to the substrate plane. No additional shielding is implemented between the sample holder and vector magnet coils. Note that the setup is identical to the one used in Ref.Borisov2020Sep
• Figure 5: IQ histogram in magnetic field: 1D histogram of the measured I quadrature (top panels) and 2D histogram of the I and Q quadrature (bottom panels). In $B_\parallel=\qty{0}{\tesla}$ (left panels) and $B_\parallel=\qty{1,2}{\tesla}$ (right panels), we extract a qubit population corresponding to a temperature of 165 and 150, respectively. These IQ histograms are illustrative for all qubit state measurements used in the main text. We use a 540 readout pulse with an equilibrium average photon number of $\bar{n} = 25$ , integrated over 540, including resonator ring-up and ring-down times corresponding to a linewidth $\kappa / 2\pi = \qty{1,2}{\mega\hertz}$.