Quantum Coherence in Superconducting Vortex States

Abstract

Abrikosov vortices, where the superconducting gap is completely suppressed in the core, are dissipative, semi-classical entities that impact applications from high-current-density wires to superconducting quantum devices. In contrast, we present evidence that vortices trapped in granular superconducting films can behave as two-level systems, exhibiting microsecond-range quantum coherence and energy relaxation times that reach fractions of a millisecond. These findings support recent theoretical modeling of superconductors with granularity on the scale of the coherence length as tunnel junction networks, resulting in gapped vortices. Using the tools of circuit quantum electrodynamics, we perform coherent manipulation and quantum non-demolition readout of vortex states in granular aluminum microwave resonators, heralding new directions for quantum information processing, materials characterization, and sensing.

Figures (15)

• Figure 1: Field cooling introduces vortex qubit states that couple to the grAl resonator.(a) When cooled to $\qty{20}{\milli\kelvin}$ in perpendicular magnetic field $B_{\rm cd} = 0\,µ T$, a $\lambda/2$ microstripline grAl resonator behaves as a quantum harmonic oscillator. The electric and magnetic field distributions are illustrated in blue and red, respectively. (b) Phase response $\arg(S_{11})$ of the resonator measured in reflection, as a function of perpendicular magnetic field $B$ applied after cooldown. The measured parabolic suppression of the resonance is given by the increase in kinetic inductance due to screening currents annunziata2010tunable, and the field range is limited by the vortex penetration threshold borisov2020superconducting. (c) When cooled down in perpendicular magnetic field $B_{\rm cd} = \qty{820}{\micro \tesla}$, vortices enter the grAl resonator and the system exhibits a behavior akin to a flux qubit coupled to a readout resonator, as illustrated in panels (d) and (e). As shown in the top right corner, the number of vortices per square ($N_\odot$) determines their spatial arrangement. (d) The measured phase response of the resonator as a function of $B$ reveals avoided level crossings, suggesting coupling to vortex states. The purple dashed line shows a fit to the asymmetric quantum Rabi model (\ref{['eq: Hamiltonian']}), yielding the coupling $g/2\pi = \qty{92.5}{\mega\hertz}$. (e) Extracted VQ frequency $f_\text{q}$ from two-tone spectroscopy (see inset) as a function of $B$. The green line corresponds to the joint fit of data in (d) and (e) to \ref{['eq: Hamiltonian']}, and the purple dashed line marks the bare resonator frequency $f_r$. Inset: Two-tone spectroscopy in the vicinity of $B_{0}$ corresponding to the minimum frequency of the VQ. The colorscale indicates the measured phase response as a function of the frequency $f_d$ of the second drive.
• Figure 2: The asymmetric quantum Rabi model describes the vortex qubit dispersively coupled to its resonator.(a) Consecutive $S_{11}$ measurements at the sweet spot show two IQ-clouds in the complex plane. The relative occurrence of points in the clouds corresponds to the population of the $\ket{g}$ (ground) and $\ket{e}$ (excited) states and yields an effective qubit temperature $T_{\rm eff} \approx \qty{74}{\milli\kelvin}$. (b) Measured IQ-clouds following a $\qty{20}{\nano\second}$ drive at $f_q$ calibrated to implement a $\pi$-pulse, show a population inversion as expected for a two-level system. The black circles have a radius of $1.5$ standard deviation. (c) Resonator phase response arg($S_{11}$), obtained from the centers of the IQ-clouds, measured versus readout frequency in the vicinity of $f_r$. A fit to the data (black solid line) yields a dispersive shift of $\chi/2\pi = \qty{-1.32}{\mega\hertz}$. The dark red ($\ket{g}$) and light red ($\ket{e}$) points correspond to the data in (a) at $f_{\rm{RO}} = \qty{7.5714}{\giga\hertz}$ (dashed line). (d) Variation of $\chi$ with magnetic field $B$, shown as triangles, with the yellow triangle corresponding to the measurement in (b). The dashed line indicates the expected values from the asymmetric quantum Rabi model \ref{['eq: Hamiltonian']} with $g_{\rm AQRM}/2 \pi = \qty{92.5}{\mega \hertz}$, and the dash-dotted line to the symmetric quantum Rabi model \ref{['eq: SQRM']} with $g_{\rm SQRM} /2 \pi = \qty{20}{\mega \hertz}$. The solid green line represents the qubit frequency (right axis), similar to \ref{['fig: Fluxon']}. (e) Gibbs free energy $G_1$ (cf. \ref{['eq:Gibbs']}, baseline) of a single vortex along the width of the resonator in units of $\varepsilon_0 = \Phi_0^2/2\pi\mu_0\Lambda \approx 2\,T Hz$, with added pinning potentials depicted as Lorentzian dips. The colors correspond to different applied magnetic fields from $B_{\rm S} = \phi_{\rm S}\Phi_0/w^2$ to $-B_0$. Top inset: A possible double-well potential arising from the energy landscape of neighboring pinning sites $\delta_{\rm LR}$ apart, offset in energy by $\epsilon$. The localized wavefunctions in these wells correspond to the two possible positions $\ket{\rm L}$ and $\ket{\rm R}$ of the vortex, coupled by tunneling amplitude $\Delta$, with an energy separation given by the VQ transition frequency $\hbar\omega_\mathrm{q}$. Bottom inset: the double-well potential is degenerate at the sweet spot, with VQ states depicted as symmetric and antisymmetric combinations of the localized wavefunctions, and $\hbar\omega_\mathrm{q}=2\Delta$.
• Figure 3: Measurement of low loss and coherence in the VQ.(a) Free energy decay measured after a $20\,n s\,\,\pi$ pulse applied selectively to the VQ measured in the ground state $\ket{g}$. The readout pulse has a duration $\tau_\text{m}=1.2 \,µ s$. The excited vortex qubit population P($\ket{e}$) as a function of wait time $t$ is fitted with an exponential corresponding to $T_1 = 186\,µ s$ (solid line). (b) Ramsey fringes exhibit a beating pattern, resulting from two frequencies separated by $f^{\rm beat} = 2\,M Hz$. We extract $T_2^*$ Ramsey coherence times of $440\,n s$. (c) Spin Hahn echo measurement with extracted $T_{2}^{\text{echo}} = 1.2\,µs$. For each panel, the corresponding pulse sequence is sketched at the top, and the insets show measured coherence times over several hours.
• Figure 4: Measurement setup. Schematic of the cylindrical copper waveguide sample holder anchored to the millikelvin stage. A Helmholtz coil pair, thermalized to the 4 stage of the cryostat, generates a perpendicular magnetic field $B$. The sample holder, directly connected to a absorber Rehammar_Gasparinetti_2023, is measured in single-port microwave reflection $S_{11}$, with a Josephson parametric amplifier (JPA) Winkel__Amplifier__2020 on the output line. Inset: A $3 \times 10 \, m m$ sapphire chip is secured with a copper dowel and hosts a grAl micro-stripline resonator (gray rectangle) positioned $\qty{0.5}{\milli \meter}$ from the bottom edge of the chip, similar to Refs. borisov2020superconductingGünzler_2025.
• Figure 5: Magnetic field calibration. Extracted field $B$ at the sample position versus bias current applied to the Helmholtz coil. For fields up to $\qty{100}{\micro \tesla}$, the flux periodicity (flux sensitivity of $\qty{22}{\micro \tesla/\Phi_0}$) of a Gralmonium flux qubit Rieger_Gralmonium_2022 is used for the calibration. The blue markers denote fields corresponding to integer multiples of $\Phi_0$ in the qubit loop. At higher fields, calibration relies on electron spin resonance (ESR) of $g=2$ spin-$1/2$ paramagnetic impurities (red marker). A linear fit to both data sets yields a conversion factor of $\qty{72.8}{\milli \tesla}/\unit{\ampere}$. Top inset: Phase response $\arg(S_{11})$ of the readout resonator plotted as a function of the drive frequency $f$ at different bias currents $I$ for the gralmonium qubit described in Ref. rieger2023fano. The avoided level crossings appear at field values where the qubit mode periodically crosses the resonator frequency. Bottom inset: Internal quality factor $Q_{\rm i}$ of a grAl resonator versus current $I$ in the coil. A dip, marked by the black line, appears at the current corresponding to ESR between the $g=2$ spins and the resonator.