GoKawiilFrom the moment of its discovery, the antagonistic relation between superconductivity and a magnetic field has provided a complex playground for experimentalists and theorists alike. The measurement of the critical field and the Meissner effect12 have anchored phase-transition theories13,14, and the trapping of quantized flux inside superconductors has provided direct evidence for the existence of Cooper pairs15. A hallmark of type II superconductivity in a magnetic field is the formation of Abrikosov vortices: regions of local gap suppression that interact to form lattices1. Vortex dynamics is detrimental for a wide range of applications15, causing heating, flux noise and magnetic hysteresis. However, pinned vortices enable quasiparticle trapping in their core, which enhances the critical current16 of superconducting films, improves micro-cooler efficiency17, boosts resonator quality factors18 and improves qubit coherence19,20. In all these cases, owing to the normal state core, vortices can be understood within semi-classical models.
Gap suppression in the vortex core stems from the crowding of supercurrent at its centre, a consequence of continuity in the superconducting medium. Recent work3 has proposed that in discretized systems, such as granular superconductors where non-superconducting regions separate superconducting islands, the vortex core can remain gapped and dissipationless; a closely related regime has also been predicted for strongly disordered superconductors, where emergent superconducting islands2 host vortices with insulating cores4. Although quantum behaviour has been revealed by tunnelling of vortices in long Josephson junctions8 and thin films9, or via the zero-point motion of pinned vortices10, direct evidence of coherent superconducting vortex states has yet to be observed.
Here we show that vortices trapped in a superconducting granular aluminium (grAl) microwave resonator form field-tunable two-level systems that behave like effective spins, strongly coupled to the resonator. They can therefore be regarded as quantum bits (qubits) that arise from vortex tunnelling in a field-modulated double-well potential formed between pinning sites. These vortex qubit (VQ) states exhibit microsecond coherence and energy relaxation times on the order of 102 μs, strikingly different from the dissipative dynamics of Abrikosov vortices. We find that VQs remain stable for weeks, enabling coherent control and quantum non-demolition readout within the framework of circuit quantum electrodynamics11.
As schematized in Fig. 1, we use a grAl micro-stripline resonator, with resistivity ρ = 3,600 μΩ cm, chosen to be within a factor of 3 below the superconducting-to-insulating transition21. In this regime, the film consists of Al grains of 3–4-nm diameter separated by amorphous AlO x barriers, resulting in a coherence length ξ ≈ 7 nm and London penetration depth of λ L ≈ 4 μm (refs. 5,22,23). The resonator is placed in a cylindrical copper waveguide (Supplementary Information section I) anchored to the 20-mK base plate of a dilution cryostat and measured in reflection. When cooled in zero magnetic field B cd = 0 μT, the grAl resonator behaves as a weakly anharmonic oscillator24, with a fundamental frequency f r = 7.572 GHz, set by its dimensions (3 μm wide, 400 μm long; Extended Data Fig. 1). Figure 1b shows the frequency decrease with perpendicular magnetic field B, as expected with the increase in kinetic inductance25,26.
Fig. 1: Field cooling introduces VQ states that couple to the grAl resonator. The alternative text for this image may have been generated using AI. Full size image a, When cooled to 20 mK in perpendicular magnetic field B cd = 0 μT, a λ/2 micro-stripline grAl resonator behaves as a quantum harmonic oscillator with resonant frequency ω r . The electric- and magnetic-field distributions are illustrated in blue and red, respectively. The grAl film has a thickness of t = 20 nm and a superconducting coherence length of ξ = 7 nm. b, Phase response arg(S 11 ) of the resonator measured in reflection, as a function of perpendicular magnetic field B applied after cooling. The measured parabolic suppression of the resonance is given by the increase in kinetic inductance owing to screening currents25, and the field range is limited by the vortex penetration threshold26. c, When cooled in perpendicular magnetic field B cd = 820 μT (see main text), vortices enter the grAl resonator and the system exhibits a behaviour akin to a flux qubit with a transition frequency ω q coupled to a readout resonator, as illustrated in d and e. 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 (equation (2)), yielding the coupling g/2π = 95 MHz. e, Extracted VQ frequency f 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 equation (2), 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 colour scale indicates the measured phase response as a function of the frequency f d of the second drive.
Following field-cooling, sweeping B reveals avoided level crossings in the grAl resonator response as illustrated in Fig. 1d, which we interpret as evidence of strong coupling with g/2π = 95 MHz to vortex states. To extract the mode’s spectrum, we sweep a second microwave drive while probing the readout resonator (Fig. 1e). We observe a minimum vortex mode frequency f q = 2 GHz at the sweet spot B 0 = 128 μT (Fig. 1e, inset), with a slope of the hyperbolic field dispersion γ = 20 GHz mT−1, reminiscent of a flux qubit27. As the field approaches the sweet spot, the resonance narrows, pointing to magnetic-field fluctuations as dominant noise source28. From measured spectra across 32 field-cooling cycles in six different resonators, we extract values of g, f q , B 0 and γ that are of similar order of magnitude but vary between cycles (Supplementary Information section II), suggesting different underlying vortex configurations. Repeated resonator reflection coefficient S 11 measurements at the sweet spot reveal two distinct clusters in the quadrature plane (Fig. 2a), indicating that the vortex state has a lifetime well beyond the 1.2-μs integration time, thereby enabling single-shot state discrimination. As demonstrated in Fig. 2b, by driving at f q , we can calibrate a 20 ns π-pulse, which inverts its thermal population (see Supplementary Information section III for the Rabi oscillations). These signatures define the VQ states \(| {\rm{g}}\rangle \) (ground) and \(| {\rm{e}}\rangle \) (excited). From their steady-state populations, we extract a 74-mK effective temperature. The VQ–resonator interaction induces a state-dependent dispersive shift \(\chi /2{\rm{\pi }}={f}_{{\rm{r}},| {\rm{e}}\rangle }-\,{f}_{{\rm{r}},| {\rm{g}}\rangle }\). As shown in Fig. 2c, fitting the resonator’s phase response to the centres of in-phase and quadrature (IQ) clouds measured versus readout frequency yields χ/2π = −1.32 MHz (see Supplementary Information section IV for all measured IQ clouds).
Fig. 2: The asymmetric quantum Rabi model describes the VQ dispersively coupled to its resonator. The alternative text for this image may have been generated using AI. Full size image 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 \(| {\rm{g}}\rangle \) (ground) and \(| {\rm{e}}\rangle \) (excited) states. The qubit excited state population P q yields an effective qubit temperature T eff ≈ 74 mK. b, Measured IQ clouds following a 20-ns drive at f q calibrated to implement a π-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 centres of the IQ clouds, measured versus readout frequency f RO in the vicinity of f r . A fit to the data (black solid line) yields a dispersive shift of χ/2π = −1.32 MHz. The dark red (\(| {\rm{g}}\rangle \)) and light red (\(| {\rm{e}}\rangle \)) points correspond to the data in a at f RO = 7.5714 GHz (dashed line). d, Variation of χ 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 equation (2) with g AQRM /2π = 92.5 MHz, and the dash-dotted line to the symmetric quantum Rabi model equation (1) with g SQRM /2π = 20 MHz. The solid green line represents the qubit frequency (right axis), similar to Fig. 1d.
For further insight into the nature of the VQ and its interaction with the grAl resonator, we measure χ versus field, as shown in Fig. 2d. We model it using the quantum Rabi model (QRM) for a spin S = 1/2 coupled via \({{\mathcal{H}}}_{{\rm{c}}}=\hbar g({\hat{a}}^{\dagger }+\hat{a}){\sigma }_{x}\) to a harmonic oscillator with frequency ω r and Hamiltonian \({{\mathcal{H}}}_{{\rm{r}}}=\hbar {\omega }_{{\rm{r}}}\left({\hat{a}}^{\dagger }\hat{a}+\frac{1}{2}\right)\) (Supplementary Information section V). Here \({\hat{a}}^{\dagger }\) and \(\hat{a}\) are the resonator bosonic operators, ħ = h/(2π) is the reduced Planck constant and σ x is the Pauli matrix for a spin S = ħ/2σ. The interaction energy between the spin and the magnetic field is \(\gamma {\bf{S}}\cdot (\widetilde{{\bf{B}}}+{{\bf{B}}}^{{\prime} })\), where γ is the gyromagnetic ratio and the field consists of two contributions: a pseudo-field \(\widetilde{{\bf{B}}}\) that sets the VQ energy at the sweet spot, and the applied magnetic field \(| {{\bf{B}}}^{{\prime} }| =B-{B}_{0}\) measured from the sweet spot. We compare joint fits of the measured VQ and resonator frequencies in field (Fig. 1d,e), using the symmetric quantum Rabi model (SQRM)
$${{\mathcal{H}}}_{{\rm{S}}{\rm{Q}}{\rm{R}}{\rm{M}}}={{\mathcal{H}}}_{{\rm{r}}}+{{\mathcal{H}}}_{{\rm{c}}}+\frac{\hbar \gamma }{2}{{\sigma }}_{z}\sqrt{{\mathop{B}\limits^{ \sim }}^{2}+{B}^{{\prime} 2}},$$ (1)
and the asymmetric quantum Rabi model (AQRM)
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