Abstract
The non-dissipative nonlinearity of Josephson junctions1 converts macroscopic superconducting circuits into artificial atoms2, enabling some of the best-controlled qubits today3,4. Three fundamental types of superconducting qubit are known5, each reflecting a distinct behaviour of quantum fluctuations in a Cooper pair condensate: single-charge tunnelling (charge qubit6,7), single-flux tunnelling (flux qubit8) and phase oscillations (phase qubit9 or transmon10). Yet, the dual nature of charge and flux suggests that circuit atoms must come in pairs. Here we introduce the missing superconducting qubit, ‘blochnium’, which exploits a coherent insulating response of a single Josephson junction that emerges from the extension of phase fluctuations beyond 2π (refs. 11,12,13,14). Evidence for such an effect has been found in out-of-equilibrium direct-current transport through junctions connected to high-impedance leads15,16,17,18,19, although a full consensus on the existence of extended phase fluctuations is so far absent20,21,22. We shunt a weak junction with an extremely high inductance—the key technological innovation in our experiment—and measure the radiofrequency excitation spectrum as a function of external magnetic flux through the resulting loop. The insulating character of the junction is manifested by the vanishing flux sensitivity of the qubit transition between the ground state and the first excited state, which recovers rapidly for transitions to higher-energy states. The spectrum agrees with a duality mapping of blochnium onto a transmon, which replaces the external flux by the offset charge and introduces a new collective quasicharge variable instead of the superconducting phase23,24. Our findings may motivate the exploration of macroscopic quantum dynamics in ultrahigh-impedance circuits, with potential applications in quantum computing and metrology.
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Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The codes that support the findings of this study are available from corresponding author upon reasonable request.
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Acknowledgements
We acknowledge funding from NSF-CAREER (1455261), Alfred P. Sloan Foundation, NSF PFC at JQI (1430094), and the ARO-LPS HiPS programme (W911NF-18-1-0146).
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R.A.M. fabricated devices and performed measurements guided by I.V.P.; I.V.P. analysed the data and co-wrote the manuscript with V.E.M.; L.B.N. and Y.-H.L. built the low-temperature microwave measurement setup; V.E.M. managed the project. All authors contributed to discussions of the results.
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Extended data figures and tables
Extended Data Fig. 1 Qubit spectroscopy.
Stitched one- and two-tone spectroscopy data as a function of the spectroscopy frequency and the normalized external flux through the loop. The fit (dashed lines) to the transition spectrum of the Hamiltonian of equation (1) is superimposed on the data. The data were collected in a patch-wise manner, with the measurement parameters optimized locally to improve the visibility of the transitions out of the ground state |0⟩. As in Fig. 3b, nonlinear colour maps are used to assign colour to the measured signal. Note that the deviation between the fit and the data is noticeable only from the |0⟩ → |6⟩ transition.
Extended Data Fig. 2 Persistent current.
Persistent current in the ground state of the device ⟨0|I|0⟩ ≡ I0 ⟨0|sin(φ − φext)|0⟩, where I0 = 9.5 nA is the junction critical current, plotted as a function of the external flux, φext. The current is calculated using the extracted device parameters.
Extended Data Fig. 3 Flux dispersion and matrix elements.
a, Zoom-in on the lowest two states of Fig. 4a. Eigenenergies of the Hamiltonians of equation (1) (dashed lines) and equation (2) (solid lines) and of a hypothetical device without the Josephson junction (grey dotted lines) as a function of the external flux, φext. The spectra are calculated using the extracted device parameters. b, Matrix element ⟨0|φ|1⟩ as a function of the external flux, φext. The dashed line corresponds to EJ/h = 4.70 GHz and should be compared to the dotted line, which corresponds to EJ = 0—that is, to the hypothetical case without the Josephson junction. In both panels, EC/h = 7.07 GHz and EL/h = 66.5 MHz.
Extended Data Fig. 4 Ground-state wavefunctions of the measured device.
a–h, Ground-state wavefunctions in the phase (φ) and integer-flux (m) bases (a–d) and in the charge (Q) and quasicharge (q) bases (f–h). a, e, Ground-state wavefunctions of the device discussed in the main text for φext = 0. The black solid line in a corresponds to the unbounded, continuous-phase φ basis and the black solid line in e to the continuous-charge Q basis, which are the natural bases for the Hamiltonian of equation (1). The stems in a correspond to the discrete integer-flux basis and the dotted grey line in e to the periodic quasicharge basis, which are the natural bases for the Hamiltonian of equation (2). EJ/h = 4.70 GHz, EC/h = 7.07 GHz and EL/h = 66.5 MHz. b, f, Same as a, e, but for a ten-times-larger inductance, that is, EL/h = 6.65 MHz. c, g, Same as a, e, but for a 100- times-larger inductance, that is, EL/h = 0.67 MHz. d, h, Cooper pair box (CPB) wavefunction in the phase φ (d) and charge Q (h) bases computed for the same values of EJ and EC used in a–c and e–g, EJ/EC = 0.66. A single period (−π, π] in d is highlighted in solid grey. The offset charge is set to zero.
Extended Data Fig. 5 Ground-state wavefunctions for modified device parameters.
Same as Extended Data Fig. 4, but for EJ/h = 4.70 GHz and EC/h = 1.18 GHz, so that EJ/EC = 4.0. Single periods (−e, e] in e–g are highlighted in solid grey.
Extended Data Fig. 6 Energy relaxation and decoherence.
Measurement of the energy relaxation time T1 and spin-echo coherence time T2 at an external flux bias point close to the half-flux quantum. The measured time traces are fitted with decaying exponents. In the spin-echo sequence, the refocusing π rotation was applied around the axis perpendicular to the axis of the two π/2 rotations.
Extended Data Fig. 7 Dephasing limit.
Estimated dephasing time Tφ due to the first-order sensitivity of the qubit transition frequency f01 to the flux noise. Here, we use \(1/{T}_{\phi }=2{\rm{\pi }}\frac{\partial {f}_{01}}{\partial \varPhi }A\sqrt{\mathrm{ln}\,2}\), where Φ is the total magnetic flux through the loop, and assume typical flux noise amplitude26 of A ≈ 1.8 × 10−6(h/2e).
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Pechenezhskiy, I.V., Mencia, R.A., Nguyen, L.B. et al. The superconducting quasicharge qubit. Nature 585, 368–371 (2020). https://doi.org/10.1038/s41586-020-2687-9
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DOI: https://doi.org/10.1038/s41586-020-2687-9
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