Abstract
Quantum sensors and qubits are usually two-level systems (TLS), the quantum analogues of classical bits assuming binary values 0 or 1. They are useful to the extent to which superpositions of 0 and 1 persist despite a noisy environment. The standard prescription to avoid decoherence of solid-state qubits is their isolation by means of extreme dilution in ultrapure materials. We demonstrate a different strategy using the rare-earth insulator LiY1−xTbxF4 (x = 0.001) which realizes a dense random network of TLS. Some TLS belong to strongly interacting Tb3+ pairs whose quantum states, thanks to localization effects, form highly coherent qubits with 100-fold longer coherence times than single ions. Our understanding of the underlying decoherence mechanisms—and of their suppression—suggests that coherence in networks of dipolar coupled TLS can be enhanced rather than reduced by the interactions.
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Data availability
Source data are provided with this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.
Code availability
The source codes used for the numerical simulations are provided with this paper.
References
Basko, D. M., Aleiner, I. L. & Altshuler, B. L. Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states. Ann. Phys. 321, 1126–1205 (2006).
Huse, D. A., Nandkishore, R. & Oganesyan, V. Phenomenology of fully many-body-localized systems. Phys. Rev. B 90, 174202 (2014).
Nandkishore, R. & Huse, D. A. Many-body localization and thermalization in quantum statistical mechanics. Annu. Rev. Condens. Matter Phys. 6, 15–38 (2015).
Nandkishore, R., Gopalakrishnan, S. & Huse, D. A. Spectral features of a many-body-localized system weakly coupled to a bath. Phys. Rev. B 90, 064203 (2014).
Yao, N. Y. et al. Many-body localization in dipolar systems. Phys. Rev. Lett. 113, 243002 (2014).
Burin, A. L. Many-body delocalization in a strongly disordered system with long-range interactions: finite-size scaling. Phys. Rev. B 91, 094202 (2015).
Gornyi, I. V., Mirlin, A. D., Müller, M. & Polyakov, D. G. Absence of many-body localization in a continuum. Ann. Phys. 529, 1600365 (2017).
De Roeck, W. & Huveneers, F. M. C. Stability and instability towards delocalization in many-body localization systems. Phys. Rev. B 95, 155129 (2017).
Abanin, D. A., Altman, E., Bloch, I. & Serbyn, M. Colloquium: many-body localization, thermalization, and entanglement. Rev. Mod. Phys. 91, 021001 (2019).
Gopalakrishnan, S. & Parameswaran, S. A. Dynamics and transport at the threshold of many-body localization. Phys. Rep. 862, 1–62 (2020).
Smith, J. et al. Many-body localization in a quantum simulator with programmable random disorder. Nat. Phys. 12, 907–911 (2016).
Roushan, P. et al. Spectroscopic signatures of localization with interacting photons in superconducting qubits. Science 358, 1175–1179 (2017).
Lukin, A. et al. Probing entanglement in a many-body–localized system. Science 364, 256–260 (2019).
Guo, Q. et al. Observation of energy-resolved many-body localization. Nat. Phys. 17, 234–239 (2021).
Chiaro, B. et al. Direct measurement of nonlocal interactions in the many-body localized phase. Phys. Rev. Res. 4, 013148 (2022).
Léonard, J. et al. Probing the onset of quantum avalanches in a many-body localized system. Nat. Phys. 19, 481–485 (2023).
Ho, W. W., Choi, S., Lukin, M. D. & Abanin, D. A. Critical time crystals in dipolar systems. Phys. Rev. Lett. 119, 010602 (2017).
Choi, J. et al. Depolarization dynamics in a strongly interacting solid-state spin ensemble. Phys. Rev. Lett. 118, 093601 (2017).
Kucsko, G. et al. Critical thermalization of a disordered dipolar spin system in diamond. Phys. Rev. Lett. 121, 023601 (2018).
Forrester, P. A. & Hempstead, C. F. Paramagnetic resonance of Tb3+ ions in CaWO4 and CaF2. Phys. Rev. 126, 923–930 (1962).
Laursen, I. & Holmes, L. M. Paramagnetic resonance of Tb3+ in LiY0.99Tb0.01F4. J. Phys. C Solid State Phys. 7, 3765 (1974).
Holmes, L. M., Johansson, T. & Guggenheim, H. J. Ferromagnetism in LiTbF4. Solid State Commun. 12, 993–997 (1973).
Youngblood, R. W., Aeppli, G., Axe, J. D. & Griffin, J. A. Spin dynamics of a model singlet ground-state system. Phys. Rev. Lett. 49, 1724–1727 (1982).
Carr, H. Y. & Purcell, E. M. Effects of diffusion on free precession in nuclear magnetic resonance experiments. Phys. Rev. 94, 630–638 (1954).
Meiboom, S. & Gill, D. Modified spin-echo method for measuring nuclear relaxation times. Rev. Sci. Instrum. 29, 688–691 (1958).
Prokof’ev, N. V. & Stamp, P. C. E. Theory of the spin bath. Rep. Prog. Phys. 63, 669–726 (2000).
Kittel, C. & Abrahams, E. Dipolar broadening of magnetic resonance lines in magnetically diluted crystals. Phys. Rev. 90, 238–239 (1953).
Burin, A. L. Localization in a random xy model with long-range interactions: intermediate case between single-particle and many-body problems. Phys. Rev. B 92, 104428 (2015).
Ghosh, S., Rosenbaum, T. F., Aeppli, G. & Coppersmith, S. N. Entangled quantum state of magnetic dipoles. Nature 425, 48–51 (2003).
Silevitch, D. M., Tang, C., Aeppli, G. & Rosenbaum, T. F. Tuning high-Q nonlinear dynamics in a disordered quantum magnet. Nat. Commun. 10, 4001 (2019).
Kindem, J. M. et al. Control and single-shot readout of an ion embedded in a nanophotonic cavity. Nature 580, 201–204 (2020).
Ruskuc, A., Wu, C.-J., Rochman, J., Choi, J. & Faraon, A. Nuclear spin-wave quantum register for a solid-state qubit. Nature 602, 408–413 (2022).
Le Dantec, M. et al. Twenty-three-millisecond electron spin coherence of erbium ions in a natural-abundance crystal. Sci. Adv. 7, eabj9786 (2022).
Mims, W. B. Envelope modulation in spin-echo experiments. Phys. Rev. B 5, 2409–2419 (1972).
Mitrikas, G. & Prokopiou, G. Modulation depth enhancement of ESEEM experiments using pulse trains. J. Magn. Reson. 254, 75–85 (2015).
Baltisberger, J. H. et al. Communication: phase incremented echo train acquisition in NMR spectroscopy. Chem. Phys. 136, 211104 (2012).
Doll, A. & Jeschke, G. Wideband frequency-swept excitation in pulsed EPR spectroscopy. J. Magn. Reson. 280, 46–62 (2017).
Grimm, M. Quantum Coherence in Rare-earth Magnets. PhD thesis, ETH Zurich (2023).
Bergli, J., Galperin, Y. M. & Altshuler, B. L. Decoherence in qubits due to low-frequency noise. New J. Phys. 11, 025002 (2009).
Schweiger, A. and Jeschke, G. Principles of Pulse Electron Paramagnetic Resonance 1st edn (Oxford Univ. Press, 2001).
Zemann, J. Crystal structures, 2nd edition. Vol. 1 by R. W. G. Wyckoff. Interscience Publishers (1965).
Holmes, L. M., Als-Nielsen, J. & Guggenheim, H. J. Dipolar and nondipolar interactions in LiTbF4. Phys. Rev. B 12, 180–190 (1975).
Acknowledgements
We thank Y. Polyhach for support with the spectrometer and M. Döbeli for Rutherford backscattering spectroscopy concentration measurements. We thank H. Sigg and J. Bailey for useful discussions. This work was financially supported by the Swiss National Science Foundation, grant nos. 200021_166271 (G.A. and M.M.) and P500PT_203179 (A.B.); Eidgenössische Technische Hochschule Zürich (grant no. ETH-48 16- 1 (G.J.)); and European Research Council under the European Union’s Horizon 2020 research and innovation programme HERO (grant agreement no. 810451 (G.A.)).
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A.B., M.G., R.T. and G.A. planned the experiments with inputs from all authors. S.G., G.M., M.M., G.J. and G.A. supervised the project. A.B., N.W. and R.T. adapted and operated the set-up. A.B. and N.W. performed the experiments and collected data. A.B., M.G. and N.W. analysed the data with inputs from all authors. M.G. and M.M. developed the theory and performed the simulations. A.B., M.G., M.M. and G.A. wrote the paper with input from all authors.
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Extended data
Extended Data Fig. 1 Experimental setup.
Microwave circuitry of the experimental setup (details see Doll & Jeschke in Methods), including a schematic of the microwave cavity. ‘AWG’ stands for the arbitrary waveform generator and ‘ADC’ for the analog to digital converter. The orientation of BDC and BAC are given with respect to the crystallographic axes of the sample.
Extended Data Fig. 2 Hahn-echo measurements of Tb3+ ions.
a Bloch sphere representation of \(\left\vert 0\right\rangle\) and \(\left\vert 1\right\rangle\), illustrating the action of a π/2 − pulse. b Hahn-echo pulse sequence. Following the initial π/2-pulse, a π-pulse is applied after a waiting time τ and the magnetic-moment-induced echo signal (red) is detected at 2τ. c Integrated echo area as a function of the external magnetic field Bz, measured at a carrier frequency of 27.75 GHz. Square π/2- and π-pulses of 12 and 24 ns duration were delayed by τ = 500 ns, respectively. The red line denotes Gaussian fits to the echo signal of the Iz = − 1/2 and − 3/2 HF states. At B3/2 fluctuators are more strongly aligned to the magnetic field which leads to a larger echo signal compared to B1/2.
Extended Data Fig. 3 Hahn-echo envelopes.
Data of Fig. 3e plotted against τ to highlight the approximately simple exponential (β = 0.9) decay of the nnn pair Hahn-echo signal (orange) as opposed to typical ions at x = 0.1% (purple) and x = 0.01% (green) with β = 1/2.
Extended Data Fig. 4 Tb–F oscillations in the CPMG experiment.
Data of Fig. 4c (right) plotted against τ3/2 = [t/(2N)]3/2 instead of t1/2, showing that the oscillations originating from Tb-F interactions fall on top of each other. The frequencies ωI match the original case (N = 1) treated by Mims34. and 2ωI for N > 1, as well as an increasing oscillation amplitude with increasing N as shown by Mitrikas et al.35.
Extended Data Fig. 5 Tb3+ - F− coupling.
a Hahn echo envelopes of the x = 0.1% crystal at 27.5 GHz with fits for a selection of fields Bz. b Extracted coupling strengths Bnn for nn coupling of F− ions to the Tb3+ ion. Parameters for three frequencies across B3/2 are shown. The black solid line is the theoretical prediction. c Fitted characteristic timescale T1/e. Detuning from the clock-condition increases the magnetic moments and thus decreases the coherence time. See main text for details. All errorbars denote fit uncertainty, extracted from the covariance matrix.
Extended Data Fig. 6 Closeup and noisefloor of the CPMG data.
a Linear scale of the typical ion data set shown in Fig. 4c. The dashed line indicates RMS noise floor. b Respective data for nnn pairs.
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Beckert, A., Grimm, M., Wili, N. et al. Emergence of highly coherent two-level systems in a noisy and dense quantum network. Nat. Phys. 20, 472–478 (2024). https://doi.org/10.1038/s41567-023-02321-y
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DOI: https://doi.org/10.1038/s41567-023-02321-y
