Abstract
In conventional Bardeen–Cooper–Schrieffer superconductors1, electrons with opposite momenta bind into Cooper pairs due to an attractive interaction mediated by phonons in the material. Although superconductivity naturally emerges at thermal equilibrium, it can also emerge out of equilibrium when the system parameters are abruptly changed2,3,4,5,6,7,8. The resulting out-of-equilibrium phases are predicted to occur in real materials and ultracold fermionic atoms, but not all have yet been directly observed. Here we realize an alternative way to generate the proposed dynamical phases using cavity quantum electrodynamics (QED). Our system encodes the presence or absence of a Cooper pair in a long-lived electronic transition in 88Sr atoms coupled to an optical cavity and represents interactions between electrons as photon-mediated interactions through the cavity9,10. To fully explore the phase diagram, we manipulate the ratio between the single-particle dispersion and the interactions after a quench and perform real-time tracking of the subsequent dynamics of the superconducting order parameter using nondestructive measurements. We observe regimes in which the order parameter decays to zero (phase I)3,4, assumes a non-equilibrium steady-state value (phase II)2,3 or exhibits persistent oscillations (phase III)2,3. This opens up exciting prospects for quantum simulation, including the potential to engineer unconventional superconductors and to probe beyond mean-field effects like the spectral form factor11,12, and for increasing the coherence time for quantum sensing.
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Data availability
The datasets generated for this study are available in a Dryad repository with the identifier https://doi.org/10.5061/dryad.7h44j100j (ref. 55).
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Acknowledgements
This material is based upon work supported by the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator. We acknowledge additional funding support from the National Science Foundation under Grant Nos. 2317149 (Physics Frontier Center) and OMA-2016244 (Quantum Leap Challenge Institutes), the National Institute of Standards and Technology, the Army Research Office of the Defense Advanced Research Projects Agency (Grant Nos. W911NF-19-1-0210 and W911NF-16-1-0576) and the Air Force Office of Scientific Research (Grant Nos. FA9550-18-1-0319 and FA9550-19-1-0275). We acknowledge helpful discussions with E. Yuzbashyan, V. Gurarie and A. Kaufman.
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D.J.Y., E.Y.S., Z.N., V.M.S. and J.K.T. collected and analysed the experimental data. A.C., D.B., D.W., R.J.L.-S. and A.M.R. developed the theoretical model. All authors discussed the results and contributed to the preparation of the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 Experimental configuration.
a, Detailed diagram of the cavity and all relevant beams. A magnetic field along \(\widehat{y}\) sets the quantization axis. The 813 nm optical lattice supported by the cavity has a tunable linear polarization. We drive a π/2 pulse with a beam polarized along \(\widehat{y}\) through the cavity, and during the experiment we probe the cavity resonance frequency using a second \(\widehat{y}\)-polarized beam to measure atom number. A 461 nm beam far-detuned from the \({|}^{1}{S}_{0}\rangle \to {|}^{1}{P}_{1}\rangle \) transition shines on the atoms from the side of the cavity, inducing a.c. Stark shifts. We probe signals transmitted through the cavity using a balanced heterodyne detector. b, Fluorescence image of the two atomic clouds used when scanning through phase III in Figs. 3 and 4. c, Frequency landscape of 689 nm beams. The atomic drive frequency ωdrive is resonant with the atomic transition. The cavity probe frequency ωcp is nominally centred with the cavity resonance frequency, 51 MHz red-detuned from the atomic transition. The local oscillator used in heterodyne detection has frequency ωLO and is 80 MHz blue-detuned from the atomic transition.
Extended Data Fig. 2 Numerical simulation of the dynamical phase diagram based on equation (3).
We identify the dynamical phases based on the long-time average (a) and the long-time standard deviation (b) of ∣ΔBCS(t)∣, normalized by its initial value Δinit ≡ ∣ΔBCS(0)∣. The white solid lines mark the corresponding dynamical phase boundaries, analytically derived from equation (1), which agree with the numerical results based on equation (3). The white dashed lines mark an extra dynamical phase transition that only exists for equation (1).
Extended Data Fig. 3 Alternative approach for phase III.
a, Simulation of an alternative experimental sequence. As described by the timing sequence at the top, we simulate an experiment that prepares the initial state using a π/2 pulse, lets the system evolve under a bimodal distribution of single-particle energy (see the inset) until ∣ΔBCS∣ reaches its minimum value and then quenches the system back to a continuous distribution of single-particle energies (inset). The theoretically predicted time trace of ∣ΔBCS∣ with χN/EW = 1.0 and δs,init/EW = 1.6 is shown at the bottom. The blue (grey dashed) line shows phase III dynamics under a continuous (bimodal) distribution. b, Long-time standard deviation of ∣ΔBCS(t)∣ after quenching to the continuous distribution shown in a. The white lines are dynamical phase boundaries for bimodal distributions (see Extended Data Fig. 2). Nearly all choices of parameters for phase III using bimodal distributions can lead to phase III behaviour after quenching to the continuous distribution.
Extended Data Fig. 4 Collective scaling in damped phase II oscillations.
a, Time dynamics of ∣ΔBCS∣ measured after engineering an initial phase spread over [0, φ0] where φ0 = 0.8π as in Fig. 2d, plotted in absolute frequency units (pink trace). The solid black curve represents a numerical simulation of the full system, whereas the dashed curve represents an ideal simulation neglecting dissipation and motional effects. We obtain a crude estimate of oscillation frequency in the experimental data by fitting a trough and peak to smoothed data (after subtracting slow-moving behaviour) within the first couple μs (magenta points), using these points to infer a half period of oscillation, and with uncertainties determined using a 90% amplitude threshold (pink bands). b, Comparing oscillation frequency estimates of experimental data (pink squares) with those of ideal simulations (black dots) for different φ0. Theory oscillation frequencies are calculated using a Fourier transform from t = 0 μs to t = 5 μs. Error bars for experimental data are set by the minimum and maximum frequencies implied by uncertainties in the half period shown in a. The two frequency estimates agree within error bars. c, Collective scaling of oscillation frequency. For each φ0 measured in the experiment, we plot the oscillation frequency against the long-time BCS gap Δ∞, calculated at t = 18 μs for ideal simulations and at t = 3 μs for experimental data. The solid black line is defined by ωosc = 2Δ∞, demonstrating the expected scaling for Higgs oscillations. The dashed pink line represents a linear fit to the experimental data. The pink band shows the uncertainty in the slope assuming correlated error in ωosc, such that its bounds are defined by linear fits to the data assuming maximum and minimum values for ωosc as defined by the error bars.
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Supplementary Information sections 1–3, including Figs. 1–5 and Table 1.
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Young, D.J., Chu, A., Song, E.Y. et al. Observing dynamical phases of BCS superconductors in a cavity QED simulator. Nature 625, 679–684 (2024). https://doi.org/10.1038/s41586-023-06911-x
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DOI: https://doi.org/10.1038/s41586-023-06911-x
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