DOI: 10.5061/dryad.7h44j100j
title: Data for: Observing dynamical phases of BCS superconductors in a cavity QED simulator

description: In conventional Bardeen-Cooper-Schrieffer (BCS) superconductors, electrons
 with opposite momenta bind into Cooper pairs due to an attractive
 interaction mediated by phonons in the material. While superconductivity
 naturally emerges at thermal equilibrium, it can also emerge out of
 equilibrium when the system's parameters are abruptly changed. The
 resulting out-of-equilibrium phases are predicted to occur in real
 materials and ultracold fermionic atoms but have not yet all been directly
 observed. Here we realise an alternate way to generate the proposed
 dynamical phases using cavity quantum electrodynamics (cavity 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 cavity. 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 subsequent dynamics of the
 superconducting order parameter using non-destructive measurements. We
 observe regimes where the order parameter decays to zero (phase I),
 assumes a non-equilibrium steady-state value (phase II), or exhibits
 persistent oscillations (phase III). 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 factor, and for increasing coherence time for quantum sensing.
description: # Data for: Observing Dynamical Phases of BCS Superconductors in a Cavity
 QED Simulator
 [https://doi.org/10.5061/dryad.7h44j100j](https://doi.org/10.5061/dryad.7h44j100j) This repository contains the data (both experimental and from simulations) plotted in the associated manuscript. ## Description of the data and file structure Files are named according to the following naming convention: `[ASSOCIATED FIGURE]_[DATA SOURCE]_[DESCRIPTION OF DATA].csv`. See the explanation below: * `[ASSOCIATED FIGURE]` references the figure of the manuscript where the data is plotted. This repository contains data from `fig2`, `fig3`, `fig4`, and `figed4`, which are the four figures containing experimental data. All of these figures contain sub-panels which are labeled alphabetically. For example, data from Fig. 3, panel f is referenced by `fig3f`. * `[DATA SOURCE]` specifies whether the data is derived from raw experimental data (`expt`) or from theoretical simulations (`simideal` or `simfull`). Note that many figure panels feature data from multiple sources. For clarity, we have uploaded data from different sources as separate files. We use two different models to simulate our system (see manuscript for a detailed description of how the models are defined). * `simideal` refers to so-called "ideal" simulations as described in the paper, which ignore single-particle dissipation and motion of our atoms along the cavity axis. * `simfull` refers to "full" simulations of the experiment, which attempt to capture all technical details of the experiment. * `[DESCRIPTION OF DATA]` is meant to be a short semantic identifier describing the type of data included in the file. We define the identifiers here: * Files labeled with `timetraces` contain time dynamics of the order parameter Δ<sub>BCS</sub>. The first column is time expressed in μs. The remaining columns describe Δ<sub>BCS</sub> for different values of our control parameters (χN, δ<sub>s</sub>, φ<sub>0</sub>). We indicate the value of any scanned parameters in the header text: for example, the header `chiN_0.109_MHz` indicates that this column was taken at χN = 2π \* 0.109 MHz. * `timestraces_finesample` files represent simulation data with a parameter sweep finer than what is realized in the experiment. These files are larger due to the many parameter values simulated. * Files labeled with `oscfreqs` or `oscamps` measure the oscillation frequency or amplitude of Δ<sub>BCS</sub>. These values are plotted against the relevant control parameter in the first column. * Files labeled with `avggap` measure the average value of Δ<sub>BCS</sub> at long times (as defined in the manuscript), plotted against the relevant control parameter. * Files labeled with `lifetime` measure the 1/e coherence time of Δ<sub>BCS</sub>. * Files labeled with `PSDs` describe the power spectra of Δ<sub>BCS</sub>. * Files labeled with `peaktrough_times` measure the detected time positions of the first peak and trough of curves in Extended Data Fig. 4. * Files labeled with `higgsfreqs` measure the oscillation frequency of transient "Higgs"-like behavior in Extended Data Fig. 4, plotted against the control parameter φ<sub>0</sub>. These frequencies are inferred from the peak and trough times from `peaktrough_times` files. * `higgsfreqs_gap` files also report the Higgs oscillation frequency, this time plotted against the quantity 2Δ<sub>∞</sub> as defined in the manuscript. *** The data is expressed in the same units present in the figures. For completeness, we describe the units here: * Time values are written in units of μs, indicated by the header label `"Time_us"`. * Frequency values are expressed in MHz and represent cycles/sec as opposed to angular units. These column headings are typically written like `"fosc_MHz"`. * The control parameter φ<sub>0</sub>, representing an initial phase spread of the atomic ensemble as defined in the manuscript, is reported in units of π as indicated in the header `phi0_pi`. For example, a reported value of 0.418 indicates that φ<sub>0</sub> = 0.418π. * Plots of the order parameter Δ<sub>BCS</sub> are typically unitless and normalized to the initial value. * The exception to this is in Extended Data Fig. 4a (`figed4a`), which plots Δ<sub>BCS</sub> as a frequency in MHz. * Data representing oscillation amplitudes in time traces of Δ<sub>BCS</sub> are also unitless and are normalized in the same way as Δ<sub>BCS</sub> itself. * The power spectral densities (PSDs) from `fig3d` are expressed in arbitrary units. *** Below, we describe how error bars are reported: * All error bars represent the standard error of the mean for the relevant quantity. * Error bars are reported a separate column and are typically indicated in the header with a `d` in front of the measured value. For example, the error bars for oscillation frequencies reported as `foscs_MHz` are labeled with a header `dfoscs_MHz`. * For some traces, we provide both statistical error bars and uncertainties to systematic corrections. These are reported as separate columns and are indicated by, for example, `dfoscs_stat_MHz` and `dfoscs_sys_MHz` respectively. * For some traces, we allow for asymmetric error bars. These are expressed with `l` or `r` characters, representing left-hand and right-hand errors respectively. For example: `dfl_MHz`, `dfr_MHz`.