DOI: 10.5061/dryad.sbcc2frgx
title: Data from: Approaching the standard quantum limit of a Rydberg-atom microwave electrometer

description: The development of a microwave electrometer with inherent uncertainty
 approaching its ultimate limit carries both fundamental and technological
 significance. However, due to the thermal motion of atoms, the
 state-of-the-art Rydberg electrometer falls considerably short of the
 standard quantum limit by about three orders of magnitude. Here, we
 utilize an optically thin medium with approximately 5.2×105 laser-cooled
 atoms to implement the microwave heterodyne detection. By mitigating
 various noises and strategically optimizing the electrometer parameters,
 our study reduces the equivalent noise temperature by a factor of 20 and
 achieves an electric-field sensitivity of 10.0nVcm-1Hz-1/2, finally
 reaching a factor of 2.6 above the standard quantum limit. Our work also
 provides valuable insights into the inherent capabilities and limitations
 of Rydberg electrometers, offering superior sensitivity in detecting weak
 microwave signals for numerous applications.
description: This is the main experimental data of this experimental work,
 including atomic spectral line diagrams, time-domain waveform diagrams
 captured by an oscilloscope, and spectral density diagrams obtained after
 digital Fourier transformation. It also includes indirect results such as
 sensitivity obtained from spectral density, with specific details detailed
 in the relevant papers. The theoretical simulation results provided in the
 paper can be calculated based on the calculation conditions and equations
 given in the article.
description: # Approaching the standard quantum limit of a Rydberg-atom microwave
 electrometer
 [https://doi.org/10.5061/dryad.sbcc2frgx](https://doi.org/10.5061/dryad.sbcc2frgx) ## Description of the data and file structure The photovoltage is recorded by a digital acquisition oscilloscope (R&S RTE1024), and the amplitude spectral density for MW sensitivity is analysed using a desktop computer. For measurements within the 3-dB bandwidth, an avalanche photodetector (Thorlabs APD130A) with a bandwidth replaces the photodetector. We conduct a discrete Fourier transform (DFT) on the photovoltage to extract the frequency and amplitude data, where a Blackman window is applied in the heterodyne time. The extracted frequency represents the intermediate frequency of the cold Rydberg-atom receiver, whilst the extracted amplitude signifies the strength of the heterodyne signal. In order to correct the amplitude of the windowed signal during the DFT, we utilize the calibration information from the balanced photodetector to determine the noise equivalent voltage (NEV). Once the amplitude correction factor is known, the photovoltage spectral density is accurately determined. The file `Fig3data.xlsx` are Microwave heterodyne detection results data. The four subtables in the file correspond to A, B, C, and D of FIG 3.  A: Time traces of the continuous MWs and square-pulsed control lasers. When the signal MW is turned off (upper), a slowly rising DC signal is evident in the probe laser transmission, indicating the decreasing OD of the atomic cloud during the detection period. With the signal MW turned on (lower), the time trace similarly displays an additional oscillation, the peak-to-peak amplitude of which is double that of the heterodyne sensing signal. B: EIT (red circle) and AT-splitting (blue triangle) spectra at a moderate probe power. The profile is measured by scanning the probe frequency within 100𝜇s (24) and the asymmetry results from the atomic heating and free expansion effects. C: Heterodyne signal amplitude versus the MW electric field 𝐸cal , calibrated with the standard antenna method. The dotted line is the linear fit. D: Sensitivity spectra, normalized to a 1 kHz resolution bandwidth, are measured for the photodetector noise (dark grey curve), the optical readout noise (red curve), and the overall noise when a signal MW at 10 kHz intermediate frequency is turned on (green curve) or turned off (blue curve). The red curve is measured by removing the atomic cloud and maintaining the probe laser beam in the same condition as the overall noise measurement. The black dashed line denotes the photodetector's noise floor level, which assists in calibrating the spectral density of the photovoltage during the DFT process. The file `Fig4data.xlsx` are shot-noise limited sensitivity data. The two subtables in the file correspond to A and B of FIG 4.  A: Sensitivity S versus the incident power 𝑃0 of the probe beam. The black dotted line represents the PSN limited sensitivity. The inset illustrates the dependence of the measured atom number on 𝑃0. B: Sensitivity S against the number of atoms at 𝑃0=7.6 𝜇W (the vertical dashed lines in A). The blue curve represents the theoretical prediction according to Eq. (4). The two dotted lines symbolize the single and triple SQL sensitivities, respectively, acquired from Eq. (1). The dashed green line exhibits the 𝑇eq =293 K sensitivity constraint S𝑡ℎ of an optimally functioning half-wave dipole antenna within the same frequency band. The dash-dotted yellow line signifies the sensitivity bound by the thermal background ( 𝑇amb =293 K ), correlating to NEFex=3.2 nVcm −1 Hz−1/2. The file `Subfig3data.xlsx` are energy-level shift and depahsing in an interacting Rydberg ensemble  data. The two subtables in the file correspond to B and C of FIG.S3.  B: Interaction-induced dephasing rate 𝛾𝑟 versus the probe Rabi frequency Ω𝑃. The solid curve is the theoretical results, and the shaded zone represents the error band generated by the 5% uncertainty of optical depth and the anisotropy of 𝐶6 interaction of the pairstate 39D5/2. The probe Rabi frequency of our experiment is indicated with the vertical dotted line. The inset depicts transmission spectrum measured at the optimal point of atomic heterodyne, and the red curve is the fit of experimental data to the approximate susceptibility. C: Dephasing rate 𝛾𝑟 versus Ω𝑃 when the different Rydberg states n𝐷5/2 are used. The file `Subfig4data.xlsx` are minimum detectable field Emin and bandwidth data. The two subtables in the file correspond to A and B of FIG.S4. A: 𝐸min  versus the integration time 𝑇′. B:  Normalized signal amplitude versus the MW detuning 𝛿s. The dotted curve is the Lorentz fit, indicating a 3−dB reduction in response at 2.3 MHz.