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# Spin squeezing of 1011 atoms by prediction and retrodiction measurements

## Abstract

The measurement sensitivity of quantum probes using N uncorrelated particles is restricted by the standard quantum limit1, which is proportional to $$1/\sqrt{N}$$. This limit, however, can be overcome by exploiting quantum entangled states, such as spin-squeezed states2. Here we report the measurement-based generation of a quantum state that exceeds the standard quantum limit for probing the collective spin of 1011 rubidium atoms contained in a macroscopic vapour cell. The state is prepared and verified by sequences of stroboscopic quantum non-demolition (QND) measurements. We then apply the theory of past quantum states3,4 to obtain spin state information from the outcomes of both earlier and later QND measurements. Rather than establishing a physically squeezed state in the laboratory, the past quantum state represents the combined system information from these prediction and retrodiction measurements. This information is equivalent to a noise reduction of 5.6 decibels and a metrologically relevant squeezing of 4.5 decibels relative to the coherent spin state. The past quantum state yields tighter constraints on the spin component than those obtained by conventional QND measurements. Our measurement uses 1,000 times more atoms than previous squeezing experiments5,6,7,8,9,10, with a corresponding angular variance of the squeezed collective spin of 4.6 × 10−13 radians squared. Although this work is rooted in the foundational theory of quantum measurements, it may find practical use in quantum metrology and quantum parameter estimation, as we demonstrate by applying our protocol to quantum enhanced atomic magnetometry.

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## Data availability

The datasets generated and analysed during this study are available from the corresponding authors upon reasonable request.

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## Acknowledgements

We thank M. Balabas for assistance in the vapour cell fabrication and V. Vuletić for discussions. This work is supported by the National Key Research Program of China under grants 2016YFA0302000 and 2017YFA0304204, and the NNSFC under grants 61675047 and 91636107. K.M. acknowledges support from the Villum Foundation. H.S. acknowledges financial support from a UK Royal Society Newton International Fellowship (NF170876).

## Author information

Authors

### Contributions

K.M., H.S. and Y.X. conceived the idea. H.B., J.D., S.J., X.L., P.L., I.N., E.E.M., H.S. and Y.X. designed the experiment, performed the measurements and analysed the data together with all other authors. K.-F.Z. helped with the fabrication and characterization of vapour cells. H.B., M.W. and H.S. carried out the theoretical analysis under K.M.’s supervision. H.B., H.S., K.M. and Y.X. wrote the manuscript with contributions from all other authors. H.S. and Y.X. supervised the project.

### Corresponding authors

Correspondence to Klaus Mølmer, Heng Shen or Yanhong Xiao.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks Julian Martinez-Rincon and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Magneto-optical resonance signal.

a, Spin response to an RF pulse. X and Y are the outputs of the lock-in amplifier, with a π/2 phase difference between them. $$R=\sqrt{{X}^{2}+{Y}^{2}}$$ is the demodulated amplitude. b, The associated Fourier transformation of the spin response signal. fLar is the centre frequency for demodulation, with the subscript ‘Lar’ representing ‘Larmor frequency’. f is the actual frequency of the signal before demodulation. f − fLar represents the frequency of the signal after demodulation, that is, at the lock-in amplifier output. Inset, energy levels of 87Rb. All the atoms are pumped into the F = 2, mF = −2 state, so that they are oriented along x. The magnetic field leads to a splitting of the magnetic sublevels by the Larmor frequency ΩL. A.U., arbitrary units.

### Extended Data Fig. 2 Measured photon shot noise with different probe powers.

Red circles are experimental data and the dashed line represents the linear fit of the data. PSN, photon shot noise.

### Extended Data Fig. 3 Spin noise versus atom number.

a, Effective coupling constant $${\tilde{\kappa }}^{2}$$ as a function of the number of atoms. The values of $${\tilde{\kappa }}^{2}$$ are derived from the spin noise of the thermal state. b, Spin noise of prepared CSS versus the number of atoms. The observed linear dependence proves that technical noise is mostly suppressed and the measured spin noise is at the projection noise limit (PNL).

### Extended Data Fig. 4 Calibration of the applied RF magnetic field.

a, Calibration using the displacement of atomic spins. b, Calibration using a small pickup coil. The amplitude of the RF output in our detection experiment is −97 dB m, which lies at the bottom left of the figure. In both curves, a slope near 1 indicates a good linear relation.

## Supplementary information

### Supplementary Information

This file contains Supplementary Information for Squeezed spin of 10^11 atoms by prediction and retrodiction measurements. The file includes the experimental detail and details on theory.

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Bao, H., Duan, J., Jin, S. et al. Spin squeezing of 1011 atoms by prediction and retrodiction measurements. Nature 581, 159–163 (2020). https://doi.org/10.1038/s41586-020-2243-7

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• DOI: https://doi.org/10.1038/s41586-020-2243-7

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