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Time-reversal-based quantum metrology with many-body entangled states

Nature Physics volume 18, pages 925–930 (2022)Cite this article

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Abstract

Linear quantum measurements with independent particles are bounded by the standard quantum limit, which limits the precision achievable in estimating unknown phase parameters. The standard quantum limit can be overcome by entangling the particles, but the sensitivity is often limited by the final state readout, especially for complex entangled many-body states with non-Gaussian probability distributions. Here, by implementing an effective time-reversal protocol in an optically engineered many-body spin Hamiltonian, we demonstrate a quantum measurement with non-Gaussian states with performance beyond the limit of the readout scheme. This signal amplification through a time-reversed interaction achieves the greatest phase sensitivity improvement beyond the standard quantum limit demonstrated to date in any full Ramsey interferometer. These results open the field of robust time-reversal-based measurement protocols offering precision not too far from the Heisenberg limit. Potential applications include quantum sensors that operate at finite bandwidth, and the principle we demonstrate may also advance areas such as quantum engineering, quantum measurements and the search for new physics using optical-transition atomic clocks.

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Fig. 1: Setup and sequence.
Fig. 2: The state evolution during the SATIN sequence.
Fig. 3: Signal amplification and metrological gain.
Fig. 4: The scaling of the sensitivity with the atom number and a comparison with previous results.

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Data availability

The datasets generated and analysed during this study are available from the corresponding authors upon reasonable request. Source data are provided with this paper.

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Acknowledgements

We thank B. Braverman, A. Kawasaki, M. Lukin and J. Ye for discussions. This work was supported by the NSF (grant no. PHY-1806765), DARPA (grant no. D18AC00037), ONR (grant no. N00014-20-1-2428), the NSF Center for Ultracold Atoms (CUA) (grant no. PHY-1734011) and NSF QLCI-CI QSEnSE (grant no. 2016244). S.C. and A.F.A. acknowledge support from the Swiss National Science Foundation (SNSF).

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Author notes

  1. Albert F. Adiyatullin

    Present address: Laboratoire PhLAM, Université de Lille, Lille, France

  2. These authors contributed equally: Simone Colombo, Edwin Pedrozo-Peñafiel, Albert F. Adiyatullin.

Authors and Affiliations

  1. Department of Physics, MIT-Harvard Center for Ultracold Atoms and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

    Simone Colombo, Edwin Pedrozo-Peñafiel, Albert F. Adiyatullin, Zeyang Li, Enrique Mendez, Chi Shu & Vladan Vuletić

  2. Department of Physics, Harvard University, Cambridge, Massachusetts, USA

    Chi Shu

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Contributions

S.C., E.P.-P., A.F.A. and Z.L. led the experimental efforts and simulations. S.C., E.P.-P., A.F.A. and Z.L. contributed to the data analysis. V.V. conceived and supervised the experiment. S.C., E.P.-P., A.F.A. and V.V. wrote the manuscript. All authors discussed the experimental implementation and the results, and contributed to the manuscript.

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Correspondence to Vladan Vuletić.

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Extended data

Extended Data Fig. 1 Experimental sequence.

The three stages of the experiment are represented in different color-shaded areas in the upper part: preparation, protocol, and detection. Bloch spheres (bottom) show the collective atomic state after the indicated process has been performed in time. The time axis is not to scale.

Extended Data Fig. 2 Single-atom cooperativity.

The dashed line represents the quadratic regression fit, with the prefactor of the linear term yielding the slope \(\eta (1+{\sigma }_\mathrm{meas}^{2})/4=2.2\pm 0.1\) and the quadratic term prefactor being consistent with zero (0 ± 3) × 10−4. The gray band denotes the expected projection noise given by the single-atom cooperativity calculated from our cavity parameters η = 7.8 ± 0.2. The linearity of the data indicates the classical sources of noise are negligible in the CSS state preparation, since they would manifest as a quadratic dependence of the measured variance on collective cooperativity Nη. Each data point corresponds to the mean value obtained from 50 to 150 experimental realizations. The error bars correspond to 1 σ and are the standard error of a Gaussian distribution: \({\sigma }_{s}^{2}/(n-1)\), where σs is the sample variance and n is the number of experimental realizations. Statistical horizontal error bars are smaller than the marker size.

Extended Data Fig. 3 Squeezed spin distribution on the generalized Bloch sphere.

The normalized shearing strength \(\tilde{Q}\) represents the angle subtended by the sheared distribution with respect to the x-axis, along which the initial CSS was prepared.

Extended Data Fig. 4 Relevant parameters for our squeezing protocol.

Excess broadening \({{{\mathcal{I}}}}\) (yellow dashed line) per scattered photon and shearing strength \(| \tilde{Q}|\) (solid line). The red and blue parts of the solid line represent positive and negative values of \(\tilde{Q}\), respectively, which lead to forward and backward evolution in time. The blue and red dashed lines represent the detuning, − 8 MHz and 8 MHz, chosen to generate \({\tilde{Q}}_{+}\) and \({\tilde{Q}}_{-}\), respectively. The duration of the entangling/disentangling pulses is about 4 ms. In this figure, we have used the experimental parameters N = 220 and η = 7.7. For illustration purposes, contrast loss has not been included.

Extended Data Fig. 5 SATIN contrast loss.

Contrast reduction in an optimized SATIN protocol as a function of the twisting strength \({\tilde{Q}}_{+}\). “Optimized SATIN” means that the entangling light detuning is chosen in order to maximize the protocol’s metrological gain. It is worth noting that, under this optimization condition, the contrast reduction is independent of the atom number.

Extended Data Fig. 6 Graphical representation of the model used to describe contrast loss due to scattering of photons into free space.

The atomic spin can be decomposed into the coherent signal (large Bloch sphere) and the signals of the sub-ensembles of atoms that due to photon scattering have been projected into the spin states \(\left|\uparrow \right\rangle\) or \(\left|\downarrow \right\rangle\), and that have lost any coherence. The radii of the generalized Bloch-spheres represent the relative populations of the states. The dashed circle indicates the size of the generalized Bloch sphere size with all spin being in the same pure state, i.e., in the absence of contrast losses. The left figure corresponds to \({\tilde{Q}}_{+}=0.3\) (mostly Gaussian distribution) while the right figure is calculated for \({\tilde{Q}}_{+}=1.3\).

Extended Data Fig. 7 Expected Metrological Gain in Optical Clocks.

Dashed lines represent fundamental quantum limits. Solid lines represent metrological gain versus atom number expected in our system with a short Ramsey time of 3.5 ms (black) and an atom loss rate of 0.1s−1., and in a state-of-the-art optical clock operated with a SATIN protocol for Ramsey times of 50 ms (red) and 1 s (blue). We notice that the transition between HS to SQL scaling occurs in the region of N = 4 × 104 and N = 300, respectively.

Source data

Source Data Fig. 2

Raw data for Fig. 2.

Source Data Fig. 3

Raw data for Fig. 3.

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Colombo, S., Pedrozo-Peñafiel, E., Adiyatullin, A.F. et al. Time-reversal-based quantum metrology with many-body entangled states. Nat. Phys. 18, 925–930 (2022). https://doi.org/10.1038/s41567-022-01653-5

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