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# Beating the classical precision limit with spin-1 Dicke states of more than 10,000 atoms

Edited by Jun Ye, National Institute of Standards and Technology, Boulder, CO, and approved May 8, 2018 (received for review August 27, 2017)

## Significance

Entanglement is central to studies in foundations of quantum mechanics, quantum information, and precision measurement. Among the variety of multipartite entangled states, Dicke states form an important class, and their realizations attract widespread interest. Most of the Dicke states produced to date are limited to pseudospin-1/2 (two-level) particles. This work reports the generation of balanced Dicke states comprising spin-1 (three-level) atoms and the subsequent demonstration of enhanced interferometric sensitivity over the standard quantum limit facilitated by them. We expect it will stimulate both experimental and theoretical research efforts on entangled states of higher-spin particles.

## Abstract

Interferometry is a paradigm for most precision measurements. Using N uncorrelated particles, the achievable precision for a two-mode (two-path) interferometer is bounded by the standard quantum limit (SQL), *M* modes. Higher precision can also be achieved using entangled particles such that quantum noises from individual particles cancel out. In this work, we demonstrate an interferometric precision of

- spin-1 Dicke state
- standard quantum limit
- three-mode interferometry
- quantum entanglement
- spinor Bose–Einstein condensate

Since it was introduced by Dicke in an effort to effectively explain superradiance in 1954 (1), Dicke state has attracted widespread attention for its potential applications in quantum information and precision measurement (2, 3). For a collection of N identical (pseudo-) spin-1/2 particles, Dicke states map onto Fock states

Dicke states are not limited to an ensemble of spin-*A*). Dicke states with

This article reports the generation of spin-1 Dicke states in the close vicinity of *Materials and Methods*). Using the prepared states, we demonstrate enhanced measurement precision beyond the SQL of three-mode interferometry.

## Generation of Spin-1 Dicke State Through QPT

In the absence of external electromagnetic fields and when the density-dependent spin-symmetric interaction dominates such that the same spatial wave function can be assumed for all spin components (^{87}Rb spin-1 BEC is ferromagnetic (33) with *Materials and Methods*).

To prepare the balanced spin-1 Dicke state *B*), with

The competition between spin-dependent interaction *C*). When

Our experiment typically starts with a condensate of *Materials and Methods*). The value of q is first linearly ramped to *C*, *Inset*); excitation is therefore unavoidable over the finite ramp time given the limited condensate lifetime of *D*, *Inset*) is optimized first by numerical simulations, and then fine-tuned experimentally (*Materials and Methods*). At the end of the ramp, the condensate is released from the optical trap and subjected to a pulsed gradient magnetic field, after which spin-resolved atomic populations

The evolution of the normalized populations, *D*. The experimental results, plotted as markers with error bars, are found to be in excellent agreement with theoretical expectations, in solid lines for the mean values and gray shaded regions for the standard deviations (SD), based on solving the Hamiltonian with the experimentally adopted ramping profile (23). In the first 425 ms of the ramp, before q reaches the QCP, the quadratic Zeeman energy prevails and very few atoms are observed in *Materials and Methods*).

## Beating the SQL Using Spin-1 Dicke State

The prepared spin-1 Dicke states enable a measurement precision beyond the three-mode SQL. The measurement sequence to show this is analogous to that applied in ref. 4 to a spin-1/2 twin-Fock state, but involves all three *A*). The rotation angle θ is then estimated from the second moment of the measured *B*, red filled circles), which depends on θ as *C*) and *D*) are fitted with polynomials of *SI Appendix*). Using the fitting results, an estimation of the interferometric sensitivity is obtained (Fig. 2*E*). The optimal measured sensitivity is found to lie at *Materials and Methods*).

The experimentally achievable best interferometric sensitivity of the prepared states is limited by *SI Appendix*). Here, *A*, *Left*). Although nonzero, this measured value is much smaller than the transverse spin uncertainty or the quantum shot noise (QSN) of

The effective spin length *A*) (*Materials and Methods*). Fig. 3*A*, *Right* shows the histogram of the measured *Materials and Methods*). With such a nearly perfect coherence, the optimal achievable phase squeezing *SI Appendix*). This value agrees with the observed phase squeezing of *B*).

We now contrast our results with three related works based also on spin-mixing dynamics of ^{87}Rb BEC (4, 38, 39). In ref. 38, a squeezed vacuum state with mean occupation of 0.75 atoms is prepared in the

## Benchmarking the Prepared Spin-1 Dicke State

We now characterize the quality of the state we prepare in comparison with the ideal balanced Dicke state *C* shows the distribution of the prepared states after (*A*, *Right*). Its main features can be understood by considering the ideal state *SI Appendix*). The experimental results show a similar structure but with a larger width. The discrepancy stems from populating the excited Dicke states *SI Appendix*). The reliability of our analysis is further confirmed in another set of experiments using a linear (less adiabatic) ramp of q (Fig. 3*D*), which show even broader distributions due to expected higher excitations.

We find that the observed peak-to-peak spread of *SI Appendix*). For the nonlinear ramp, we infer the highest excitation with *E*).

In conclusion, we report the deterministic preparation of high-quality balanced spin-1 Dicke states, by driving a condensate of ^{87}Rb atoms through a QCP. The prepared states are used to demonstrate a rotation measurement sensitivity of

## Materials and Methods

### Main Experimental Sequence.

A condensate of about 1.2×10^{4 87}Rb atoms in the 5-s

### Calibrating q and c 2 .

The effective quadratic Zeeman shift ^{87}Rb atoms. Here q varies linearly with the microwave power according to a setup-specified slope that is precisely calibrated as in ref. 23. Another important parameter

### Ramping Profile.

The ramping profile of q we use is designed with the main aim of minimizing the excitation of the system and atom loss. The basic idea is to ramp slower across the QCP, where the energy gap is the smallest and where excitations occur most easily. For our case, the ramping profile is optimized based on the following form:**1**, i.e., the steepness of the tangent function β and the start point of the ramp **1**, which is optimized experimentally. Eventually, the parameters we use are

### Measurement of L e ff 2 .

The squared effective spin length is

### Balanced Spin-1 Dicke State and Coherent Spin State.

The balanced spin-1 Dicke state can be represented as a superposition of coherent spin states equally distributed on the equator of the Bloch sphere,

### Squeezing Limitations.

The number squeezing is mainly limited by detection noise and atom loss. The largest contribution is the atom-independent detection noise which arises mostly from the photon shot noise of the probe light and amounts to

### SQL of a M-Mode Interferometer.

The optimal phase sensitivity for an interferometer is given by

### Phase Sensitivities for Dicke State and Polar State.

The QFI for the Dicke state

### Operators Versus Measured Values.

In this report, A without a hat represents measured values of the operator Â.

## Acknowledgments

This work is supported by National Natural Science Foundation of China (NSFC) (Grants 91421305, 91636213, 11654001, 91736311, and 11574177) and by the National Key Basic Research Program of China (Grants 2014CB921403 and 2018YFA0306504).

## Footnotes

↵

^{1}Y.-Q.Z. and L.-N.W. contributed equally to this work.↵

^{2}Present address: Max-Planck-Institut für Physik Komplexer Systeme, Dresden 01187, Germany.↵

^{3}Present address: Max-Planck-Institut für Quantenoptik, Garching 85748, Germany.- ↵
^{4}To whom correspondence may be addressed. Email: lyou{at}mail.tsinghua.edu.cn or mengkhoon_tey{at}mail.tsinghua.edu.cn.

Author contributions: M.K.T. and L.Y. designed research; Y.-Q.Z., L.-N.W., Q.L., X.-Y.L., S.-F.G., J.-H.C., M.K.T., and L.Y. performed research; Y.-Q.Z., L.-N.W., and M.K.T. analyzed data; and Y.-Q.Z., L.-N.W., M.K.T., and L.Y. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1715105115/-/DCSupplemental.

Published under the PNAS license.

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