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# Nonlinear interferometry beyond classical limit enabled by cyclic dynamics

## Abstract

Time-reversed evolution has substantial implications in physics, including applications in refocusing of classical waves or spins and fundamental studies such as quantum information scrambling. In quantum metrology, nonlinear interferometry based on time-reversal protocols supports entanglement-enhanced measurements without requiring low-noise detection. Despite the broad interest in this topic, it remains challenging to reverse the quantum dynamics of an interacting many-body system, which is typically realized by an (effective) sign flip of the system’s Hamiltonian. Here we present an approach that is broadly applicable to cyclic systems for implementing nonlinear interferometry without invoking time reversal. As time-reversed dynamics drives a system back to its starting point, we propose to accomplish the same by forcing the system to travel along a ‘closed loop’ instead of explicitly tracing back its antecedent path. Utilizing the quasiperiodic spin mixing dynamics in a three-mode 87Rb atomic spinor condensate, we implement such a closed-loop nonlinear interferometer and achieve a metrological gain of $$5.0{1}_{-0.76}^{+0.76}$$ decibels over the classical limit for a total of 26,500 atoms. Our approach unlocks the potential of nonlinear interferometry by allowing the dynamics to penetrate into the deep nonlinear regime, which gives rise to highly entangled non-Gaussian states.

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## Data availability

Source data (including raw experimental data) are provided with this paper. Source data for Supplementary Figs. 5 and 6 are given in Supplementary Data 1 and 2. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

## Code availability

The codes are available from the corresponding author upon reasonable request.

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## Acknowledgements

We thank F. Chen, Y. Q. Zou, J. L. Yu and M. Xue for helpful discussions. This work is supported by the National Natural Science Foundation of China (NSFC; grant numbers 11654001, U1930201, 91636213 and 91836302), by the Key-Area Research and Development Program of GuangDong Province (grant number 2019B030330001) and by the National Key R&D Program of China (grant nnumbers 2018YFA0306504 and 2018YFA0306503). L.-N.W. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG) via grant no. 163436311.

## Author information

Authors

### Contributions

L.-N.W. and L.Y. conceived this study. Q.L., J.-H.C., T.-W.M. and S.-F.G. performed the experiment and analysed the data. Q.L., L.-N.W. and X.-W.L. conducted the numerical simulations. Q.L., L.-N.W., M.K.T. and L.Y. wrote the paper.

### Corresponding author

Correspondence to Li You.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature Physics thanks Paul Griffin 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

### Extended Data Fig. 1 Calibration of quadratic Zeeman shift qt for phase imprinting.

For qt much larger than spin-exchange rate c2, the state will accumulate a spinor phase proportional to interrogation time τ without undergoing noticeable population changes in the three Zeeman sublevels. Following the short-term spin dynamics for 5 ms at q ≈ c2, the encoded phase is transformed into oscillations of 〈ρ0(τ)〉 at frequency 2qt. Blue (orange) dots denote data points measured at microwave detuning of Δ = − 3.1(3.1) MHz for the ‘dressing’ (‘pre-dressing’) pulse. Fitting them with sinusoidal function Eq. (2) in Methods enables the value of qt to be determined. Solid lines display the corresponding fitting curves.

### Extended Data Fig. 2 Implementation of the three-mode linear interferometer.

a, The experimental sequence consists of four steps: initialization of the system in the polar state, linear path splitting with π/4 Rabi rotation around collective spin Lx, phase encoding with generator − N0/2 (N0 denotes the number of atoms in the $$\left|1,0\right\rangle$$ component), and linear path recombining with π/2 rotation around − Ly. The dependence of magnetization Lz/N on the phase ϕ is measured in the end. b, Data in the main figure shows results averaged over three measurements, while the solid line is the fitted sinusoidal function curve. Two horizontal black dashed lines denote the bounds of ± 1. The inset presents measured mean value and fluctuation of Lz/N in the vicinity of ϕ = 0 from 50 repetitions. Error bars (invisible in the main figure) denote one standard deviation and the spread of shaded area corresponds to the quantum projection noise for ideal atomic coherent state. Solid line in the inset denotes the result from a linear fit, which is used to calculate the metrological gain.

## Supplementary information

### Supplementary Information

Supplementary Figs. 1–6 and Discussion.

### Supplementary Data 1

Experimental data in Supplementary Fig. 5.

### Supplementary Data 2

Experimental data in Supplementary Fig. 6.

## Source data

### Source Data Fig. 1

Statistical source data

### Source Data Fig. 2

Statistical source data

### Source Data Fig. 3

Statistical source data

### Source Data Extended Data Fig. 1

Statistical source data

### Source Data Extended Data Fig. 2

Statistical source data

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Liu, Q., Wu, LN., Cao, JH. et al. Nonlinear interferometry beyond classical limit enabled by cyclic dynamics. Nat. Phys. 18, 167–171 (2022). https://doi.org/10.1038/s41567-021-01441-7

• Accepted:

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• DOI: https://doi.org/10.1038/s41567-021-01441-7

• ### Measure in circles

• Philipp Kunkel
• Monika Schleier-Smith

Nature Physics (2022)