# Twisting the noise away

*This is a Perspective on "Ramsey interferometry with generalized one-axis twisting echoes" by Marius Schulte, Victor J. Martínez-Lahuerta, Maja S. Scharnagl, and Klemens Hammerer, published in Quantum 4, 268 (2020).*

**By Luca Pezzè (QSTAR, INO-CNR and LENS, Largo Enrico Fermi 2, 50125 Firenze, Italy).**

Published: | 2020-05-15, volume 4, page 36 |

Doi: | https://doi.org/10.22331/qv-2020-05-15-36 |

Citation: | Quantum Views 4, 36 (2020) |

### Entanglement-enhanced sensors

Improving the performance of sensors by engineering specific quantum correlations between probe particles is a research frontier in quantum technology. Outstanding experimental results have been obtained in the recent years [1], clearly demonstrating phase sensitivities overcoming the standard quantum limit, $\Delta \theta = 1/\sqrt{N}$, that is achieved with $N$ uncorrelated particles. However, noise sources pose notable challenges [1,2,3]. For instance, poor detection efficiency is one of the main limitations in many current experiments exploring entanglement-enhanced phase sensitivity.

Interestingly, the requirement of high detection efficiency can be relaxed when interaction between the probe particles is applied after the phase imprinting and before the readout measurement. This strategy — generally called** interaction-based readout (IBR)** [4,5,6,7,8] — can make phase estimation robust against detection noise, removing the requirement of single-particle resolution to reach the Heisenberg limit $\Delta \theta = 1/N$. The advantage offered by IBR has been demonstrated experimentally in atomic [9,10] and trapped ions [11,12] ensembles.

### Encoding phase in spin parity

The basic working principle of IBR is nicely illustrated by considering a Greenberger-Horne-Zeilinger (GHZ) state of $N$ qubits, $\vert {\rm GHZ} \rangle = (\vert \uparrow \rangle^{\otimes N} + \vert \downarrow \rangle^{\otimes N})/\sqrt{2}$ [11]. Following the standard readout procedure [13], after phase encoding, described by the transformation $e^{-i \hat{J}_z \theta}$ (where $\theta$ is the unknown phase to be determined), the state undergoes a collective rotation by $\pi/2$ around the $x$ axis, $e^{-i \tfrac{\pi}{2} \hat{J}_x}$, which converts the rotation angle $\theta$ into a measurable signal given by the spin population [14]. The probability to measure $n_\uparrow$ spin-up qubits is

\begin{equation} \label{P1}

P(n_\uparrow \vert \theta) = \big\vert \langle n_\uparrow \vert e^{-i \tfrac{\pi}{2} \hat{J}_x} e^{-i \hat{J}_z \theta} \vert {\rm GHZ} \rangle \big\vert^2 =

f_{n_{\uparrow},+}\cos^2 \frac{N\theta}{2} + f_{n_{\uparrow},-} \sin^2 \frac{N\theta}{2},

\end{equation}

where $f_{n_{\uparrow},\pm} = \binom{N}{n_{\uparrow}} \tfrac{1 \pm (-1)^{n_{\uparrow}}}{2^N}$ are binomial distributions in which odd (in the case of $f_{n_{\uparrow},+}$) or even (in the case of $f_{n_{\uparrow},-}$) values of $n_{\uparrow}$ are suppressed.

The information about $\theta$ is thus contained in the **parity** of $n_{\uparrow}$: the probability to detect an even [odd] value of $n_{\uparrow}$ is $\cos^2 (N\theta/2)$ [$\sin^2 (N\theta/2)$]. This fast-oscillating signal, with period $2\pi/N$, allows for an estimate of $\theta$ with a sensitivity at the Heisenberg limit [13].

### Twisting to amplify spin population differences

However, measuring the parity of $n_{\uparrow}$ requires a detection system with single-particle resolution in order to distinguish $n_{\uparrow}$ from $n_{\uparrow} \pm 1$: this is experimentally prohibitive, especially when $N$ is large. IBR overcomes this limitation by letting the qubits interact before readout. When replacing the the local transformation $e^{-i \tfrac{\pi}{2} \hat{J}_x} $ by the non-local **one-axis twisting (OAT)** evolution $e^{-i \tfrac{\pi}{2} \hat{J}_x^2}$ [15], the probability to measure $n_\uparrow$ spin-up qubits becomes

\begin{equation} \label{P2}

P(n_{\uparrow} \vert \theta) = \big\vert \langle n_\uparrow \vert e^{-i \tfrac{\pi}{2} \hat{J}_x^2} e^{-i \hat{J}_z \theta} \vert {\rm GHZ} \rangle \big\vert^2 = \delta_{n_{\uparrow}, N} \cos^2 \frac{N\theta}{2} + \delta_{n_{\uparrow}, 0} \sin^2 \frac{N\theta}{2},

\end{equation}

where $\delta_{n_{\uparrow},\nu}$ is the Kronecker delta.

Only two detection events are possible: the $N$ qubits are measured **all spin-up** ($n_{\uparrow} = N$) or **all spin-down** ($n_{\uparrow} = 0$), with probabilities $\cos^2(N\theta/2)$ and $\sin^2(N\theta/2)$, respectively. Two nearby values of $\theta$, such as $0$ and $\pi/N$, for instance, are “magnified” by the OAT evolution, leading to two detection events that can be distinguished without requiring single-particle resolution. Notice that OAT generates the state $\vert {\rm GHZ} \rangle$ from $N$ uncorrelated qubits, namely $\vert {\rm GHZ} \rangle = e^{i \tfrac{\pi}{2}\hat{J}_x^2} \vert \downarrow \rangle^{\otimes N}$ [11,16]: therefore, IBR realizes an **echo protocol** [4,6,10] where the unitary operation $e^{i \tfrac{\pi}{2}\hat{J}_x^2}$ that

entangles the $N$ qubits is inverted, by $e^{-i \tfrac{\pi}{2}\hat{J}_x^2}$, before readout.

### Optimizing that twist

The paper of Schulte at al. [17] investigates the optimal use of IBR based on twisting dynamics, where the optimization involves interaction strength as well as rotation and spin-projection directions, and addresses the different regimes of OAT evolution. This provides a comprehensive overview on the possibilities offered by OAT echo protocols, which apply to a variety of experimental systems [1,9]. An interesting prediction is that over-squeezed (or non-Gaussian) states produced with OAT can be detected using IBR, with error propagation predicting a sensitivity with Heisenberg scaling $\Delta \theta \sim 1/N$. Non-Gaussian spin states have been only partially explored experimentally [18], and Ref. [17] provides a feasible and robust approach to the optimal detection and use of these states [19].

In a broader context, the work of Schulte at al. [17], belongs to a significant research direction aimed at strengthening entanglement-enhanced sensing against experimental noise and imperfections, making the Heisenberg limit less elusive. In addition to IBR, interesting possibilities are also offered, for instance, by decoherence-free protocols [21,22], error correction techniques [23,24,25] and the use of robust probes states [20].

### ► BibTeX data

### ► References

[1] L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Quantum metrology with nonclassical states of atomic ensembles, Rev. Mod. Phys. 90, 035005 (2018) 10.1103/RevModPhys.90.035005.

https://doi.org/10.1103/RevModPhys.90.035005

[2] B. M. Escher, R. L. de Matos Filho, and L. Davidovich. General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology. Nat. Phy. 7, 406 (2011) 10.1038/nphys1958.

https://doi.org/10.1038/nphys1958

[3] R. Demkowicz-Dobrzański, J. Kolodyѝski, and M. Guta. The elusive Heisenberg limit in quantum-enhanced metrology, Nat. Comm. 3, 1063 (2012) 10.1038/ncomms2067.

https://doi.org/10.1038/ncomms2067

[4] E. Davis, G. Bentsen, and M. Schleier-Smith, Approaching the Heisenberg Limit Without Single-Particle Detection, Phys. Rev. Lett. 116, 053601 (2016) 10.1103/PhysRevLett.116.053601.

https://doi.org/10.1103/PhysRevLett.116.053601

[5] F. Fröwis, P. Sekatski, and W. Dür, Detecting large quantum Fisher information with finite measurement precision, Phys. Rev. Lett. 116, 090801 (2016) 10.1103/PhysRevLett.116.090801.

https://doi.org/10.1103/PhysRevLett.116.090801

[6] T. Macrí, A. Smerzi, and L. Pezzè, Loschmidt echo for quantum metrology, Phys. Rev. A 94, 010102 (2016) 10.1103/PhysRevA.94.010102.

https://doi.org/10.1103/PhysRevA.94.010102

[7] S. P. Nolan, S. S. Szigeti, and S. A. Haine, Optimal and robust quantum metrology using interaction-based readouts. Phys. Rev. Lett. 119, 193601 (2017) 10.1103/PhysRevLett.119.193601.

https://doi.org/10.1103/PhysRevLett.119.193601

[8] F. Anders, L. Pezzè, A. Smerzi, and C. Klempt. Phase magnification by two-axis countertwisting for detection-noise robust interferometry. Phys. Rev. A 97, 043813 (2018) 10.1103/PhysRevA.97.043813.

https://doi.org/10.1103/PhysRevA.97.043813

[9] O. Hosten, R. Krishnakumar, N. J. Engelsen, and M. A. Kasevich. Quantum phase magnification. Science 352 1552 (2016) 10.1126/science.aaf3397.

https://doi.org/10.1126/science.aaf3397

[10] D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. J. Lewis- Swan, K. V. Kheruntsyan, and M. K. Oberthaler, Quantum-Enhanced Sensing Based on Time Reversal of Nonlinear Dynamics, Phys. Rev. Lett. 117, 013001 (2016) 10.1103/PhysRevLett.117.013001.

https://doi.org/10.1103/PhysRevLett.117.013001

[11] D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, Creation of a six-atom ‘Schrödinger cat’ state, Nature 438, 639 (2005) 10.1038/nature04251.

https://doi.org/10.1038/nature04251

[12] S. C. Burd, R. Srinivas, J. J. Bollinger, A. C. Wilson, D. J. Wineland, D. Leibfried, D. H. Slichter, and D. T. C. Allcock, Quantum amplification of mechanical oscillator motion, Science 364, 1163 (2019) 10.1126/science.aaw2884.

https://doi.org/10.1126/science.aaw2884

[13] J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, Optimal frequency measurements with maximally correlated states, Phys. Rev. A 54, R4649 (1996) 10.1103/PhysRevA.54.R4649.

https://doi.org/10.1103/PhysRevA.54.R4649

[14] Here, $\hat{J}_{x,yz} = \sum_{k=1}^N \hat{\sigma}_{x,y,z}^{(k)}/2$ are collective spin operators, $\hat{\sigma}_{x,y,z}^{(k)}$ are Pauli matrices for particle $k$ and we consider, without loss of generality, even values of $N$.

[15] M. Kitagawa and M. Ueda, Squeezed spin states, Phys. Rev. A 47, 5138 (1993) 10.1103/PhysRevA.47.5138.

https://doi.org/10.1103/PhysRevA.47.5138

[16] Here and in Eq. (2) we consider a GHZ state with a $\pi/2$ relative phase, $\vert {\rm GHZ} \rangle = (\vert \uparrow \rangle^{\otimes N} + i\vert \downarrow \rangle^{\otimes N})/\sqrt{2}$.

[17] Marius Schulte, Victor J. Martínez-Lahuerta, Maja S. Scharnagl, and Klemens Hammerer, Ramsey interferometry with generalized one-axis twisting echoes, Quantum 4, 268 (2020) 10.22331/q-2020-05-15-268.

https://doi.org/10.22331/q-2020-05-15-268

[18] H. Strobel et al. Fisher information and entanglement of non-Gaussian spin states, Science 345, 424 (2014) 10.1126/science.1250147.

https://doi.org/10.1126/science.1250147

[19] M. Gessner, A. Smerzi, and L. Pezzè, Metrological nonlinear squeezing parameter, Phys. Rev. Lett. 122, 090503 (2019) 10.1103/PhysRevLett.122.090503.

https://doi.org/10.1103/PhysRevLett.122.090503

[20] M. Oszmaniec, R. Augusiak, C. Gogolin, J. Kolodyński, A. Acín, and M. Lewenstein, Random Bosonic States for Robust Quantum Metrology, Phys. Rev. X 6, 041044 (2016) 10.1103/PhysRevX.6.041044.

https://doi.org/10.1103/PhysRevX.6.041044

[21] C. F. Roos, M. Chwalla, K. Kim, M. Riebe and R. Blatt, ‘Designer atoms’ for quantum metrology, Nature 443, 316 (2006) 10.1038/nature05101.

https://doi.org/10.1038/nature05101

[22] M. Landini, M. Fattori, L Pezzè, and A,. Smerzi, Phase-noise protection in quantum-enhanced differential interferometry, New J. Phys. 16, 113074 (2014) 10.1088/1367-2630/16/11/113074.

https://doi.org/10.1088/1367-2630/16/11/113074

[23] W. Dür, M. Skotiniotis, F. Fröwis, and B. Kraus, Improved quantum metrology using quantum error correction, Phys. Rev. Lett. 112, 080801 (2014) 10.1103/PhysRevLett.112.080801.

https://doi.org/10.1103/PhysRevLett.112.080801

[24] E. M. Kessler, I. Lovchinsky, A. O. Sushkov, M. D. Lukin, Quantum Error Correction for Metrology, Phys. Rev. Lett. 112, 150802 (2014) 10.1103/PhysRevLett.112.150802.

https://doi.org/10.1103/PhysRevLett.112.150802

[25] S. Zhou, M. Zhang, J. Preskill, and L. Jiang, Achieving the Heisenberg limit in quantum metrology using quantum error correction Nature Communications 9, 78 (2018) 10.1038/s41467-017-02510-3.

https://doi.org/10.1038/s41467-017-02510-3

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