Butterfly Echo Protocol for Axis-Agnostic Heisenberg-Limited Metrology

Abstract

The extreme sensitivity of chaotic systems to external perturbations makes them natural candidates for sensing applications. We propose a single-shot echo-based protocol for estimating small rotations about an unknown axis that leverages random symmetric probe states prepared via chaotic dynamics. In contrast to previous protocols for this axis-agnostic rotation sensing problem that depend on difficult-to-prepare anticoherent states, the random probe states used in our protocol can be prepared via constant-depth chaotic circuits composed of random one-axis twisting pulses. We demonstrate analytically that our protocol achieves Heisenberg scaling relative to an arbitrary rotation axis that need not be a priori known. We also investigate the effects of collective and single-particle dephasing in our protocol using analytical and numerical tools. While the requirements on dephasing rates to maintain Heisenberg sensitivity are strict, they are achievable in near-term experiments, for instance, for magnetometric rotosensing with high-spin lanthanide atoms such as dysprosium-164.

Figures (12)

• Figure 1: The butterfly echo protocol reveals information about a rotation angle $\theta$ with Heisenberg-scaling without revealing information about an arbitrary, a priori unknown, rotation axis $\hat{n}$ (a). An initially spin-polarized state (b) is scrambled by chaotic dynamics $U$ into a random probe state (c). Absent a rotation (d) the probe state returns to its original spin-polarized state (e) under time reversal $U^{\dagger}$. By contrast, a nonzero rotation (f) leads to imperfect time-reversal and a commensurate reduction in spin polarization $\langle S_z \rangle$. Bloch spheres show representative Wigner functions for the state of the probe at various points in the protocol, with colors denoting the quasiprobability.
• Figure 2: Spin polarization signal$\langle S_z \rangle / S$ for $N = 100$ spins and 8 random OAT steps, each with twisting strength $\sim 1/\sqrt{N}$. Numerical simulations, averaged over 100 circuit realizations with a fixed rotation axis $\hat{n}=\hat{y}$, and analytic calculations (solid black) show a sharply decaying spin polarization signal as a function of $\theta$, with a useful metrological bandwidth scaling like $\theta_{\mathrm{bw}} \sim N^{-1}$. Shot-to-shot FWHM fluctuations in the signal (numerics in blue, analytics in grey) are subleading in the large-$N$ limit. Inset: Metrological gain for $N=32,64,\ldots,1024$ as a function of rotation angle within the bandwidth.
• Figure 3: Probe state preparation requires only a handful of random twists. Successive applications of random one-axis twisting dynamics yield approximately Haar-random states (a) with a mean QFI (b, dots) that rapidly approaches the ideal value $\overline{\mathrm{QFI}} = N(N+1)/3$ (dashed) on a timescale that is independent of the system size $N$. Fluctuations in the QFI (c, dots) also converge to their Haar-random values (dashed) on the same timescale. Numerical data was averaged over 100 random OAT circuits (300 for $N=12$) and $10^3$ randomly-chosen rotation axes $\hat{n}$. Error bars are shrunk by a factor of 3 for visual clarity.
• Figure 4: Decoherence reduces gain and bandwidth of the butterfly echo protocol. We consider increasing rates of (a) collective and (b) single-particle dephasing, expressed in terms of the system size, $N$, and the total random one-axis twisting evolution time $T{\sim} \frac{\pi}{2\chi\sqrt{N}} \times \mathcal{O}(1)$. The results in (a) are analytic for $N=10^3$ and scale universally with $N$, whereas the results in (b) are numerical for $N=24$ and $T=\frac{\pi}{2\chi \sqrt{N}}\times 8$.
• Figure S1: Visual representation of the Choi-Jamioł kowski isomorphism (CJI) depicted using tensor-network notation. Notation is defined in panel 1 with: (1a) identity, (1b-c) states, (1d-e) products, and (1f) operators. The CJI acts turns the bras or kets that make up an operator into their counterpart. For instance (2a) shows the transformation on the identity for a spin-1/2 system, which maps $\left|\mspace{0.5mu} 0 \mspace{0.5mu}\right\rangle\left\langle\mspace{0.5mu} 0 \mspace{0.5mu}\right| + \left|\mspace{0.5mu} 1 \mspace{0.5mu}\right\rangle\left\langle\mspace{0.5mu} 1 \mspace{0.5mu}\right| \rightarrow \left\langle\mspace{0.5mu} 0 \mspace{0.5mu}\right|_L \otimes \left\langle\mspace{0.5mu} 0 \mspace{0.5mu}\right|_R + \left\langle\mspace{0.5mu} 1 \mspace{0.5mu}\right|_L \otimes \left\langle\mspace{0.5mu} 1 \mspace{0.5mu}\right|_R = \left\langle\mspace{0.5mu} EPR \mspace{0.5mu}\right|_{LR}$. (2b) shows how expectation values of operators on states transform under the isomorphism: $\left\langle\mspace{0.5mu} \psi \mspace{0.5mu}\right| \mathcal{O} \left|\mspace{0.5mu} \psi \mspace{0.5mu}\right\rangle \rightarrow \left\langle\mspace{0.5mu} EPR \mspace{0.5mu}\right|_{LR} (\mathcal{O}_L \otimes \mathbb{I}_R) \left|\mspace{0.5mu} \psi \mspace{0.5mu}\right\rangle_L \otimes \left|\mspace{0.5mu} \psi \mspace{0.5mu}\right\rangle_R$. (3) When taking expectation values averaged over Haar random unitaries $U$, we can use the CJI to isolate the $U$ operators, absorbing any other operators into our boundary conditions. (4) shows a more complicated version of the same idea. Here, $k$ is the number of pairs of replicas of $U$.