Minimal-noise estimation of noncommuting rotations of a spin

Jakub Czartowski1,2, Karol Życzkowski2,3,4, and Daniel Braun5

1Doctoral School of Exact and Natural Sciences, Jagiellonian University, ul. Łojasiewicza 11, 30-348 Kraków, Poland
2Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, ul. Łojasiewicza 11, 30-348 Kraków, Poland
3Centrum Fizyki Teoretycznej PAN, Al. Lotników 32/46, 02-668 Warszawa, Poland
4National Quantum Information Center (KCIK), University of Gdańsk, Poland
5Institute of Theoretical Physics, University of Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany

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We propose an analogue of $\text{SU}(1,1)$ interferometry to measure rotation of a spin by using two-spin squeezed states. Attainability of the Heisenberg limit for the estimation of the rotation angle is demonstrated for maximal squeezing. For a specific direction and strength an advantage in sensitivity for $all$ equatorial rotation axes (and hence non-commuting rotations) over the classical bound is shown in terms of quadratic scaling of the single-parameter quantum Fisher information for the corresponding rotation angles. Our results provide a method for measuring magnetic fields in any direction in the $x$-$y$-plane with the same optimized initial state.

Interferometric methods in their varied spectrum of flavours have long stood as a hallmark of physics, offering a window into phenomena such as the elusive gravitational waves. Recent developments in the understanding of $\text{SU}(1,1)$ interferometry in terms of a Squeeze-Displace-Unsqueeze protocol, which depends heavily on squeezed states of light, promise novel approaches to search for dark matter. In parallel, spin squeezing has been investigated since 1971, finding its use for Bose-Einstein condensates, quantum rotosensors and other problems. In this work the ideas from both fields are merged by introducing Squeeze-Rotate-Unsqueeze protocol for single-axis squeezed spin systems. We demonstrate its advantage and adaptability from the vantage point of Quantum Fisher Information, offering a novel theoretical tool useful for sensing of magnetic fields or rotations in the polarization space.

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