The theory of angular momentum connects physical rotations and quantum spins together at a fundamental level. Physical rotation of a quantum system will therefore affect fundamental quantum operations, such as spin rotations in projective Hilbert space, but these effects are subtle and experimentally challenging to observe due to the fragility of quantum coherence. We report on a measurement of a single-electron-spin phase shift arising directly from physical rotation, without transduction through magnetic fields or ancillary spins. This phase shift is observed by measuring the phase difference between a microwave driving field and a rotating two-level electron spin system, and it can accumulate nonlinearly in time. We detect the nonlinear phase using spin-echo interferometry of a single nitrogen-vacancy qubit in a diamond rotating at 200 000 rpm. Our measurements demonstrate the fundamental connections between spin, physical rotation, and quantum phase, and they will be applicable in schemes where the rotational degree of freedom of a quantum system is not fixed, such as spin-based rotation sensors and trapped nanoparticles containing spins.
With the creation of logic gates and algorithms for quantum computers and entering our lives, it is predicted that great developments will take place in this area and important efforts are made. Spin rotation processors are quantum mechanical rotation processors and have no classic counterparts. The rotation operators of spin 1/2 are well known and can be found in related textbooks. But rotation operators of other spins greater than 1/2 can be found numerically by evaluating the series expansions of exponential operator obtained from Schr?dinger equation, by evaluation of Wigner-d formula or by recently established expressions in polynomial forms discussed in the text. In a previous paper, rotation operators for spins 1/2 to 2 were published. In this work, explicit symbolic expressions of x, y and z components of rotation operators for spin 5/2, 3 and 7/2 are worked via exponential operator for each element of related spin operators and utilizing simple linear curve fitting process. The procedures gave out exact expressions of each element of the rotation operators.
Benchmarking a high-precision quantum operation is a big challenge for many quantum systems in the presence of various noises as well as control errors. Here we propose an
O(1) benchmarking of a dynamically corrected rotation by taking quantum advantage of a squeezed spin state in a spin-1 Bose-Einstein condensate (BEC). Our analytical and numerical results show that tiny rotation infidelity, defined by $1\ensuremath{-}F$ with
F the rotation fidelity, can be calibrated in the order of
1/N2 by only several measurements of the rotation error for
N atoms in an optimally squeezed spin state. Such an
O(1) benchmarking is possible not only in a spin-1 BEC but also in other many-spin or many-qubit systems if a squeezed or entangled state is available.