Measurable Quantum Geometric Phase from a Rotating Single Spin

We demonstrate that the internal magnetic states of a single nitrogen-vacancy defect, within a rotating diamond crystal, acquire geometric phases. The geometric phase shift is manifest as a relative phase between components of a superposition of magnetic substates. We demonstrate that under reasonable experimental conditions a phase shift of up to four radians could be measured. Such a measurement of the accumulation of a geometric phase, due to macroscopic rotation, would be the first for a single atom-scale quantum system.

Figure 1

(a) Schematic of system: a diamond crystal containing NV centres, on a spinning table, with the microwave control polarized along the axis of rotation and the 532 nm optical pulse used for readout. As the NV centre rotates around the axis the $m=\pm 1$ states can accumulate a geometric phase. An example of such phase accumulation is schematically shown for the $m=1$ state with the NV axis in the plane of rotation, i.e., $\theta =\pi /2$. (b) Energy level diagram of the NV center. (c) Geometry of the NV center. Defining the magnetic field direction of the microwave pulses as the $z$ direction, $z^{\prime }$ is the instantaneous direction of the NV axis, defined with respect to the lab frame, unprimed coordinate system, by $\theta$ and $\varphi$.

Figure 2

(a) Proposed Ramsey geometry. The crystal is attached to a rotating spindle with the NV ($z^{\prime }$) axis of one of its NV centers at an angle $\theta$ to the spindle ($z$) axis. Rabi pulses are produced by a microwave field linearly polarised with its magnetic field oscillating along the $z$ axis. The magnitude of the geometric phase after a complete rotation of the spindle is given by the solid angle subtended by the $z^{\prime }$ axis. (b) Proposed spin echo geometry. The spindle is at an angle ${\theta }_0$ to the fixed $z$ axis of the microwave field, and the NV axis is perpendicular to the spindle axis. The actual path of the NV ($z^{\prime }$) axis encloses a solid angle of $2\pi$, which is not observable. The $\pi$ pulse in the spin echo geometry, however, rectifies the alternating Berry phase accumulation, producing a total phase of $4{\theta }_0$ after a full spindle rotation, the solid angle enclosed by the effective trajectory shown.