Observation of a Quantum Phase from Classical Rotation of a Single Spin

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.

Figure 1

(a) Experimental setup depicting the NV center in a (100) diamond mounted to an electric motor that rotates about an axis $z$ with azimuthal coordinate $\varphi$. We use 532 nm laser light to optically pump the NV spin, and microwaves resonant with the $m_S=0\leftrightarrow m_S=-1$ transition are applied via a $20\mu \mathrm{m}$ diameter copper wire above the surface of the diamond. (b) The position of the microwave (mw) wire determines the angle the microwave polarization vector makes to the rotation axis, ${\theta }_{\mathrm{mw}}$. Moving the microwave wire by up to $100\mu \mathrm{m}$ along the $y$ axis varies ${\theta }_{\mathrm{mw}}$ from 28° to 67°. (c) The microwave vector at tilt angles of ${\theta }_{\mathrm{mw}}=30°$ (purple), $54.7°$ (green), and 90° (orange) in the (i) stationary lab frame, (ii) NV frame, with ${\theta }_{\mathrm{NV}}=54.7°$, and (iii) in the NV frame but rotating at the microwave frequency. The geometric path of the microwave vector, which reflects the physical rotation, is projected onto the azimuthal plane of the NV Bloch sphere under the rotating-wave approximation, where the azimuthal coordinate represents the overall effective phase ${\varphi }_{\mathrm{eff}}$ of the microwave field. (d) The effective phase ${\varphi }_{\mathrm{eff}}$ as a function of rotation angle $\varphi$ (equivalently, time $t$) for various ${\theta }_{\mathrm{mw}}$, highlighting the nonlinear accumulation with rotation angle $\varphi$. The phase of the first $\pi /2$ pulse, $\pi$ pulse, and final $\pi /2$ pulse in a spin-echo sequence applied at times $t_0$, $t_1$, and $t_2$, respectively, to the rotating qubit is set as ${\varphi }_{\mathrm{eff}}$, the linear component of the effective phase ${\varphi }_{\mathrm{lin}}$ is eliminated while the phase nonlinearity $\delta \varphi$ introduced by rotation in the presence of the tilted microwave field is reflected in the spin-echo signal as ${\cos }^2\delta \varphi$.

Figure 2

Reconstruction of effective phase accumulation from stationary Rabi frequency. The motor is held constant at different park angles ${\varphi }_p$ (defined with respect to an arbitrary reference point), allowing the angular dependence of the Rabi frequency to be measured. (Top) Measured normalized Rabi frequency vs ${\varphi }_p$ for different microwave tilt angles (average angles across the NV trajectory). Lines denote the model derived from Eq. (2), and error bars denote uncertainty in the calculated Rabi frequency derived from a Fourier transform of time-domain Rabi oscillations. (Bottom) Reconstructed effective phase using the same model. Vertical lines denote the Rabi frequency and effective phase sampled where microwave pulses are applied—for minimum (blue) and maximum (orange) nonlinearity of ${\varphi }_{\mathrm{eff}}$.

Figure 3

Detection of electron-spin phase shifts from physical rotation. (Top panels) Spin-echo fringes with $\tau =100\mu \mathrm{s}$ (spanning 120°), as a function of the applied $x$ field for different average microwave tilt angles, for (a) the linear measurement, where the trigger delay time sets the initial park angle to ${\varphi }_p=357°$, and (b) a measurement maximally sensitive to the nonlinear part of ${\varphi }_{\mathrm{eff}}$, ${\varphi }_p=160°$. Spin-echo signal error bars denote uncertainty in the measured spin state, which is dominated by the photon counting statistics. (c) Fitted fringe phase shift vs microwave tilt angle for linear (blue) and nonlinear (orange) regions. Error bars denote uncertainty in the fitted phase.