Matter-Wave Interferometry of a Levitated Thermal Nano-Oscillator Induced and Probed by a Spin

We show how the interference between spatially separated states of the center of mass (c.m.) of a mesoscopic harmonic oscillator can be evidenced by coupling it to a spin and performing solely spin manipulations and measurements (Ramsey interferometry). We propose to use an optically levitated diamond bead containing a nitrogen-vacancy center spin. The nanoscale size of the bead makes the motional decoherence due to levitation negligible. The form of the spin-motion coupling ensures that the scheme works for thermal states so that moderate feedback cooling suffices. No separate control or observation of the c.m. state is required and thereby one dispenses with cavities, spatially resolved detection, and low-mass-dispersion ensembles. The controllable relative phase in the Ramsey interferometry stems from a gravitational potential difference so that it uniquely evidences coherence between states which involve the whole nanocrystal being in spatially distinct locations.

Figure 1

Setup: An optical trap holds a diamond bead with a NV center with both weakest confinement and spin quantization along the $z$ axis. A magnetized sphere at $z_0$ produces spin-dependent shifts to the center of the harmonic well. An angle $\theta$ between the vertical and the $z$ axis places the centers of the wells corresponding to the $|+1⟩$ and $|-1⟩$ states in different gravitational potentials. A random coherent state of the center of mass of the bead oscillates as different coherent states in the two wells (gray filled circles), accumulating a relative gravitational phase difference due to superpositions. At $t_0=2\pi /{\omega }_z$ this phase can be read from spin measurements.

Figure 2

The phase space evolution of two arbitrary initial coherent states $\beta =0$ (solid circle) and $\beta =2+i1.5$ (solid square) with $\lambda =0.5\hslash {\omega }_z$ and $\Delta \lambda =0.05\hslash {\omega }_z$. The arrows show the direction of evolution of the coherent states with increasing time for a whole time period $t_0=2\pi /{\omega }_z$. The return of each component to the initial coherent state at $t_0$ is evident. A relative phase $e^{i\Delta {\varphi }_{\mathrm{grav}}}$ appears between the $|+1⟩$ and $|-1⟩$ component due to the gravitational potential difference between the centers of the phase space trajectories for $|-1⟩$ (more towards the left of the figure) and $|+1⟩$ (more towards the right of the figure).