Tolerance in the Ramsey interference of a trapped nanodiamond

In the scheme recently proposed by M. Scala et al. [Phys. Rev. Lett. 111, 180403 (2013)], a gravity-dependent phase shift is induced on the spin of a nitrogen-vacancy (NV) center in a trapped nanodiamond by the interaction between its magnetic moment and the quantized motion of the particle. This provides a way to detect spatial quantum superpositions by means of only spin measurements. Here, the effect of unwanted coupling with other motional degrees of freedom is considered, and we show that it does not affect the validity of the scheme. Both this coupling and the additional error source due to misalignment between the quantization axis of the NV center spin and the trapping axis are shown not to change the qualitative behavior of the system, so that a proof-of-principle experiment can be neatly performed. Our analysis, which shows that the scheme retains the important features of not requiring ground-state cooling and of being resistant to thermal fluctuations, can be useful for several schemes which have been proposed recently for testing macroscopic superpositions in trapped microsystems.

Figure 1

An optical trap holds a diamond bead with an 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 $z$ axes places the centers of the wells corresponding to the $|+1\rangle$ and $|-1\rangle$ states in different gravitational potentials. Starting with an arbitrary coherent state, the c.m. of the bead oscillates as different coherent states in the center-shifted, spin-dependent well (red solid and dashed lines), accumulating a relative gravitational phase difference due to the superpositions. At $t_0=2\pi /{\omega }_z$ this phase can be read from Ramsey measurements on spin. The blue shaped zone shows a generic orientation of the NV center' s axis $z^{\prime }$ with respect to magnetic direction $z$.

Figure 2

(a) Experimental schematic of the optical dipole trap. A 1064-nm laser beam is tightly focused by a high-numerical-aperture ($0.95$) objective. The polarization of the trapping light can be rotated by a half-wave plate. Scattered light from levitated nanodiamonds is collected by a lens and sent to a balanced photodiode in an interferometric scheme described in Ref. [25], providing a position-dependent signal from the levitated nanodiamond. (b) Power spectral density (PSD) as a function of $\omega$ at approximately 10 mB using 200 mW of trapping power. Fourier transforming the position-dependent signal yields the PSD of the trapped nanodiamond. For the measurement of the $z$ frequency, the amplitude has been increased by a factor of 20 for clarity.

Figure 3

Fringes of spin-zero population $P(s_z=0)$ as a function of the orientation $\theta$ of the trapping axis $z$, with respect to the direction of the gravitational acceleration, and of the direction cosine, where $c_x=0$ corresponds to the NV center being parallel to the trapping axis and $c_x=1$ corresponds to the case in which the NV center is orthogonal to it. The initial motional state has been taken to be equal to the vacuum state of the quantum oscillator. The other parameters are such that $\lambda =0.01$ J and $\mathrm{\Delta }\lambda /cos\theta =11.9$ J.

Figure 4

Fidelity $F=|〈{\mathrm{\Psi }}^{\left(2\right)}\left(t_0\right)|{\mathrm{\Psi }}^{\left(0\right)}\left(t_0\right)〉|$ against the magnitude of $|{\beta }_x|=|{\beta }_y|=|{\beta }_z|=|{\beta }_0|$ under realistic parameters $\lambda =0.01\hslash {\omega }_z$ and ${\gamma }_x=0.4,{\gamma }_y=0.5$.