Probing mechanical quantum coherence with an ultracold-atom meter

We propose a scheme to probe quantum coherence in the state of a nanocantilever based on its magnetic coupling (mediated by a magnetic tip) with a spinor Bose Einstein condensate (BEC). By mapping the BEC into a rotor, its coupling with the cantilever results in a gyroscopic motion whose properties depend on the state of the cantilever: the dynamics of one of the components of the rotor angular momentum turns out to be strictly related to the presence of quantum coherence in the state of the cantilever. We also suggest a detection scheme relying on Faraday rotation, which produces only a very small back-action on the BEC and is thus suitable for a continuous detection of the cantilever's dynamics.

Figure 1

Sketch of the setup for BEC-based probing of mechanical coherences. A BEC is placed in close proximity to a nanomechanical cantilever endowed with a magnetic tip. The coupling between the magnetic field generated by the mechanical quantum antenna and the ultracold atoms embodies a mechanism for the effective probing of coherences in the state of the mechanical system.

Figure 2

Mean value of ${\widehat{\mathcal{L}}}_x^{}\left(t\right)$ for a cantilever in the initial state as given by Eq. (10) with $C_0=C_1/\alpha =1/\sqrt{1+{\alpha }^2}$ and $C_n=0$ otherwise. The BEC consists of $N={10}^3$${}^{87}\text{Rb}$ atoms and $\langle {\widehat{\mathcal{L}}}_{x,y}^{}\left(0\right)\rangle =0,\langle {\widehat{\mathcal{L}}}_z^{}\left(0\right)\rangle =100$. We have used $B_z^0=3\times {10}^{-6}\mu \text{T}$ and $G_c\approx 1.8\times {10}^3\mu \text{T}/\mu$m.

Figure 3

Mean value of ${\widehat{\mathcal{L}}}_x^{}\left(t\right)$ for a cantilever in the initial state as given by Eq. (10) with $C_0=C_1=C_2/\alpha =1/\sqrt{2+{\alpha }^2}$ and $C_n=0$ otherwise. The BEC parameters are the same as in Fig. 2. The inset shows that the change in $\left|\alpha \right|$ amounts to a shift of the oscillations [we have taken $=e^{i\pi /6}(0.5,1,2)$].

Figure 4

Time evolution of ${\widehat{\mathcal{L}}}_x$ for a coherent initial state of the cantilever with (a) ${\left|\alpha \right|}^2=1$, (b) 5, (c) 15, and (d) 20. For the same parameter as in (d), the plot (e) shows that the carrier frequency ${\omega }_L$ is not significantly affected.