Nonclassicality tests and entanglement witnesses for macroscopic mechanical superposition states

We describe a set of measurement protocols for performing nonclassicality tests and the verification of entangled superposition states of macroscopic continuous variable systems, such as nanomechanical resonators. Following earlier works, we first consider a setup where a two-level system is used to indirectly probe the motion of the mechanical system via Ramsey measurements and discuss the application of this method for detecting nonclassical mechanical states. We then show that the generalization of this technique to multiple resonator modes allows the conditioned preparation and the detection of entangled mechanical superposition states. The proposed measurement protocols can be implemented in various qubit-resonator systems that are currently under experimental investigation and find applications in future tests of quantum mechanics at a macroscopic scale.

Figure 1

(a) Coupling between the two-level atom and the nanomechanical oscillator and its mechanical analog. (b) Ramsey measurement consisting of three steps: (I) an initial $\pi /2$ rotation, (II) evolution under the Hamiltonian in Eq. (1) for a time ${\tau }_1$, and (III) final $\pi /2$ rotation and a successive measurement of the atom population $Z$. (c) State evolution under the Ramsey sequence I–III as given by Eq. (3).

Figure 2

Nonclassicality detection for the state ${\rho }_n=(1-p)|n\rangle \langle n|+p|0\rangle \langle 0|$. The shaded parts indicate the regions where the criterion given by Eq. (19) is violated. The upper two panels show the result for $n=1$ and the lower two panels for $n=2$. The values for $p$ are $p=0.1$ in (a) and (c) and $p=0.75$ in (b) and (d) and ${\alpha }_i\in \mathbb{R}$. The red dashed circle indicates the value of $\left|\alpha \right|$ that is required to detect nonclassicality using the criterion given in Eq. (18).

Figure 3

Nonclassicality of a Schrödinger cat state as defined in Eq. (20). (a) Time dependence of ${\chi }_N(\alpha ,t)=\langle :\mathcal{D}\left(\alpha \right):\rangle \left(t\right)$ in the presence of decoherence and for $\alpha ={\xi }_0=2,\theta =0$. (b) and (c) Regions of nonclassicality that can be detected using the criteria (18) and (19), respectively. In both plots the different shadings represent the nonclassical regions evaluated at times (i) $\gamma t=0$, (ii) $\gamma t=0.02$, (iii) $\gamma t=0.04$, and assuming $N_{\mathrm{th}}=10$.

Figure 4

Minimal eigenvalue ${\lambda }_{\min }$ of the partial transpose of the matrix of moments $M^{\Gamma }$, for an entangled cat state $|{\psi }_{+}\rangle$ given in Eq. (23) and for values of ${\alpha }_i$ and ${\beta }_i$ as defined in Eq. (33).

Figure 5

Expectation value of the entanglement witness $W$ defined in Eq. (37) for the state $|{\psi }_{+}\rangle$ (23). The plot is shown for different values of the cat size ${\xi }_0$ and for fixed $\varepsilon =\pi /2$ and $w\approx 0.4247$.