Ground-state cooling of a nanomechanical oscillator with $N$ spins

It is typical of modern quantum technologies employing nanomechanical oscillators to demand few mechanical quantum excitations, for instance, to prolong coherence times of a particular task or to engineer a specific nonclassical state. For this reason, we devote the present work to exhibiting how to bring an initially thermalized nanomechanical oscillator to near its ground state. Particularly, we focus on extending the novel results of D. D. B. Rao et al. [Phys. Rev. Lett. 117, 077203 (2016)], where a mechanical object can be heated up, squeezed, or cooled down to near its ground state through conditioned single-spin measurements. In our work, we study a similar iterative spin-mechanical system when $N$ spins interact with the mechanical oscillator. Here, we have also found that the postselection procedure acts as a discarding process; i.e., we steer the mechanics to the ground state by dynamically filtering its vibrational modes. We show that when considering symmetric collective spin postselection, the inclusion of $N$ spins in the quantum dynamics is highly beneficial—in particular, decreasing the total number of iterations to achieve the ground state, with a success rate of probability comparable with the one obtained from the single-spin case.

Figure 1

Schematic depiction of the iterative cooling process for the NMO. $N$ spins are coupled to a single-mode NMO. After the spin-mechanical system has evolved a time $t$, we proceed to postselect the $N$ spins. If the postselection is successful (“True”), we then repeat the process, where in the next step the NMO state will correspond to the one collapsed by the spin postselection.

Figure 2

Finding the parameters $\{t,\lambda \}$ to optimally cool-down the NMO. In (a), we show the ratio of the phonon number on average between the postselected state (${\langle \widehat{n}\rangle }_{\mathrm{post}}$) and the initial state (${\langle \widehat{n}\rangle }_0=\overline{n}$). In (b) we illustrate the mechanical variances after spin postselection ($\mathrm{\Delta }{\widehat{x}}_{\mathrm{post}}/\mathrm{\Delta }{\widehat{y}}_{\mathrm{post}}$). Interestingly, only at $t=\pi /2$ both quadratures decrease simultaneously. Once $t=\pi /2$ is fixed, the optimal scaled coupling strength can be obtained from the top panel, being $\lambda \approx 0.12$.

Figure 3

Ratio of the phonon number on average (${\langle \widehat{n}\rangle }_{\mathrm{post}}/{\langle \widehat{n}\rangle }_0$) as a function of the coupling parameter ($\lambda$) for several spins coupled to the NMO, where $N$ stands for the number of the interacting spins. We have considered the spin postselection as in Eq. (19).

Figure 4

In the top panel, we show the ratio of the phonon number on average (${\langle \widehat{n}\rangle }_{\mathrm{post}}/{\langle \widehat{n}\rangle }_0$) against the number of iterations for several spins coupled to the NMO. In the bottom panel, we illustrate the success probability at each iteration.

Figure 5

We compare the ratio of the phonon number occupation at each iteration when $N=2,N=3$, and $N=4$ spins are postselected independently, with the case when two and three spins are postselected in a joint basis, $N=2\left(corr\right)$ and $N=3\left(corr\right)$, respectively. We also show that although the ratio ${\langle \widehat{n}\rangle }_{\mathrm{post}}/{\langle \widehat{n}\rangle }_0$ for $N=3\left(corr\right)$ can in fact surpass $N=4$, its negligible success probability (${\mathrm{Pr}}_s$) at the tenth iteration makes it impracticable. Other values are $t=\pi /2,\lambda =0.12$, and ${\langle \widehat{n}\rangle }_0=10$.

Figure 6

Cooling rate (left $y$ axis, blue dots) and its related success probability (right $y$ axis, green crosses) as a function of the number of spins coupled to the NMO. We have fixed the total number of iterations $\left(\tau \right)$ as $\tau =5$ (therefore, $t=\tau \times \pi /2$, corresponds to the total time of the protocol). In the top panel (a), we considered a flat distribution for the parameters $\{c^{\prime }s,d^{\prime }s\}$ as shown in Eq. (47). In the bottom panel (b), we use the symmetric coherent spin state ${\left|\mathrm{CSS}\right.〉}_{+}$ both for the preselection and postselection of the spins—a state which proves to be the optimal case. In both cases, going beyond $N=10$ spins does not improve the cooling performance substantially.

Figure 7

Phonon occupation probability [$\mathrm{Prob}\left(n\right)$] for the postselected mechanical state. In (a) we solved the unitary evolution and (b) the open quantum case, where we have considered both local and collective channels of dissipation $\{\gamma =\mathrm{\Gamma },{\gamma }_{\varphi }={\mathrm{\Gamma }}_{\downarrow }={\mathrm{\Gamma }}_{\uparrow }={\gamma }_{\mathrm{\Phi }},{\langle \widehat{n}\rangle }_0\}=\{{10}^{-3},{10}^{-2},10\}$. As before, we simulated the protocol up to five iterations ($t=\tau \times \pi /2=5\times \pi /2$) when $N=10$ spins are coupled to the NMO. Preparation and postselection of the spins have been performed in the state ${\left|\mathrm{CSS}\right.〉}_{+}$.