Cooling phonons with phonons: Acoustic reservoir engineering with silicon-vacancy centers in diamond

We study a setup where a single negatively-charged silicon-vacancy center in diamond is magnetically coupled to a low-frequency mechanical bending mode and via strain to the high-frequency phonon continuum of a semiclamped diamond beam. We show that under appropriate microwave driving conditions, this setup can be used to induce a laser-cooling-like effect for the low-frequency mechanical vibrations, where the high-frequency longitudinal compression modes of the beam serve as an intrinsic low-temperature reservoir. We evaluate the experimental conditions under which cooling close to the quantum ground state can be achieved and describe an extended scheme for the preparation of a stationary entangled state between two mechanical modes. By relying on intrinsic properties of the mechanical beam only, this approach offers an interesting alternative for quantum manipulation schemes of mechanical systems, where otherwise efficient optomechanical interactions are not available.

Figure 1

(a) Sketch of a single ${\mathrm{SiV}}^{-}$ center embedded near the freely vibrating end of a diamond cantilever of length $\ell$, width $w$ and thickness $t$. In the presence of a strong magnetic field gradient produced by a nearby magnetized tip, the motion of the beam modulates the local magnetic field and results in a magnetic coupling between the electronic spin of the ${\mathrm{SiV}}^{-}$ center and the fundamental bending mode with frequency ${\omega }_{\mathrm{b}}$. In addition, the orbital states of the defect are coupled via strain to a continuum of compression modes propagating along the beam (indicate by the curvy arrow). (b) Level scheme of the ${\mathrm{SiV}}^{-}$ ground state and the relevant microwave transitions used for ground state cooling of the bending mode.

Figure 2

Spectral density of the high frequency compression modes. The blue (short-) and red (long-) dashed lines represent the spectral density in the limit of a semi-infinite 1D cantilever along $x$. The density of state of the phonons is linear in frequency while the sinusoidal envelope is due to the position dependence of the strain field. The blue and red curves are obtained for a ${\mathrm{SiV}}^{-}$ defect positioned at $x_{\mathrm{SiV}}={\lambda }_{\mathrm{\Delta }}/2$ and $x_{\mathrm{SiV}}={\lambda }_{\mathrm{\Delta }}/8$ of the free end, respectively. The black dotted line represents the opposite limit of an infinite 3D crystal. In that case, transverse modes play an important role and the phonon density of state goes as ${\omega }^3$. For a realistic cantilever, the 1D limit should be corrected toward the 3D limit, due to contributions of transverse modes. In the inset, we show a finite size ($\ell =25\mu \mathrm{m}$) 1D cantilever, where the frequencies of each mode can be resolved. The finite width of each peak is due to finite lifetime modeling losses into the bulk of the support. We chose $w=t=0.1\mu \mathrm{m}$ (1D limit), $Q_{\mathrm{\Delta }}=250$, and $g_1=g_2=2\pi \times 1$ PHz.

Figure 3

The force fluctuation spectrum $S_{\mathrm{FF}}\left(\omega \right)$ defined in Eq. (23) is plotted for an ${\mathrm{SiV}}^{-}$ center driven on the red sideband, $\delta =-{\omega }_{\mathrm{b}}$, and for $N_{\mathrm{c}}=0.5$ and ${\mathrm{\Gamma }}_{\mathrm{ph}}/{\omega }_{\mathrm{b}}=0.15$. The dashed line corresponds to the approximate analytic result given in Eq. (27). For comparison, the red dotted line illustrates the fluctuation spectrum $S_{\mathrm{FF}}^{\mathrm{eq}}\left(\omega \right)\sim \omega \left[coth(\hslash \omega /\left(2k_{B}T\right))+1\right]$ for an ohmic environment in equilibrium with a temperature $k_{B}T/\left(\hslash {\omega }_{\mathrm{b}}\right)=5$.

Figure 4

Final occupation of the bending mode ${\overline{n}}_{\mathrm{b}}$ as a function of the beam length $\ell$ for different values of ${\mathrm{\Gamma }}_{\mathrm{ph}}$. We considered a diamond cantilever at $T=4\text{K}$ (dotted-dashed red line), which corresponds to an occupation of the compression mode of frequency $\mathrm{\Delta }/2\pi =50$ GHz of $N_{\mathrm{c}}\approx 1.2$. The blue solid line shows the results for ${\mathrm{\Gamma }}_{\mathrm{ph}}/2\pi =250$ kHz and a lower temperature $T=100\text{mK}\left(N_{\mathrm{c}}\simeq 0\right)$ while the dotted black line corresponds to an intermediate temperature of $T=1\text{K}\left(N_{\mathrm{c}}\simeq 0.1\right)$. In the inset, we plot the initial occupation ($g_{\mathrm{m}}=0$) of the bending mode. It shows that even for initial occupancy $N_{\mathrm{b}}\sim {10}^4$, it is possible to reach the ground state (${\overline{n}}_{\mathrm{b}}<1$) using experimentally accessible parameters. These results are obtained using a driving strength $\epsilon =0.2$ [cf. Eq. (30)] and the full expression of $S_{\mathrm{FF}}\left(\omega \right)$ that includes the drive at every order; we use the corresponding optimal detuning $\delta \approx -{\omega }_{\mathrm{b}}/\sqrt{1+4\epsilon }$. The remaining parameters are the bending mode quality factor $Q_{\mathrm{b}}={10}^6$, the transverse beam dimension $w=t=0.1\mu \mathrm{m}$, and a magnetic field gradient of ${10}^7$ T/m.

Figure 5

Setup for the dissipative preparation of a mechanical two-mode squeezed state. (a) Two possible configurations, where two vibrational modes, representing either orthogonal vibration modes of the same beam (left) or the fundamental bending modes of two different beams (right), are coupled magnetically to the same ${\mathrm{SiV}}^{-}$ center. (b) Energy level diagram of the ${\mathrm{SiV}}^{-}$ defect. The green and blue arrows indicate the four driving fields which are detuned from resonance by $\pm {\omega }_a$ and $\pm {\omega }_b$. See text for more details.

Figure 6

The separability parameter $\xi$ is plotted as a function of the squeezing parameter $r$ for three values of the ratio ${\mathrm{\Gamma }}_{\text{th}}/{\mathrm{\Gamma }}_{\text{eff}}$, in the limit ${\gamma }_{a,b}\ll {\mathrm{\Gamma }}_{\mathrm{sq}}$ and ${\gamma }_a{N}_a={\gamma }_b{N}_b={\mathrm{\Gamma }}_{\mathrm{th}}$. Vertical dotted line indicates the threshold below which the two bending modes are entangled, as given by Eq. (32). For this plot ideal sideband resolved conditions and $N_{\mathrm{c}}=0$ have been assumed.

Figure 7

Contour plot of the minimal achievable separability parameter ${\xi }_{\min }=\min \left\{\xi \left(r\right)\right|r\}$ as a function of the ratio ${\mathrm{\Gamma }}_{\text{eff}}/{\mathrm{\Gamma }}_{\text{th}}$ and the effective occupation $N_{\text{eff}}$. The red line corresponds to the threshold given by the condition ${\xi }_{\min }=1$. For this plot the parameter $\xi \left(r\right)$ has been evaluated using Eq. (49), where equal parameters for the two modes and ${\gamma }_a={\gamma }_b\ll {\mathrm{\Gamma }}_{\mathrm{sq}}$ have been assumed. The values of ${\mathrm{\Gamma }}_{\text{eff}}$ and $N_{\text{eff}}$ are plotted in (b) and (c) as a function of the length $\ell$ of the beam. For both plots the values of $T=100$ mK, ${\mathrm{\Gamma }}_{\text{ph}}/\left(2\pi \right)=250$ kHz (solid line), $T=100$ mK, ${\mathrm{\Gamma }}_{\text{ph}}/\left(2\pi \right)=1$ MHz (dashed line), $T=1$ K, ${\mathrm{\Gamma }}_{\text{ph}}/\left(2\pi \right)=1$ MHz (dotted line), and $T=1$ K, ${\mathrm{\Gamma }}_{\text{ph}}/\left(2\pi \right)=250$ kHz (dashed-dotted line) have been assumed. All the other parameters are the same as in Fig. 4.