Entangled-state generation and Bell inequality violations in nanomechanical resonators

We investigate theoretically the conditions under which a multimode nanomechanical resonator, operated as a purely mechanical parametric oscillator, can be driven into highly nonclassical states. We find that when the device can be cooled to near its ground state, and certain mode matching conditions are satisfied, it is possible to prepare distinct resonator modes in quantum entangled states that violate Bell inequalities with homodyne quadrature measurements. We analyze the parameter regimes for such Bell inequality violations, and while experimentally challenging, we believe that the realization of such states lies within reach. This is a re-imagining of a quintessential quantum optics experiment by using phonons that represent tangible mechanical vibrations.

Figure 1

Schematic illustration of a conceptual nanomechanical resonator with two homodyne measurement setups that probe two different modes of oscillation. The beam can oscillate in a large number of vibrational and flexural modes with different frequencies. The two homodyne detectors measure the mode quadratures $X_1\left(\varphi \right)$ and $X_2\left(\theta \right)$ with frequencies ${\omega }_1/2\pi$ and ${\omega }_2/2\pi$, respectively. By analyzing the correlations between the $X_1\left(\varphi \right)$ and $X_2\left(\theta \right)$ quadratures, it is possible to determine whether or not the mode states are quantum-mechanically entangled.

Figure 2

A visualization of the mode-matching condition required to obtain an effective three-mode system. The modes are represented as solid vertical lines, and the driving frequency and the sum of the signal and idler frequencies, which should sum up to a frequency close to ${\omega }_0$, are represented by dashed vertical lines.

Figure 3

Visualization of the steady state of the effective two-mode system. (a) The Fock state distribution of modes $a_1$ and $a_2$. (b) The single-mode Wigner function of modes $a_1$ and $a_2$. Both the Fock state distribution and the Wigner function are identical for both modes $a_1$ and $a_2$, due to the symmetric two-phonon processes and equal dissipation rates and initial states. The single-mode Wigner function is positive, and the single-mode state can therefore be considered classical. However, strong correlations exists between quadratures of two different modes, as shown in the joint quadrature probability distribution $P_{X_1,X_2}(0,0)$ in (c). Here we have used the parameters $\kappa =0.15,E=0.094,{\gamma }_0=1.0,{\gamma }_1={\gamma }_2=0$, and ${\Delta }_0={\Delta }_L=0$. These parameters were chosen to produce a steady state that closely corresponds to the ideal state [49] for quadrature Bell inequality violation, i.e., with $r=1.12$ (see Sec. 4).

Figure 4

The time evolution of the phonon number (a), single-mode quadrature variances (b), and the variances of the two-mode quadrature differences (c), the logarithmic negativity (d), for the case when the state is initially in the zero-temperature ground state. In the large-time limit the state approaches the steady state that is visualized in Fig. 3. The single-mode variances increase above the vacuum limit as time increases, but the variance of cross-quadrature difference $\mathrm{Var}(x_1-x_2)$ decrease below the vacuum limit, which is a characteristic of two-mode squeezing, and the nonzero logarithmic negativity demonstrates that the steady state is nonclassical. Here we used the same parameters as those given in the caption of Fig. 3.

Figure 5

(a) The normalized CH and CHSH Bell quantities (violation above 1) as a function of time, for the pumped two-mode nanomechanical resonator, under the ideal condition without signal and idler mode dissipation (dashed line) and for the case including signal and idler dissipation. The initial state is the vacuum state, which we assume can be prepared to a good approximation using cooling. At $t=0$, the parametric amplification is turned on by the activation of the driving field with amplitude $E$. In the ideal case, the steady state violates both the CH and CHSH Bell inequalities, but there is no steady state violation when single-phonon dissipation is included. However, there is a period of time during the transient where both inequalities are violated. (b) The angle $\phi$ dependence for the normalized CH and CHSH Bell quantities for $t{\kappa }^2/{\gamma }_0=1.8$, where solid lines include dissipation and dashed lines are the ideal case. The optimal value of $\phi$ for the states produced in the model we investigate here is $\phi =\pi /4$, which was used in (a). The parameters used here are the same as in Fig. 3, and for the solid lines we used ${\gamma }_1={\gamma }_2=0.001$.

Figure 6

Violation of the normalized quadrature CHSH Bell inequality (redish region) as a function of time $t$ and the intermode coupling $\kappa$ (a) and (d), the pump mode driving amplitude $E$ (b) and (e), and the pump-mode dissipation rate ${\gamma }_0$ (c) and (f). The ideal case without signal and idler mode dissipation is shown in (a)–(c), and (d)–(f) include signal and idler mode dissipation with equal dissipation rates ${\gamma }_1={\gamma }_2=0.001$. In (a)–(c) there is a parameter window for $\kappa$ and $E$ which results in a violation for sufficiently large $t$, as well as in the steady state. However, in (d)–(f) there is no violation in the steady state, but during a transient time a violation may still occur for suitably chosen parameters. Apart from the parameters on the vertical axes, the parameters were kept fixed at the same values as given in Fig. 3, and denoted by a bar over the symbol in the axes.

Figure 7

The regions where a transient Bell inequality violation can be achieved, as a function of nonlinearity $\kappa$ and time $t$, and for different temperatures. The region of violation at zero temperature is shown in redish color. The contours mark the regions of violation for finite signal and idler mode temperatures (assumed equal), labeled by the average number $N$ of thermal phonons. We note that even a very small number of initial thermal phonons inhibits the Bell inequality violation, which suggest that excellent ground-state cooling of both modes is a prerequisite to obtaining a violation.