Entangling Two Macroscopic Mechanical Resonators at High Temperature

At high temperature, thermal decoherence dominates so that the entanglement of quantum states is difficult to preserve. Realizing high-temperature entanglement is, therefore, a challenge to the current quantum technologies. Here, we demonstrate that considerable degrees of continuous-variable entanglement between two macroscopic objects placed in an environment of high temperature can be created through the medium of properly prepared light fields coupled to them. There are two steps to make such entanglement. First, by pumping an optical cavity field pressuring on a mechanical resonator as a macroscopic object with a blue-detuned drive field, the competition between the induced squeezing effect due to the blue-detuned drive and the existing thermal decoherence leads to a stable entanglement between the cavity field and mechanical resonator. A condition for realizing field-resonator entanglement is determined at any temperature and for any given optomechanical system. The second step is to entangle two distant mechanical resonators through a procedure of entanglement swapping. A detailed example of illustrating this entanglement swapping shows that a considerable degree of entanglement between the two mechanical resonators can be created. The current study proposes a route toward high-temperature entanglement in a realistic physical system.

Figure 1

Scheme for realizing the entanglement between two mechanical resonators, which independently vibrate with the displacement $x_m$ and $y_m$, respectively. Here each optical cavity coupled to a mechanical resonator is driven by a strong pump field blue detuned by the frequency ${\omega }_m$ of the mechanical resonator. As indicated inside the eclipse, under the blue-detuned driving field, the nonlinear optomechanical coupling due to the radiation pressure can be well approximated with a two-mode squeezing action of the intensity $g_{m}E/{\omega }_m$. The lower panel synopsizes the main features in the process. Each mechanical resonator is initially placed in thermal equilibrium with its environment, which can be at high temperature. The cavity fields coupled to them are driven by a blue-detuned external field, respectively, so that there is no time-independent steady state in the end (due to the induced squeezing-type coupling). By employing an entanglement swapping procedure, which is realized by measuring the leaked light out of the cavities, the two mechanical resonators are entangled.

Figure 2

Real-time evolutions of cavity photon number, purity, and entanglement in thermal environment. In (a) we consider the room temperature $T=300\mathrm{K}$ for a mechanical resonator with the frequency ${\omega }_m/2\pi =100\mathrm{MHz}$, and the system is driven by a blue-detuned ($\mathrm{\Delta }/{\omega }_m=-1$) cw pump field with the amplitude $E/\kappa ={10}^5$. The inset in (a) shows the stabilized cavity photon number evolved according to the nonlinear dynamical equations, in contrast to the growing cavity photon in the linear dynamical regime. In (b) and (c), two initial temperatures corresponding to the indicated different thermal occupations are used for the evolutions of the purity and entanglement. The parameters for the system are taken as $g_m/\kappa ={10}^{-4}$, ${\omega }_m/\kappa =10$, and $Q={\omega }_m/{\gamma }_m={10}^6$.

Figure 3

Evolved entanglement for the setups with ${\omega }_m/\kappa \gg 1$. The system parameters are $g_m/\kappa ={10}^{-4},Q={10}^6$. The drive intensity and the thermal reservoir occupation are set to satisfy the relations $J/\kappa =(g_m/{\omega }_m)(E/\kappa )=2.5$ and $({\gamma }_m/\kappa )n_{\mathrm{th}}=1$.

Figure 4

Comparison of the entanglement values with and without quantum-noise effects. Here we calculate the logarithmic negativity for the entanglement evolved with only the cavity noise, with only the mechanical noise and under both the cavity and mechanical noises, respectively. The other system parameters, which are not indicated here, are the same as those in Fig. 2.

Figure 5

Evolved entanglement with a series of gradually increased drive intensity. Following the order from the upper left frame to the lower right frame, the drive intensity $E/\kappa$ grows from ${10}^5$ to $2.4\times {10}^6$ with a gap of $2\times {10}^5$ for each step. The system parameters are the same as those in Fig. 4 with a fixed $Q={10}^6$. The colors of the curves are in one-to-one correspondence with those in Fig. 4.

Figure 6

Relations between the achieved peak value entanglement with the temperature and mechanical quality factor. (a) The ratio $Q/n_{\mathrm{th}}$ for the existence of optomechanical entanglement. For all the given drive intensities, the entanglement appears with the ratio in the order of $Q/n_{\mathrm{th}}\sim 1$. The inset shows a saturation of the values of the stabilized entanglement. With the even higher drive intensities, the entanglement may exist at a point $Q/n_{\mathrm{th}}<1$. (b) The tendency of the stabilized peak values of entanglement, which are obtained given the drive intensity $E/\kappa ={10}^5$, with the temperature up to the degree higher than 900 K for a technically available quality factor $Q={10}^6$. In both (a) and (b) we have ${\omega }_m/2\pi =100\mathrm{MHz}$, and $g_m/\kappa ={10}^{-4},{\omega }_m/\kappa =10$, $\mathrm{\Delta }/{\omega }_m=-1$ as in Fig. 2.

Figure 7

(a) Stabilized value of entanglement with Eq. (11) in the space of the parameters $J/\kappa =g_{m}E/({\omega }_m\kappa )$ and $({\gamma }_m/\kappa )n_{\mathrm{th}}$, with $g_m/\kappa ={10}^{-4}$, ${\omega }_m/\kappa =10$, $Q={10}^6$. The optomechanical coupling intensity on the general level is indicated by the first parameter, and the general thermal decoherence rate is measured by the second parameter. (b) The analytical entanglement with Eq. (19) in the space of the same parameters as in (a).

Figure 8

Entanglement that can be achieved with the experimental setup in Ref. [65], which operates at optical frequency and has the parameters ${\omega }_m/2\pi =3\times {10}^9\mathrm{Hz}$, $\kappa /2\pi =5\times {10}^8\mathrm{Hz}$, $g_m/2\pi =9\times {10}^5\mathrm{Hz}$, and $Q={10}^5$. Here we consider its performance at the room temperature $T=300\mathrm{K}$ corresponding to $n_{\mathrm{th}}=1.6\times {10}^3$.

Figure 9

Entanglement swapping. (a) The output two cavity modes ($C_1$ and $C_2$ as the intrinsic cavity modes multiplied by $\sqrt{\kappa }$) interfere through a 50:50 beam splitter (BS) and two of the output modes are measured. (b) The two output modes after the first 50:50 BS are transmitted into two 1:1000 BS, respectively. The weak modes are measured while the strong modes are projected out.

Figure 10

Correlation functions of the thermal noise at different temperatures. The functions are calculated with the form in Eq. (A15) divided by the factor $2{\gamma }_m/(\pi {\omega }_m)$. The convergence factor is chosen as $\lambda =\hslash /(2\pi k_{B}T)$.

Figure 11

Comparison of the optomechanical entanglement evolutions under the colored (non-Markovian) and the white (Markovian) mechanical noises, which are calculated in the limit ${\omega }_m/\kappa \to \mathrm{\infty }$. The evolutions take place at room temperature, and the system parameters are chosen to be $g_{m}E/{\omega }_m=2.5\kappa$ and ${\gamma }_m=5\times {10}^{-4}\kappa$.