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Abstract
Quantum entanglement in macroscopic systems is not only essential for practical quantum information processing, but also valuable for the study of the boundary between quantum and the classical world. However, it is very challenging to achieve the steady remote entanglement between distant macroscopic systems. We consider two distant nanoparticles, both of which are optically trapped in two cavities. Based on the coherent scattering mechanism, we find that the ultrastrong optomechanical coupling between the cavity modes and the motion of the levitated nanoparticles could be achieved. The large and steady entanglement between the filtered output cavity modes and the motion of nanoparticles can be generated if the trapping laser is under the red sideband. Then through entanglement swapping, the steady motional entanglement between the distant nanoparticles can be realized. We numerically simulate and find that the two nanoparticles with 10 km distance can be entangled for the experimentally feasible parameters, even in room temperature environments. The generated continuous variable multipartite entanglement is the key to realizing the quantum enhanced sensor network and the sensitivity beyond the standard quantum limit.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Quantum entanglement is widely considered to be a fundamental resource in quantum computation [1], quantum network [2], quantum metrology [3], and quantum enhanced sensing [4,5]. The entanglement has been generated in various systems, such as atoms [6,7], atomic assembles [8,9], nitrogen-vacancy centers [10], superconducting qubits [11], and mechanical oscillators [12–15]. The deterministic quantum entanglement has been realized between two systems with the distance around 1 meter or less [13,14,16,17]. In order to achieve the remote entanglement between distant quantum systems, the pre- and/or post-selections are usually applied with the cost of the low success possibility [9,15,18,19]. As the decoherence rate is proportional to the size of the system, it is extremely challenging to achieve the steady entanglement between the distant macroscopic systems.
Because of the ultra-high Q factor ($>10^{10}$), the levitated optomechanical system is one of the best testbed for the macroscopic quantum mechanics [20–24]. Many quantum phenomena have been predicted in this macroscopic system, such as quantum superpositions and matter-wave interference [25–28], gravity induced entanglement [29,30], quantum time crystal [31,32], etc. Recently, the center-of-mass (CoM) motion of the optically levitated nanoparticle has been cooled to the quantum ground state [33–36], which is the first step towards the macroscopic quantum phenomena. The strong coupling between the CoM motion of the levitated nanoparticle and the cavity mode has also been achieved via the coherent scattering mechanism [37–40]. Inspired by these breakthroughs, the mechanical squeezing and entanglement for the optically levitated nanoparticle within a single cavity have been theoretically studied [41–46].
In this paper, we propose a practical scheme to realize the steady entanglement between the motional modes of two distant optically levitated nanoparticles, which are coupled with two cavities respectively [47–49]. Based on the coherent scattering mechanism [50], we calculate the optomechanical coupling $g_{s\phi }$ ($g_{sy}$) between the libration (CoM) modes and cavity modes. We find that the ultra-strong coupling regime ($g_{s\phi (sy)}/\omega _{\phi (y)}>0.1$) could achieve for the current experimental conditions [51]. The rotating-wave approximation is no longer valid here. If the trapping lasers are in red sideband of the cavity modes, the libration (CoM) motion of the levitated nanoparticles is not only cooled down, but also strongly entangled with the cavity modes [52]. We further propose to unconditionally entangle the libration (CoM) motion of the distant nanoparticles via entanglement swapping [47–49,53,54]. The resulting steady entanglement between two remote nanoparticles is robust to the thermal environment in a high vacuum. We numerically calculate the steady entanglement between two levitated nanoparticles with the experimentally feasible parameters [55–57], and find that maximal distance between the nanoparticles could be larger than $10$ km.
2. Dynamics of the system
As shown in Fig. 1(a), we theoretically consider two optically levitated nano-ellipsoids in distant cavity $\xi$ ($\xi =A,B$). By the coherent scattering interaction, the motion of the levitated nano-ellipsoid is strongly coupled with the cavity field. The output modes are filtered and then measured by the Bell-like detection. In this way, two levitated cavity optomechanical systems are integrated into a quantum network in which the two distant motional states can be entangled.
Fig. 1. (a) Schematic diagram for the scheme that contains two remotely levitated nanoparticles locating in the center of cavity. The optomechanical coupling between the cavity mode $\hat a_\xi$ and libration (or CoM) mode $\hat b_\xi$ is induced through the coherent scattering mechanism. The output modes of $\xi$-th $(\xi =A,B)$ cavity are selected by the filtering operation and then measured by the Bell-like detection. (b) The scheme for the orientation of the ellipsoid. By rotating the axes $z$, $y'$ and $z''$ with the Euler angles $\phi$, $\theta$, and $\gamma$, the ellipsoid coordinate system ${\{x_E,y_E,z_E\}}$ maps to the fix experimental Cartesian coordinate system ${\{x,y,z\}}$. In this configuration, $\theta$ ($\phi$) describes the libration motion in $x-y$ ($x-z$) plane, and the $\gamma$ describes the rotation angle around the axis $z''$.
The nano-ellipsoid is assumed to be much smaller than the wavelength of the linearly polarized optical tweezers. The dipole moment of the nano-ellipsoid is given by $\hat {P}=\hat {\alpha }'(\hat j,\hat o)\hat {E}(\hat r),$ where $\hat {\alpha }'(\hat j,\hat o)$ is the polarization tensor charged by the semiaxes $\hat j=\{a,b,c\}$ and the orientation angle $\hat o=\{\theta,\phi,\gamma \}$. $\hat {E}(\hat r)$ denotes the amplitude of the electrical field in position $\hat r = \{x,y,z\}$. As shown in Fig. 1(b), the nano-ellipsoid is prolate ellipsoid with the semiaxes $a>b=c$. Due to the breaking of the spherical symmetry, the polarization components along the principal axes depend on both the position and orientation of the nano-ellipsoid in electrical field. The optical potential wells are induced on both the position ($\hat r$) and orientation ($\hat o$). In order to minimize the energy of the system, the levitated nano-ellipsoid has equilibriums in both the position $\hat r$ and the orientation $\hat o$, leading to the CoM motion and libration motion respectively [55]. It is noted that the polarization components in ellipsoid coordinate system $\{x_E, y_E, z_E\}$ are complex for evaluating both the CoM motion and libration motion. By rotating the ellipsoid coordinate system $\{x_E, y_E, z_E\}$ with the angles $\{\theta, \phi, \gamma \}$, the polarization tensor in fix experimental Cartesian coordinate system can be constructed (see Section 1 in Supplement 1). In this way, both the CoM motion and libration motion can be characterized via the fix experimental coordinate system.
In this work, we take the example for either CoM motion in $y$ or the libration motion around $\phi$. Based on the coherent scattering interaction, the levitated cavity optomechanical Hamiltonian has the form
The CoM (libration) mode frequency $\omega _{\phi (y)}$ and the optomechanical coupling $g_{s,\phi (sy)}$ are calculated and shown in Fig. 2. We find that the ratio $g_{s,\phi (sy)}/\omega _{\phi (y)}$ is proportional to $P_t^{1/4}$, and could be larger than $0.1$ for the optical tweezers power in focus $P_t$ from $10^{-3}$ W to $1$ W. Therefore, the optomechanical coupling in our scheme is in the ultrastrong coupling regime [51]. For example, if we set $P_t=0.01$ W ($P_t=0.41$ W), the ratio between the coupling strength $g_{s\phi }=53$ kHz ($g_{sy}=56$ kHz) and the motional frequency $\omega _\phi =128$ kHz $(\omega _y=139$ kHz) is around $0.4$. The rotating-wave approximation is no longer valid here. Both of the rotating and counter-rotating terms of the Hamiltonian must be considered. Moreover, in the ultrastrong coupling regime, no matter the detuning is on red sideband $\Delta \approx \omega _{m}$ or on blue sideband $\Delta \approx -\omega _m$, both the beam-splitter (BS) interaction terms $\hat a^{{\dagger }}\hat b + \hat a \hat b^{{\dagger }}$ and the two-mode squeezing (TMS) terms $\hat a\hat b + \hat a^{{\dagger }} \hat b^{{\dagger }}$ in Eq. (1) have to be considered in this paper. The motional modes could entangle with the cavity output modes under either red or blue sideband. By the consideration of system dynamics, the optical tweezers frequency should be adjusted to the near resonant region where the anti-Stokes process is enhanced ($\Delta \approx \omega _{m}$). In this region, the nanoparticle can be cooled and steadily levitated in a high vacuum. Therefore, we perform the system to the red sideband.
Fig. 2. The libration (CoM) frequency $\omega _{\phi (y)}$ (a), the coherent scattering coupling strength between the cavity mode and libration (CoM) mode $g_{s\phi (sy)}$ (b), and the ratio $g_{s\phi (s,y)}/\omega _{\phi (y)}$ (c) as a function of the optical tweezers power $P_t$. $\omega _{\phi }$ ($\omega _{y}$) and $g_{s,\phi }$ ($g_{s,y}$) are theoretically calculated via cavity length $L=1$ mm ($L=10$ mm) and semiaxis of the ellipsoid $a=100$ nm, $b=c=50$ nm ($a=150$ nm, $b=c=60$ nm). Other parameters used in this figure are listed as follow: the wavelength of the optical tweezers $\lambda _t=1550$ nm, the waist of the optical the tweezers $w_0=1$ $\mu$m, the relative permittivity of the ellipsoid $\varepsilon =2.1$, the density of the ellipsoid $\rho =2200$ kg/m$^3$ [55,57].
By placing the position of the nano-ellipsoid to the antinode (node) of the cavity field, the coherent scattering coupling between the CoM (libration) mode and cavity mode is maximized. Under that condition, the CoM and libration motion can be decoupled from each other. Beside, due to the lack of the cavity photons, the cavity enhanced coupling between the libration (CoM) mode and the cavity mode can be safely neglected [33]. Therefore, we can independently consider the coherent scattering coupling between the cavity mode and the libration mode, or CoM mode. Now, we take $\varphi =n\pi /2$ for $n=0,2,4,\ldots$ ($n=1,3,5,\ldots$). Only the coherent scattering coupling between the cavity mode and libration mode (CoM) is considered. Based on Hamiltonian (1), the Langevin equations can be written down as follows
Next, we define the dimensionless quadratures $\hat {Q}_m = {\hat {b}^{\dagger } _m} + \hat {b}_m,\hat {P }_m= i\left ( {{\hat {b}^{\dagger } _m} - \hat {b}_m} \right ),\hat {X} = {\hat {a}^{\dagger } } + \hat {a},\hat {Y}= i\left ( {{\hat {a}^{\dagger } } - \hat {a}} \right )$, and the corresponding noise quadratures ${\hat {Q}^{in}}_m = {\hat {b}^{in{\dagger } }_m} + {\hat {b}^{in}_m},{\hat {P}^{in}_m} = i\left ( {{\hat {b}^{in{\dagger } }}_m - {\hat {b}^{in}}_m} \right ),{\hat {X}^{in}} = {\hat {a}^{in{\dagger } }} + {\hat {a}^{in}},{\hat {Y}^{in}} = i\left ( {{\hat {a}^{in{\dagger } }} - {\hat {a}^{in}}} \right )$. Equations (2) and (3) can be rewritten as a compact matrix form
where ${\hat {n}^T} = \left \{ {\sqrt {\gamma } {\hat {Q}^{in}_m},\sqrt {\gamma _m} {\hat {P}^{in}_m},\sqrt \kappa {\hat {X}^{in}},\sqrt \kappa {\hat {Y}^{in}}} \right \}$, ${\hat {u}^T}= \left \{ {\hat {Q_m},\hat {P_m},\hat {X},\hat {Y}} \right \}$, and the drift matrixThen, it is convenient to characterize the fluctuation of two-mode Gaussian state by the $4\times 4$ covariance matrix (CM) $V$, whose components are given by ${V_{i,j}} = {{\left \langle {{u_i}\left ( \infty \right ){u_j}\left ( \infty \right ) + {u_j}\left ( \infty \right ){u_i}\left ( \infty \right )} \right \rangle } \mathord {\left / {\vphantom {{\left \langle {{u_i}\left ( \infty \right ){u_j}\left ( \infty \right ) + {u_j}\left ( \infty \right ){u_i}\left ( \infty \right )} \right \rangle } 2}} \right. } 2}$ $(i,j=1,2,3,4)$. In the steady state, the corresponding CM $V$ of intra-cavity can be numerically calculated by the following Lyapunov equation
where $D = diag\left \{ {\frac {\gamma }{2}\left ( {2\bar n_m + 1} \right ),\frac {\gamma }{2}\left ( {2\bar n _m+ 1} \right ),\frac {\kappa }{2},\frac {\kappa }{2}} \right \}$ denotes the correlations of different noise [58]. Then the steady entanglement between the caivty mode and the motional mode can be evaluated by the logarithmic negativity [47,48,58] where ${\eta ^ - } \equiv \left ( {{1 \mathord {\left / {\vphantom {1 {\sqrt 2 }}} \right. } {\sqrt 2 }}} \right ){\left [ {\sum {\left ( V \right ) - \sqrt {\sum {{{\left ( V \right )}^2}} - 4\det V} } } \right ]^{{1 \mathord {\left / {\vphantom {1 2}} \right. } 2}}}$ is the smallest partially-transposed symplectic eigenvalue of CM $V \equiv \left \{ {{A_1},{A_3};A_3^T,{A_2}} \right \}$ with $\sum {\left ( V \right )} \equiv \det A_1 + \det A_2 - 2\det A_3$. $A_1$, $A_2$, and $A_3$ are the $2\times 2$ block matrixes of V.3. Output entanglement in a levitated optomechanical system
Although the entanglement between the intra-cavity field and motional mode can be theoretically evaluated from Eqs. (6) and (7). it is still very difficult to detect in experiment. By contrast, measuring the output cavity mode is an available way to research the non-classical correlation in experiment. However, the output field has broad bandwidth which covers both blue and red sidebands regime. In order to distinguish those output photons, a filtering operation is introduced to decompose the continuous traveling light into different temporal modes and code it in a time sequence [59]. The output photons in different frequency are filtered into either the two-mode-squeezing (TMS) or beam-splitting (BS) temporal modes in which the TMS temporal modes carry the two-mode-squeezing interaction and the BS temporal modes origin from the beam-splitting interaction. In the bad cavity limit $\kappa \gg \omega _m$, the temporal modes for the TMS and BS can be easily distinguished in time [59]. And there is almost no overlap between them [49]. The coherence can be preserved during those temporal modes [59]. On the other hand, the dynamics of the intra-cavity field can be adiabatically eliminated in bad cavity limit, leading to the effectively coupling between the motional mode and output field [59]. Based on this, we can investigate the entanglement between the motional mode and output temporal mode in a bad cavity. Note that the cavity decay should not be too large, or the correlations between the output temporal modes and the motional modes are difficult to construct since the cavity photons quickly loss in cavity. It is also notable for the demand of the system dynamics. The smaller cavity decay could amplify the motion of the optically levitated nano-ellipsoid in a high vacuum environment or in a ultra-strong coupling regime, resulting the instability of the system. In this manuscript, the cavity decay is set to $\kappa =3\omega _m$.
The filtering operation can be depicted mathematically by [59]
where $\Gamma$ is the filtering width [59,60]. $F_t$ is responsible for filtering the TMS temporal modes at early time, while $F_b$ takes the BS temporal modes at later time. The selected output mode is given as $\hat a^{\mathrm {f}} (t)= \int {{F_{t(b)}}\left ( {t - s} \right )} {\hat a^{out}}\left ( s \right )\mathrm {d}s$ [59]. Combining with the standard input-output relation ${\hat a^{out}}(t) = \sqrt \kappa \hat a(t) - {\hat a^{in}(t)}$, the output CM $V^{out}$ can be calculated in frequency domain [59,60]$I$ is the identity matrix. $T(\omega )$ is the Fourier transform of
As an example, we calculate the entanglement for the libration motion in the following, while the entanglement for the CoM motion has the similar results. The levitated nano-ellipsoid and cavity field are initial in thermal state and vacuum state respectively. Driving by the optical tweezers, the levitated optomechanical system can be characterized by the multi-mode Gaussian state if the system being linear and stable [61]. Under the filtering operation, the output modes are filtered into two kinds of temporal modes, which one for the contribution with two-mode-squeezing interaction and another for the contribution with beam-splitting interaction. Then the output $6\times 6$ CM of motional mode, TMS temporal mode, and BS temporal mode is given by Eq. (10). The corresponding steady output entanglement between the TMS/BS temporal mode and libration mode, and between the BS and TMS temporal modes can be evaluated via Eq. (7).
Previously, most of studies focused on the weak coupling regime where the rotating-wave approximations was valid. As shown in Fig. 3(a), $-ln(2\eta ^{-})$ as a function of the filtering width $\Gamma$ with $g_{s\phi }=0.04\omega _\phi$. Note that the BS interaction will not contribute to the BS-LIB entanglement. It conforms the view that the BS interaction cannot generate the optomechanical entanglement in a weak coupling regime [60]. By contrast, the TMS temporal mode and libration mode are entangled via the two-mode squeezing interaction. With the state swapping by the beam-splitter interaction, TMS-BS entanglement is induced by the TMS-LIB entanglement. Obviously, the induced TMS-BS entanglement is smaller than the TMS-LIB entanglement. Next, we consider the coherent scattering coupling is enhanced from $g_{s\phi }=0.04\omega _\phi$ to $g_{s\phi }=0.4\omega _\phi$. In this condition, the system arrives the ultrastrong coupling regime. Both of the rotating-wave terms and counter-rotating wave terms are involved to generate the entanglement. As shown in Fig. 3(b), the steady entanglement can be generated between the BS temporal mode and libration mode. The steady entanglement also depends to the filtering width $\Gamma$. $\Gamma$ cannot be too large, or the incoherent spectral components would be collected and the interaction time is not enough to generate the entanglement [59]. The optimum $\Gamma$ is related to the effective damping rate of motional mode, which would approximately fit the width of Stokes and anti-Stokes peaks in output spectral [60]. It enables to attain a more entangled state by appropriately filtering the output light.
Fig. 3. $- \ln \left ( {2{\eta ^ - }} \right )$ as a function of the filtering width $\Gamma$ in a single levitated optomechanical system. The legend TMS/BS-LIB denotes the output entanglement between the filtered TMS/BS temporal mode and libration mode (LIB). Another legend TMS-BS represents the output entanglement between the TMS and BS temporal modes. The coherent scattering coupling in pictures (a) and (b) are given as $g_{s\phi }=0.04\omega _\phi$ and $g_{s\phi }=0.4\omega _\phi$, respectively. Parameters are listed below: the optical tweezers power $P_t=0.01$ W, the pressure of residual gas $P=10^{-4}$ Pa, the temperature of residual gas $T_a=300$ K, the temperature for libration mode $T=300$ K, the accommodation efficient $\gamma _{ac}=0.9$, the decay rate $\kappa =3\omega _\phi$. Other parameters are the same with Fig. 2. Note that the parameters in this paper fulfill the demand of steady trapping according to the Mie scattering theory [62] and the Routh-Hurwitz criterion [58].
4. Remote entanglement between two distant nanoparticles
The steady entanglement between the motional mode and output mode can be generated via the ultra-strong optomechanical coupling, in which either the TMS modes or BS modes are filtered appropriately. Based on the above discussion, it is possible to induce the motional entanglement between two distant levitated optomechanical systems via the Bell-like detection [15,16,48,54]. In this manner, the filtered modes of the both levitated optomechanical system are interfaced by the beam splitter, allowing the entanglement swapping in a continuous variable system. Then the conjugate homodyne detections are applied to measure the position and momentum quadratures, respectively [54,63]. One mode for the detection of the $\hat {p}$ quadrature while another is measured in $\hat {p}$ quadrature (see Section 5 in Supplement 1). Communicating the results of the Bell-like detection to both remote nanoparticles dissipatively drives them into an entangled state.
The output CM of each levitated cavity optomechanical system can be written as the following expression
whereIn terms of the non-ideal Bell-like detection, the retaining CM of libration modes is given by [54]
where the elements of matrix $K$ are listed by$T_r$ is the transmissivity of the beam splitter. $\eta ^c_l=\eta _l 10 ^{-\alpha _0 d/10}$ is the detector efficiency of the $l$-th detector, which includes the quantum efficiency $\eta _l$ and the channel loss [64]. $\alpha _0$ and $d$ are the loss coefficient and the length of the channel.
Now, we study the remote entanglement between two distant levitated nano-ellipsoids. Without loss of generality, we assume that the two levitated optomechanical systems are identical and perform the same filtering operations to extract the TMS or BS temporal modes. The steady entanglement between two distant libration modes can be generated by Bell-like detection and evaluated by Eqs. (7) and (19). As depicted in Fig. 4, the remote entanglement obtained by performing the ideal Bell-like detection on BS temporal modes is larger than the entanglement induced by measuring the TMS temporal modes. This seems contradict to the results shown in Fig. 3, where the entanglement of TMS-LIB is lager than that of the BS-LIB [47–49]. However, as we known, most photons are scattered into the BS temporal mode in the red sideband ($\Delta _\phi =\omega _\phi$). In terms of the output intensity, the BS temporal mode, which contains the stronger entanglement, is much greater than that of the TMS temporal mode [61].
Fig. 4. $- \ln \left ( {2{\eta ^ - }} \right )$ as a function of the filtering width $\Gamma$ in two remote levitated system. The legend BS-BS (TMS-TMS) denotes that the remote entanglement between libration modes induced by measuring the filtered output BS (TMS) temporal modes. The detector efficiency is assumed to be ideal $\eta ^c=1$ ($\eta =1,\alpha _0=0$). Other parameters are the same with Fig. 3.
By comparing the other optomechanical systems, the optically levitated nanoparticle has no contact damping [61]. With reduction of the residual gas pressure, the collisions between the gas molecules (or atoms) and optically levitated nanoparticle are reduced. On the one hand, the absorbed heat from the optical tweezers is difficult to release since the reduction of gas molecule collisions. The internal temperature of the nanoparticle will be increased with the accumulative heat. With such, the thermal motion of the optically levitated nanoparticle will be quickly amplified, resulting the instability in dynamics. On the other hand, The damping on the motional nanoparticle will decrease with the decrease of the residual gas pressure for the lack of gas molecules collisions. It will give rise to an ultra-high mechanical quality factor. According to theoretical estimation [55,65], the quality factor of the libration mode exceeds $10^9$ if the pressure of the residual gas reaches $P=10^{-4}$ Pa (see Section 3 in Supplement 1). It enables the system to have a very long coherence time in a highly isolated environment [55]. Therefore, as shown in Fig. 5(a), the steady entanglement of two distant libration modes is robust to the temperature of the thermal bath. Even under the room temperature, the relatively high quantum entanglement between two libration modes can still achieve. As the residual gas pressure increases, the soaring collisions of the gas molecules will ultimately heat the libration motion of the nano-ellipsoid, leading to a dramatic decoherence. The motional entanglement will be diminished when the residual gas pressure increases.
Fig. 5. (a) The steady entanglement of two distant libration modes $E_n$ as a function of the thermal bath temperature $T$ with different pressure of the residual gas $P$. (b) The steady entanglement of two distant libration modes $E_n$ as a function of the quantum efficiency $\eta$. The optical fiber is assumed to be ideal ($\alpha _0$=0 km/dB). The BS temporal modes of each independent levitated optomechanical system are filtered with $\Gamma =1.5\times 10^5$ rad$\cdot$Hz. Other parameters are the same with Fig. 4.
The above results are based on the ideal Bell-like detection. The corresponding quantum efficiency of homodyne detector and loss coefficient of the optical fiber are $\eta =1$ and $\alpha =0$ dB/km respectively. However, the received signals are not ideal since the optical signals can not fully be converted to the current or voltage signals and the signals inevitably couple to the environment during transmission. Both the quantum efficiency of the detector and the signal loss of the optical fiber should be taken into consideration when two distant nano-ellipsoids are designed to generate the remote entanglement. Theoretically, the quantum efficiency of the homodyne detector and the signal lose of the optical fiber can be modeled as a fictitious beam-splitter in front of the ideal homodyne detector. Both the quantum efficiency and the loss coefficient can be included into the detector efficiency, corresponding to the transmissivity of the fictitious beam-splitter $0<\eta ^c<1$ [66]. As shown in Fig. 5(b), the steady entanglement between the two distant libration modes decreases to zero when the quantum efficiency $\eta$ is around $0.8$. It indicates that the highly efficient detector is available to explore the quantum properties of the light. In the presence of the remote entanglement, the maxim distance between two remote subsystems can be evaluated by the state-of-the-art detector and optical fiber. With the realistic superconducting detector ($\eta$=0.98@1550 nm) and optical fiber ($\alpha _0=0.14$ dB/km) [67,68], the maximum distance between the two entangled nano-ellipsoids could be $12$ km.
5. Discussion on experiment
Inspired by the experimental advances on levitated optomechanics, i.e., ground sate cooling [33–36], strong optomechanical coupling [37–40], and squeezing light by quantum control [69,70], we proposed a implementable scheme to generate the motional entanglement between two distant nanoparticles [15,50,59,71]. Although the motional entanglement has been observed in trapped atomic irons [12]. As the optically levitated particle size increases, thermal decoherence on quantum motional state will be remarkable [23,24,72]. The optomechanical entanglement is hard to be generated. In the levitated optomechanical systems, residual gas collisions and photon recoil are the main sources of thermal decoherence [23,24]. Levitating the nanoparticle in a higher vacuum environment is an effective way to reduce the residual gas collisions, thus high degree of isolation from the environment [37]. When the residual gas pressure is low enough, the photon recoil becomes dominant, and the optically levitated nanoparticle will be heated via the incoherent scattering [73]. According to the parameters in Fig. 3(b), the heating rates for the photon recoiling and gas collisions are corresponding to $\Gamma _p=0.25$ kHz and $\Gamma _g=4.69$ kHz, both of which are smaller than the coherent scattering coupling $g_{s,\phi }=53$ kHz (see Section 4 in Supplement 1). With the coherent scattering coupling exceeds the decoherence rate, it allows to create and manipulate the non-classical states before decoherence [37,72], which is a prerequisite for our scheme to achieve the remote entanglement. Then two remote levitated optomechanical systems are integrated in a way of Bell-like homodyne detection. The motional entanglement between two distant nanoparticles is available via the entanglement swapping [15,16,48,54]. Applying the nearly quantum-limited measurement on position and momentum quadratures of both nanoparticles [69,70], the corresponding covariance matrix can be directly observed via state tomography [72]. Thereby, we can perform the logarithmic negativity to evaluate the steady motional entanglement in realistic experimental setup.
6. Conclusion
In summary, we have proposed a scheme to achieve the steady motional entanglement between two distant optically levitated nanoparticles in a unconditional way. The ultra-strong coupling between the motion of the nanoparticle and the cavity mode is feasible under the coherent scattering mechanism. Therefore, the rotating-wave approximation is no longer valid here. The optical tweezers are in red sideband of the cavity modes, and the strong and robust entanglement between the nanoparticle and the output cavity mode can be generated. Furthermore, by the Bell-like detection, two distant nanoparticles can be entangled unconditionally through entanglement swapping. The generated remote steady entanglement is a valuable resource for the matter-based quantum sensor network, which can achieve the measurement precision beyond the standard quantum limit [4,5,74–76]. Besides, in an ultra-strong optomechanical coupling regime, a lot of new phenomena and applications become possible, such as the single photon blockade [77,78], quantum simulations [51,79], etc.
Funding
National Natural Science Foundation of China (61771278).
Disclosures
The authors declare no conflicts of interests.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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