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Abstract
An optomechanical system is a promising platform to connect different “notes” of quantum networks. Therefore, entanglements generated from it is also of great importance. In this paper, the parameter dependence of optomechanical and optical-optical entanglements generated from the double-longitudinal-mode cavity optomechanical system are discussed and two quadrapartite entanglement generation schemes based on such a system are proposed. Furthermore, 2N and 4N-partite entangled states of optical modes can be obtained by coupling N cavities that used in the above two schemes with N-1 beamsplitters, respectively. Certain ladder or linear entanglement structures are included in the finally obtained entangled state, which are important for its application in one-way quantum computing.
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1. Introduction
Quantum entanglement, as a profound feature of quantum theory, is the prerequisite of quantum teleportation [1], quantum dense coding [2], quantum computing [3], quantum communication [4], and quantum networks [5]. With the development of quantum information technology, multipartite entanglement is required for realizing measurement-based quantum computing and building practical quantum networks. One conventional way to generate multipartite entangled state is combining optical parametric processes and beamsplitter(BS)s [6–8], but the partite number is limited by the complex experimental setup and technical difficulty. Another effective way is using frequency/time multiplexing [9,10], by which multipartite entanglement of ultra-large scale can be achieved with a more compact setup. However, hybrid quantum networks depend on the generation and distribution of entanglement among different physical systems (with different frequencies), such as atoms [11,12], ions [13,14], quantum dots [15], superconducting circuits [16,17], etc., which is a big challenge for the above-mentioned methods. Therefore, the optomechanical system can be considered as a promising candidate for connecting different components in hybrid quantum systems. Thus, various entangled states have been generated from cavity optomechanical systems theoretically [18–37] and experimentally [38–41], including entangled states between light (microwave) modes [18–21], light (microwave) and mechanical modes [22–25,38,39], and mechanical modes [26–28,40,41]. In particular, entanglement between light and/or microwave modes generated from optomechanical system is of great importance, because it is valuable for building hybrid quantum networks. In the past ten years, entanglement between two light modes, such as two spatial modes [29–31], two longitudinal modes in a single cavity optomechanical system [32], two cavity modes in a double-cavity optomechanical system [19], were investigated and discussed. A scheme of entangling optical and microwave cavity modes by means of a nanomechanical resonator was also proposed, which indicates that the mechanical resonator can mediate the robust entanglement between the optical and microwave cavity modes [20]. The generation of microwave-optical entanglement from a piezo-optomechanical system and microwave-microwave entanglement by entanglement swapping with the generated entangled microwave-optical sources was studied recently [21]. In addition, generating multipartite entanglement through interaction with the mechanical mode, including three optical cavity modes, two optical and one microwave modes, one optical and two microwave modes, were further proposed and discussed [33–37]. However, in the above mentioned works, the partite numbers of the entangled state among optical or/and microwave modes were no more than three, which is not enough for connecting many different “notes” composed of different physical systems, in a hybrid quantum network. In this paper, we discuss the double-longitudinal-mode cavity optomechanical system in detail and propose two schemes for generating multipartite continuous variable (CV) entangled states by coupling some certain output modes of N such systems.
2. Double-longitudinal-mode cavity optomechanical system
The driven double-longitudinal-mode cavity optomechanical system is shown in Fig. 1(a). The cavity includes a fixed mirror and a movable mirror (with frequency $\omega _m$ and decay $\gamma _m$) and is driven by an input laser (with frequency $\omega _L$). Considering the condition that the mechanical frequency $\omega _m$ is equal to half of the free spectral range (FSR), the corresponding Stokes and anti-Stokes side-bands of the input driven laser can resonate with two longitudinal modes of the cavity, and the frequency relation is shown in Fig. 1(b). The Hamiltonian of this system can be described as [32,42]:
Fig. 1. The considered double-longitudinal-mode cavity optomechanical system. (a) Schematic diagram. (b) Frequency relation.
From the Hamiltonian described by Eq. (1), the linearized quantum Langevin equations of the quadrature fluctuations ($\delta q$, $\delta p$, $\delta X_{1}$, $\delta Y_{1}$, $\delta X_{2}$, $\delta Y_{2}$) can be obtained as $\dot {u}=\mathcal {A} u + n(t)$ (details are shown in Appendix 5.1), where $\delta X_{j}=(\delta a_{j}+\delta a_{j}^{\dagger })/\sqrt {2}$, and $\delta Y_{j}=i(\delta a_{j}^{\dagger }-\delta a_{j})/\sqrt {2}, j=1, 2$. Then the corresponding covariance matrix ($CM$) can be obtained by solving the Lyapunov equation, and the optical-optical and optomechanical entanglements were studied carefully in Ref. [32], with the criterion of logarithmic negativity. $\Delta _{j}=\omega _{j}-\omega _{L} (j=1, 2)$ is the frequency detuning of the cavity field $j$ with respect to the input laser field in the frame rotating at the input field frequency $\omega _L$. However, the steady state solutions used in Ref. [32] are approximate and some special characteristics were absent. Therefore, we check the model again and further investigate the parameter dependence (including the detuning $\Delta _{2}$, cavity decay $\kappa$, mechanical decay $\gamma _{m}$ and temperature $T$) of the entanglements, based on the general steady state solutions of the quantum Langevin equations. The optical-optical entanglement $E_{12}$, the optomechanical entanglements $E_{01}$ and $E_{02}$, versus detuning $\Delta _2$, are shown in Fig. 2. As expected, $E_{12}$, $E_{01}$ and $E_{02}$ reach their maximum at $\Delta _{2}=\omega _{m}$ ($\Delta _{1}=-\omega _{m}$), due to the perfect collaboration of the two-mode squeezing (entanglement generation) and the beam-splitter (state transfer) interaction mechanisms. With the increase of detuning $\Delta _{2}$, these two interaction mechanisms become weaker and unbalanced, and $E_{12}$ decreases. Note that $E_{01}$ and $E_{02}$ are not identical, which is different from that indicated in Ref. [32]. In particular, $E_{02}$ is not monotonous. It experiences a decreasing process with smaller entanglement and then an increasing process with larger entanglement, compared with $E_{01}$. This is understandable: for $E_{01}$, the entanglement generation mechanism changes from strong to weak; while for $E_{02}$, the cooling mechanism becomes to dominate under proper red-detuning [43], which can react the entanglement to be better, during the increase of detuning $\Delta _2$. It is also noticed that the entanglement mainly exists in the zone $\Delta _{2}\geq \omega _{m}$. This may because that the effective mechanical damping rate $\Gamma _{eff}$ is negative in a wide range of $\Delta _{2}<\omega _{m}$, which leads to amplification of thermal fluctuations and finally to an instability and the prediction of entanglement is breakdown [43]. The definition of the effective mechanical damping rate is $\Gamma _{eff} =\Gamma _{opt} +\Gamma _{m}$, where $\Gamma _{opt}$ is the optomechanical damping rate and $\Gamma _{m}$ is the mechanical damping rate. Under the conditions in Fig. 2 $\Gamma _{eff}<0$ at $\Delta _{2}<\omega _{m}$, because $\Gamma _{opt}$ is negative and $\Gamma _{m} \ll \lvert \Gamma _{opt} \rvert$. The optomechanical entanglements $E_{01}$ and $E_{02}$ versus cavity delay $\kappa$, mechanical delay $\gamma _m$, and detuning $\Delta _{2}$, are shown in Fig. 3(a)-(c). It is obvious that optomechanical entanglements decrease with the increase of cavity decay $\kappa$, while increasing $\gamma _m$ can slow down this decrease of entanglement, as is shown in Fig. 3(a). In Fig. 3(b), $E_{01}$ and $E_{02}$ decrease with the increase of detuning $\Delta _{2}$ and increasing $\kappa$ can fasten this decrease of entanglement. In Fig. 3(c), increasing $\gamma _m$ can keep the entanglement robust when $\Delta _{2}=\omega _m$ (maximum entanglement condition), and can promote the entanglement to bigger value when $\Delta _{2}>\omega _m$. These results means that bigger detuning or cavity decay will destroy the entanglements, but bigger mechanical decay $\gamma _m$ may help to increase the entanglements and there exists an optimized mechanical decay, as is further shown in Fig. 4. The optical-optical entanglement $E_{12}$ versus cavity delay $\kappa$, mechanical delay $\gamma _m$, and detuning $\Delta _{2}$, are shown in Fig. 3(d)-(f). In Fig. 3(d), $E_{12}$ decrease when $\kappa$ and $\gamma _m$ increase. It is interesting that there is a dispersion-like shape (corresponding to the two-phonon resonance) at a certain point of $\kappa$, because the Lorentz line shape of two cavity modes cannot be ignored if $\kappa$ is big enough. Increasing $\gamma _m$ makes the position of the certain point of $\kappa$ change to a bigger value, because faster decaying phonon number decreases the probability of the two-phonon resonance. Fig. 3(e) shows that bigger detuning $\Delta _2$ makes the dispersion-like region wider, because frequency difference between cavity mode 2 and pump mode increases and two cavity modes are no longer symmetric to the pump mode any more. Note that entanglement existing range can cover part of $\Delta _2<\omega _m$ zone. Because $\Gamma _{opt}$ will change from negative to positive and more positive when $\kappa$ becomes bigger and bigger. Fig. 3(d)-(f) prove that detuning, mechanical or cavity decays, are all negative factors for $E_{12}$. However, mechanical decay can be a positive factor for $E_{01}$ and $E_{02}$, which is counter-intuitive. Therefore, larger ranges of $\gamma _m$ are considered and shown in Fig. 4(a) and Fig. 4(b), with different detunings. The positive role of mechanical decay $\gamma _m$ is limited, and optimized values (changes with detuning) exist for the strongest entanglements.
Fig. 2. The optical-optical entanglement $E_{12}$, the optomechanical entanglements $E_{01}$ and $E_{02}$, versus $\Delta _{2}$. The relative parameters are: $L=0.01m$, $T= 0.01 K$, $\lambda =1.33\mu m$, $\gamma _{m}=0.1MHz$, $\kappa _{1}=\kappa _{2}=\kappa =1MHz$, $m=5\times 10^{-9} kg$, $P=20mW$, $\Delta _{1}=\Delta _{2}-2\omega _{m}$.
Fig. 3. The optomechanical entanglements $E_{01}$ and $E_{02}$ ((a),(b),(c)), and the optical-optical entanglement $E_{12}$ ((d),(e),(f)), vary with detuning $\Delta _2$, cavity decay $\kappa$ and mechanical decay $\gamma _m$. In some regions of subgraph (a), (b) and (c), only one color is shown and the visible color stands for stronger entanglement. $\gamma _{m}=0.01MHz$. Other parameters are the same as that in Fig. 2.
Fig. 4. Optomechanical entanglements $E_{01}$ and $E_{02}$ versus mechanical decay $\gamma _m$. (a) $\Delta _{2}=\omega _m$. (b) $\Delta _{2}=1.005\omega _m$. $\kappa =10MHz$. Other parameters are the same as that in Fig. 2.
In addition, other than $E_{01}$ and $E_{02}$, $E_{12}$ is robust enough even at room or higher temperatures, as is shown in Fig. 5, which is the same as the corresponding conclusion in Ref. [32]. However, the difference between $E_{01}$ and $E_{02}$, the dispersion-like shape of $E_{12}$ versus $\kappa$, and the positive role of $\gamma _m$, are the characteristics that was absent in Ref. [32].
Fig. 5. Entanglements versus temperature $T$, with $\Delta _{2} = \omega _{m}$ and $\gamma _{m}=0.1MHz$. (a) Optomechanical entanglements. (b) Optical-optical entanglement. Other parameters are the same as that in Fig. 2.
3. Multipartite optical-optical entanglement generation
Multipartite entanglement can be realized by increasing the dimension of entanglement, because the quantity of entanglement is finite in each dimension. Based on the robust bipartite optical-optical entanglement, one can further expand the partite number (which is very important to the ability of quantum computing) generally in two ways: coupling them with BS(s) [44], and using double pumps with orthogonal polarization [24,25]. In the following sections, these two ways are discussed separately to generate multipartite optical-optical entanglement. To make it clearer and more understandable, schemes for generating quadrapartite entanglements are considered first.
3.1 Scheme 1: coupling with BS(s)
The scheme of generating quadrapartite entanglement by using a BS, is shown in Fig. 6(a). Output beams $a^{\prime }_1$ and $a^{\prime }_3$ (same frequency) from two identical double-longitudinal-mode cavity optomechanical systems, are coupled through a BS, and the corresponding frequency relations are shown in Fig. 6(b). Here $a^{\prime }_1$ and $a^{\prime }_2$ are the two output modes of $a_1$ and $a_2$, and $a^{\prime }_3$ and $a^{\prime }_4$ represent the output modes of the added optomechanical cavity, where $\omega _1=\omega _3$, $\omega _2=\omega _4$, and $\omega _{m1}=\omega _{m2}=\omega _m$. In this case, the Hamiltonians of the two cavity optomechanical systems can be described separately:
Fig. 6. The scheme of generating quadrapartite entanglement using a BS. (a) Schematic diagram. (b) Frequency relation.
Here $g_0=\sqrt {\hbar \omega _{1} \omega _{2}/ m \omega _{m1}} / L =\sqrt {\hbar \omega _{3} \omega _{4}/ m \omega _{m2}} / L$, and $\eta _{j}=\sqrt {2P\kappa _{j}/\hbar \omega _{j}}$. Other parameters are defined the same as the above. The third terms represent the optomechanical couplings, and the last terms are the driving terms of the cavity modes. As is well known, the BS transformation can be expressed by matrix $U _{BS}=\left (\begin {matrix} cos\theta e^{i\phi } & sin\theta e^{i(\phi +\pi )} \\ sin\theta & cos\theta \\ \end {matrix} \right )$, where $\phi$ and $cos\theta$ represent the relative phase and the transmission coefficient of the BS. Therefore, the generated quadrapartite entanglement of the filtered output modes versus BS parameters ($\theta$ and $\phi$) can be obtained (details can be found in Appendix 5.2), as is shown in Fig. 7. It is interesting that one can obtain different quadrapartite entanglement structures by changing the BS parameters. When $\theta =\pi /8$ (85:15) and $\theta =3\pi /8$ (15:85), quadrapartite entanglements with linear structure can be obtained at about $2\pi /5<\phi <3\pi /5$ (described by the gray shadow areas), as are shown in Fig. 7(a) and Fig. 7(c), and note that the two linear structures are not the same. If $\theta =\pi /4$ (50:50), a square structure entanglement can be generated, regardless of the relative phase $\phi$ takes, as is shown in Fig. 7(b). Therefore, this scheme may help to realize controllable multipartite entanglement, whose entanglement structure can be adjusted by changing the BS parameters.
Fig. 7. The quadrapartite entanglement of the filtered output modes versus BS parameters $\phi$ with different $\theta$. $L=0.01m$, $T= 0.01 K$, $\omega _{m}=\frac {\pi c}{2L}$, $\lambda =1.33\mu m$, $\kappa =100\gamma _{m}=1MHz$, $m=5\times 10^{-9} kg$, $P=20mW$, $\Delta _{1}=-\Delta _{2}=\Delta _{3}=-\Delta _{4}=-\omega _{m}$, $\tau _j=\frac {1}{\omega _m}$, $\Omega _1=\Omega _3=-\Omega _2=-\Omega _4=-\omega _m$, where $\tau _j^{-1}$ and $\Omega _{j}$ are the bandwidth and central frequency of the $j$-th filter, respectively.
Moreover, 2N-partite entanglement generation scheme based on the above scheme is shown in Fig. 8. By combining many identical double-longitudinal-mode cavity optomechanical systems and coupling output beams with same frequencies through several BSs, a 2N-partite entangled state can be generated, and the corresponding entanglement structures with different BS parameters are shown in Fig. 9. As is shown in Fig. 9(b), the entanglement structure changes from bipartite (one cavity) to quadrapartite (two cavities), hexapartite (three cavities), octapartite (four cavities),…, and 2N-partite (N cavities), by using 0, 1, 2, 3,…, N-1 50:50 BSs, respectively. Here red bars describe the original entanglements from double-longitudinal-mode cavity optomechanical systems. Blue and green bars mean the generated entanglements between modes of adjacent and next-adjacent cavities, respectively, after using BS(s). The generated 2N-partite entanglement has a special “double ladder” structure, including ladders formed by red-blue and blue-green bars. In Fig. 9(a), different BS parameter ($\theta =\pi /8$) is considered, and a linear entanglement structure can be obtained. Note that this linear structure can be viewed as removing those blue bars in the “double ladder” structure shown in Fig. 9(b), and the yellow bar describes the entanglement generated from the last added BS. Finally, 2N-partite entanglements of different structures can be obtained by coupling N cavities with N-1 BSs. Therefore, manipulation of the entanglement structure can be realized by changing the BS parameters, which may be helpful for further applications of quantum computing.
Fig. 9. Generation of linear (a) and “double ladder” (b) entanglement structure. (a) $\theta =\pi /8$ and $\phi =\pi /2$. (b) $\theta =\pi /4$ and $\phi =\pi /2$. Other parameters are the same as that in Fig. 7. Note that the blue balls stand for those optical modes, and each column corresponds to two modes from the same cavity. The column formed by two yellow balls stand for the middle part that are not shown separately.
3.2 Scheme 2: using double pumps with orthogonal polarization
The other scheme of generating quadrapartite entanglement by using double pumps with orthogonal polarizations is shown in Fig. 10(a), and Fig. 10(b) indicates the corresponding frequency relation. This time two sets of longitudinal modes with orthogonal polarizations both resonate in the cavity. The intracavity modes $a_1$ ($a_2$) and $a_3$ ($a_4$) have same frequencies, $\omega _1=\omega _3$ ($\omega _2=\omega _4$), but their polarizations are different ($a_1$ ($a_2$): H, $a_3$ ($a_4$): V). Then, the Hamiltonian of this system can be described as [42,45,46]:
Fig. 10. The scheme of generating quadrapartite entanglement by using double pumps with orthogonal polarizations. (a) Schematic diagram. (b) Frequency relation.
Here $g_0=\sqrt {\hbar \omega _{1} \omega _{2}/ m \omega _{m}} / L =\sqrt {\hbar \omega _{3} \omega _{4}/ m \omega _{m}} / L$, and $\eta _{j}=\sqrt {2P\kappa _{j}/\hbar \omega _{j}}$. Other parameters are defined the same as the above.
Based on scheme 2, entanglements between cavity modes can be obtained (details can be found in Appendix 5.3), as are shown in Fig. 11. Fig. 11(a) and Fig. 11(b) show that entanglement can be generated between each two of the four optical modes, i.e., a quadrapartite GHZ entangled state can be obtained. Specifically, entanglements between those modes with different frequencies $E_{12}$, $E_{34}$, $E_{14}$ and $E_{23}$ are the same and have the same value as those in single double-londitudinal-mode cavity optomechanical system, due to the same entanglement generation mechanism. However, entanglements between modes with the same frequency ($E_{13}$ and $E_{24}$) have totally opposite trend and are not limited by the effective mechanical damping rate. This may result from different mechanisms: connecting modes with same frequency by mechanical mode is different from the Stokes and anti-Stokes scattering processes that connecting modes with different frequencies. It seems that the mechanical mode (the movable mirror) is not so sensitive to polarization, but has a strong dependence on frequency. Note that the entanglements $E_{13}$ and $E_{24}$ always fluctuate with the detuning $\Delta _2$, which is similar to that in Ref. [19]. The fluctuating amplitude is related to the cavity and mechanical decays, and a larger decay will lead to more violent fluctuations. Similarly, by coupling certain output beams from several such kind optomechanical cavities pumped with orthogonal polarizations, a 4N-partite entangled state can be obtained, as is shown in Fig. 12 and Fig. 13. Here red bars still describe the original entanglements from orthogonal-polarization-pumped optomechanical systems, and blue and green bars still mean the generated entanglements between modes of adjacent and next-adjacent cavities, respectively, similar to that in Fig. 9. Here the generated 4N-partite entanglement by using 50:50 BSs, as is shown in Fig. 13(b), contains 6 sets of ladder entanglement structures including 5 ladders formed by red-blue bars and 1 ladder formed by blue-green bars. In Fig. 13(a), different BS parameter ($\theta =\pi /8$) is considered, and a special entanglement structure can be obtained, which can be viewed as removing some blue bars in the structure shown in Fig. 13(b), and the yellow bar still describes the entanglement generated from the last added BS. Finally, 4N-partite entanglements of different structures can be obtained by coupling N cavities with N-1 BSs. Thus, one can control this kind of three-dimensional entanglement structure by changing BS parameters, which provides rich possibilities for building some special-structured 3D multipartite entangled states towards quantum computing and quantum networks.
Fig. 11. The optical-optical entanglements with different frequency ($E_{12}$, $E_{34}$, $E_{14}$ and $E_{23}$) and with same frequency ($E_{13}$ and $E_{24}$) versus detuning $\Delta _{2}$. The relative parameters are the same as that in Fig. 2.
Fig. 13. (a) $\theta =\pi /8$ and $\phi =\pi /2$. (b) $\theta =\pi /4$ and $\phi =\pi /2$. Other parameters are shown as Fig. 11. Note that the blue balls stand for those optical modes, and each slice corresponds to four modes from the same cavity. The slice formed by four yellow balls stand for the middle part that are not shown separately.
4. Conclusion
In this paper, we investigate the parameter dependence of optomechanical and optical-optical entanglements generated from a double-longitudinal-mode cavity optomechanical system and propose two multipartite entanglement generation schemes based on such systems. Compared with the conventional method of combining optical parametric processes and BSs, our schemes require less BSs to obtain the same or twice partite number of entanglement, and this may simplify the experimental setup. Rich entanglement structure can also be obtained, including certain ladder or linear structures, which may be promising choices to be applied in quantum computing and quantum networks in the future. Our schemes of increasing entanglement-partite-number have high scalability, and the obtained entanglement can be further enhanced by cascading, feedback and nonlinear gain medium. Considering this optomechanical coupling together with multiplexing of new degrees of freedom, such as frequency, orbital angular momentum, etc., it is also possible to build large-scale entangled states, which is very important for quantum computing ability. Our work provides a new thought of generating multipartite entangled states applied to connecting different physical systems with different frequencies in practical quantum networks.
5. Appendix
Here we provide the details on how we obtain the steady state entanglement. The entanglement is calculated based on the covariance matrix of the cavity modes and mechanical mode. The covariance matrix can be achieved by solving the linearized quantum Langevin equations, which can be rewritten in the following form:
where u(t) is the vector of quadrature fluctuation operators of cavity modes and mechanical mode. $\mathcal {A}$ is the drift matrix, and n(t) is the vector of noise quadrature operators associated with the noise terms.The steady-state covariance matrix $V(t\rightarrow \infty )$ of the system quadratures, with its entries defined as $V_{ij} = \frac {1}{2} \langle {\{u_i(t),u_j(t)\}}\rangle$, which can be obtained by solving the Lyapunov equation:
where $\mathcal {D}$ is the diffusion matrix, with its entries defined as $\frac {1}{2} \langle {n_i(t) n_j(t^{\prime }) +n_j(t^{\prime }) n_i(t)}\rangle =\mathcal {D}_{ij}\delta (t-t^{\prime })$. $a_{in}$ and $\xi$ are the optical noise operators of the cavity mode and the mechanical mode, respectively, which are zero mean and characterized by the following correlation function:Once the covariance matrix $V$ is obtained, the entanglement can then be quantified by means of logarithmic negativity [47,48]:
where $\nu _{-} = min \thinspace eig\lvert i\Omega _2V_m\rvert$ ($\Omega _2= \oplus i \sigma _y$ is the so-called symplectic matrix and $\sigma _y$ is the y Pauli matrix) is the minimum symplectic eigenvalue of the covariance matrix $V_m= PVP$, with $V_m$ being the $4 \times 4$ covariance matrix associated, and $P = diag\begin {pmatrix} 1, 1, 1, -1 \end {pmatrix}$ the matrix that inverts the sign of momentum, $p_2\rightarrow -p_2$, realizing partial transposition at the level of covariance matrices.5.1 Single double-longitudinal-mode cavity optomechanical system
From the Hamiltonian of a single double-longitudinal-mode cavity optomechanical system, which is described in Eq. (1), the corresponding linearized $QLEs$ of quadrature fluctuations ($\delta q$, $\delta p$, $\delta X_{1}$, $\delta Y_{1}$, $\delta X_{2}$, $\delta Y_{2}$) can be obtained as:
5.2 Coupling two double-longitudinal-mode cavity optomechanical systems by a BS
In the frame rotating at the input field frequency ($\omega _{L}$), the quantum Langevin equations ($QLEs$) describing the system shown in Fig. 6 can be written as:
The correlation between the output optical modes can be analyzed in terms of thermal modes, appropriately filtered from the field with specific filter functions. Simple and explicit filter functions in time and frequency domain are given by
and where $\tau _j^{-1}$ and $\Omega _{j}$ are the bandwidth and central frequency of the $j$-th filter, respectively. $\theta _{j}(t)$ is the Heaviside step function. Correspondingly, the field’s filtered modes can be written by the bosonic annihilation operatorThe transformation matrix of filter function can be expressed as
The total $CM$ of all output modes can be written as:
where $V_i (i=1,2)$ is the $CM$ of the $i$-th cavity and $U$ is the BS matrix, which can be described as5.3 Pumping the double-longitudinal-mode cavity optomechanical system with two orthogonal polarizations
In the frame rotating at the input field frequency $\omega _L$, the quantum Langevin equations ($QLEs$) describing the system that shown in Fig. 10, can be written as:
The corresponding $\mathcal {D}$ matrix of Eq. (6) is
Funding
National Key Research and Development Program of China (2021YFA1402002); National Natural Science Foundation of China (11874248, 11874249, 11974225).
Disclosures
The authors declare no conflicts of interest.
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.
References
1. S. Pirandola and S. L. Braunstein, “Physics: Unite to build a quantum internet,” Nature 532(7598), 169–171 (2016). [CrossRef]
2. Y. Guo, B.-H. Liu, C.-F. Li, and G. Guo, “Advances in quantum dense coding,” Adv. Quantum Technol. 2(5-6), 1900011 (2019). [CrossRef]
3. T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. R. Monroe, and J. L. O’Brien, “Quantum computers,” Nature 464(7285), 45–53 (2010). [CrossRef]
4. N. Gisin and R. Thew, “Quantum communication technology,” Electron. Lett. 46(14), 965–967 (2010). [CrossRef]
5. K. Fang, J. Zhao, X. Li, Y. Li, and R. Duan, “Quantum network: from theory to practice,” arXiv, arXiv:2212.01226 (2022). [CrossRef]
6. X. Su, A. Tan, X. Jia, J. Zhang, C. Xie, and K. Peng, “Experimental preparation of quadripartite cluster and greenberger-horne-zeilinger entangled states for continuous variables,” Phys. Rev. Lett. 98(7), 070502 (2007). [CrossRef]
7. X. Su, Y. Zhao, S. Hao, X. Jia, C. Xie, and K. Peng, “Experimental preparation of eight-partite cluster state for photonic qumodes,” Opt. Lett. 37(24), 5178 (2012). [CrossRef]
8. M. Yukawa, R. Ukai, P. van Loock, and A. Furusawa, “Experimental generation of four-mode continuous-variable cluster states,” Phys. Rev. A 78(1), 012301 (2008). [CrossRef]
9. J. Zhang, Y. Wang, X.-Y. Liu, and R. Yang, “Parallel generation of 31 tripartite entangled states based on optical frequency combs,” Chin. Phys. B 26(12), 124205 (2017). [CrossRef]
10. P. Du, Y. Wang, K. Liu, R. Yang, and J. Zhang, “Generation of large-scale continuous-variable cluster states multiplexed both in time and frequency domains,” Opt. Express 31(5), 7535–7544 (2023). [CrossRef]
11. Z.-S. Yuan, Y.-A. Chen, B. Zhao, S. Chen, J. Schmiedmayer, and J.-W. Pan, “Experimental demonstration of a bdcz quantum repeater node,” Nature 454(7208), 1098–1101 (2008). [CrossRef]
12. W. Qin, A. Miranowicz, P.-B. Li, X.-Y. Lü, J. Q. You, and F. Nori, “Exponentially enhanced light-matter interaction, cooperativities, and steady-state entanglement using parametric amplification,” Phys. Rev. Lett. 120(9), 093601 (2018). [CrossRef]
13. B. B. Blinov, D. L. Moehring, L. Duan, and C. R. Monroe, “Observation of entanglement between a single trapped atom and a single photon,” Nature 428(6979), 153–157 (2004). [CrossRef]
14. D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, “Quantum dynamics of single trapped ions,” Rev. Mod. Phys. 75(1), 281–324 (2003). [CrossRef]
15. W. Gao, P. Fallahi, E. Togan, J. Miguel-Sánchez, and A. Imamoglu, “Observation of entanglement between a quantum dot spin and a single photon,” Nature 491(7424), 426–430 (2012). [CrossRef]
16. X. Gu, A. F. Kockum, A. Miranowicz, Y. xi Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Phys. Rep. 718-719, 1–102 (2017). [CrossRef]
17. J. Q. You and F. Nori, “Atomic physics and quantum optics using superconducting circuits,” Nature 474(7353), 589–597 (2011). [CrossRef]
18. Q. Cai, J. Liao, and Q. Zhou, “Entangling two microwave modes via optomechanics,” Phys. Rev. A 100(4), 042330 (2019). [CrossRef]
19. Z. X. Chen, Q. Lin, B. He, and Z. Y. Lin, “Entanglement dynamics in double-cavity optomechanical systems,” Opt. Express 25(15), 17237–17248 (2017). [CrossRef]
20. S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84(4), 042342 (2011). [CrossRef]
21. C. Zhong, X. Han, and L. Jiang, “Microwave and optical entanglement for quantum transduction with electro-optomechanics,” Phys. Rev. Appl. 18(5), 054061 (2022). [CrossRef]
22. T. Wang, L. Wang, Y. Liu, C.-H. Bai, D.-Y. Wang, H. Wang, and S. Zhang, “Temperature-resistant generation of robust entanglement with blue-detuning driving and mechanical gain,” Opt. Express 27(21), 29581–29593 (2019). [CrossRef]
23. D.-G. Lai, J.-Q. Liao, A. Miranowicz, and F. Nori, “Noise-tolerant optomechanical entanglement via synthetic magnetism,” Phys. Rev. Lett. 129(6), 063602 (2022). [CrossRef]
24. Y. Li, Y.-F. Jiao, J.-X. Liu, A. Miranowicz, Y.-L. Zuo, L.-M. Kuang, and H. Jing, “Vector optomechanical entanglement,” Nanophotonics 11(1), 67–77 (2022). [CrossRef]
25. V. Giovannetti, S. Mancini, and P. Tombesi, “Radiation pressure induced einstein-podolsky-rosen paradox,” Europhys. Lett. 54(5), 559–565 (2001). [CrossRef]
26. T. G. Tesfahannes, “Generation of the bipartite entanglement and correlations in an optomechanical array,” J. Opt. Soc. Am. B 37(11), A245–A252 (2020). [CrossRef]
27. X. Z. Hao, X. Y. Zhang, Y. H. Zhou, W. Li, S. C. Hou, and X. X. Yi, “Dynamical bipartite and tripartite entanglement of mechanical oscillators in an optomechanical array,” Phys. Rev. A 104(5), 053515 (2021). [CrossRef]
28. R.-X. Chen, C.-G. Liao, and X.-M. Lin, “Dissipative generation of significant amount of mechanical entanglement in a coupled optomechanical system,” Sci. Rep. 7, 1 (2017). [CrossRef]
29. H. Miao, C. Zhao, L. Ju, and D. G. Blair, “Quantum ground-state cooling and tripartite entanglement with three-mode optoacoustic interactions,” Phys. Rev. A 79(6), 063801 (2009). [CrossRef]
30. C. Zhao, Q. Fang, S. Susmithan, H. Miao, L. Ju, Y. Fan, D. G. Blair, D. J. Hosken, J. Munch, P. J. Veitch, and B. J. J. Slagmolen, “High-sensitivity three-mode optomechanical transducer,” Phys. Rev. A 84(6), 063836 (2011). [CrossRef]
31. Y.-H. Ma and X.-F. Zhang, “Genuine quadripartite macroscopic entanglement generated in two-mode optomechanical systems,” Appl. Phys. B 112(4), 485–489 (2013). [CrossRef]
32. X. Yang, Y. Ling, X. Shao, and M. Xiao, “Generation of robust tripartite entanglement with a single-cavity optomechanical system,” Phys. Rev. A 95(5), 052303 (2017). [CrossRef]
33. Z. J. Deng, X.-B. Yan, Y.-D. Wang, and C.-W. Wu, “Optimizing the output-photon entanglement in multimode optomechanical systems,” Phys. Rev. A 93(3), 033842 (2016). [CrossRef]
34. C.-G. Liao, X. Shang, H.-Y. Xie, and X. Lin, “Dissipation-driven entanglement between two microwave fields in a four-mode hybrid cavity optomechanical system,” Opt. Express 30(7), 10306–10316 (2022). [CrossRef]
35. C. Jiang, S. Tserkis, K. Collins, S. Onoe, Y. Li, and L. Tian, “Switchable bipartite and genuine tripartite entanglement via an optoelectromechanical interface,” Phys. Rev. A 101(4), 042320 (2020). [CrossRef]
36. X.-B. Yan, Z. J. Deng, X. Tian, and J.-H. Wu, “Entanglement optimization of filtered output fields in cavity optomechanics,” Opt. Express 27(17), 24393–24402 (2019). [CrossRef]
37. X.-B. Yan, “Enhanced output entanglement with reservoir engineering,” Phys. Rev. A 96(5), 053831 (2017). [CrossRef]
38. T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342(6159), 710–713 (2013). [CrossRef]
39. S. T. Velez, V. Sudhir, N. Sangouard, and C. Galland, “Bell correlations between light and vibration at ambient conditions,” Sci. Adv. 6(51), 1 (2020). [CrossRef]
40. C. F. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, M. Asjad, A. A. Clerk, F. Massel, M. J. Woolley, and M. A. Sillanpää, “Stabilized entanglement of massive mechanical oscillators,” Nature 556(7702), 478–482 (2018). [CrossRef]
41. R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, “Remote quantum entanglement between two micromechanical oscillators,” Nature 556(7702), 473–477 (2017). [CrossRef]
42. A. Mari and J. Eisert, “Cooling by heating: Very hot thermal light can significantly cool quantum systems,” Phys. Rev. Lett. 108(12), 120602 (2012). [CrossRef]
43. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014). [CrossRef]
44. H. Tan, Y. hua Wei, and G. xiang Li, “Building mechanical greenberger-horne-zeilinger and cluster states by harnessing optomechanical quantum steerable correlations,” Phys. Rev. A 96(5), 052331 (2017). [CrossRef]
45. K. Zhang, F. Bariani, Y. Dong, W. Zhang, and P. Meystre, “Proposal for an optomechanical microwave sensor at the subphoton level,” Phys. Rev. Lett. 114(11), 113601 (2015). [CrossRef]
46. H. K. Cheung and C. K. Law, “Nonadiabatic optomechanical hamiltonian of a moving dielectric membrane in a cavity,” Phys. Rev. A 84(2), 023812 (2011). [CrossRef]
47. G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65(3), 032314 (2002). [CrossRef]
48. M. B. Plenio, “Logarithmic negativity: A full entanglement monotone that is not convex,” Phys. Rev. Lett. 95(9), 090503 (2005). [CrossRef]
49. C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust entanglement of a micromechanical resonator with output optical fields,” Phys. Rev. A 78(3), 032316 (2008). [CrossRef]
50. J. Zhang, X.-Y. Liu, R. Yang, and T.-C. Zhang, “Scheme for enhancing quadripartite entangled optical modes from an opto-mechanical system,” J. Opt. Soc. Am. B 35(12), 2945–2951 (2018). [CrossRef]
