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Abstract
The generation and manipulation of highly pure and strongly entangled steady state in a quantum system are vital tasks in the standard continuous-variable teleportation protocol. Especially, the manipulation implemented in integrated devices is even more crucial in practical quantum information applications. Here we propose an effective approach for creating steady-state entanglement between two microwave fields in a four-mode hybrid cavity optomechanical system. The entanglement can be achieved by combining the processes of three beam-splitter interactions and two parametric-amplifier interactions. Due to the dissipation-driven and cavity cooling processes, the entanglement obtained can go far beyond the entanglement limit based on coherent parametric coupling. Moreover, our proposal allows the engineered bath to cool both Bogoliubov modes almost simultaneously. In this way, a highly pure and strongly entangled steady state of two microwave modes is obtained. Our finding may be significant for using the hybrid opto-electro-mechanical system fabricated on chips in various quantum tasks, where the strong and pure entanglement is an important resource.
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1. Introduction
The generation and manipulation of highly pure and strongly entanglement in hybrid quantum systems are crucial to the accomplishment of tasks in quantum computer and quantum communication networks. Recently, a lot of attentions have been paid to hybrid cavity optomechanical system involving phonon, which characterizes a vibrational mode of the mechanical harmonic oscillator. It has been confirmed that cavity optomechanical system not only can be used for highly sensitive detection of various physical quantities [1–4] but also provides a promising platform to study quantum properties of the macroscopic object [5,6] and the applications in quantum information science and engineering [7]. Especially, the cavity optomechanical system possesses the ability to build hybrid quantum devices which combine otherwise incompatible degrees of freedoms of different physical systems [8]. All of the above together drive the rapidly growing interest in the system.
The quadripartite opto-electro-mechanical hybrid system consisted of an optical cavity mode, a mechanical resonator, and two microwave cavity modes can be fabricated on chips [9–11], which provides a new approach to realize an integrated devices for quantum information. The system has several advantages. For instance, the optical cavity mode can be used as a flying qubit to interface with other distant nodes via fibers for its unsusceptible to thermal noise. Additionally, the entanglement between subsystems can be distributed to other network nodes through the entanglement swapping protocol [12–15]. We notice that, on the basis of the quadripartite opto-electro-mechanical hybrid system mentioned above, Cai and his coworkers proposed a scheme to generate stationary quantum entanglement between two microwave modes [16], which can be used in quantum illumination radar protocols. Recently, the other study on microwave-microwave entanglement has also been carried out by using the nonlinear magnetostrictive interaction in a ferrimagnet [17]. In particular, realization of the prototype quantum radar has shown robustness to the ambient background noise and loss, and highlighted the opportunities and challenges in the way toward a first room-temperature application of microwave quantum circuits [18,19].
In this work, we investigate the quantum entanglement of two microwave modes in a quadripartite opto-electro-mechanical hybrid system. We show that the entanglement can be achieved by combining the processes of three beam-splitter interactions and two parametric-amplifier interactions. Due to the dissipation-driven and cavity cooling processes [20–24], the entanglement between two microwave modes can go far beyond the entanglement limit (i.e., $\ln 2$) [25–27] based on coherent parametric coupling. Moreover, our proposal allows the engineered bath to cool both Bogoliubov modes almost simultaneously. In this way, a highly pure and strongly entangled steady state of two microwave modes is obtained. Our finding may be significant if one tends to use the hybrid opto-electro-mechanical system fabricated on chips in various quantum tasks, where the strong and pure entanglement is an important resource.
2. Model and dynamics
The hybrid system includes an optical cavity mode, a mechanical resonator, and two microwave cavity modes, which is depicted as follows: a movable mirror at one end of a Fabry-Perot optical cavity is treated as a quantum-mechanical harmonic oscillator $B$ with the characteristic frequency ${\omega _{\rm {b}}}$ and damping coefficient $\gamma _{\rm {b}}$, which interacts with a single optical mode $A_1$ with frequency ${\omega _1}$ and is simultaneously capacitively coupled to two microwave cavity modes $A_2$ and $A_3$ of superconducting circuits with resonate frequencies ${\omega _2}$, ${\omega _3}$, respectively. Here, $B$ and $A_j$ separately represent the annihilation operators of the corresponding modes. The Hamiltonian of the system can be written as ($\hbar =1$) [16]
The system dynamics is governed by the quantum Langevin equations (QLEs) [28]
In the presence of strong external pumping, the system operators can be written as $O=\langle O(t)\rangle +o$ $(O=A_j,B)$, where $o$ is quantum fluctuation operator with zero mean value around classical $c$-number mean amplitude $\langle O(t)\rangle$ . Under the strong coherent driving regime $\langle O(t)\rangle \gg 1$, standard linearization techniques [26] can be applied to Eq. (3). As a result, it leads to a set of differential equations for the mean values
3. Effective Hamiltonian and the mechanism
When we focus on the weak optomechanical coupling regime, i.e., $|g_j/\omega _{\rm {b}}|\ll 1$, approximately analytical solutions for Eq. (5) can be found by expanding the classical mean values in powers of $g$ as [20,21,29]
Equation (18) can be rewritten as
Equation (20) finally becomes
Obviously, it is a group of beam-splitter-like interaction terms, which is well known from optomechanical sideband cooling [30,31]. For the small leakage rate of the microwave cavities, the dissipations of the optical cavity mode $a_1$ and the mechanical mode $b$ can be exploited to cool both of the sum and difference modes $\beta _{\rm {sum}}$ and $\beta _{\rm {diff}}$, which means the two Bogoliubov modes $\beta _1$ and $\beta _2$, can be effectively cooled, generating two-mode squeezing between two microwave modes $a_2$ and $a_3$. The approach is unlike that in Refs. [21] and [32], where only one of the two Bogoliubov modes of the target modes is cooled while the other is a dark mode that cannot be cooled because it is not coupled to the engineered bath, and the obtained steady state is a two-mode squeezed thermal state, i.e., mixed state. But our proposal allows both Bogoliubov modes to be cooled almost simultaneously, and can obtain a highly pure and strongly entangled steady state which is vital in the standard continuous-variable teleportation protocol.
4. Evolution equation of the covariance matrix
When the system is stable, the linearized Hamiltonian of the system ensures that an initial Gaussian state will remain Gaussian [33] whose information-related properties can be fully described by the $8\times 8$ covariance matrix (CM) $\sigma$ with components defined as [33–35]
Here, $R=(q_{\rm {1}},p_{\rm {1}},q_{2},p_{2},q_{3},p_{3},q_{\rm {b}},p_{\rm {b}})^{T}$ is a column vector of dimensionless quadrature operators related to bosonic modes $o$ ($o\in \{a_1,a_2,a_3,b\}$) via $q_j=(o+o^{\dagger })/\sqrt {2}$ and $p_j=(o-o^{\dagger })/(i\sqrt {2})$. By further introducing the column vector of input noise quadrature operators
Equation (6) can be written in a compact matrix form as
withIn the following, we will utilize Eq. (29) to numerically simulate the time evolution of the entanglement between two microwave cavity fields. It should be note that the only approximation of the coefficient matrices in Eq. (28) is correspond to the linearization techniques of the system Hamiltonian in Eq. (7), which is commonly used in optomechanics. Besides, based on Floquet’s theorem [36], the stability conditions will be carefully checked in all simulations throughout this paper.
For the two-mode Gaussian state, it is convenient to use the logarithmic negativity $E_{\rm {N}}$ [37,38] to gauge its level of entanglement, which can be computed from the reduced $4\times 4$ CM ${\sigma _{\rm {m}}}(t)$ for two microwave fields (i.e., by taking the partial trace of the optical field and the mechanical oscillator, leaving the the third to sixth rows and the third to sixth columns of CM)
5. Numerical simulation results and discussion
In order to verify the feasibility of our scheme, we first make a comparison between the numerical simulation and the desired results of the effective optomechanical coupling strength in Fig. 1. Here, the numerical simulation results are evaluated with $G_j(t)=-g_j(\langle A_j(t)\rangle$, where the time-dependent mean values $\langle A_j(t)\rangle$ are obtained by applying driving lasers $E_j(t)$ to integrate the differential Eq. (5). The desired results of the effective optomechanical coupling strength $\bar G_j(t)$ are directly calculated through Eqs. (13) and (14). Evidently, the numerical solutions of the effective optomechanical coupling strengths $G_2(t)$ and $G_3(t)$ agree well with the desired solutions of the effective optomechanical coupling strengths $\bar G_2(t)$ and $\bar G_3(t)$ in the long time limit. It is worth emphasizing that the periodic behavior for $\bar G_2(t)$ and $\bar G_3(t)$ can be described as a superposition of cosine functions [see Eq. (13)], then beating in $\bar G_2(t)$ and $\bar G_3(t)$ dynamics can be generated when $\omega _{ij}$ are slightly different. It is an important and interesting phenomenon and similar results have been previously studied by pumping an especially tuned optical degenerate parametric amplifier (DPA) inside the cavity [39].
Fig. 1. Time evolution of the real and imaginary parts of the effective optomechanical coupling strength. The chosen parameters in units of $\omega _{\rm {b}}$ are: $\kappa _{1}=\gamma _{{\rm {b}}}=0.07$, $\kappa _{2}=\kappa _{3}=0.001$, $\Delta _1=1$, $\Delta _2=\Delta _3=2$, $\delta =0.021$, $g_1=1\times 10^{-5}$, $g_2=g_3=0.1g_1$, $G_-=0.03$, $G_+=0.67G_-$, and $\bar G_1=0.02G_-$.
Figure 2 displays the time evolution of the entanglement $E_{\rm {N}}$ between two microwave modes $a_2$ and $a_3$ when all cavity modes and mechanical mode initially are in thermal equilibrium with their baths. The results are numerically evaluated with the full linearized QLEs. (6) for the quantum fluctuations (or the equivalent linear differential Eq. (29) for the CM) including the non-resonant terms. To do this, we need to first numerically integrate the differential Eq. (5) by applying driving lasers $E_j(t)$ to get the time-dependent mean values for Eq. (6). Obviously, the entanglement gradually increases and eventually tends to be saturated with a fixed period. The reason is that the two Bogoliubove modes $\beta _1$ and $\beta _2$ have been sufficiently cooled after some time. The obtained maximal entanglement ($E_{\rm {N}}\sim 0.9859$, see Fig. 2 (a)) is much larger than that ($E_{\rm {N}}\sim 0.69$) generated in Ref. [16], which is based on the coherent parametric interaction and subjected to the stability constraint. For larger mean thermal phonon number, the higher effective temperature of mechanical bath reduces the amount of steady-state entanglement as illustrated in Fig. 2 (b).
Fig. 2. Time evolution of entanglement $E_{\rm {N}}$ between two microwave modes $a_2$ and $a_3$. The mean thermal occupation numbers are chosen as (a) $\bar n_{1}=\bar n_{2}=\bar n_{3}=\bar n_{\rm {b}}=0$, (b) $\bar n_{1}=\bar n_{2}=\bar n_{3}=0$, $\bar n_{\rm {b}}=0.5$, and all the other parameters are the same as those in Fig. 1.
The generation of entanglement in our scheme is related to the cascaded dissipative cooling process of the target modes and more subtle than that in Refs. [21,22,24]. If there is no optimization, the entanglement will remain small. To find the optimal parameters, one can recall the Hamiltonian under the rotating-wave approximation in Eq. (24). As it has already been detailed in Refs. [21,22,24], the leakage rate $\kappa _{j}$ of the cavities, the damping coefficient $\gamma _{\rm {b}}$, the interaction strengths $\delta$, $\bar G_1$, and $\sqrt {2}\tilde {G}$ (i.e., $G_+$ and $G_-$) play an important role in the generation of entanglement. For fixed coupling $\delta$, $\bar G_1$, $G_+$, and $G_-$, one can optimize the entanglement as a function of the leakage rate $\kappa _{j}$ of the cavities and the damping coefficient $\gamma _{\rm {b}}$. Since the entanglement generation is largely relied on cooling the Bogoliubov modes via the cascaded dissipative dynamics of the optical cavity mode $a_1$ and the mechanical mode $b$, one expect that strong leakage rate $\kappa _{1}$ and $\gamma _{\rm {b}}$ of $a_1$ and $b$ while simultaneously weak leakage rates $\kappa _{2}$ and $\kappa _{3}$ of $a_2$ and $a_3$ should increase the level of entanglement. Experimentally, one are more interested in how to optimize entanglement by adjusting coupling strengths when $\kappa _{j}$ and $\gamma _{\rm {b}}$ are fixed. Apparently, the steady state entanglement is a nonmonotonic function of $G_+/G_-$ in most sets of parameters and takes the maximum for a specific $G_+/G_-$. On the one hand, the growth of $G_+/G_-$ can increase the squeezing parameter $r=\tanh ^{-1}(G_+/G_-)$ of the two-mode squeezed state for the microwave fields in the stationary regime, which can in turn enhance the stationary entanglement between two microwave fields. On the other hand, it will weaken the cooling effect of the sum mode $\beta _{\rm {sum}}$ due to the declining coupling strength $\tilde {G}=\sqrt {G_-^{2}-G_+^{2}}$ between the sum mode and the mechanical mode. Therefore, the optimal entanglement is obtained when the two competing effects balance. For fixed coupling $\tilde {G}$, one also expect a optimal $\delta$ that corresponds to the maximum entanglement. Due to the fact that the sum mode $\beta _{\rm {sum}}$ is simultaneously coupled to the difference mode $\beta _{\rm {diff}}$ and the mechanical mode $b$ with beam-splitter-like interaction strengths $\delta$ and $\sqrt 2 \tilde {G}$, respectively, one of the interaction terms induces the cooling of the sum mode $\beta _{\rm {sum}}$, while the other is responsible for cooling the difference mode $\beta _{\rm {diff}}$. As a result, when $\delta$ is small compared with $\tilde {G}$, the difference mode $\beta _{\rm {diff}}$ cannot be effectively cooled by the sum mode $\beta _{\rm {sum}}$. On the contrary, when $\delta$ is large (i.e., $\beta _{\rm {sum}}$ is strongly coupled to $\beta _{\rm {diff}}$), the quanta are confined and swap rapidly between the sum and difference modes, which implies that both the sum and difference modes cannot be effectively cooled by the mechanical mode $b$. Besides, the mechanical mode $b$ is simultaneously coupled to the sum mode $\beta _{\rm {sum}}$ and the optical cavity mode $a_1$ with beam-splitter-like coupling strengths $\sqrt 2 \tilde {G}$ and $\bar G_1$, respectively. Similar to the analyses above, for a given parameter $\bar G_1$, one expect some moderate values of $\tilde {G}$ that correspond to the maximum entanglement. Thus, a slight change in one parameter may affect the situation as a whole.
Figure 3 displays the peak values of stationary entanglement $E_{\rm {N}}$ and purity for each time period in the long-time limit as functions of $G_+$ and $\delta$, which confirms the entanglement $E_{\rm {N}}$ is indeed a nonmonotonic function of $G_+$ and $\delta$ as mentioned above and take the maximum for a specific $G_+$ and $\delta$. It’s worth noting that the appropriate non-zero value of $\delta$ can substantially enhance the peak value of stationary entanglement $E_{\rm {N}}$ without decreasing the purity significantly. Consequently, our proposal allows a highly pure and strongly entangled steady state to be generated, which is crucial to the accomplishment of tasks in quantum computer and quantum communication networks.
Fig. 3. Maximum entanglement and purity for each time period in the asymptotic regime as functions of $G_+$ and $\delta$. The mean thermal occupation numbers are chosen as $\bar n_{1}=\bar n_{2}=\bar n_{3}=\bar n_{\rm {b}}=0$, and all the other parameters are the same as those in Fig. 1.
Finally, it should be noted that low-Q mechanical resonator as well as low-loss microwave cavity in our scheme would be ideal. The enhanced mechanical damping rate resulting from optical cavity cooling is actually highly beneficial. This requires an atypical cavity optomechanical system. Except for these, the used parameters in the numerical simulation are within reach of current technology.
6. Conclusions
In summary, we have proposed a theoretical scheme to generate highly pure and strongly entangled steady state between two microwave fields in a quadripartite opto-electro-mechanical hybrid system, which can be fabricated on chips. The scheme is based on the cascaded dissipative dynamics of the optical cavity mode $a_1$ and the mechanical mode $b$. The mechanical mode $b$ is first cooled to near its ground state by the optical cavity mode $a_1$. Then, by optimally tuning the couplings of the other two beam-splitter-like interactions, both the sum and difference modes $\beta _{\rm {sum}}$ and $\beta _{\rm {diff}}$ can be cooled by the mechanical mode $b$. Due to the fact that our proposal allows both Bogoliubov modes to be cooled almost simultaneously, a highly pure and strongly entangled steady state between two microwave fields can be obtained. The essential physics is analyzed and the optimal parameters are obtained by numerically simulation. Our results may be significant for using the hybrid opto-electro-mechanical system fabricated on chips in various quantum tasks, where the strong and pure entanglement is an important resource.
Funding
National Natural Science Foundation of China (12004336, 12074067, 12075205, 12174054, 62071430); Natural Science Foundation of Fujian Province (2020J01191, 2021J011228); Zhejiang Gongshang University.
Acknowledgments
We acknowledge supports from the National Natural Science Foundation of China, the Natural Science Foundation of Fujian Province of China, and funds from Zhejiang Gongshang University.
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.
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