Quantum synchronization and quantum state sharing in an irregular complex network

We investigate the quantum synchronization phenomenon of the complex network constituted by coupled optomechanical systems and prove that the unknown identical quantum states can be shared or distributed in the quantum network even though the topology is varying. Considering a channel constructed by quantum correlation, we show that quantum synchronization can sustain and maintain high levels in Markovian dissipation for a long time. We also analyze the state-sharing process between two typical complex networks, and the results predict that linked nodes can be directly synchronized, but the whole network will be synchronized only if some specific synchronization conditions are satisfied. Furthermore, we give the synchronization conditions analytically through analyzing network dynamics. This proposal paves the way for studying multi-interaction synchronization and achieving effective quantum information processing in a complex network.

Figure 1

Two hybrid electro-optomechanical systems are coupled via a phonon tunneling and a linear resistor. For each subsystem, a charged oscillator is placed at wave node of a Fabry-Pérot cavity, and it couples with the cavity field via a linear optomechanical interaction. An electric potential difference exists between walls of a cavity which is provided by the inductance of the Duffing circuit.

Figure 2

(a), (b) Expectation values of the oscillator coordinates corresponding to existing (a) or disconnecting (b) of coupling. Here black solid lines denote $\langle q_1\rangle$, and yellow dotted lines denote $\langle q_2\rangle$. (c), (d) Expectation values of the coordinate error operator corresponding to $K=2$, $\mu =0.02$ (c, black solid); $K=0$, $\mu =0$ (c, yellow dotted); $K=2$, $\mu =0$ (d, black solid); and $K=0$, $\mu =0.02$ (d, yellow dotted). (e) Expectation values of the momentum error operator corresponding to $K=2$, $\mu =0.02$ (black solid) and $K=0$, $\mu =0$ (yellow dotted). (f) Effective frequencies of system 1 (yellow dotted) and 2 (black solid) when $K=0$. In these simulations, the oscillator frequency is set ${\omega }_{m1}=1$ as a unit, and other parameters are ${\omega }_{m2}=1$, ${\omega }_0=0.8$, ${\mathrm{\Delta }}_j={\omega }_{mj}=1$, $g=0.005$, $\kappa =0.15$, $\gamma =0.005$, $\eta =0.01$, $\varepsilon =0.18$, $\nu =1$, $E=10$, and $\mathbb{E}=26.7$. The initial state of the cavity field is vacuum state which corresponds to $A_j\left(0\right)=0$, and other initial conditions are all random. All horizontal coordinates denote time $t$.

Figure 3

Simulations of the expectation value error $\langle q_{-}\rangle$, quantum synchronization measure $S_c^{\prime }\left(t\right)$, and Gaussian fidelity $\mathcal{F}$. The inset in (b) is the partial enlarged drawing of $S_c^{\prime }$ in $t\in [1900,2000]$. In (b) and (c), the yellow (pale) lines express the local average synchronization measure and Gaussian fidelity by calculating $\xi \left(\overline{t}\right)={\mathrm{\Delta }t}^{-1}{\int }_t^{t+\mathrm{\Delta }t}\xi dt$ in the time window $\mathrm{\Delta }t\left(\xi \in \{S_c^{\prime },\mathcal{F}\}\right)$; Here all the other parameters are the same as in Fig. 2.

Figure 4

(a) Quantum synchronization measures under different conditions. Here black (dark) lines denote $S_c^{\prime }$ when ${\overline{n}}_b=0$ (solid), ${\overline{n}}_b=0.25$ (dotted), and ${\overline{n}}_b=2.5$ (circle). Yellow (pale) lines denote $S_c^{\prime }$ under $K=0$, $\mu =0.02$ (solid), and $K=0$, $\mu =0$ (dotted). (b) Evolution of logarithmic negativity. (c) Local averaged fidelities, respectively, corresponding to $K=2$, $\mu =0.02$ (red square); $K=2$, $\mu =0$ (yellow (pale) dotted); $K=0$, $\mu =0.02$ (black (dark) dashed); and $K=0$, $\mu =0$ (black (dark) solid). Here all the other parameters are the same as in Fig. 2.

Figure 5

(a) Initial state dependency of fidelity. The initial states of cavity fields are respectively selected as two vacuum states: $|0\rangle$ and $|0\rangle$ (yellow solid); coherent states with identical photon number: $|\alpha \rangle$ and $|\alpha e^{i\varphi }\rangle$ (black dark dashed); and coherent states with different photon number: $|{\alpha }_1\rangle$ and $|{\alpha }_2\rangle$ (green light dotted). Here $|\alpha \rangle =exp(\alpha a^{†}-{\alpha }^{*}a)|0\rangle$, $\alpha ={\alpha }_1=1$, ${\alpha }_2=10$, and $\varphi =\pi /2$. (b) Gaussian fidelities with varied phonon number difference. Here we set ${\overline{n}}_{b1}+{\overline{n}}_{b2}=20$, $\mathrm{\Delta }n={\overline{n}}_{b1}-{\overline{n}}_{b2}$, and all the other parameters are the same as in Fig. 2.

Figure 6

Each node represents an eletro-optomechanical system. The nodes connect to each other through different coupling forms and dissipate into different baths, including a separate bath (SB), common bath (CB), and local bath (LB). Here (a) represents the structure of the SW network consisting of primary and secondary nodes and (b) shows a SW network with varying structure.

Figure 7

(a) Average trace distance $\overline{D}$ with varied coupling intensity $\mu$ under $N=12$, $M=2$, and $P=0.1$. The inset in (a) shows change of $\overline{D}$ with varied edge-adding probability $P$ under $N=12$, $M=2$, and $\mu =0.02$. (b) Fidelities between two nodes in different linear motifs. Here link $i$ refers to the connection between nodes $i$ and $i^{\prime }$ in Fig. 6. (c) and (d) Entanglement measures of the quantum states in different cavities. (e) and (f) Fidelities and synchronization measures between two nodes when the network structure varies. The yellow (dotted) line represents the link disconnects at $t_1$ and reconnects at $t_2$. The initial state of each cavity field is a vacuum state or random coherent mix state. Other initial conditions are all random, and all the other parameters are the same as in Fig. 2.

Figure 8

(a) Structure diagram of an SF network with 18 nodes. (b) An SF network with a variable structure. (c) A simple example of network synchronization with an auxiliary node. Here the red solid lines represent the quantum couplings.

Figure 9

(a) Network-averaged fidelity (black dark) and its local time average value (yellow pale). (b) Fidelities between two nodes when the network structure varies. (c) Quantum synchronization measures when the network structure varies and the time interval corresponds to the green box in (b). Here the initial state of each cavity field is vacuum state and other initial conditions are all random, and all the other parameters are the same as in Fig. 2.

Figure 10

(a) Entanglement measures of the quantum states in each cavity. (b) Initial state dependency of fidelity. Here the initial states of cavity fields are respectively selected as random coherent mix states. All the other parameters are the same as in Fig. 2.