Universal quantum control over bosonic network

Abstract

Perfect transfer of {\em unknown} states across distinct nodes is a basic function in bosonic quantum networks. Here we develop a general theory to construct an $N$-node bosonic network governed by the time-dependent Hamiltonian, as the universal quantum control theory for continuous-variable systems. In particular, we can activate nonadiabatic passages superposed of initial and target modes by the commutation condition about the Hamiltonian's coefficient matrix and projection operator in the representation of time-independent ancillary modes, which serves as the necessary and sufficient condition to solve the time-dependent Schrödinger equation of the full Hamiltonian. To exemplify the versatility of our theory on the Heisenberg-picture passages, we perform arbitrary state exchange between two nodes, chiral entanglement transfer among three bosonic nodes, and chiral Fock-state transfer among three of four bosonic nodes. Our work provides a promising avenue toward the universal control of any pair of nodes or modes as well as the entire bosonic network.

Figures (7)

• Figure 1: Fidelity dynamics $F(t)$ for the state exchange in the two-mode system about (a) the Fock state $|5,0\rangle\rightarrow|0,5\rangle$, (b) the product of coherent state and Fock state $|\alpha,5\rangle\rightarrow|5,\alpha\rangle$ with $\alpha=5$, (c) the cat state $|{\rm cat},0\rangle\rightarrow|0,{\rm cat}\rangle$, where $|\rm cat\rangle=(|\alpha\rangle+|-\alpha\rangle)$ with $\alpha=5$, and (d) the thermal state $\rho_{\rm th}\otimes|0\rangle\langle0|\rightarrow|0\rangle\langle0|\otimes\rho_{\rm th}$, where $\rho_{\rm th}=\sum_np_n|n\rangle\langle n|$ with $p_n=(\bar{n}^n)/(1+\bar{n})^{n+1}$ and $\bar{n}=1$. The coupling strength $J$ and the detuning $\Delta(t)$ are set as Eq. (\ref{['ConditionInv']}) with $\theta_1(t)=\pi t/(2\tau)$ and $f(t)=0$ in (a) and (b), or $f(t)=3\theta_1(t)$ in (c) and (d).
• Figure 2: Fidelity dynamics $F(t)$ for the target state $|0,\rm cat\rangle$ during the exchange of cat state $|\rm cat,0\rangle\rightarrow|0,\rm cat\rangle$: (a) for various environmental decay rates $\kappa$; and (b) for various derivation coefficients $\epsilon$ in the imperfect parameter setting $\theta_1(t)\rightarrow(1+\epsilon)\theta_1(t)$ and $\theta_1(t)=(\pi t)/(2\tau)$. The evolution period is set as $\tau=100$ ns and the coupling strength $J/2\pi\sim25$ MHz. The other parameters are the same as Fig. \ref{['PopuTwo']}(c).
• Figure 3: Sketch of a tripartite system comprising three bosonic modes $a_1$, $a_2$, and $a_3$, which are coupled by the exchange interactions with the coupling strengths $J_1$, $J_2$, and $J_3$, and the phases $\varphi_1$, $\varphi_2$, and $\varphi_3$, respectively.
• Figure 4: Fidelity dynamics $\mathcal{F}_{jk}(t)$ about the chiral transfer of the NOON state in the three-mode system of a triangular configuration (see Fig. \ref{['modelThree']}) along (a) the counterclockwise direction and (b) the clockwise direction. Under the conditions in Eqs. (\ref{['detuning']}) and (\ref{['CouplingStrength']}), the parameters $\Delta_a(t)$, $\Delta(t)$, $J_a(t)$, and $J(t)$ are set according to Eqs. (\ref{['Mu0Inverse']}) and (\ref{['Mu12Inverse']}) with $f_1(t)=0$ and $f(t)=3\theta_2(t)$. In (a) $\theta_1(t)$ and $\theta_2(t)$ are set by Eq. (\ref{['parathree']}) and in (b) $\theta_1(t)$ and $\theta_2(t)$ are set by Eq. (\ref{['parathreeCount']}).
• Figure 5: Sketch of the bosonic network comprising $N$ bosonic modes, in which a central bosonic mode $a_N$ is coupled to the other uncoupled bosonic modes $a_n$ with $1\leq n\leq N-1$ via the exchange interaction that is characterized by the coupling strength $J_n$ and the phase $\varphi_n$.