Optimal transfer of entanglement in oscillator chains in non-Markovian open systems

TL;DR

The paper addresses transferring continuous-variable entanglement in chains of coupled oscillators subject to non-Markovian environments. It employs Krotov's gradient-based optimization to design control fields that tune oscillator frequencies, and extends the method to non-Markovian dynamics via a time-dependent O-operator derived from the quantum state diffusion framework. Demonstrations on linear and X-shaped chains show high-fidelity entanglement transfer with smooth, experimentally feasible controls, and reveal that memory effects can enhance transfer performance compared to memoryless cases. Importantly, the approach can target a range of initial states and entanglement levels, making it robust to unknown parameters, with practical implications for implementations in superconducting circuits, SQUID arrays, and LC-circuit chains.

Abstract

We considered the transfer of continuous-variable entangled states in coupled oscillator chains embedded in a generic environment. We demonstrate high-fidelity transfer via optimal control in two configurations - a linear chain and an X-shaped chain. More specifically, we use the Krotov optimization algorithm to design control fields that achieve the desired state transfer. Under the environmental memory effects, the Krotov algorithm needs to be modified, since the dissipative terms in non-Markovian dynamics are generally governed by the time-dependent system Hamiltonian. Remarkably, we can achieve high-fidelity transfer by simply tuning the frequencies of the oscillators while keeping the coupling strength constant, even in the presence of open-system effects. For the system under consideration, we find that quantum memory effects can aid in the transfer of entanglement and show improvement over the memoryless case. In addition, it is possible to target a range of entangled states, making it unnecessary to know the parameters of the initial state beforehand.

Figures (9)

• Figure S1: Schematic of the models under consideration. We have considered two types of oscillator chains, a linear chain and an X-shaped chain, with the goal of transferring entangled states through the coupled chains.
• Figure S2: Transferring the entangled state through a linear chain as a closed system. Panel (a): The control fields applied as functions of time, with the control targeting both fidelity and negativity. Different colors correspond to the control field applied to each oscillator. Panel (b): First 10 discrete Fourier transform frequencies of controls targeting just the fidelity (circle markers) and controls
• Figure S3: Transferring the entangled state through an X-shaped chain as a closed system. Panel (a): Control fields as functions of time. Panel (b): Fidelity and negativity dynamics under control, illustrating the transfer of entanglement from the head of the chain to the tail.
• Figure S4: Optimized control under open-system effects. Panel (a): Control fields for the non-Markovian open system. Panel (b): Modified Krotov control iteration for non-Markovian open-system dynamics. Panel (c): Entanglement dynamics under different scenarios: deriving a new control under non-Markovian open-system effects (blue solid line), deriving new a control under Markov noise (green solid line), and applying the closed-system controls in (non-)Markovian open systems as red (purple) solid lines. The ideal closed-system dynamics is also shown as a black dashed line for reference. Panel (d): Logarithmic of the final residuals of the target function; left solid ones are for the negativity and right fainter ones are for the fidelity.
• Figure S5: Dynamics of the fidelity (dashed lines) and entanglement (solid lines). Compare the case where the control is calculated under the Markov open system (green lines) with applying the closed-system control in a non-Markovian setting (purple lines); we can see that while the non-Markovian case here has better fidelity than the Markov case at $t=T$, it is actually less entangled, illustrating that a higher value of fidelity against the target entangled state does not necessarily mean a higher degree of entanglement is achieved.