Navigating entanglement via Ruderman-Kittel-Kasuya-Yosida exchange: Snake, bouncing, boundary-residing, pulse, and damping-stabilized time-frozen trajectories

TL;DR

The paper tackles the challenge of shaping time resolved entanglement in solid-state spin qubits with long range coupling. It introduces the Exchange-Time Integral $I(t)$ as a unifying control parameter that maps the spatial motion of two qubits A and B, coupled to a central spin qudit C via RKKY exchange, into a time dependent evolution of their joint state. By defining boundary-proximal initial states and exploring both cyclic (in-phase and antiphase) and non-cyclic (out-of-phase) vibrational regimes, the authors classify entanglement trajectories into snake, bouncing, boundary-residing and pulse types, and show how damping can stabilize complex, high entanglement dynamics into time frozen configurations. The framework also suggests built-in error correction through alternating ferromagnetic and antiferromagnetic exchange, scalability to qubit chains, and practical relevance for quantum computation, cryptography, and metrology.

Abstract

Entanglement dynamics are fundamental to quantum technologies, yet navigating their temporal profiles (trajectories) remains challenging. Here, we propose a scalable solid-state platform based on RKKY exchange, where two spin qubits couple to a central spin qudit that oscillatorily spin-polarizes the surrounding conduction electrons. We introduce the exchange-time integral (ETI), which maps the spatial motion of the qubits to a time-dependent exchange interaction and serves as an effective "trajectory clock" governing the system evolution. We focus specifically on entanglement trajectories initially near the entanglement-unentanglement boundary, with the distance to this boundary quantified by concurrence extended to include negative values. By alternating the sign changes of the exchange, implemented through vibrational motion of qubits, the ETI enables programmable entanglement trajectories. For in-phase and antiphase vibrations, including scenarios with controlled stopping at the RKKY exchange-free nodes, we identify distinctive trajectories: snake (repeatedly crossing the boundary), bouncing (immediately reversing upon reaching the boundary), boundary-residing (remaining at the transition point), and pulse (controllable entanglement intervals). The vibration phase creates asymmetric shifts to the trajectories. The proposed device offers built-in error correction against dephasing by utilizing both ferromagnetic and antiferromagnetic regimes. Out-of-phase vibrations drive trajectories away from the boundary, accessing larger entanglement values but with irregular/unsteady final states. To stabilize these trajectories, we introduce a damping mechanism. Our framework offers a systematic method for navigating and engineering entanglement dynamics in quantum systems, with potential applications in quantum computation, cryptography, and metrology.

Figures (8)

• Figure 1: System schematic and entanglement dynamics. (a) Top view of suspended vibrating spin qubits $A$ and $B$ (orange) coupled via an effective, alternating RKKY exchange $J$ to a central spin qudit $C$ (green). This coupling is mediated by the interaction between the qubits and the local electron spin polarization induced by the central qudit. The environment comprises $C$ and a two-dimensional electron gas $e$ (2DEG, yellow), and the spatial RKKY profile (blue) defines the exchange nodes $R_{n}$. (b) Side view illustrating the motion-driven exchange modulation scheme. Here, the exchange coupling to the underlying local electron spins is modulated by the distance between the qubits and the $e$-spins (non-contact proximity). Alternatively, a gate-driven scheme (gray dashed box), using an applied dynamical voltage $V_{G}=V_{G}\left( t\right)$, can generate an equivalent sign-alternating exchange interaction; in this scheme, stationary spins are confined within the $V_{G}$-defined quantum dots. (c) Conceptual illustration of the environment (gray dotted box) and the qubits. The Exchange-Time-Integral (ETI)---combining the exchange modulation (blue lines) and time (blue clock icon)---serves as the effective control parameter that determines the qubit evolution along a trajectory in Hilbert space. (d) Schematic representation of entanglement trajectories, where the horizontal gray line denotes the boundary separating the entangled (upper) and separable (lower) subspaces, highlighting distinct dynamical regimes including snake, bouncing, pulse, and boundary-residing trajectories.
• Figure 2: Entanglement trajectories for mixed (marked by bold italic M) and pure (marked by bold italic P) states under in-phase vibrations. The extended concurrence $\mathcal{C}_{E}(t)$ is plotted versus time $t$ for various Bell-state weightings $W_{1\text{--}14}$ (see corresponding color text labels). The upper panels (a)--(i) display the dynamics for mixed states: (a)--(c) show snake trajectories ($T=1.25$$T^{\ast }$), (d)--(f) show bouncing trajectories ($T=1.25$$T^{\ast }$), and (g)--(i) exhibit entangled/separable trajectories ($T=1.25$$T^{\ast }$). The lower panels (j)--(o) show the corresponding dynamics for pure states: snake trajectories in (j) and (k), bouncing trajectories in (l) and (m), and entangled/separable trajectories in (n) and (o). For the bouncing trajectories, solid lines denote initially entangled states (${ \if@compatibility \mathchar"0122 {} \mathchar"0122 } >0$), while dashed lines denote separable states (${ \if@compatibility \mathchar"0122 {} \mathchar"0122 } <0$).
• Figure 3: Entanglement trajectories originating from abruptly halted vibrational motion (schematic in the upper right) for (a)--(c) mixed states and (d)--(e) pure states with different weightings. The qubits remain at the entanglement-separability boundary after $t\geq T^{\ast }/4$, producing a boundary-residing trajectory. The insets show zoomed views near the stopping time $t=T^{\ast }/4$. All trajectories approach the boundary tangentially (smoothly), despite the abrupt cessation of motion.
• Figure 4: Entanglement pulses generated by controlled stopping and restarting of qubit cyclic motion, as shown schematically in the upper-right panel. The qubits halt temporarily at the exchange nodes, producing entangled (solid) and separable (dashed) pulses. Panels (a)--(c) show mixed states, while panels (d) and (e) show pure states. The trajectories include repeated boundary-residing segments, with the qubits departing from and returning to the boundary between these segments, thereby forming pulse trains.
• Figure 5: Entanglement trajectories for qubits initialized with a finite vibrational phase ${ \if@compatibility \mathchar"011E {} \mathchar"011E } \neq 0$. Qubits vibrating toward (${ \if@compatibility \mathchar"011E {} \mathchar"011E } =10{{}^\circ}$) or away (${ \if@compatibility \mathchar"011E {} \mathchar"011E } =-10{{}^\circ}$) from the exchange nodes with period $T=T^{\ast }$ in Table \ref{['tab:Tstar']} produce snake trajectories. Panels (a) and (b) show mixed states with weightings $W_{1\text{--}5}$, and panels (c) and (d) show pure states with weightings $W_{7\text{--}10}$. A finite ${ \if@compatibility \mathchar"011E {} \mathchar"011E }$ causes asymmetric vertical shifts, while reversing its sign introduces horizontal phase shifts. Solid and dashed lines represent positive (${ \if@compatibility \mathchar"0122 {} \mathchar"0122 } >0$) and negative (${ \if@compatibility \mathchar"0122 {} \mathchar"0122 } <0$) ESP, respectively.