Chiral Quantum Entanglement Transfer with Giant Atoms

Abstract: We investigate entanglement transfer in a multi-giant-atom waveguide system. By tailoring chiral spontaneous emission and exploiting dark-state dynamics, the setup enables perfect, unidirectional sequential or selective transfer of quantum states and their associated entanglement. The distance between two entangled atoms, i.e., the entanglement length, can be dynamically adjusted, allowing robust conversion between long-range and short-range entanglement during propagation. The system inherently converges to a dark state, guaranteeing high-fidelity directional transfer. When the additional phase is modulated as a periodic piecewise function, spatially separated giant atoms exhibit stable, nearly lossless state exchange and maintain steady entanglement even under non-Markovian conditions. This behavior mimics conventional braided architectures without suffering from propagation delays or spatial restrictions. Our proposal offers a scalable pathway for continuous long-distance entanglement transport and resilient state exchange in quantum networks.

• The paper demonstrates near-perfect unidirectional state and entanglement transfer by combining phase-engineered chiral emission with dark-state dynamics in giant-atom waveguides.
• Time-dependent coupling pulses enable sequential and selective transfer across atom chains, while dynamically tuning entanglement length from short-range to separations of seven lattice spacings.
• Periodic reversal of the emission direction enables sustained back-and-forth exchange between two distant atoms despite non-Markovian delays, with atomic dissipation identified as the primary fidelity limit.

Overview

This paper investigates entanglement transfer in a chain of giant atoms coupled to a one-dimensional waveguide. Giant atoms—quantum emitters that couple to their environment at multiple spatially separated points—exhibit self-interference effects that enable decoherence-free interactions, chiral spontaneous emission, and non-Markovian retardation. The authors exploit these properties to achieve perfect unidirectional sequential transfer of quantum states and entanglement along the chain, dynamic control of the entanglement length, and sustained back-and-forth state exchange between two distant atoms under non-Markovian conditions.

The central mechanism combines two ingredients: chiral spontaneous emission, engineered via an additional phase $\varphi_l$ between the two coupling points of each atom so that emission into one propagation direction is completely suppressed; and dark-state dynamics, in which pairs of atoms form a dark state satisfying $J_R^N|\Psi(t)\rangle = 0$, guaranteeing lossless transfer. The system spontaneously evolves into and remains within this dark-state manifold, which is what ensures high-fidelity directional transfer throughout the protocol.

Model

The system consists of $N$ identical two-level giant atoms with transition frequency $\omega_a$, each coupled at two lattice sites ($n_{2l-1}$ and $n_{2l}$) of an infinite one-dimensional tight-binding lattice with hopping rate $\eta$. The atom–waveguide coupling coefficients $g_l(t)$ are time-dependent, and a phase difference $\varphi_l$ between coupling points breaks time-reversal symmetry. The dynamics are treated exactly in the single-excitation subspace via coupled Schrödinger equations for atomic amplitudes $u_l(t)$ and photonic amplitudes $J_R^N|\Psi(t)\rangle = 0$0.

Setting $J_R^N|\Psi(t)\rangle = 0$1, the relevant photon modes are left- and right-propagating waves with wave vectors $J_R^N|\Psi(t)\rangle = 0$2 and $J_R^N|\Psi(t)\rangle = 0$3. The squared coupling amplitude,

$J_R^N|\Psi(t)\rangle = 0$4

with $J_R^N|\Psi(t)\rangle = 0$5, shows that choosing $J_R^N|\Psi(t)\rangle = 0$6 such that $J_R^N|\Psi(t)\rangle = 0$7 suppresses left-propagating emission entirely while leaving right-propagating emission finite—and vice versa. This phase-matching condition is the foundation of the chiral behavior exploited throughout the paper.

Dark-state construction and two-atom state transfer

With only right-propagating photons present, the jump operator $J_R^N|\Psi(t)\rangle = 0$8 must annihilate the system state, imposing the constraint

$J_R^N|\Psi(t)\rangle = 0$9

which simplifies considerably when $N$0. For two atoms with $N$1, $N$2, and $N$3, this reduces to $N$4. Combined with boundary conditions requiring complete population inversion between the two atoms over a critical time $N$5 (where $N$6 is the propagation delay), this is satisfied by imposing time-reversal symmetry on the couplings: $N$7, using hyperbolic-tangent-like pulse shapes controlled by a parameter $N$8.

Numerical results demonstrate complete state transfer from atom 1 to atom 2 after $N$9, with generation of the maximally entangled state $\omega_a$0 near the midpoint $\omega_a$1. A key finding is that the final fidelity $\omega_a$2 and maximum concurrence $\omega_a$3 exhibit nearly identical dependence on $\omega_a$4 and $\omega_a$5: state-transfer fidelity directly determines achievable entanglement. The protocol requires both the dark-state condition and adiabaticity—the mixing angle $\omega_a$6 must vary slowly enough. For large $\omega_a$7, reducing $\omega_a$8 compensates by increasing the waveguide relaxation rate $\omega_a$9; for small $n_{2l-1}$0, adiabaticity is naturally satisfied and efficiency becomes largely independent of $n_{2l-1}$1.

Sequential and selective entanglement transfer

Extending to four atoms with periodically modulated couplings $n_{2l-1}$2 and $n_{2l-1}$3 (period $n_{2l-1}$4, with roles swapped between intervals), the excitation transfers strictly sequentially: $n_{2l-1}$5, with concurrence propagating as $n_{2l-1}$6. Each transfer cycle creates and then annihilates entanglement between the involved pair. Small residual nonzero concurrences arise from incomplete intermediate transfers—a minor imperfection the authors acknowledge explicitly.

A seven-atom configuration demonstrates selective transfer: alternating atoms emit only right- or left-propagating photons, so starting from $n_{2l-1}$7 the dynamics are confined to the odd-atom subspace $n_{2l-1}$8, transferring entanglement as $n_{2l-1}$9; starting from $n_{2l}$0, propagation proceeds leftward through even atoms as $n_{2l}$1. Transitions to the opposite chirality's subspace are forbidden by the jump-operator structure.

For initially entangled states such as $n_{2l}$2 in a nine-atom chain, the paper shows sequential entanglement transfer $n_{2l}$3, passing transiently through four-body entangled configurations where $n_{2l}$4, $n_{2l}$5, $n_{2l}$6, and $n_{2l}$7 all reach 0.5 simultaneously. Reversing the sign of $n_{2l}$8 reverses the chirality. Notably, the entanglement length—the distance between entangled atoms—is dynamically tunable: in one process it increases from one to seven lattice spacings, in the reverse process from seven back to one, enabling robust interconversion between long-range and short-range entanglement during propagation.

Back-and-forth transfer under periodic phase modulation

When the additional phase is itself modulated as a periodic piecewise function, $n_{2l}$9 for $\eta$0 and $\eta$1 for $\eta$2, the chiral direction alternates with period $\eta$3. The dark-state constraint correspondingly alternates between right- and left-propagating forms, producing coherent state transfer from atom 1 to atom 2 and back, indefinitely. Because the propagation delay $\eta$4 is absorbed into the coupling coefficient design, increasing the interatomic distance does not degrade performance: simulations with $\eta$5 versus $\eta$6 both yield nearly perfect long-range state exchange and stable entanglement.

This result is the paper's strongest claim relative to prior work: conventional braided giant-atom architectures suffer reduced transfer fidelity under non-Markovian delays and require structurally complex multi-atom networks, whereas here two spatially separated giant atoms suffice for persistent, nearly lossless long-distance state exchange mimicking decoherence-free braided interactions without their structural constraints or delay-induced losses.

Robustness against dissipation

An appendix quantifies robustness to realistic losses by adding decay rates $\eta$7 (atomic) and $\eta$8 (lattice). Lattice dissipation at $\eta$9 leaves transfer quality essentially intact. Atomic dissipation at $g_l(t)$0 causes population and concurrence to decay as $g_l(t)$1, identifying atomic dissipation as the dominant limiting factor. Importantly, because only one effective transfer process occurs at any instant—at most two atoms participate in dissipation simultaneously—the fidelity depends only on total evolution time, not on the number of atoms or transfer events. Shortening individual transfer windows to fit more cycles into the same total time yields essentially identical fidelities.

Limitations and open questions

Several caveats bear on the results. First, the protocol requires precisely engineered time-dependent coupling pulses satisfying time-reversal symmetry and exact phase-matching conditions ($g_l(t)$2, $g_l(t)$3, $g_l(t)$4); the sensitivity of the scheme to fabrication disorder or pulse-shape errors is not analyzed. Second, the maximally entangled state generally appears not exactly at $g_l(t)$5 but slightly offset, and residual nonzero concurrences appear during nominally "off" intervals due to incomplete intermediate transfers—imperfections that would accumulate in longer chains. Third, the analysis is confined to the single-excitation subspace; multi-excitation dynamics, relevant for genuine quantum-network operation, remain unexplored. Fourth, the claimed advantage over braided architectures rests on idealized piecewise phase modulation whose experimental implementation speed relative to $g_l(t)$6 is not addressed. Finally, the dissipation study assumes uniform decay rates and does not consider dephasing, which may be significant in superconducting-circuit implementations.

Conclusion

This work establishes that chiral spontaneous emission combined with dark-state engineering in giant-atom waveguide systems enables perfect unidirectional sequential transfer of quantum states and entanglement, tunable entanglement length, and non-Markovian-resilient back-and-forth state exchange using only two distant atoms. The identification of atomic dissipation—not lattice loss—as the fidelity bottleneck, together with the demonstration that fidelity depends on total evolution time rather than transfer count, provides concrete design guidance. Open questions include robustness to disorder and dephasing, extension beyond the single-excitation regime, and experimental realization of the required fast periodic phase modulation.