Quantum Contact Processes on a Topological Lattice

Abstract: Contact processes play an important role in classical non-equilibrium dynamics, describing the spreading of diseases, the dynamics of earthquakes and forest fires, and the distribution of information through the internet. Here we show that their quantum counterpart, where the spreading occurs through coherent couplings, displays even richer dynamics and offers new means of control. A quantum contact process on a topologically non-trivial lattice can be confined to a protected subspace corresponding to either a single site or a fully excited lattice. Furthermore, excitation spreading can be controlled to occur in quantized steps and on demand when employing topological pumps. We show that the many-body dynamics of excited domains can be mapped to an effective single-particle model, which also determines the topological properties. Throughout this work, we consider a specific type of contact process corresponding to coherent Rydberg facilitation in a tweezer array of trapped atoms in a one-dimensional lattice.

• The paper demonstrates that mapping quantum contact processes onto topological lattice models yields a robust framework for controlled excitation dynamics.
• It utilizes a one-dimensional Rydberg facilitation model and TEBD simulations to expose topologically protected edge states and coherent oscillations.
• The study implements topological pumping protocols to achieve deterministic, quantized control over excitation domain sizes, highlighting clear transitions between regimes.

Quantum Contact Processes on Topological Lattices: Control and Dynamics

Introduction and Context

The study "Quantum Contact Processes on a Topological Lattice" (2604.03184) establishes a comprehensive framework for understanding the non-equilibrium dynamics of quantum contact processes (QCPs) through the lens of topological lattice models. The classical contact process, a cornerstone in modeling various spreading phenomena (epidemics, information diffusion, etc.), is known for emergent phase transitions between active and absorbing states, typically described by the directed percolation universality class. The quantum extension, characterized by the interplay of coherence and kinetic constraints, fundamentally enriches the dynamical landscape, yielding phenomena inaccessible in classical systems.

This work focuses on a specific realization: a 1D quantum excitation process (QXP) modelled as coherent Rydberg facilitation in a tweezer array, subject to kinetic constraints whereby excitations can only spread in the presence of a single excited neighbor. By mapping the system dynamics onto key topological paradigms, notably the Su-Schrieffer-Heeger (SSH) and Aubry-André-Harper (AAH) models, the study demonstrates robust topological control over spreading, confinement, and quantized domain growth.

Model and Kinetic Constraints

The quantum contact process is defined for a chain where each site is either in the ground or excited state. The kinetic constraint enforces that excitations only propagate when exactly one neighbor is excited. The Hamiltonian, incorporating site-specific coherent excitation rates and detunings, maps physical Rydberg atom arrays into an effective lattice model:

• The QXP dynamics translate under stringent facilitation conditions: detuning and interaction strengths ensure only facilitated transitions are resonant.
• Time-evolving block decimation (TEBD) is used for simulating the nontrivial quantum many-body dynamics.

  Figure 1: QXP quantum contact process on a one-dimensional lattice; schematic of coherent facilitation mechanism, domain mapping, and contrasting classical vs quantum spreading in a small system.

This mapping uniquely enables tracking the evolution not just in real space, but in an effective domain-size basis. The domain size $m$—the number of contiguous excited sites—serves as a central variable, allowing reduction to a single-particle effective model for specific initial conditions.

Topological Mapping: The SSH Model

By imposing an alternation in the site-dependent excitation rates, the lattice inherits the structure of the SSH model, well-known for its topological edge states and bulk-boundary correspondence. The study establishes:

• For strong dimerization (alternating large/small hopping amplitudes), two topologically protected edge states emerge: one localized at the initially excited boundary ($\Psi_L$), the other at the opposite, fully excited configuration ($\Psi_R$).
• In the trivial regime, the excitation diffuses and reflects, manifesting as periodic de-facilitation cascades.

  Figure 2: Time-resolved dynamics in both real space and domain basis for trivial and topological SSH regimes, evidencing coherent edge-state oscillations only in the latter.

The SSH mapping dictates that, in the topologically nontrivial phase ($\lambda_w > \lambda_v$), the system oscillates coherently between the two edge states. The period of oscillation $T_{\mathrm{hyb}}$ is set by the energy splitting due to finite-size hybridization, scaling exponentially with system size. This dynamics is inherently protected by topological invariants of the SSH chain.

Figure 3: Hybridization of edge states, with oscillations between $\Psi_L$ and $\Psi_R$ clarifying the edge-to-edge coherent transfer as a function of the dimerization ratio.

Numerically and analytically, the authors confirm the selective confinement of dynamics to a topologically protected subspace, robust against perturbations that preserve the gap and symmetries.

Quantum Control via Topological Pumps

The reduction to an effective single-particle model further enables the extension to time-dependent, topological pumping protocols. The Hamiltonian is modulated in time according to the AAH model, facilitating a Thouless pump:

• Time-periodic modulation of onsite energies and tunneling amplitudes generates quantized transport in domain space, i.e., deterministic and quantized control over the domain size.
• High-order tunneling suppression ensures minimal dispersion of the wavepacket during pumping, required for sharp, quantized dynamics.

  Figure 4: Quantum control of domain size via AAH topological pump; modulation of system parameters enables precise, quantized cycling of the excitation domain, matching domain Fock states over cycles.

Key findings indicate that for sufficiently large detuning and well-separated resonance conditions, the full many-body dynamics can be adiabatically mapped to the single-particle pumping process. Experiments with sub-critical detuning disrupt this mapping, resulting in loss of coherent control.

Mapping Validity: Regimes and Breakdown

The correspondence between the many-body Rydberg Hamiltonian and the single-particle QXP model is sensitive to the detuning regime:

• In the facilitation regime ($\Delta_0$ much larger than modulation amplitudes), individual excitations are suppressed, preserving effective mapping and topological transport.
• For smaller detunings, the mapping breaks down due to non-facilitated (unconstrained) processes, seen as pronounced deviations in excitation dynamics both in real-space and domain basis.

  Figure 5: Accuracy of the single-particle QXP model as a function of detuning; breakdown of domain description for insufficient detuning highlighted in both representations.

Implications and Future Directions

This work demonstrates that quantum coherent spreading processes, when mapped onto topological lattice models, exhibit a suite of controllable, robust, and quantized dynamical phenomena unavailable in classical analogs. The integration of kinetic constraints, topological structure, and quantum coherence yields:

• Confinement of dynamics to topologically protected subspaces, even in highly coarse-grained public health or information diffusion analogs.
• Deterministic, quantized and robust spreading facilitated by driving protocols inherited from topological pumping theory.
• Direct implications for the design and analysis of synthetic quantum matter, including Rydberg atom quantum simulators and platforms exploiting topological protection for robust state transfer and manipulation.

From a theoretical perspective, these results offer a blueprint for integrating non-equilibrium statistical mechanics with topological band theory, potentially guiding new approaches to quantum control, robust state preparation, and nontrivial steady states in driven open quantum systems. Open questions include the impact of dissipation, disorder, and higher-dimensional generalizations on the confinement and control found in this study.

Conclusion

"Quantum Contact Processes on a Topological Lattice" (2604.03184) provides a rigorous and multifaceted exploration of quantum spreading dynamics engineered with topological control. The reduction of complex many-body facilitation processes to effective single-particle topological models underpins robust mechanisms for the quantum control of excitation domains, with immediate relevance for both experimental realization in Rydberg atom arrays and broader theoretical understanding of topological phenomena in out-of-equilibrium quantum systems. The findings open several avenues for future research in quantum simulation, non-equilibrium phase transitions, and fault-tolerant control in many-body networks.