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Efficient Simulation of Nonreciprocal Many-body Physics via Quantum Feedback

This paper proposes a scalable experimental scheme to efficiently simulate nonreciprocal many-body spin models and their exotic phenomena, such as the Liouvillian skin effect, by utilizing a single quantum emitter with coherent delayed feedback and classical reset mechanisms.

Original authors: Kseniia Vodenkova, Andrew Pocklington, Fan Yang, Andrew Lingenfelter, Aashish A. Clerk, Hannes Pichler

Published 2026-09-03

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CC BY 4.0

Original authors: Kseniia Vodenkova, Andrew Pocklington, Fan Yang, Andrew Lingenfelter, Aashish A. Clerk, Hannes Pichler

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, particles often behave in ways that defy our everyday intuition, particularly when they are driven by external forces and constantly losing energy to their surroundings. These "driven-dissipative" systems are not closed boxes where energy is conserved; instead, they are open environments where excitations are constantly pumped in and leaked out. When such systems are arranged in chains, they can exhibit strange behaviors where the direction of movement matters deeply. In a standard system, a disturbance might ripple equally in both directions, but in a nonreciprocal system, excitations might flow easily one way while being blocked in the other. This one-way traffic creates unique states of matter and new types of critical behavior that are difficult to study because building large, perfect chains of interacting quantum particles is an immense engineering challenge. Scientists have long sought a way to simulate these complex, one-way interactions to understand phenomena like the "skin effect," where particles pile up at the edges of a system, but existing methods often struggle to scale up beyond a few particles or require intricate, hard-to-build hardware.

A team of researchers has proposed a clever workaround that bypasses the need for a massive physical chain of particles. Instead of building a long line of many quantum emitters, they suggest using a single quantum emitter—a tiny source of light or matter—coupled to a waveguide, which is essentially a path for guiding waves. The key to their method is a mirror placed at the end of this path. When the emitter sends out a signal, it travels down the waveguide, hits the mirror, and bounces back to the emitter after a specific delay. This creates a feedback loop where the emitter interacts with its own past self. By carefully timing this delay, the researchers show that the single emitter effectively becomes a chain of many emitters, with each "link" in the chain representing the emitter at a different moment in time. The signal from the past acts on the present, creating a simulated flow of information that moves in only one direction, mimicking the behavior of a long, nonreciprocal spin chain without ever needing to build one.

The researchers demonstrated that this time-delayed feedback setup can reproduce the complex dynamics of these exotic many-body models with surprising accuracy. In their simulations, they observed that the system exhibits the "Liouvillian skin effect," a phenomenon where the relaxation of the system takes an anomalously long time because all the internal modes of the system become localized at one boundary. In a normal system, disturbances might fade away quickly, but here, the one-way nature of the interaction causes the system to hold onto its state for much longer, with the time it takes to settle down scaling with the size of the simulated chain. They also found that the system displays a "quasi-long-range order," where correlations between different parts of the chain decay slowly in a specific mathematical pattern, rather than vanishing quickly as they would in a standard system. This suggests that the light emitted from the system carries a unique, structured signature of these many-body interactions.

Crucially, the team showed that this simulation is robust against the imperfections that plague real-world experiments. They accounted for factors like the loss of energy in the waveguide and the random jittering of the quantum emitter, known as dephasing. Their analysis indicates that even with these realistic flaws, the distinctive algebraic patterns and the slow relaxation dynamics remain visible, provided the chain is not too long. They calculated that using a superconducting circuit coupled to a specialized metamaterial waveguide, which can slow down light significantly to create the necessary time delays, could allow for the simulation of a chain with a length of about ten to eleven sites. This is a significant step forward, as it moves the study of these complex nonreciprocal models from purely theoretical speculation into the realm of near-term experimental realization.

To access the results of this simulation, the researchers outlined how to measure the system. By observing the quantum emitter itself and the light field that leaks out of the waveguide, one can reconstruct the properties of the entire simulated chain. For instance, measuring the correlations between photons in the output field reveals the hidden order within the chain. The team also described a method to reset the emitter rapidly, allowing them to simulate how the system evolves from a specific starting point, rather than just settling into a steady state. This capability opens the door to studying the dynamic history of these systems, such as how they react to sudden changes. The work establishes a feasible path to scaling up nonreciprocal many-body systems, offering a new route to generate and study exotic states of light that were previously out of reach, all by using the simple, elegant trick of letting a single particle talk to its own reflection.

Technical Summary: Efficient Simulation of Nonreciprocal Many-body Physics via Quantum Feedback

Problem Statement
Driven-dissipative many-body systems exhibit emergent phenomena absent in closed systems, particularly nonreciprocal models where excitations propagate asymmetrically. Key theoretical concepts in this regime include the Liouvillian skin effect (LSE), new types of quantum criticality, and quasi-long-range order. While these phenomena have been studied theoretically, experimental realization of large-scale, nonreciprocal many-body systems remains challenging. Existing approaches, such as synthetic dimensions or chiral waveguide quantum electrodynamics (QED), are often limited to the single-particle regime, require complex waveguide designs, or struggle with the inhomogeneity of artificial quantum emitters, hindering scalability.

Methodology
The authors propose a scalable analog quantum simulation scheme that maps a nonreciprocal many-body spin chain onto a single physical quantum emitter (QE) coupled to a 1D waveguide with coherent time-delayed feedback.

Core Mechanism: A single QE is coupled to a waveguide terminated by a mirror. The emitted field propagates, reflects, and is fed back to the emitter after a time delay $\tau$ . This creates a coherent interaction where the emitter at time $t$ affects itself at time $t+\tau$ .
Temporal Multiplexing: The system effectively maps the single QE evolving over time to a cascaded spin chain of $N$ sites. The site index $n$ in the effective spin chain corresponds to the time interval $[(n-1)\tau, n\tau]$ .
Nonreciprocity: Causality in the feedback loop naturally enforces nonreciprocal interactions (excitations hop $i \to i+1$ but not $i+1 \to i$ ), eliminating the need for engineered chirality or complex waveguide geometries.
Generalization: The scheme can be extended to engineer nonlocal interactions (e.g., $K$ -local couplings or long-range tails) by utilizing multiple feedback loops with commensurate delays or Bragg mirrors with optimized transmittance.
State Preparation and Measurement:
- Steady State: Without intervention, the system relaxes to a steady state where initial correlations decay, allowing access to steady-state observables via two-time correlations of the QE.
- Quench Dynamics: To simulate dynamics from arbitrary product states, the authors propose a "stroboscopic resetting" protocol. At intervals of $\tau$ , the QE is rapidly reset to a target state (e.g., via projective measurement and conditional preparation), effectively initializing each "spin" in the chain independently.
- Observables: Local observables and correlation functions of the effective spin chain are extracted by measuring the QE at specific times or by analyzing the output photonic field correlations using input-output relations.

Key Contributions and Results
The paper demonstrates the feasibility of this scheme through numerical simulations and theoretical analysis of a 1D cascaded spin chain with nearest-neighbor nonreciprocal hopping.

Liouvillian Skin Effect (LSE): The simulation of a quench from a fully excited state reveals an anomalously long global relaxation time scaling as $N\kappa^{-1}$ , despite a finite Liouvillian gap. This is identified as a direct manifestation of the LSE, where nonreciprocity causes eigenmodes to localize at a boundary.
Anomalous Relaxation Dynamics: The local excitation probability exhibits algebraic decay during the transient phase, attributed to a gapless pseudospectrum, followed by exponential decay once a wavefront traverses the chain.
Steady-State Density Profiles: Under weak incoherent pumping, the system exhibits a highly asymmetric steady-state spin density distribution. The authors derive an analytical approximation using a free-fermion model (Jordan-Wigner transformation), predicting algebraic growth of spin density $\langle \sigma^z_i \rangle_{ss} \sim \sqrt{i}$ , which is confirmed by numerical simulations.
Quasi-Long-Range Order: The scheme successfully probes the algebraic decay of correlation functions in the near-critical regime. Both atomic correlations and output field correlations display power-law scaling, though the output field exponents differ due to destructive interference between emissions at distinct times.
Robustness to Imperfections: The authors analyze the impact of parasitic dissipation (qubit $T_1$ loss, waveguide photon loss) and dephasing. They show that the algebraic scaling of spin density persists up to a length scale determined by the ratio of the coupling rate to the parasitic loss rates.
Experimental Feasibility: The proposal is tailored for superconducting circuit experiments using transmon qubits coupled to metamaterial waveguides with high group indices ( $n_g \sim 1000$ ). Calculations suggest that a 12 cm waveguide section could simulate a chain of length $N \approx 10$ , limited primarily by waveguide loss and dispersion.

Significance and Claims
The paper claims to establish a "feasible way to scale up a nonreciprocal many-body system" using a surprisingly simple setup involving a single quantum emitter. By circumventing the need for multiple inhomogeneous qubits and complex chiral waveguides, the method offers a scalable route to study nonreciprocal many-body physics.

The authors position their work as providing a "new route towards generation of exotic states of light," specifically correlated photonic many-body states exhibiting quasi-long-range order. They emphasize that the scheme is robust to realistic experimental imperfections and is amenable to near-term realization in existing superconducting circuit platforms. The work bridges the gap between theoretical models of nonreciprocal driven-dissipative systems and experimental implementation, enabling the direct probing of phenomena like the Liouvillian skin effect and anomalous relaxation dynamics that were previously inaccessible in large-scale systems.

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