Aperiodic Dissipation as a Mechanism for Steady-State Localization

Dissipation is traditionally regarded as a disruptive factor in quantum systems because it often leads to decoherence and delocalization. However, recent insights into engineered dissipation reveal that it can be tuned to facilitate various quantum effects, from state stabilization to phase transitions. In this work, we identify aperiodic dissipation as a mechanism for inducing steady-state localization, independent of disorder or a quasiperiodic potential in the Hamiltonian. This localization arises from long-range phase correlations introduced by a spatially varying dissipation phase parameter, which enables nontrivial interference in the steady-state. By systematically comparing two classes of aperiodic dissipation (defined as commensurate and incommensurate cases), we find that incommensurate modulation plays the most efficient role in stabilizing a localized steady-state. Our analysis, based on coherence measures, purity, and participation ratio, reveals a direct link between eigenstate coherence and real-space localization, showing that dissipation can actively shape localization rather than simply causing decoherence. These findings highlight aperiodic dissipation as a viable approach to controlling localization in open quantum systems, potentially enabling new ways to manipulate quantum states and design dissipation-driven phases.

💡 Research Summary

In this paper the authors investigate whether engineered dissipation alone can induce steady‑state localization in a clean, one‑dimensional tight‑binding lattice whose Hamiltonian supports only extended plane‑wave eigenstates. Using the Lindblad master equation, they introduce a set of nearest‑neighbor jump operators Sₙ that contain a site‑dependent phase αₙ. The phase is modulated according to αₙ = α₀ + α₁ cos(2πβ n^ν), where α₀ is a uniform offset, α₁ controls the modulation amplitude, β determines whether the modulation is commensurate (rational β) or incommensurate (irrational β), and ν governs the spatial rate of variation (slow for small ν, fast for ν≈1). This construction creates long‑range phase correlations without any disorder in the Hamiltonian.

The authors focus on three quantitative diagnostics of the steady state ρ_ss: (i) the relative entropy of coherence C_re, which measures how much off‑diagonal coherence remains; (ii) the purity Tr(ρ_ss²), indicating the mixedness of the state; and (iii) the participation ratio PR = 1/∑ₙ(ρ_nn)², which counts the effective number of lattice sites occupied. Large C_re and small PR together signal a coherent, spatially localized steady state.

Two families of modulation are examined. In the incommensurate case they set β to the inverse golden ratio (β = (√5 − 1)/2), a classic irrational that yields quasiperiodic structure. Numerical simulations show that for small ν (e.g., ν = 0.1) the phase varies slowly across the lattice, establishing long‑range correlations. When the modulation strength α₁ exceeds a threshold (≈ 4 in their units), C_re rises sharply while PR drops dramatically, indicating that the dissipative dynamics selects a “dark” mode that is not only protected from loss but also spatially confined to a few sites. The steady state therefore resembles Anderson localization, yet it arises purely from engineered dissipation rather than from a random potential. Increasing ν (e.g., ν = 0.6) makes the phase vary rapidly, effectively acting like random dephasing; coherence collapses, purity falls, and PR increases, destroying localization.

In the commensurate case (β rational, e.g., β = 1/3) the phase pattern repeats periodically. Even with large α₁, the participation ratio remains relatively high and C_re stays modest. The periodic repetition fails to generate the same long‑range interference needed for strong localization; the steady state remains extended, similar to the uniform‑α₀ scenario. Thus, the irrational nature of β is crucial for the emergence of robust localization.

The paper also revisits the known result that uniform phase dissipation (αₙ = α₀) stabilizes a single momentum eigenstate (a dark state) which is delocalized. By introducing spatial variation, the authors demonstrate that dissipation can be turned from a decohering agent into a constructive one that engineers interference patterns. The key physical insight is that the phase‑modulated jump operators create an effective non‑Hermitian potential with spatially varying loss, and the eigenmode with minimal loss becomes localized due to destructive interference elsewhere.

Beyond the numerical findings, the authors discuss experimental feasibility. The required phase profile could be implemented in photonic waveguide arrays, superconducting qubit lattices, or ultracold atoms with site‑dependent laser‑induced losses. The parameters α₁, β, and ν are all tunable in such platforms, allowing direct observation of the predicted transition from delocalized to localized steady states.

In summary, the work establishes a new mechanism—aperiodic, phase‑modulated dissipation—for inducing steady‑state localization without any disorder in the Hamiltonian. The mechanism relies on long‑range phase correlations introduced by an irrational modulation, which preserve quantum coherence while enforcing spatial confinement. This challenges the conventional view that dissipation inevitably destroys interference, opening avenues for dissipation‑engineered quantum phases, robust quantum memories, and novel non‑equilibrium phase transitions.