Dissipation-tunable extended and localized steady states in a non-disordered lattice

AI summary

Overview

Research area: Open quantum systems / quantum transport — specifically, disorder-free (clean) one-dimensional tight-binding lattices coupled to an engineered Markovian environment, studied through the Lindblad master equation.

Technical level: Advanced. The paper assumes familiarity with Anderson localization, tight-binding models, bound states in the continuum (BICs), the Lindblad/GKSL master equation, Liouville–Fock vectorization, and participation-ratio measures of localization.

Scope (one sentence): The paper shows, numerically, that a nonlocal phase-selective "bond" jump operator can steer the long-time density matrix of a strictly non-disordered 1D lattice toward either localized boundary states or extended bulk states purely by tuning a relative phase α — without changing any Hamiltonian parameter.

What This Paper Is About

Dissipation is normally treated as the enemy of quantum interference: environment-induced dephasing is expected to erase the delicate phase relations that allow waves to stay localized, pushing the system toward a featureless delocalized steady state. This paper asks the opposite question for a clean (disorder-free) lattice: can dissipation be engineered so that it selects which kind of state survives — extended or localized — rather than destroying localization outright? The authors build a lattice that already hosts both extended bulk states and localized boundary states (including an algebraically localized bound state in the continuum), then couple it to a nonlocal dissipative channel whose phase α acts as a tunable "filter" over the spatial phase correlations of the eigenstates.

Key Contributions

1. A dissipation-only control knob for localization. The authors demonstrate that the steady state of the open system can be redirected between localized-dominated and extended-dominated sectors by changing a single dissipative parameter — the relative phase α in the jump operator — for both bond ranges studied (l = 1 and l = 2), with no modification of any Hamiltonian parameter.

2. A microscopic, quantitative explanation of the selection rule. The selection is traced to the fraction of site pairs separated by distance l whose wavefunction phase difference vanishes, P^in_{n,l} = N^in_{n,l}/N_tot with N_tot = L − l. This quantity is shown to vary monotonically with eigenstate index for l = 1 and to take a V-shaped profile (peaked at both band edges, vanishing at the spectrum center) for l = 2.

3. Demonstration of robustness via finite-size scaling. The localized and extended sectors remain well separated across system sizes L = 201, 401, 601, 801, 1001 (for l = 1) and L = 200, 400, 600, 800, 1000 (for l = 2), with no appreciable systematic drift, indicating the separation is not a finite-size artifact.

4. A dissipative quench protocol with post-dissipation stability. Using quantum fidelity between the time-evolved and initial states, the authors show that the selected character persists, and argue that once dissipation is removed the diagonal elements of the density matrix stay constant under unitary evolution, freezing the prepared localization character.

Main Findings

• The clean lattice already supports both state types. With hopping t_j = t except at sites j = κN where t_j = ((κ+1)/κ)^β t (N = M+1, κ = 1, 2, 3, …), the model supports M surface bound states in the continuum with E_σ = −2t cos k_σ and k_σ = πσ/N. For N = 2 and β = 1 (M = 1), a single surface BIC sits at E = 0 with an envelope decaying algebraically as |ψ_n| ∝ 1/(j+1), while additional states lie outside the bulk band −2t < E < 2t, the outermost near E ≈ ±2.56t.

• Eigenstate-resolved steady state concentrates near p, q ≃ L/2. For l = 1, the steady-state density matrix in the Hamiltonian eigenbasis has its dominant weight centered near p, q ≃ L/2, showing substantial overlap with a localized eigenstate sector; in real space it concentrates along the diagonal m = n with a spatially extended off-diagonal structure for both α = 0 and α = π.

• Phase reversal leaves the sector separation intact. For α = 0, the localized branch stays at P_e ≃ 0.85–0.9 and the extended branch near P_e ≃ 0.1 over the full size range; the same well-separated branches appear at α = π. The paper describes this as consistent with the phase-selective nature of the jump operator — α changes the selected relative phase, not the underlying localized/extended distinction.

• The in-phase pair fraction explains which sector wins. For l = 1, P^in_{n,1} rises monotonically from approximately zero at the low-energy band edge to unity at the high-energy band edge. Because α selects eigenstates with a specific in-phase structure, the steady state at α = π (α = 0) is predominantly composed of localized boundary states from the low-energy (high-energy) region. For l = 2, P^in_{n,2} is V-shaped, maximal near both band edges and vanishing at the spectrum center.

• Fidelity quench distinguishes preserved versus erased initial states. Starting with l = 2 and Γ = 1, for α = 0 the fidelity with an extended initial state rapidly approaches zero while it tends to a nonzero value for a localized initial state (consistent with P_l ≫ P_e); for α = π the behavior reverses. The reported example uses L = 201, an extended eigenstate labeled n = 1 (blue) and a localized eigenstate labeled n = 90 (red).

• Localization is read out in the Hamiltonian eigenbasis, and it survives. The authors emphasize that the steady state's localization is best characterized in the eigenbasis of H, and that it can be detected by measuring the system after dissipation is removed: the diagonal elements then remain constant because the dynamics reduce to dρ/dt = −i[H, ρ].

Methodology in Plain English

The strategy has three layers.

1. Build a lattice with two kinds of states, without disorder. Instead of random on-site energies, the authors modulate the hopping amplitudes deterministically: at regularly spaced bonds (j = κN) the hopping is boosted by a factor ((κ+1)/κ)^β, while everywhere else it equals t. Because the modulation weakens away from the boundary, the lattice becomes asymptotically uniform in the bulk, so the bulk band stays −2t < E < 2t. The modulation nevertheless produces boundary states whose energies sit inside the band (the algebraically localized BIC) and outside it (exponentially localized states). The participation ratio PR(E) = (Σ_j |ψ_j|²)² / Σ_j |ψ_j|⁴ is used to sort eigenstates: PR = O(1) means localized, PR = O(L) means extended. Figure 1 illustrates this with L = 201, N = 2, β = 1, showing an extended bulk state at E = −1.46 and localized boundary modes at E = ±2.56.

2. Couple the lattice to a designed environment. The dynamics follow the Lindblad equation dρ/dt = −i[H, ρ] + Γ Σ_j (O_j ρ O_j† − ½{O_j†O_j, ρ}). The unusual ingredient is the jump operator O_j = (c_j† + e^{iα} c_{j+l}†)(c_j − e^{iα} c_{j+l}), which acts on two sites a distance l apart and conserves total particle number. It is a pure decoherence channel that locks the relative phase of the two sites. At α = 0 the annihilation factor vanishes on the antisymmetric single-particle mode (c_j† − c_{j+l}†)|0⟩ while the creation factor projects onto the symmetric mode, driving in-phase synchronization; at α = π the rule is reversed and the antisymmetric configuration is favored.

3. Solve for the steady state and watch the dynamics. Vectorizing the density matrix via the Choi–Jamiołkowski isomorphism turns the master equation into a linear problem d|ρ⟩⟩/dt = ℒ|ρ⟩⟩ with an explicit non-Hermitian Liouvillian matrix; the long-time state is the zero-eigenvalue eigenvector when that subspace is one-dimensional. The authors then compare the steady-state density matrix in real space and in the Hamiltonian eigenbasis, decompose it into extended (P_e) and localized (P_l) weights from the diagonal elements, check how those weights scale with system size, count phase-matched site pairs to explain the outcome, and finally compute the quantum fidelity F[ρ(t), ρ₀] = Tr[√(ρ(t)^{1/2} ρ₀ ρ(t)^{1/2})] after quenching on the dissipation at t = 0.

Why This Matters

Impact on research. The work inverts the standard intuition that dephasing destroys localization, and does so in a strictly non-disordered lattice — a regime the authors explicitly identify as largely unexplored compared with quasiperiodic systems featuring mobility edges, where dissipative couplings have already been shown to mediate extended-to-localized transitions. It reframes the dissipation as a resource: the jump operator does not alter the Hamiltonian spectrum, it only redistributes asymptotic weight among spectral sectors according to phase matching. It also supplies a concrete, analytically tractable diagnostic (the in-phase pair fraction P^in_{n,l}) that predicts which sector a given α will favor, and shows that the preparation is stable after the environment is switched off.

Potential real-world applications (as motivated by the paper's framing):

• Controllable state preparation in engineered optical Raman lattices and 1D Bose–Hubbard chains, the two platforms the authors cite as hosting nonlocal jump operators of this form.
• Transport manipulation and switching in disorder-free photonic or acoustic lattices, where avoidance of disorder removes a major fabrication constraint.
• Phase-locking and synchronization protocols between distant lattice sites, given that the jump operator explicitly enforces in-phase (α = 0) or out-of-phase (α = π) bonding.
• Sensing and readout schemes that exploit the fact that the prepared localization character is frozen — and therefore measurable — after dissipation is removed.

Industry relevance. The results are theoretical and numerical; no experimental demonstration or device is reported in the paper. The relevance is therefore prospective, aimed at quantum simulation and synthetic-lattice hardware where bond-range and bond-phase couplings can be engineered, rather than at an existing commercial process.

Future Directions

• Higher-dimensional lattices. The conclusion explicitly proposes extending the framework to higher-dimensional systems, where the geometry of the bond pattern and the structure of the boundary states differ from the 1D case studied here.
• Interacting many-body mechanisms. All results reported are single-particle; the authors identify interacting many-body settings as an open extension, where the clean-lattice BIC and boundary modes would need to be reconsidered in the presence of interactions.
• Non-Markovian reservoirs. The treatment uses the Born–Markov approximation throughout; relaxing it to non-Markovian environments is named as a route beyond the current single-particle Markov limit.
• Experimental realization. The paper notes that nonlocal jump operators of this form have been proposed in optical Raman lattices and 1D Bose–Hubbard chains, but reports no experimental implementation, leaving the practical demonstration of dissipation-tunable localization selection as an open step.

Target Audience

Best suited to researchers in open quantum systems, non-Hermitian and dissipative quantum dynamics, and quantum transport in synthetic lattices (cold atoms, photonic waveguide arrays, acoustic lattices). It will also interest theorists working on bound states in the continuum in clean lattices and on dissipation-engineered state preparation. Readers need a working command of the Lindblad formalism and localization measures to follow the derivations; the conceptual message about phase-selective dissipation is accessible at an intermediate level, but the numerical machinery is advanced.

Authors’ abstract

Dissipation is usually regarded as a source of decoherence that suppresses quantum interference and localization. Here we show that suitably engineered dissipation can instead be used to select localized or extended states in a strictly non-disordered one-dimensional lattice. The underlying clean lattice has spatially inhomogeneous hopping and supports both extended bulk states and localized boundary states, including an algebraically localized bound state in the continuum. We introduce a nonlocal bond jump operator with a tunable relative phase and show that this phase selectively favors eigenstates with different spatial phase correlations. As a result, the long-time density matrix can be steered toward sectors dominated by localized or extended Hamiltonian eigenstates without changing any Hamiltonian parameter. The microscopic origin of the selection is quantified by the fraction of site pairs separated by a distance $l$ that are phase matched with the dissipative channel. We further characterize the dissipative quench through the quantum fidelity and show that the selected character of the steady state can persist after the dissipation is removed. Our results establish phase-selective bond dissipation as a route to controllable state preparation and transport manipulation in non-disordered lattices.