Coherence controls the visibility of frozen disorder in an open quantum chain

Technical Summary: Coherence Controls the Visibility of Frozen Disorder in an Open Quantum Chain

Problem Statement
The paper addresses a fundamental ambiguity in open quantum transport: when a particle's position distribution appears nearly Gaussian, does this imply that static (frozen) disorder has become irrelevant to the dynamics, or does the disorder's influence merely manifest at larger length scales and later times? While dephasing is known to suppress coherent localization and induce diffusion, the specific interplay between coherent propagation, frozen return biases, and the resulting position statistics remains unresolved. The authors investigate whether the "visibility" of frozen disorder depends on the observation scale relative to the coherence time.

Methodology
The authors study a resonant quantum chain where ground states $|j\rangle$ couple to auxiliary states $|e_j\rangle$ located between lattice sites. The coupling is alternating ($g$ and $-g$), and auxiliary states decay back to the ground chain with rates that depend on a frozen, random bias $b_j = \varepsilon \xi_j$. The system is governed by a Lindblad master equation incorporating:

1. Coherent coupling: $H = g \sum_j (|e_j\rangle\langle j| - |e_j\rangle\langle j+1| + \text{h.c.})$.
2. Dissipative jumps: Asymmetric return rates to $|j\rangle$ or $|j+1\rangle$ determined by $b_j$, and ground-state dephasing at rate $\gamma$.
3. Measurement protocol: The authors define a "terminal readout" $Y = f(cX)$ (where $f=\tanh$) at a fixed time $T$. To distinguish sample-to-sample variations from intrinsic quantum noise, they calculate the covariance of two independent trials ($A$ and $B$) conditioned on the same frozen disorder realization. This covariance, $\text{Cov}(Y_A, Y_B) = \text{Var}_\xi F_\xi(T; c)$, measures the variance of the conditional mean across different disorder fields.

The analysis proceeds by taking the limit of vanishing dephasing ($\gamma \downarrow 0$) while keeping the disorder amplitude and coupling fixed. The authors define two distinct observation scales:

• Packet Scale: Spatial resolution $a_\gamma \sim \gamma^{1/2}$ and time $R/\gamma$.
• Long Scale: Spatial resolution $h_\gamma \sim \gamma$ and time $T_\gamma \sim \gamma^{-2}$.

They employ renewal theory to decompose the evolution into coherent segments between quantum jumps (dephasing or auxiliary decay) and derive exact expressions for segment duration and endpoint distributions.

Key Contributions and Results

1. Two-Scale Separation of Disorder Visibility:
   The primary result is the demonstration that frozen disorder becomes invisible on the packet scale but remains visible on the long scale as $\gamma \to 0$.

   • Packet Scale Limit: For almost every frozen disorder realization, the rescaled position distribution converges to a universal Gaussian. The covariance of the readout across samples vanishes ($C_p \to 0$). This occurs because coherent propagation spreads the effect of a local bias over many sites, and the "unfinished" coherent packet at the measurement time averages out the fluctuations.
   • Long Scale Limit: On the diffusive time scale, the position distribution remains sample-dependent. The rescaled process converges to a diffusion in a random potential $V(x) = -\Sigma W(x)$, where $W(x)$ is a Brownian motion. The covariance across samples remains strictly positive ($C_l > 0$), indicating that the disorder is still "visible."

2. Mechanism of Coherent Suppression:
   The paper elucidates that the alternating coupling ($g, -g$) creates destructive interference for long-wavelength modes, suppressing auxiliary excitation when ground-state amplitudes vary slowly. As dephasing decreases, the lifetime of these coherent modes increases, allowing a local return bias to influence a wider spatial region. This "smearing" of the bias leads to the Gaussian limit on the packet scale.

3. Distinct Response to Local Perturbations:
   The authors compare the coherent chain to a "memoryless" random-force model (a continuous-time random walk) matched to the same clean diffusion constant and leading random-potential strength.

   • While both models share the same clean Gaussian limit, their responses to a local change in return probabilities differ.
   • The coherent chain exhibits a logarithmic peak in its response kernel near the initial site ($H_1(x) \sim \log(1/|x|)$), whereas the memoryless model's response remains finite at the origin.
   • This difference arises from the correlation between segment duration and displacement, and the contribution of the unfinished packet, which are absent in the memoryless model.

4. Numerical Verification:
   Simulations on an infinite chain with binary disorder ($b_j = \pm 0.2$) at three decreasing dephasing rates confirm the theoretical predictions. Over a sixteenfold decrease in the dephasing rate, the packet-scale covariance point estimate is reduced by approximately 85% (falling from $2.07 \times 10^{-3}$ to $0.31 \times 10^{-3}$), while the long-scale covariance remains statistically stable near $2 \times 10^{-3}$.

Significance and Claims
The paper claims to resolve the ambiguity of Gaussian profiles in disordered quantum systems by showing that the "irrelevance" of disorder is scale-dependent. A Gaussian profile on the packet scale does not imply the absence of disorder effects; rather, those effects are hidden until the system evolves to the long scale where diffusion in a Brownian potential dominates.

The work highlights that coherence acts as a filter for frozen disorder. By extending the coherent excursion length, the system averages out local biases, rendering them invisible to measurements with finite spatial resolution and short observation times. However, this averaging is incomplete on the long scale, where the accumulated potential retains its sample-specific signature.

Furthermore, the paper establishes that matching the clean spreading law and the leading potential strength is insufficient to characterize a system's sensitivity to disorder. The coherent chain's unique response kernel (with its integrable logarithmic singularity) distinguishes it from classical memoryless models, even when they share the same macroscopic transport parameters. This suggests that the "unfinished" nature of coherent segments and the correlation between time and displacement are critical factors in determining the observable impact of frozen disorder.

The authors conclude that the distinction between Born spread (intrinsic quantum variance) and sample variation (disorder-induced variance) is measurable via the covariance of independent trials, providing a concrete observable to probe the interplay of coherence and disorder without monitoring the environment.