🔬 condensed matter

Fermions on a 1D lattice: localized sources and sinks with dephasing

This paper investigates a one-dimensional spinless fermionic system with localized sources and sinks under bulk dephasing, revealing that dephasing suppresses coherent quantum features like Friedel oscillations and secondary density peaks while altering transport dynamics from ballistic to diffusive regimes.

Original authors: Elka Bhattacharya, Sushant Saryal, Soumya Ghosh, Kabir Ramola

Published 2026-07-27

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CC BY 4.0

Original authors: Elka Bhattacharya, Sushant Saryal, Soumya Ghosh, Kabir Ramola

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

Imagine a world where tiny particles, like electrons, don't just bounce around randomly like marbles in a box. Instead, they behave like waves, rippling through space and interfering with each other, creating complex patterns of high and low density. This is the strange, beautiful realm of quantum mechanics, specifically focusing on "open quantum systems." These are systems that aren't isolated; they are constantly interacting with their environment, like a musician playing in a noisy room. In the real world, nothing is perfectly quiet. Noise, heat, and random jitters cause these delicate quantum waves to lose their "phase coherence"—a fancy way of saying they stop marching in step and start acting more like a chaotic crowd. Scientists care deeply about this because understanding how quantum systems turn into classical, predictable ones is key to building future technologies like quantum computers, which need to stay "in step" to work, and understanding how energy moves through materials.

This paper takes a look at a one-dimensional line of fermions (a specific type of particle, like electrons, that refuses to share the same space) moving along a lattice, or a grid of spots. The researchers set up a scenario where particles are being injected at one spot (a "source") and removed at another (a "sink"), while the whole line is subjected to "dephasing." Think of dephasing as a constant, gentle tapping on the shoulder of every particle, reminding them to forget their quantum wave nature and just act like ordinary, diffusive particles. The team uses a mathematical framework called the Lindblad master equation to simulate how the density of these particles changes over time. They are essentially asking: "If we keep tapping these quantum particles to make them lose their rhythm, how does the flow of traffic change, and do the cool quantum patterns survive?"

The story begins with a single "sink" (a drain) in a line of particles. Without any tapping (dephasing), the particles flow away from the drain in a perfect, wave-like burst. This creates a phenomenon called "Friedel oscillations," which are like ripples in a pond spreading out from where a stone was dropped. The paper shows that in this quiet, coherent world, the rate at which particles disappear follows a very specific script with three distinct chapters: a quick initial drop, a long, steady middle phase where the wave travels ballistically (like a bullet), and a final phase where the wave hits the ends of the line and bounces back, creating a temporary spike in the loss rate.

However, the moment the researchers introduce dephasing—the constant tapping—the story changes dramatically. The beautiful ripples (Friedel oscillations) vanish instantly, replaced by a smooth, featureless hill. The three-chapter script collapses into just two. The "ballistic" middle phase, where particles zoomed away in a coordinated wave, is destroyed. Instead, the particles start moving like a drop of ink in water, spreading out slowly and randomly in a "diffusive" manner. The paper demonstrates that as the tapping gets stronger, the system transitions smoothly from this quantum wave behavior to a classical, messy diffusion. Interestingly, they find a "sweet spot" where a little bit of tapping actually helps particles reach the drain faster (a phenomenon called the anti-quantum Zeno effect), but if you tap too hard, you freeze them in place, and they can't reach the drain at all (the quantum Zeno effect).

The plot thickens when they add a "source" (a faucet) at one end and a "sink" (a drain) at the other, separated by a distance $\Delta$ . In the quiet, coherent world, the particles don't just flow from A to B; they interfere with themselves. This creates a series of secondary peaks and valleys in the particle density at regular intervals—specifically, at integer multiples of the distance $\Delta$ . It's as if the particles are playing a game of echo, creating standing waves that leave a signature of peaks every time they travel the distance between the source and the sink. These peaks appear regardless of whether the line started full, empty, or half-full, proving they are a fundamental result of quantum coherence.

But when the dephasing taps are turned on, these magical secondary peaks disappear. The interference pattern is washed out, and the density profile becomes a simple, straight line sloping from the source to the sink, exactly as you would expect in a classical, non-quantum world. The paper concludes that dephasing acts as a powerful control knob: it can systematically destroy the exotic, wave-like features of quantum transport, forcing the system to behave like a classical diffusive fluid. While the simulations show this crossover clearly, the authors suggest that future work could look deeper into the statistical noise of these particles to fully map out the transition from the quantum to the classical world.

Technical Summary: Fermions on a 1D Lattice with Localized Sources, Sinks, and Dephasing

Problem Statement
This study investigates the non-equilibrium dynamics of a one-dimensional spinless fermionic system coupled to localized particle sources and sinks in the presence of bulk dephasing. The work addresses the interplay between boundary-driven coherent injection/loss and bulk incoherent dephasing, a scenario relevant to mesoscopic transport and solid-state devices. Specifically, the authors analyze how local dephasing—a mechanism modeling phase coherence loss without particle exchange—alters transport regimes, suppresses quantum interference features (such as Friedel oscillations), and drives the system from ballistic to diffusive behavior.

Methodology
The system is modeled on a 1D periodic lattice with nearest-neighbor hopping amplitude $J$ . The dynamics are governed by the Lindblad master equation, which incorporates:

Unitary Evolution: Defined by the tight-binding Hamiltonian $H = -J \sum_i (a^\dagger_i a_{i+1} + h.c.)$ .
Localized Dissipation: A source (injection rate $\Gamma_G$ ) and a sink (extraction rate $\Gamma_L$ ) acting at specific lattice sites.
Bulk Dephasing: A Lindblad superoperator acting uniformly on all sites with rate $\gamma$ , modeling local measurements or environmental noise.

The authors compute the time evolution of the two-point correlation functions $C_{i,j}(t) = \langle a^\dagger_i(t) a_j(t) \rangle$ and the local density $n_i(t) = C_{i,i}(t)$ . By vectorizing the correlation matrix, the equations of motion are reduced to linear differential equations solvable via matrix exponentiation and diagonalization. The study explores three initial conditions: a Fermi sea, a completely filled lattice, and an empty lattice. Analytical solutions are derived for the strong-dephasing (Zeno) limit using an adiabatic approximation.

Key Results and Contributions

Suppression of Quantum Coherence: The presence of bulk dephasing systematically suppresses off-diagonal elements of the density matrix. This leads to the complete erasure of Friedel oscillations around the impurity sites, which are prominent in the coherent (unitary) limit.
Dynamical Regime Crossover:
- Coherent Case (No Dephasing): The particle loss/gain rate exhibits three distinct dynamical regimes: (i) an initial transient due to local depletion/injection, (ii) a ballistic plateau where the perturbation propagates at a constant rate, and (iii) a finite-size regime where boundary reflections cause transient enhancements due to constructive interference.
- Dephasing Case: As $\gamma$ increases, the intermediate ballistic plateau is destroyed. The system transitions to a two-regime dynamics: an initial transient followed by a diffusive regime. The system-size dependence of the steady current vanishes, and the curves for different lattice sizes collapse onto a single universal curve.
Quantum Zeno and Anti-Zeno Effects:
- Sink (Loss): The particle loss rate exhibits a non-monotonic dependence on $\gamma$ . For weak dephasing, the rate increases (Anti-Zeno Effect) as dephasing suppresses destructive interference that hinders particles from reaching the sink. For strong dephasing, the rate decreases (Quantum Zeno Effect) due to Zeno localization hindering transport.
- Source (Gain): The particle growth rate is monotonically suppressed by increasing $\gamma$ , providing a clear signature of the Quantum Zeno effect where frequent "measurements" inhibit the injection dynamics.
Strong-Dephasing Limit (Zeno Limit): In the limit $\gamma \gg J, \Gamma$ , the system's dynamics reduce to a classical diffusion equation with a localized sink/source. The authors derive an exact analytical solution for the density profile, $n(x,t) \propto \text{erf}(|x|/\sqrt{t})$ , confirming diffusive scaling ( $n \sim f(x/\sqrt{t})$ ).
Source-Sink Interference: When a source and sink are separated by a distance $\Delta$ , the coherent steady state exhibits secondary density peaks at integer multiples of $\Delta$ . These peaks arise from multipath quantum interference. Under dephasing, these secondary peaks are systematically washed out, leaving a smooth, linear density profile characteristic of classical diffusion. This effect persists regardless of whether the initial state is a coherent Fermi sea or an incoherent filled/empty lattice, provided the system reaches the steady state.

Significance and Claims
The paper claims that local dephasing acts as a control parameter that drives a smooth crossover from quantum ballistic transport to classical diffusive transport. The primary significance lies in demonstrating how dephasing fundamentally alters the transport mechanisms of open quantum systems:

It destroys the system-size-dependent steady currents characteristic of coherent ballistic transport.
It erases non-local quantum interference features, such as Friedel oscillations and secondary interference peaks in source-sink setups, effectively restoring classical diffusive behavior.
It provides a unified framework to observe both Quantum Zeno and Anti-Zeno effects in particle transport depending on the specific observable (loss vs. gain) and the strength of the dephasing.

The authors conclude that while the transient dynamics depend on initial conditions, the steady-state properties in the presence of dephasing are robustly classical, governed by macroscopic diffusion equations, whereas the coherent limit retains distinct quantum signatures dependent on the initial state's coherence. Future directions suggested include investigating higher cumulants and full counting statistics (FCS) to further quantify the quantum-to-classical noise crossover.

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