Dynamics of two particles with quasiperiodic long-range interactions

This study utilizes numerical exact diagonalization to demonstrate that strong quasiperiodic modulation of long-range interactions in a one-dimensional lattice of two identical spinless fermions induces a robust correlated dynamical regime characterized by constant inter-particle distance, suppressed entanglement, and distinct separation behaviors dependent on boundary conditions and initial states.

The Gist

Imagine a tiny, one-dimensional dance floor made of stepping stones. On this floor, two identical dancers (our particles) are trying to move around. Usually, in the quantum world, these dancers would wander off in random directions, getting tangled up with each other and spreading out until they cover the whole floor.

But in this study, the researchers added a special, invisible "music" to the dance floor. This music isn't a simple beat; it's a quasiperiodic rhythm—a pattern that repeats but never quite gets the same twice, like a melody that shifts slightly every time it loops. Furthermore, this music doesn't just affect the dancers individually; it changes how they feel about each other depending on how far apart they are.

Here is what happens when these two dancers try to move under this strange, shifting music:

1. The "Shadow Partner" Effect

Normally, if you push two people on a frictionless floor, they might drift apart or crash into each other. But here, when the "music" (the interaction) is loud enough, something magical happens. The two dancers start moving together as if they are holding hands, even though they aren't touching.

No matter where they start, they maintain a roughly constant distance between them. If they start 10 steps apart, they stay 10 steps apart as they glide across the floor. It's like they are locked in a synchronized dance where their relative position is fixed, even though the whole pair is moving.

2. Three Different Dance Styles

The researchers found that by slightly changing the "phase" of the music (like shifting the start time of the song), the dancers could perform three distinct routines:

• The Frozen Statue (Localization): Sometimes, the music is so specific that the dancers get stuck. They start at a certain distance, and the "pull" of the music cancels out any movement. They just stand there, frozen in place, refusing to wander. It's as if the floor has turned into quicksand for them.
• The Bouncing Ball (Nearest-Neighbor Oscillation): In other cases, the dancers don't stay perfectly still, but they don't drift apart either. Instead, they bounce back and forth between two specific distances. Imagine they are trying to walk 10 steps apart, but the music pushes them to 9 steps, then pulls them back to 10, then 9 again. They oscillate between these two spots, like a spring that can't quite settle.
• The Giant Leap (Next-Nearest-Neighbor Transition): This is the rarest move. Usually, the dancers are stuck at a specific distance. But occasionally, the music creates a "shortcut" where jumping two steps away (e.g., from 10 steps apart to 12 steps apart) costs almost no energy. So, while they mostly stay put, they occasionally make a sudden, synchronized leap to a new distance, skipping the step in between.

3. The "Entanglement" Mystery

In the quantum world, when two particles interact, they become "entangled." Think of this like two dancers who have memorized each other's moves so perfectly that you can't describe one without describing the other. Usually, as they dance, this connection gets stronger and messier, making the system very complex.

However, the researchers found that in this special "constant distance" regime, the dancers refuse to get messy. Their connection remains simple and stable. It's as if, instead of a chaotic tango, they are performing a very disciplined, predictable waltz. This "suppression of entanglement" means the system stays orderly and predictable for a long time, which is very rare in quantum systems.

Why Does This Matter?

Think of this research as discovering a new rule of physics for how particles behave when they are "talking" to each other over long distances in a weird, shifting environment.

• For Technology: This could help scientists build better quantum computers. Quantum computers are very fragile; they lose their information (decoherence) easily. If we can find ways to make particles stay in a stable, synchronized state (like our dancing pair), we might be able to protect quantum information for longer.
• For Understanding Nature: It shows us that even in a chaotic, non-repeating environment (quasiperiodic), nature can find ways to create order and stability.

In a nutshell: The paper shows that if you put two quantum particles on a line with a specific type of "weird music" playing, they will stop acting like random wanderers and start acting like a coordinated team, staying a fixed distance apart, staying simple, and refusing to get lost.

Technical Summary

1. Problem Statement

The paper investigates the non-equilibrium dynamics of two identical spinless fermions on a one-dimensional lattice with Open Boundary Conditions (OBC). The system is subject to quasiperiodic long-range interactions, a regime that differs from standard short-range models and the well-studied Aubry-André model (where the potential is quasiperiodic, not the interactions).

The core scientific question is how quasiperiodic modulation of the interaction strength between particles affects their correlated motion. Specifically, the authors aim to understand if and how these interactions can induce a robust dynamical regime where particles move together while maintaining a fixed separation, and how this behavior depends on interaction strength, initial conditions, and the phase of the modulation.

2. Methodology

• Model Hamiltonian: The system is described by a Hamiltonian $\hat{H}$ consisting of nearest-neighbor hopping ($J$) and a quasiperiodic long-range interaction term ($U$):
  $$\hat{H} = -J \sum_{i} (\hat{c}^\dagger_i \hat{c}_{i+1} + \text{H.c.}) + \sum_{1 \le i < j \le L} \Delta \cos(2\pi\beta|i-j| + \phi) \hat{n}_i \hat{n}_j$$
  Where:
  • $\Delta$: Modulation amplitude (interaction strength).
  • $\beta$: Spatial frequency (chosen as the irrational number $(\sqrt{5}-1)/2$ to ensure quasiperiodicity).
  • $\phi$: Initial phase of the modulation.
  • $L$: System size (ranging from 8 to 71 in simulations).
• Numerical Approach: The authors employ Exact Diagonalization (ED) to solve the time-dependent Schrödinger equation for the continuous-time quantum walk.
• Key Observables:
  • Single-particle probability density $P(i, t)$: To track particle localization.
  • Two-particle correlation function $\Gamma(i, j, t)$: To visualize joint probability and separation.
  • Expected inter-particle distance $\langle r(t) \rangle$: To quantify the stability of the separation.
  • Loschmidt Echo $L(t)$: To measure the return probability and oscillatory dynamics.
  • Entanglement Entropy $S_1(t)$: Calculated via the von Neumann entropy of the single-particle reduced density matrix.

3. Key Contributions and Results

A. Discovery of a Correlated Dynamical Regime

The primary finding is the emergence of a robust regime where two fermions "walk together," maintaining an approximately constant inter-particle distance.

• Mechanism: This occurs when the interaction strength $\Delta$ is sufficiently strong ($\Delta/J \gg 1$). The quasiperiodic modulation creates an alternating pattern of attractive and repulsive forces as the distance changes by one lattice unit.
  • If particles are at distance $r$ (attractive), they tend to move to $r-1$.
  • At $r-1$ (repulsive), they are pushed back to $r$.
  • This creates an effective energy barrier ($\Delta E_1$) that suppresses deviations from the initial separation.
• Eigenstate Analysis: Exact diagonalization reveals that the system's eigenstates are predominantly composed of configurations with fixed inter-particle distances. The initial state projects heavily onto a single (or pair of) eigenstate(s), leading to long-lived correlation.

B. Three Distinct Manifestations via Phase Tuning

By tuning the phase $\phi$ of the quasiperiodic modulation, the authors identify three distinct dynamical behaviors depending on the initial separation $r_0$:

1. Sustained Localization:

   • Occurs when the interaction energy $U(r_0)$ is close to zero.
   • Particles remain localized at their initial positions (or boundaries) with minimal motion.
   • Localization is stronger when particles start at the boundaries compared to the interior.

2. Nearest-Neighbor Separation Oscillations:

   • Occurs when adjacent distances ($r$ and $r \pm 1$) have opposite interaction signs but similar magnitudes, or when the system is dominated by two eigenstates with a specific energy difference.
   • The inter-particle distance oscillates between two values (e.g., $r$ and $r-1$).
   • Quantitative Finding: The authors established a linear relationship between the angular frequency of the Loschmidt echo oscillation ($\omega$) and the energy difference between the dominant eigenstates ($\Delta E_1$): $\Delta E_1 \approx 1.4656 \omega - 3.2525$ ($R^2 = 0.9622$).

3. Next-Nearest-Neighbor Separation Transitions:

   • A rare phenomenon occurring under specific phase conditions where the interaction energies for distances $r$ and $r+2$ are nearly equal ($U(r) \approx U(r+2)$), while $U(r)$ and $U(r+1)$ differ significantly.
   • Despite the strong barrier preventing $r \to r \pm 1$ transitions, there is a finite probability for the separation to jump by two lattice units ($r \to r \pm 2$).

C. Suppression of Entanglement Entropy

The study reports a significant suppression of entanglement entropy ($S_1$) in this regime.

• In the localization regime, $S_1$ remains low and stable over time.
• In the oscillatory regimes, $S_1$ exhibits oscillatory behavior that correlates with the Loschmidt echo, but the amplitude remains far below the theoretical maximum ($\log_2 L$).
• This indicates that the strong quasiperiodic interactions prevent the system from thermalizing or reaching a maximally mixed state, preserving quantum coherence and correlations.

4. Significance and Implications

• Few-Body Physics: The work extends the understanding of few-body quantum dynamics in long-range systems, demonstrating that quasiperiodic interactions can induce robust, non-ergodic behavior even in small systems.
• Experimental Relevance: The model is proposed to be realizable in current quantum simulators, including Rydberg atom arrays, dipolar systems, trapped ions, and superconducting circuits, where long-range interactions can be engineered.
• Quantum Information: The suppression of entanglement entropy suggests these systems could be useful for preserving quantum information or creating stable correlated states (quasi-particles) in noisy environments.
• Generalizability: While the study focuses on fermions with OBC, the authors suggest the underlying mechanism (energy barriers created by alternating interaction signs) likely applies to infinite chains and hard-core bosons as well.

In conclusion, the paper demonstrates that quasiperiodic long-range interactions act as a powerful control knob to engineer specific dynamical phases in few-body quantum systems, ranging from static localization to coherent oscillations, all characterized by the preservation of inter-particle distance and low entanglement.