How compactness curbs entanglement growth in bosonic systems

This paper demonstrates that while non-compact zero modes in bosonic systems cause unbounded logarithmic growth of entanglement entropy, compact zero modes in systems like quantum rotors halt this spreading and dephasing, thereby capping entanglement at a finite value and revealing the necessity of compact descriptions for accurately modeling late-time dynamics in ultracold-atom realizations.

The Gist

The Big Idea: The "Infinite Hallway" vs. The "Roundabout"

Imagine you are watching two friends, Alice and Bob, who are connected by a spring. They are in a room, and you want to know how much they are "entangled" (how much their actions are correlated).

In the world of quantum physics, there is a rule: if you suddenly change the rules of the game (a "quench"), the amount of entanglement between Alice and Bob usually grows.

The Old Theory (The Infinite Hallway):
For a long time, physicists thought that if you removed the walls holding Alice and Bob in place (removing the "confining potential"), their connection would grow forever.

• The Analogy: Imagine Alice and Bob are in an infinite, endless hallway. If you let them go, they start running away from each other. Because the hallway never ends, they can run forever. As they run further and further apart, the "entanglement" (the complexity of their relationship) grows and grows, theoretically becoming infinite.
• The Problem: In math, this looks like a "logarithmic divergence." It means the entanglement keeps getting bigger and bigger without stopping.

The New Discovery (The Roundabout):
This paper argues that the "infinite hallway" is a lie. In reality, the universe is often more like a giant roundabout (a circle).

• The Analogy: Imagine Alice and Bob are on a circular track. If they start running, they will eventually run all the way around the track and bump into each other again. They can't run "forever" in one direction because the track loops back on itself.
• The Result: Because they are stuck on a circle, their running (spreading) eventually stops making them more complex. The entanglement grows for a while, but then it hits a "ceiling" and stops. It saturates.

The Key Characters: The "Zero Mode"

To understand why this happens, we need to talk about a specific character in the story called the Zero Mode.

• What is it? Think of the Zero Mode as the "Center of Mass" of the system. It's the part of the system that isn't being pushed or pulled by any springs. It's the "free rider."
• In the Non-Compact World (Harmonic Oscillators): This free rider is on the infinite hallway. It can drift forever. This causes the entanglement to explode.
• In the Compact World (Quantum Rotors): This free rider is on the roundabout. It can drift, but only until it wraps around the circle. Once it wraps around, it can't get any "more spread out" than that.

The Experiment: Two Scenarios

The authors tested this idea by comparing two very similar setups:

1. The Harmonic Oscillators (The Infinite Hallway):

   • They simulated two particles connected by springs.
   • They removed the springs holding them to the center.
   • Result: The particles spread out forever. The entanglement grew logarithmically (slowly but forever).

2. The Quantum Rotors (The Roundabout):

   • They simulated two particles that are actually angles on a circle (like hands on a clock).
   • They removed the "springs" holding the hands in place.
   • Result: The hands spun freely, but because they are on a clock face, they couldn't go past 12 o'clock and keep going up. They just kept looping.
   • The Twist: The entanglement grew at first (just like the hallway), but once the hands had spun around enough to cover the whole clock face, the growth stopped. It hit a maximum limit.

Why Does This Matter? (The Real-World Connection)

You might ask, "Who cares about two particles on a clock?"

This is huge for Ultra-Cold Atom Experiments.

• Scientists use lasers to trap atoms and make them behave like waves.
• Often, they use a math model called the Klein-Gordon model (the infinite hallway) to predict what happens.
• The Paper's Warning: This model works great for the beginning of the experiment. But if you wait long enough, the atoms will "wrap around" the circle (due to the nature of quantum phases).
• If you keep using the "infinite hallway" math, you will predict that the entanglement keeps growing forever. But in the real lab, it will stop growing.

The "Phase Diffusion" Problem

The paper also discusses a practical headache for scientists trying to measure this.

• The Issue: When scientists measure the position of these atoms, they only see the angle (0 to 360 degrees). They don't see how many times the atom has spun around the circle.
• The Analogy: Imagine watching a runner on a track. If you only look at a camera that shows the track, you see the runner at the 100m mark. You don't know if they just started, or if they've run 10 laps.
• The Consequence: Once the atoms have spread out enough to cover the whole circle, it becomes impossible to tell if they are still spreading or if they have just wrapped around. The "entanglement growth" becomes hidden. The paper suggests that to see the "saturation" (the stopping point), scientists need new ways to measure that can track these "laps" (winding numbers).

Summary in One Sentence

Just because a particle is "free" to move doesn't mean it can run forever; if the space it lives in is a loop (compact), it will eventually run out of room to spread, causing its quantum connections (entanglement) to stop growing and settle at a maximum limit.

The Takeaway for the Future

This paper tells us that geometry matters. The shape of the space where quantum particles live (a line vs. a circle) fundamentally changes how they behave over time. It corrects a long-held assumption that entanglement always grows forever in these systems, showing instead that nature has a built-in "speed limit" for complexity when the space is compact.

Technical Summary

1. Problem Statement

The paper addresses a fundamental discrepancy in the theoretical understanding of entanglement dynamics in bosonic systems following a quantum quench.

• The Paradox: In standard Gaussian (non-compact) models, such as coupled harmonic oscillators or the massless Klein–Gordon field, a global quench that removes the on-site confining potential creates a zero mode (a free-particle degree of freedom). This zero mode possesses a continuous spectrum and an unbounded configuration space ($\mathbb{R}$), leading to unbounded, logarithmic growth of entanglement entropy at late times.
• The Question: In physical realizations, such as ultra-cold atom experiments, the phase field is inherently compact (defined on a circle $S^1$ modulo $2\pi$), described by the sine-Gordon or Tomonaga-Luttinger liquid (TLL) theories. Does the logarithmic divergence predicted by non-compact Gaussian approximations persist indefinitely, or does the compact nature of the field eventually halt this growth?
• The Gap: While the initial state of many experiments is well-described by non-compact Gaussian theories, the long-time dynamics are governed by the compact field theory. The paper investigates whether the compactness of the zero mode acts as a structural bound on entanglement growth, preventing the dynamical divergence seen in Gaussian models.

2. Methodology

The authors employ a multi-scale approach, progressing from minimal models to many-body systems and continuum field theories:

• Minimal Models (Two Sites):
  • Non-compact: Two coupled harmonic oscillators (CHO) with a global frequency quench ($\omega \to 0$). The system is solved analytically using the Lewis-Riesenfeld invariant method and covariance matrix formalism.
  • Compact: Two coupled quantum rotors (CR) with a similar quench. The Hamiltonian involves cosine potentials ($1-\cos x$). The system is solved numerically using momentum-space truncation (Fourier basis) and analytical estimates via the Generalized Gibbs Ensemble (GGE).
• Many-Body Systems (Chains):
  • Comparison of an $N$-site harmonic chain (Neumann boundary conditions) with an $N$-site coupled-rotor chain.
  • The rotor chain is simulated using Tensor Network methods (Matrix Product States/Operators) with the Time-Dependent Variational Principle (TDVP) to handle the infinite local Hilbert space via controlled truncation.
• Continuum Field Theory:
  • Analysis of the ultra-cold-atom experimental setting (tunnel-coupled 1D Bose gases).
  • Derivation of the compactness timescale ($t_c$) by estimating when the variance of the zero-mode phase distribution reaches the scale of the compact domain.
  • Discussion of experimental challenges in reconstructing the "unwrapped" phase (tracking winding numbers) versus the "wrapped" phase (modulo $2\pi$).

3. Key Contributions and Results

A. Mechanism of Entanglement Capping

The central finding is that compactness curbs entanglement growth.

• Non-Compact (CHO): The zero mode spreads unboundedly in position space ($\sigma_x \sim t$) and undergoes indefinite dephasing in momentum space due to its continuous spectrum. This leads to a logarithmic divergence of entanglement entropy: $S(t) \sim \log t$.
• Compact (CR): The zero mode is confined to a bounded domain ($S^1$).
  • Position Space: Spreading is halted once the wave packet wraps around the circle.
  • Momentum Space: The spectrum becomes discrete (integer momenta), preventing indefinite dephasing.
  • Result: The entanglement entropy grows initially (coinciding with the CHO dynamics) but eventually saturates at a finite ceiling ($S_{max}$).

B. Analytical Estimates of the Entropy Ceiling

The authors derive an analytic estimate for the maximum entanglement entropy in the two-rotor model using the Generalized Gibbs Ensemble (GGE).

• The GGE density matrix is constructed based on conserved energies of the decoupled center-of-mass and relative modes, subject to a global parity constraint ($p - q \equiv 0 \mod 2$).
• The estimated ceiling is given by:
  $$S_{1, \text{GGE}} \approx \frac{1}{2} \ln \left[ \frac{\pi e}{2} \left( \omega + \sqrt{\omega^2 + 2\kappa} \right) \right]$$
• Numerical benchmarks show that the GGE (and block-diagonal ensembles) provide a tight upper bound on the actual dynamical maximum, confirming that the saturation is a robust feature of the compact dynamics.

C. Many-Body Scaling

• In $N$-site chains, the harmonic chain exhibits unbounded logarithmic growth.
• The rotor chain exhibits sub-extensive growth followed by saturation. The saturation value $S_{N/2, \text{max}}$ and the time to reach it increase with system size $N$, but the growth remains bounded, unlike the non-compact case.

D. Experimental Relevance and Timescales

• The paper applies these findings to recent ultra-cold-atom experiments (e.g., Ref. [13]) involving a quench from a massive Klein-Gordon state to a massless TLL.
• Compactness Timescale ($t_c$): Using experimental parameters (e.g., $L=49\mu m$, $T=49 nK$), the authors estimate $t_c \approx 12$ ms. This is the time required for the zero-mode variance to spread across the compact domain.
• Observation Challenges:
  • Standard tomographic reconstruction assumes Gaussian (non-compact) statistics and fails at late times.
  • Experimental measurements yield phase modulo $2\pi$. Once the distribution wraps around the circle, "unwrapping" the phase to track the winding number becomes impossible without non-destructive, continuous tracking (which is currently not feasible in these destructive measurement protocols).
  • The paper argues that the saturation of entanglement growth is the "experimental fingerprint" of compactness, but observing it requires reconstruction schemes adapted to the compact phase variable.

4. Significance

• Theoretical Correction: The work corrects the assumption that zero modes always lead to divergent entanglement. It establishes that compactness is the decisive factor: non-compact zero modes diverge, while compact zero modes saturate.
• Experimental Guidance: It provides a concrete prediction for ultra-cold-atom experiments: the entanglement entropy should not grow indefinitely but should saturate. It identifies the timescale ($\sim 10$ ms) where this transition occurs, suggesting that current experiments are already probing the onset of this regime.
• Methodological Advance: It demonstrates the necessity of using compact field theories (TLL) rather than non-compact approximations (Klein-Gordon) for long-time dynamics, even when the initial state is Gaussian.
• Broader Implications: The mechanism (compact configuration space acting as a bound on information spreading) is proposed as a universal constraint applicable to lattice gauge theories, quantum gravity models, and other systems with topological or angular variables.

Conclusion

The paper concludes that while Gaussian non-compact models are excellent for describing early-time dynamics and initial states, they fundamentally fail to capture the late-time behavior of bosonic systems with compact degrees of freedom. The compactness of the zero mode acts as a "structural bound," halting the unbounded spreading and dephasing that drive logarithmic entanglement growth, thereby ensuring the system reaches a finite, saturated entanglement entropy. This insight bridges the gap between idealized Gaussian theories and realistic compact quantum field theories in experimental settings.