Tunable many-body burst in isolated quantum systems

The authors propose a numerical method to engineer low-entangled initial states in isolated quantum systems that generate a tunable, transient "burst" of non-equilibrium behavior, such as magnetization, which persists until quantum scrambling dominates.

The Gist

The Big Idea: The "Time-Traveling" Quantum Trick

Imagine you have a cup of hot coffee in a cold room. Physics tells us that the coffee will cool down, and the heat will spread out until the coffee and the room reach the same temperature. This is thermalization. Once things reach equilibrium, they usually just stay there. You don't expect the coffee to suddenly get hot again on its own.

In the world of quantum physics (the rules governing tiny particles), things are supposed to behave the same way: if you start with a specific state, the system should eventually settle into a messy, random equilibrium and stay there.

This paper discovered a loophole. The researchers found a way to prepare a quantum system so that, at a specific time you choose, it suddenly "bursts" out of equilibrium. It's like putting your coffee in the fridge, waiting 20 minutes, and then—poof—it suddenly boils for a split second before cooling down again.

The Cast of Characters

1. The System: Think of a long line of tiny magnets (spins) linked together. This is the "Mixed-Field Ising Chain." It's a chaotic system, meaning it's very good at scrambling information, like a blender mixing ingredients.
2. The Goal: To make one of these magnets (or the average of all of them) point in a specific direction at a specific time, even though the system is trying to randomize itself.
3. The Problem: Usually, to make a system do something weird like this, you need a "perfect" initial state that is incredibly complex and tangled (highly entangled). Creating such a state in a real lab is nearly impossible. It's like trying to fold a piece of paper into a perfect origami crane with your eyes closed while wearing oven mitts.

The Solution: The "Smart Search" (Method 2)

The researchers didn't just guess a starting state. They used a powerful computer algorithm (called DMRG) to search for a simple starting state that would work.

• The Analogy: Imagine you want to throw a ball so it hits a specific target on a wall at exactly 5 seconds.
  • Old way: You throw the ball randomly until you get lucky.
  • This paper's way: You use a computer to calculate the exact angle and force needed, but with a twist: you force the throw to be simple (low entanglement).
  • The Result: They found a "simple" starting state (like a neatly folded paper) that, when the system runs its course, creates a massive spike (a "burst") in magnetization at the exact time they wanted.

The "Burst" Phenomenon

When they ran the simulation, here is what happened:

1. Start: They set the magnets in a simple, low-entanglement state.
2. Wait: They let the system evolve. Usually, the "messiness" (entanglement) of the system grows linearly, like a ball of yarn getting bigger and bigger.
3. The Burst: At the designated time (say, $t=20$), the magnetization suddenly spiked far away from its normal average.
4. The Surprise: While the burst was happening, the "messiness" (entanglement) didn't grow fast. In fact, it grew slowly or even decreased.
   • Metaphor: It's like a chaotic party where everyone is dancing wildly, but suddenly, for a few seconds, everyone stops dancing and stands in perfect formation, then goes back to dancing.

Why This Matters: The "Scrambling" Race

The paper explains a race between two forces:

1. The Burst: The ability to keep the system organized and out of equilibrium.
2. Scrambling: The system's natural tendency to mix everything up and forget its initial state.

The Finding:

• Short Term: If you pick the right simple starting state, you can win the race for a while. You can keep the system "awake" and out of equilibrium for a surprisingly long time, even in a large system.
• Long Term: If you wait too long, scrambling wins. The paper proves mathematically that if you try to make a burst happen too far in the future, it becomes statistically impossible. The system will just forget how to do the trick.

The "Low-Entanglement" Secret

The most exciting part is that the starting state didn't need to be a super-complex, impossible-to-make quantum monster. It was a Matrix Product State (MPS).

• Analogy: Think of a complex 3D sculpture (high entanglement) vs. a simple stack of flat cards (low entanglement).
• The researchers showed that a simple stack of cards, if arranged just right, can still perform a magic trick (the burst) that you'd expect only a complex sculpture could do.
• Real-world implication: Because the starting state is simple, we can actually build this in a lab using current technology (like quantum computers or simulators). We don't need magic; we just need the right recipe.

Summary in a Nutshell

1. The Myth: Quantum systems always settle down and stay settled.
2. The Discovery: You can trick a chaotic quantum system into having a temporary "burst" of order at a specific time.
3. The Method: Use a computer to find a simple, low-complexity starting state that triggers this burst.
4. The Catch: It only works for a limited time. Eventually, the system's natural chaos (scrambling) takes over and the burst disappears.
5. The Future: This could help us build better quantum sensors or test how well quantum computers are working, because we can predict exactly when and how this "burst" should happen.

In short: The authors found a way to make a chaotic quantum system "hold its breath" and stay out of equilibrium for a while, using a simple starting state that we can actually create in a lab.

Technical Summary

1. Problem Statement

The paper addresses the fundamental question of how irreversible macroscopic thermalization emerges from reversible microscopic quantum laws in isolated many-body systems. While the Eigenstate Thermalization Hypothesis (ETH) guarantees that observables will eventually reach thermal equilibrium in the infinite-time limit for non-integrable systems, it leaves the finite-time dynamics largely unconstrained.

Specifically, the authors investigate the possibility of a "burst": a transient, significant deviation of an observable's expectation value from its thermal equilibrium value.

• The Challenge: Previous theoretical proposals for bursts relied on:
  1. Non-local observables (experimentally difficult).
  2. Quantum recurrence times (exponentially long, $e^{e^L}$).
  3. Time-reversed evolution or tailored superpositions that result in highly entangled initial states, making them physically unrealistic for macroscopic systems and offering little insight into thermalization from low-complexity conditions.
• The Goal: To demonstrate that a burst can be generated at a designated time starting from a low-entangled initial state in a generic, non-integrable system, and to determine the timescales and conditions under which this is possible.

2. Methodology

The authors propose a numerical framework to construct specific Matrix Product States (MPS) that act as initial states to trigger a burst.

A. Numerical Construction (DMRG Optimization)

The core of their method involves using the Density Matrix Renormalization Group (DMRG) algorithm to minimize a custom cost function.

• Target: Find an initial MPS $|\psi_0\rangle$ with a fixed, small bond dimension $\chi$ (independent of system size $L$) such that the expectation value of a local observable $O$ at time $\tau$, $\langle O(\tau) \rangle$, deviates maximally from its thermal equilibrium value $\langle O \rangle_{eq}$.
• Cost Function:
  $$H_{\text{DMRG}} = O(\tau) + \lambda_L (H - \langle H \rangle_\beta)^2$$
  Where:
  • $O(\tau) = e^{iH\tau} O e^{-iH\tau}$ is the Heisenberg-evolved observable.
  • $\lambda_L$ is a penalty weight to suppress energy fluctuations around a target thermal energy $\langle H \rangle_\beta$.
  • $\beta$ is the inverse temperature.
• Two Methods:
  1. Method 1 (Heuristic): Time-reverse evolution from a ground state and truncate. Efficient but lacks control over energy fluctuations.
  2. Method 2 (Optimization): Directly optimizes the cost function using DMRG. This allows precise control over the target energy and yields larger burst amplitudes.

B. Analytical Framework (Local Random Circuits)

To understand the limits of burst creation, the authors employ a probabilistic analysis using local random quantum circuits (approximating time-dependent 2-local Hamiltonians). They utilize the theory of approximate unitary $k$-designs to derive upper bounds on the probability of finding a burst from the manifold of low-entangled states ($M_\chi$).

3. Key Contributions and Results

A. Demonstration of Bursts in Non-Integrable Systems

The authors applied Method 2 to a mixed-field Ising chain (a known non-integrable, chaotic system) with Hamiltonian:
$$H = \sum J_z S_i^z S_{i+1}^z + \sum (h_x S_i^x + h_z S_i^z)$$

• Observables: Average magnetization $M^y$ and $M^z$.
• Result: They successfully generated a large burst in the expectation value of magnetization at a designated time $\tau = 20$ for system sizes up to $L=40$.
• Low Entanglement: The initial states required only a bond dimension $\chi = 10$, which is exponentially smaller than the volume-law entanglement ($\chi \sim 2^{L/2}$) typical of thermal states. This proves the burst is accessible via quantum quenches from ground states or shallow quantum circuits.

B. Anomalous Entanglement Dynamics

A critical finding is the behavior of entanglement entropy ($S_A$) prior to the burst:

• Typical Behavior: In chaotic systems, entanglement usually grows linearly with time (ballistic spreading).
• Burst Behavior: For the constructed initial states, the entanglement entropy shows slow growth or even negative growth (decrease) before the burst time.
• Implication: This indicates that local information is temporarily retained, and the reduced density matrix remains far from the Gibbs state, defying the typical "scrambling" narrative for a specific window of time.

C. Finite-Size Scaling and Thermodynamic Limit

• System Size Independence: For short burst times ($\tau \lesssim 20$), the burst amplitude remains large and even increases with system size $L$, provided the bond dimension $\chi$ is fixed.
• Thermodynamic Limit: Using infinite MPS (iMPS) formalism, they confirmed that the burst survives in the limit $L \to \infty$ for short times, decaying only as quantum scrambling eventually overwhelms the low-complexity state.

D. Analytical Bounds and Time Scales

Using the local random circuit model, the authors derived a probabilistic bound on the existence of bursts:

• Short Time ($\tau \ll L(\ln L)^{11}$): The probability of finding a burst does not decay with system size; bursts are stable.
• Long Time ($\tau \gg L$): The probability of finding a burst from a low-entangled state decays exponentially (or double-exponentially for very long times) with system size.
• Conclusion: A burst is a transient phenomenon. It is possible to engineer a state that stays out of equilibrium for a time comparable to the onset of scrambling, but eventually, the complexity of the eigenstates exceeds the representational capacity of low-bond-dimension MPSs.

4. Significance and Applications

1. Redefining Thermalization Timescales: The work challenges the notion that thermalization is immediate or monotonic. It shows that for "appropriately chosen" low-complexity initial states, a system can maintain a nonequilibrium state (a burst) for a significant duration before scrambling dominates.
2. Experimental Feasibility: Unlike previous theoretical proposals requiring non-local observables or highly entangled states, this method relies on low-entangled MPSs. These can be prepared experimentally using:
   • Quantum quenches from ground states.
   • Shallow quantum circuits (polynomial depth).
   • Programmable quantum simulators (e.g., cold atoms, trapped ions, superconducting qubits).
3. Quantum Metrology: The authors propose that the burst phenomenon offers a new avenue for quantum metrology. Unlike thermal fluctuations which scale as $O(L^{-1/2})$, the burst amplitude of an intensive observable scales as $O(L^0)$ (constant). This high signal-to-noise ratio could be used to benchmark quantum simulators or estimate system parameters with high precision.
4. Variational Quantum Algorithms: The study demonstrates the utility of variational methods (like DMRG) not just for finding ground states, but for engineering specific non-equilibrium trajectories (revivals, oscillations, bursts) by modifying cost functions.

Summary

The paper provides a constructive proof that transient deviations from thermal equilibrium (bursts) can be engineered in generic chaotic quantum systems using low-entangled initial states. By combining numerical optimization (DMRG) with rigorous probabilistic bounds, the authors establish that while quantum scrambling eventually drives systems to equilibrium, there exists a tunable window where macroscopic observables exhibit anomalous dynamics, offering new insights into the interplay between complexity, entanglement, and thermalization.