Dynamics of Loschmidt echoes from operator growth in noisy quantum many-body systems

This paper investigates the dynamics of Loschmidt echoes in noisy quantum many-body systems by establishing an equivalence to dissipative operator norms, proposing a universal two-regime decay model (Gaussian for weak noise and exponential for strong noise) linked to operator growth, and rigorously validating these findings using a solvable chaotic circuit.

The Gist

Imagine you are trying to remember a secret recipe you wrote down on a piece of paper. You write it down, then you try to "undo" the writing process by erasing the ink in reverse order, hoping to get back to a blank sheet of paper.

In the world of quantum physics, this "recipe" is a piece of quantum information, and the "erasing" is called a Loschmidt Echo. It's a test to see how well a system can reverse time and return to its original state.

However, in the real world, nothing is perfect. There is always noise—like a gust of wind blowing dust on your paper, or a shaky hand smudging the ink. This paper by Takato Yoshimura and Lucas S´a asks a big question: What happens to our ability to reverse time when the system is messy and noisy?

Here is the story of their discovery, broken down into simple concepts.

1. The Setup: The "Butterfly" and the "Smudge"

In a perfect, quiet quantum world (a "closed system"), if you push a butterfly (a tiny piece of information), it spreads out through the whole forest (the system) in a predictable way. If you could reverse time perfectly, the butterfly would fly back to your hand, and the forest would return to normal. This is the Loschmidt Echo.

But in the real world, the forest is windy and chaotic. The "noise" acts like a smudge that randomly changes the ink on your paper. When you try to reverse time, the smudge doesn't go away; it gets worse. The paper never returns to being blank. The "echo" (the signal that you successfully reversed time) fades away.

2. The Two Phases of Fading

The authors discovered that this fading doesn't happen in just one way. It depends on how much time has passed and how strong the noise is. They found two distinct "modes" of decay:

Phase A: The "Gaussian" Blur (The Slow Start)

• When: This happens when the noise is weak or the time is short (specifically, when Time × Noise Strength is small).
• The Analogy: Imagine dropping a single drop of ink into a glass of water. At first, the ink spreads out slowly and smoothly. The "fuzziness" grows gradually.
• What it means: In this phase, the quantum information is still growing and spreading through the system (like the ink spreading), but the noise hasn't had enough time to destroy it completely yet. The "echo" fades in a smooth, curved way (mathematically called Gaussian decay).

Phase B: The "Exponential" Crash (The Sudden Drop)

• When: This happens when the noise is strong or a lot of time has passed (when Time × Noise Strength is large).
• The Analogy: Now imagine you are in a room where the lights are flickering violently. No matter how hard you try to remember the recipe, the flickering makes it impossible to focus. The information doesn't just blur; it gets shredded.
• What it means: Once the noise has had enough time to act, the system enters a new regime. The information is being destroyed at a constant, rapid rate. The "echo" crashes down in a straight line on a graph (mathematically called Exponential decay).

The Big Surprise: The authors found that the "switch" between these two phases isn't a fixed point in time. It depends on the product of time and noise. If the noise is very weak, you can wait a long time before the "crash" happens. If the noise is strong, the crash happens almost immediately.

3. The "Growing String" Metaphor

To understand why this happens, the authors used a clever idea called Operator Growth.

Imagine the quantum information isn't a static dot, but a growing string.

• In a perfect world, this string grows longer and longer, weaving through the entire system.

• In a noisy world, the noise acts like scissors that randomly snip the string.

• Early on: The string is short. The scissors haven't cut much yet. The string is still growing, and the "echo" fades slowly because the string is just getting bigger.

• Later on: The string has grown very long. Now, the scissors are cutting it constantly. The string can't grow anymore; it just gets chopped up. The "echo" fades rapidly because the string is being destroyed faster than it can grow.

4. The "Magic" Proof

The authors didn't just guess this; they proved it using a specific, mathematically perfect model called the Dissipative Random Phase Model (DRPM). Think of this model as a "toy universe" where they can run the experiment millions of times on a computer to see exactly what happens.

They showed that their "growing string" theory works perfectly in this toy universe. They proved that:

1. The "echo" always fades.
2. It fades slowly at first (Gaussian).
3. It fades fast later (Exponential).
4. The point where it switches depends entirely on how much noise there is.

Why Does This Matter?

This isn't just about abstract math. It's crucial for the future of Quantum Computers.

• The Problem: Quantum computers are incredibly fragile. Noise (heat, radiation, interference) destroys their calculations.
• The Insight: This paper tells us how that destruction happens. It tells us that if we want to keep quantum information safe, we need to act before the "Exponential Crash" phase kicks in.
• The Takeaway: It gives scientists a new rulebook for understanding how long quantum information can survive in a noisy environment. It suggests that even in a chaotic, noisy world, there are universal patterns to how information dies.

In a nutshell: The paper explains that when you try to reverse time in a noisy quantum world, the signal doesn't just fade away evenly. It starts with a slow, smooth blur, and then suddenly hits a wall and crashes. The moment it hits that wall depends on how "noisy" your world is.

Technical Summary

1. Problem Statement

The paper addresses the challenge of understanding quantum information flow and operator growth in noisy (open) quantum many-body systems, specifically those without conservation laws.

• Context: In closed chaotic systems, local operators grow linearly in time (butterfly effect), spreading information across the system. However, environmental noise causes decoherence, leading to qualitatively different dynamics in open systems.
• Gap: While the Loschmidt echo (LE) is a standard measure of sensitivity to perturbations and quantum chaos in closed systems, its behavior in noisy, dissipative many-body systems is less understood. Specifically, the interplay between unitary operator growth and bulk dissipation, and how this affects the decay of the LE, requires a unified theoretical framework.
• Goal: To characterize the dynamics of the operator Loschmidt echo in noisy Floquet systems, establish the relationship between operator growth, Out-of-Time-Order Correlators (OTOCs), and dissipation, and provide exact analytical results for a solvable model.

2. Methodology

The authors employ a combination of heuristic physical arguments and rigorous analytical proofs using a specific solvable model.

• Model Setup:
  • System: One-dimensional locally interacting Floquet circuits on a spin lattice of size $L$.
  • Dynamics: A forward evolution $U$ and a backward evolution $U'$ (imperfect time reversal). The system evolves via a unitary gate $W$ followed by random noise $V_\tau$ at each step.
  • Noise Model: A depolarizing channel where, with probability $p$, a random local basis operator is applied, and with probability $1-p$, the identity is applied.
• Key Transformation:
  • The authors demonstrate that the noise-averaged operator Loschmidt echo is mathematically equivalent to the operator norm of the operator evolved under the corresponding dissipative (non-unitary) Floquet dynamics.
  • $E_V[M_O(t)] = \langle O(t)O(t) \rangle$, where the average is over noise realizations.
• Analytical Tool:
  • Dissipative Random Phase Model (DRPM): An exactly solvable chaotic many-body circuit. The authors utilize diagrammatic techniques and transfer matrix methods in the large-$q$ limit (where $q$ is the local Hilbert space dimension).
  • They map the calculation of the operator norm to the spectral properties of a $(2t+1)$-dimensional transfer matrix modified by a diagonal dissipation matrix.

3. Key Contributions

1. Equivalence Proof: Established that the noise-averaged operator Loschmidt echo equals the operator norm under dissipative dynamics, linking LE directly to open-system operator growth.
2. Universal Dynamical Regimes: Identified two distinct dynamical regimes for the decay of the Loschmidt echo based on the dimensionless parameter $pt$ (noise strength $\times$ time):
   • Weak Dissipation ($pt \ll 1$): Gaussian decay.
   • Strong Dissipation ($pt \gg 1$): Exponential decay with a noise-independent rate.
3. Operator Size Saturation: Demonstrated that in open systems, the average operator size saturates to a finite, non-zero value at late times, contrasting with previous findings that suggested unbounded growth or different saturation behaviors depending on the model.
4. Exact Solvability: Provided a rigorous proof of the heuristic picture using the DRPM, deriving exact closed-form expressions for the operator norm and average operator size.

4. Detailed Results

A. Heuristic Picture (Generic Floquet Systems)

The authors propose that the dynamics are governed by a competition between unitary operator growth (driven by butterfly velocity $v_B$) and bulk dissipation.

• Regime 1: $pt \ll 1$ (Early Time/Weak Noise)

  • Dissipation is too weak to significantly alter the operator growth trajectory.
  • The operator size $\Sigma(t)$ grows linearly: $\Sigma(t) \approx 2v_B t$.
  • The Loschmidt echo (operator norm) decays as a Gaussian (or stretched exponential) because the survival probability is the product of survival probabilities for an operator string of growing length:
    $$\langle O(t)O(t) \rangle \sim (1-p)^{2 \sum \Sigma(t')} \approx \exp\left[ -2p v_B t^2 \right]$$
  • Note: The decay rate depends on time, leading to a quadratic exponent.

• Regime 2: $pt \gg 1$ (Late Time/Strong Noise)

  • Dissipation dominates. The operator size saturates to a constant value $\Sigma_{sat}$.
  • The growth of the operator is controlled by single-site physics; any extension beyond the saturation size is immediately dissipated.
  • The Loschmidt echo decays exponentially with a rate determined by the single-site decay and the intrinsic chaotic decay rate $\lambda$:
    $$\langle O(t)O(t) \rangle \sim ((1-p)e^{-\lambda})^{2t}$$
  • Crucially, the decay rate becomes independent of the noise strength $p$ in the exponent's leading order (though the prefactor depends on $p$), as the system is dominated by the intrinsic chaotic relaxation.

B. Exact Results for the DRPM

Using the large-$q$ limit of the Dissipative Random Phase Model, the authors derived exact formulas:

• Operator Norm:
  $$N(t) = (\omega \rho^2)^{t-1} \left[ (1 - \rho^{-1}; \omega)_{t-1} \right]^2$$
  where $\omega = (1-p)^2$, $\rho = e^{-2\varepsilon}$ (related to interaction strength), and $(a;q)_n$ is the $q$-Pochhammer symbol.
• Verification of Regimes:
  • For $t \ll 1/p$, the exact result matches the Gaussian decay prediction.
  • For $t \gg 1/p$, the $q$-Pochhammer symbol converges to a constant, confirming the exponential decay regime.
• Operator Size:
  • The average operator size $\Sigma(t)$ is calculated exactly and shown to saturate to a finite value, validating the heuristic assumption that dissipation halts operator growth at late times.

5. Significance

• Theoretical Insight: The paper resolves a long-standing ambiguity regarding the functional form of Loschmidt echo decay in noisy systems. It corrects the assumption that the transition between decay regimes is time-independent, showing instead that the critical timescale is dynamically determined by $t \sim 1/p$.
• Open System Chaos: It provides a rigorous framework for "dissipative quantum chaos," linking operator growth, OTOCs, and noise in a unified manner.
• Experimental Relevance: Since Loschmidt echoes and OTOCs are measurable in quantum simulators (e.g., cold atoms, superconducting qubits), these predictions offer specific targets for experimental verification of how noise alters quantum information scrambling.
• Methodological Advance: The successful application of large-$q$ diagrammatic techniques to open systems (DRPM) opens the door for exact calculations in other dissipative quantum many-body models.

In summary, the paper establishes that noise does not merely add a constant decay rate to quantum chaos; it fundamentally alters the time-dependence of the decay, creating a crossover from Gaussian to exponential behavior as the system evolves, driven by the saturation of operator growth due to dissipation.