Towers of quantum many-body scars under stochastic resetting

This paper demonstrates that interspersing scarred quantum dynamics with stochastic resets to simple product states enables the experimental preparation of local properties of highly entangled many-body scar eigenstates, yielding a stationary state that is locally equivalent to a single pure scarred eigenstate in the sparse-resetting limit.

The Gist

Imagine a crowded dance floor where everyone is trying to move to a chaotic, unpredictable beat. In the world of quantum physics, this is what usually happens: if you start a system of particles in a specific pattern, they quickly get confused, mix up their energy, and eventually settle into a boring, "thermal" state where all the original information is lost. This is the rule of the road, known as thermalization.

However, some special dance floors have a few "ghosts" in the machine. These are called Quantum Many-Body Scars. Unlike the chaotic crowd, these specific quantum states remember their original moves perfectly. They don't get confused; instead, they keep dancing in a perfect, repeating loop forever, refusing to settle down.

The problem? These "ghosts" are incredibly hard to catch. They are like high-wire acts made of pure, tangled energy. To create them in a lab, you usually need to perform a perfect, delicate maneuver that is almost impossible to do without the system falling apart.

Enter the "Reset Button."

This paper proposes a clever, counter-intuitive trick to catch these ghosts: Stochastic Resetting.

The Analogy: The Forgetful Gardener

Imagine you are trying to grow a very specific, rare flower (the Scar) in a garden. The flower has a habit of growing wild and chaotic if left alone.

• The Old Way: You try to guide the plant perfectly every second. One tiny mistake, and it grows into a weed.
• The New Way (This Paper): You let the plant grow for a while. Then, randomly, you gently cut it back to a simple, un-tangled seedling (a "product state") and let it start growing again.

You do this at random intervals. Sometimes you cut it back after 1 second, sometimes after 10.

What Happens?

1. Damping the Chaos: If you reset too often, the plant never gets a chance to grow (this is the "Quantum Zeno effect"—too much watching stops the action). If you never reset, it grows wild and chaotic.
2. The Sweet Spot: If you reset just right, something magical happens. The plant doesn't just grow wild, nor does it stay a seed. It settles into a steady state that looks and acts like the rare, perfect flower you wanted, even though you only ever planted simple seeds.

The Key Discoveries

The authors of this paper used math to prove that this "resetting" trick works for quantum systems in two surprising ways:

• It Creates Order from Chaos: Even though you are constantly resetting the system to a simple, unentangled state, the average behavior of the system over time develops a deep, complex structure. It's like taking a pile of scattered Lego bricks, randomly snapping them together, and finding that, on average, they form a perfect castle.
• The "Local" Magic: The most important part is that you don't need to build the whole complex castle at once. You only need to prepare the simple seeds. The resetting process naturally "stitches" them together into the complex, entangled state locally.
• The "Logarithmic" Secret: In physics, complex states usually have "entanglement" (a measure of how connected the particles are) that grows huge as the system gets bigger. But these scars are special: their entanglement grows very slowly (logarithmically). The paper shows that the "reset" method creates a state that mimics this slow growth perfectly.

Why This Matters

Think of it like trying to record a perfect symphony.

• The Hard Way: You need a perfect orchestra, a perfect hall, and a perfect conductor, all at once. If one violinist sneezes, the recording is ruined.
• The Reset Way: You record a single violinist playing a simple note. Then, you stop, reset, and record them again. You do this thousands of times. By mixing all these simple recordings together, you accidentally create a perfect, complex symphony that sounds like the whole orchestra played at once.

The Bottom Line:
This paper shows that we don't need to be perfect quantum engineers to create these exotic, high-energy states. We just need to be a bit "forgetful" and hit the reset button at random times. By doing so, we can trick nature into creating complex, entangled quantum states using only simple, easy-to-prepare starting materials. This could be a game-changer for building future quantum computers and simulators.

Technical Summary

1. Problem Statement

Quantum many-body scars (QMBS) are atypical, highly excited eigenstates in non-integrable Hamiltonians that violate the Eigenstate Thermalization Hypothesis (ETH). They exhibit slow entanglement growth (logarithmic rather than volume law) and persistent oscillations (revivals) in time. However, preparing these states experimentally is challenging because they are highly entangled and typically require precise initialization or post-selection of measurement outcomes, which introduces severe overheads.

The central problem addressed is: Can the local properties of these complex, entangled scarred states be prepared and stabilized using a simpler, experimentally feasible protocol that avoids post-selection?

The authors investigate the interplay between stochastic resetting (reinitializing the system to a simple state at random times) and the unitary dynamics of Hamiltonians hosting towers of quantum scars.

2. Methodology

The authors employ a theoretical framework combining quantum many-body physics with stochastic processes.

• Models: They analyze two paradigmatic models known to host towers of scars:

  1. Deformed Fermi-Hubbard Model: Featuring $\eta$-pairing states (doublon condensates).
  2. Spin-1 XY Chain: Featuring bimagnon excitations.
     Both models possess a "restricted spectrum-generating algebra" where a quasiparticle creation operator $\hat{Q}^\dagger$ generates equally spaced eigenstates $|\psi_n\rangle \propto (\hat{Q}^\dagger)^n |\psi_0\rangle$ with energy spacing $\omega$.

• Resetting Protocol:

  • The system evolves unitarily under the scarred Hamiltonian $\hat{H}$.
  • At random times drawn from an exponential distribution (rate $r$), the system is reset to a specific coherent-like state $|\alpha\rangle$.
  • Crucially, $|\alpha\rangle$ is chosen to be a product state (unentangled in real space) but entirely contained within the scarred subspace. It is defined as a superposition of scar eigenstates: $|\alpha\rangle \propto \exp(\alpha \hat{Q}^\dagger)|\emptyset\rangle$.

• Mathematical Tools:

  • Renewal Theory: The density matrix $\hat{\rho}_r(t)$ is derived using the renewal equation, accounting for trajectories with and without reset events.
  • Analytical Derivations: Exact solutions are obtained for the time evolution of fidelity, local observables, correlation functions, and Rényi entropies.
  • Asymptotic Analysis: Saddle-point methods are used to analyze the thermodynamic limit ($L \to \infty$) and the sparse-resetting limit ($r \to 0^+$).

3. Key Contributions

1. Experimental Route to Scarred States: The paper proposes a protocol to prepare the local properties of highly entangled scarred eigenstates using only simple product states and stochastic resets, eliminating the need for global post-selection.
2. Generation of Off-Diagonal Long-Range Order (ODLRO): The authors demonstrate that stochastic resetting, which typically damps oscillations, paradoxically generates spatial ODLRO and quasiparticle condensation in the Non-Equilibrium Stationary State (NESS), a feature usually absent in thermal states.
3. Logarithmic Mixedness Scaling: They prove that the mixedness (entropy) of the stationary state scales logarithmically with system size ($S \sim \frac{1}{2}\log L$). This is a direct signature of the underlying scarred structure, distinguishing it from thermal states (volume law) and pure states (zero entropy).
4. Local Equivalence to Pure States: In the limit of sparse resetting, the mixed NESS is shown to be locally indistinguishable from a single pure scarred eigenstate, uniquely determined by the parameter $\alpha$ of the reset state.

4. Key Results

A. Dynamics and Fidelity

• Damping of Revivals: In unitary dynamics, scarred states show perfect revivals (fidelity returns to 1 periodically). Stochastic resetting introduces damping, causing the fidelity to decay to a stationary value.
• Zeno Regime: In the limit of frequent resets ($r \to \infty$), the system is frozen in the initial state (Quantum Zeno effect).
• Sparse Resetting: In the limit $r \to 0$, the system behaves like a diagonal ensemble over the scarred subspace.

B. Observables and Correlations

• Local Observables: Diagonal operators in the scar subspace remain constant. Off-diagonal operators (e.g., quasiparticle creation) exhibit damped oscillations, settling to a stationary value determined by $r$ and $\omega$.
• Spatial ODLRO: While the initial product state $|\alpha\rangle$ has no long-range order, the stationary state induced by resetting exhibits non-vanishing connected correlations for the quasiparticle operator $\hat{Q}$:
  $$\lim_{L\to\infty} \frac{1}{L^2} \langle \hat{Q}\hat{Q}^\dagger \rangle_c \neq 0$$
  This indicates the formation of a condensate of quasiparticles with momentum $\pi$, a hallmark of scarred physics.
• Temporal Correlations: Conversely, resetting destroys the time-crystalline order (long-range temporal correlations) typical of scars, as reset events erase the memory of past dynamics.

C. Entropy and Mixedness

• Logarithmic Scaling: The Rényi-2 entropy of the stationary state scales as $S_2 \sim \frac{1}{2} \log L$.
• Physical Interpretation: This scaling matches the entanglement entropy of a single scarred eigenstate. The authors show that the NESS density matrix has the same structure as the reduced density matrix of a scar eigenstate. Thus, the classical noise of resetting effectively "simulates" the entanglement structure of the quantum scars.

D. Local Equivalence (Weak Resetting Limit)

• In the limit $r \to 0$ and $L \to \infty$, the diagonal ensemble concentrates around a single scar eigenstate $|\psi_{n^*}\rangle$, where $n^* \approx L \frac{|\alpha|^2}{1+|\alpha|^2}$.
• For any local observable $\hat{O}$, the expectation value in the NESS equals that of the pure state $|\psi_{n^*}\rangle$:
  $$\lim_{L\to\infty} \text{Tr}[\hat{O}\hat{\rho}_{NESS}] = \langle \psi_{n^*} | \hat{O} | \psi_{n^*} \rangle$$
• Limitation: This equivalence holds only for local observables. For non-local operators (like the fidelity projector), the equivalence breaks down due to the exponential dependence on system size.

5. Significance

• Experimental Feasibility: The protocol offers a viable path to engineer correlated quantum states on current quantum hardware. It requires only the preparation of simple product states and the implementation of stochastic resets (which can be simulated or realized via measurement-feedback loops), avoiding the "overhead" of post-selection.
• New Phase of Matter: It identifies a new class of Non-Equilibrium Stationary States (NESS) that inherit the "athermal" properties of quantum scars (low entanglement, ODLRO) despite being generated by a dissipative, stochastic process.
• Theoretical Insight: The work bridges the gap between stochastic resetting (a classical concept) and quantum many-body scars, revealing that classical noise can be used to "harvest" the specific structural properties of quantum scars without needing to isolate the pure quantum eigenstates directly.

In summary, the paper demonstrates that stochastic resetting acts as a filter, suppressing thermalization and time-crystalline order while selectively amplifying the spatial correlations and low-entanglement structure characteristic of quantum many-body scars, providing a robust method for their experimental preparation.