Orthogonalization speed-up from quantum coherence after a sudden quench

This paper introduces a nonequilibrium phenomenon where quantum coherence in an initial superposition state accelerates the orthogonalization of a quantum system following a sudden interaction quench, manifesting as a power-law decay in work distribution and a reduction in the minimal orthogonalization time, with a proposed experimental verification using Ramsey interferometry in trapped cold atoms.

The Gist

The Big Idea: The "Quantum Speed-Up"

Imagine you have a perfectly still pond (this is your quantum system). Suddenly, you drop a heavy stone into it (this is the sudden quench or the defect).

In the classical world, the ripples spread out predictably. But in the quantum world, things get weird. The authors of this paper discovered that if the water in the pond was already "wiggling" in a specific, coordinated way before you dropped the stone, the ripples will spread out and cancel each other out much faster than if the water was just sitting still.

They call this "Coherence-Enhanced Orthogonalization." That's a fancy way of saying: "Quantum memory makes the system forget its past faster."

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The Characters in Our Story

1. The Harmonic Trap (The Pond): Imagine a ball bouncing back and forth in a bowl. This is a particle trapped in a specific energy state.
2. The Defect (The Stone): A sudden, sharp disturbance (like a delta-function spike) that hits the particle.
3. The Superposition (The Coordinated Wiggle): Before the stone hits, the particle isn't just in one state; it's in a "superposition." Think of this as the particle being in many different states of motion at once, all perfectly synchronized (like a choir singing the same note perfectly in tune).
4. The Diagonal State (The Random Wiggle): This is the same particle, but instead of being synchronized, it's just a random mix of states (like a choir where everyone is singing different notes at different times).

The Main Discovery: The "Orthogonality Catastrophe"

In physics, there's a famous concept called Anderson's Orthogonality Catastrophe. It usually happens when you have a huge crowd of particles (like a Fermi sea). If you poke one, the whole crowd reacts so violently that the new state of the crowd is completely unrecognizable compared to the old one. It's like trying to recognize a friend in a crowd after they've all changed their clothes, hair, and posture instantly.

The Twist in this Paper:
Usually, you need a massive crowd of particles to get this "catastrophe" effect. But the authors found that you can get a similar effect with just one single particle, provided that particle is in a coherent superposition (the synchronized choir).

• Without Coherence (The Random Choir): When the stone drops, the particle changes, but it takes a long time to become "unrecognizable" from its starting state.
• With Coherence (The Synchronized Choir): When the stone drops, the particle changes instantly. It becomes "orthogonal" (completely different) much faster. The more states you mix into the superposition, the faster this happens.

The Analogy: The Spinning Top

Imagine a spinning top.

• Scenario A (No Coherence): The top is wobbling randomly. If you poke it, it wobbles a bit more, but it takes a while to fall over completely.
• Scenario B (With Coherence): The top is spinning perfectly upright, but it's also vibrating in a very specific, complex pattern (the superposition). If you poke it, the vibration amplifies the poke. The top doesn't just wobble; it flips over and becomes a completely different shape almost instantly.

The "speed-up" is the time it takes for the top to flip. The paper shows that quantum coherence acts like a lever, making that flip happen in record time.

The "Work" Distribution: The Negative Money

The paper also looks at "Work." In physics, work is energy transferred.

• The Weird Part: They calculated the "quasiprobability" of how much work the defect did. In the quantum world, probabilities can be negative.
• The Metaphor: Imagine a bank account. Usually, you have positive money. But in this quantum scenario, if the system is coherent, the "work" distribution looks like a bank statement with negative numbers.
• Why it matters: These "negative probabilities" are a signature of pure quantum magic. The more coherent the system is, the more "negative money" (non-classical behavior) appears. This negativity is directly linked to why the system forgets its past so quickly.

The Experiment: How to Test This

The authors propose a real-world experiment using ultracold atoms (atoms cooled until they almost stop moving) trapped in laser beams (optical tweezers).

1. Setup: Trap a single atom (or a few) in a laser bowl.
2. Preparation: Put the atom into a "superposition" state using lasers (making it the synchronized choir).
3. The Quench: Suddenly change the trap or introduce a second atom to act as the "defect."
4. Measurement: Use a technique called Ramsey Interferometry (basically a quantum stopwatch) to see how fast the atom's state changes.

They predict that if the atom was prepared with quantum coherence, the "stopwatch" will show the state changing much faster than if it was just a random mix.

Why Should You Care?

This isn't just about abstract physics; it has practical implications for the future of technology:

1. Faster Quantum Computers: If we can control how fast a quantum state changes (orthogonalizes), we might be able to process information faster or detect errors more quickly.
2. Super-Sensitive Sensors: Because the system is so sensitive to the "defect" when it's coherent, we could build sensors that detect tiny changes in the environment (like a single magnetic field or a tiny mass) with incredible precision.
3. Understanding the Quantum World: It bridges the gap between single-particle physics and the complex behavior of huge crowds of particles, showing that "quantumness" (coherence) is a powerful force even in small systems.

Summary

The paper reveals a secret superpower of quantum mechanics: If you prepare a system with the right kind of "quantum coordination" (coherence) before you disturb it, the system will react and change its identity much faster than nature usually allows. It's like having a quantum shortcut that lets a system "forget" its past and become something new in the blink of an eye.

Technical Summary

1. Problem Statement

The paper investigates the non-equilibrium dynamics of a quantum system subjected to a sudden interaction quench with a localized defect. Specifically, it addresses a scenario often overlooked in standard literature: the initial state of the system is not an eigenstate of the pre-quench Hamiltonian, but rather a coherent superposition of energy eigenstates.

The central problem is to characterize how quantum coherence in the initial state influences the system's dynamical response (specifically, the rate at which the state becomes orthogonal to its initial configuration) when perturbed by a localized defect (modeled as a delta-function potential). The authors aim to determine if this non-commutativity between the initial state and the Hamiltonian leads to a "speed-up" in state orthogonalization, analogous to the many-body Anderson Orthogonality Catastrophe (OC), but occurring even in single-particle or few-particle systems.

2. Methodology

The authors employ a combination of analytical derivations, numerical simulations, and statistical mechanics tools:

• Physical Model: A quantum particle (or a few fermions) confined in a 1D harmonic trap. At $t=0$, a localized defect is introduced via a sudden quench of the Hamiltonian: $\hat{H}' = \hat{H} + k\delta(\hat{x})$, where $k$ is the defect strength.
• Initial States:
  • Coherent Superposition: A pure state $|\psi(0)\rangle$ constructed as a superposition of $N$ lowest-energy even eigenstates of the harmonic oscillator, with specific phase modulation ($(-1)^n$) to maximize interference.
  • Incoherent Diagonal State: A mixed state $\hat{\rho}_{diag}(0)$ with the same populations as the superposition but no off-diagonal coherence terms.
• Key Observables:
  • Loschmidt Echo (LE): Defined as $\nu(t) = \text{Tr}[\hat{U}'(t)\hat{\rho}(0)\hat{U}(-t)]$, measuring the overlap between the time-evolved state and the initial state. The decay of $|\nu(t)|$ indicates orthogonalization.
  • Kirkwood-Dirac Quasiprobability (KDQ): The Fourier transform of the LE yields the work distribution $P(w)$, analyzed via its real part (Margenau-Hill quasiprobability) to detect non-classical features (negative probabilities).
  • Quantum Speed Limit (QSL): Calculated using the Bures angle and the average work $\langle w \rangle$ to determine the minimal time $\tau_{QSL}$ required for the state to orthogonalize.
• Numerical Approach: The perturbed Hamiltonian is diagonalized by projecting onto the basis of the unperturbed harmonic oscillator. Due to the singularity of the delta potential, the authors use a spectral method with large basis truncation ($N_b \approx 8000$) and logarithmic arithmetic to handle factorial overflows in the wavefunction overlaps at the origin.

3. Key Contributions

1. Discovery of Coherence-Enhanced Orthogonalization: The paper demonstrates that quantum coherence in the initial state acts as a catalyst for state orthogonalization. While a standard Anderson OC is a many-body effect arising from a Fermi sea, this work shows a similar power-law decay scaling in single-particle and few-fermion systems driven purely by initial-state coherence.
2. Scaling Law Derivation: The authors derive a specific scaling law for the decay of the Loschmidt Echo modulus:
   $$|\nu(t)| \sim 1 - \beta(t) N^{\gamma(t)}$$
   They identify the exponent $\gamma(t)$ as the critical indicator:
   • $\gamma > 0$: Occurs for coherent superpositions, leading to a rapid decay (speed-up) as $N$ increases.
   • $\gamma < 0$: Occurs for incoherent diagonal states, leading to a slowing down of the decay as $N$ increases.
3. Link Between Work Statistics and Orthogonalization: The authors establish a direct link between the non-positivity of the work quasiprobability distribution (negative probabilities) and the reduction of the Quantum Speed Limit. The presence of coherence induces an "infinite discontinuity" in the work distribution, which correlates with the accelerated orthogonalization.
4. Experimental Proposal: A concrete protocol is proposed using ultracold alkaline-earth-like atoms in optical tweezers. The setup utilizes state-selective potentials and Ramsey interferometry to prepare the superposition, induce the quench, and measure the decay.

4. Key Results

• Decay Dynamics: For a coherent superposition of $N$ states, increasing $N$ drastically accelerates the decay of $|\nu(t)|$, driving the system toward orthogonality much faster than in the incoherent case. The orthogonalization time scales roughly as $1/N$.
• Work Distribution:
  • The work distribution for coherent states exhibits negative probabilities (non-positivity), quantified by the functional $N_{Re}$. This negativity grows with $N$.
  • The average work $\langle w \rangle$ done by the defect increases with $N$ for coherent states (scaling as $\sqrt{N}$), whereas it decreases for incoherent states.
  • The variance of the work, $\text{Var}(w)$, is shown to diverge in the limit of strong perturbations due to the ultraviolet behavior of the delta potential.
• Quantum Speed Limit: The minimal time for orthogonalization, $\tau_{QSL} \propto (1 - |\nu(\tau)|)/|\langle w \rangle|$, decreases linearly with $N$ for coherent states, confirming a quantum speed-up. Conversely, for incoherent states, the speed-up is absent or reversed.
• Two-Fermion System: The effect persists and is enhanced in a system of two fermions initialized in an antisymmetric superposition. The Hilbert space dimensionality and fermionic statistics further enrich the interference effects, leading to a deeper "plateau" in the LE decay, mimicking the macroscopic Anderson OC.
• Cusp Singularities: The squared modulus of the LE, $|\nu(t)|^2$, exhibits periodic cusps (singularities in the derivative) at times $t = j\pi/\omega$, corresponding to quantum recurrences.

5. Significance

• Fundamental Physics: The work bridges the gap between single-particle quantum dynamics and many-body phenomena like the Anderson Orthogonality Catastrophe. It proves that "catastrophic" orthogonalization is not exclusive to many-body ground states but can be engineered in small systems via quantum coherence.
• Quantum Thermodynamics: It highlights the role of coherence as a thermodynamic resource. Coherent states allow for greater energy extraction (work) and faster state evolution compared to incoherent mixtures, challenging classical thermodynamic intuitions.
• Quantum Sensing and Control: The sensitivity of the orthogonalization dynamics to the initial coherence suggests a new paradigm for quantum sensing. A probe prepared in a coherent superposition could detect localized defects with higher sensitivity and speed than classical probes.
• Experimental Feasibility: By proposing a realistic implementation with ultracold atoms and Ramsey interferometry, the paper moves these theoretical concepts from abstract models to testable experiments, potentially enabling the study of non-equilibrium quantum thermodynamics in controlled settings.

In summary, the paper establishes that quantum coherence is a driver of dynamical speed-up, transforming the interaction with a local defect from a perturbative event into a rapid orthogonalization process, with distinct signatures in work statistics and quantum speed limits.