⚛️ quantum physics

Phase transitions in first-detection statistics of monitored long-range quantum walks

This paper demonstrates that the first-detection return probability of long-range quantum walks undergoes a continuous phase transition at a critical hopping exponent $\alpha=1$ , separating recurrent and transient behaviors, with nonanalytic decay exponents arising from measurement-induced interference between infrared and ultraviolet energy modes.

Original authors: Sayan Roy, Shamik Gupta, Giovanna Morigi, Gabriele Perfetto

Published 2026-09-10

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CC BY 4.0

Original authors: Sayan Roy, Shamik Gupta, Giovanna Morigi, Gabriele Perfetto

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a traveler moving through a vast, empty city, stepping from one intersection to the next. In the classical world, if this traveler moves randomly, they are guaranteed to eventually return to their starting point, provided the city is not too large or complex. This certainty of return is a fundamental rule of probability that governs everything from the spread of a scent in a room to the movement of stock prices. However, when we enter the quantum realm, where particles behave like waves and can exist in multiple places at once, the rules change. Here, the act of checking to see if the traveler has returned actually changes their path. If you look for them too often, you can freeze their motion entirely, a phenomenon known as the quantum Zeno effect. The question physicists have long asked is how these two forces—the natural tendency to wander and the disruptive act of observation—interact when the traveler is allowed to make giant leaps across the city, skipping over many blocks at once.

A team of researchers has now mapped out exactly how this interaction plays out in a system where a quantum particle hops between sites on a ring, with the ability to jump long distances. They discovered that the likelihood of the particle returning to its starting point depends entirely on how quickly the strength of these long jumps fades as the distance increases. The researchers found a sharp dividing line in this behavior. When the jumps remain strong even over long distances, the particle is effectively trapped near its starting point, and a detector placed there will eventually find it with absolute certainty. But as the ability to jump far weakens, the system undergoes a sudden shift. The particle stops being guaranteed to return and instead begins to wander off, with the probability of finding it at the start dropping over time. This transition happens at a specific threshold where the decay of the jump strength follows a precise mathematical rule, separating a world of guaranteed returns from one of permanent departure.

The researchers simulated this scenario by modeling a single quantum walker on a ring of sites, where the walker could hop to any other site, but the chance of making a long hop decreased according to a specific power law. They monitored the starting site at regular intervals, asking a simple question: has the walker returned yet? If the answer was yes, the process stopped. If the answer was no, the walker continued to evolve until the next check. By varying the rate at which the long-distance hopping strength decayed, they observed a continuous phase transition. On one side of the transition, the total probability of eventually detecting the walker was one, meaning the walker was recurrent and would always come home. On the other side, the total probability dropped below one, meaning the walker was transient and could vanish forever without ever being caught again.

What makes this discovery particularly striking is the behavior of the walker in the transient regime. The researchers found that the probability of detecting the walker at later times does not simply fade away smoothly. Instead, it decays in a complex pattern that changes depending on the specific strength of the long-range hopping. In some regions, the decay follows a simple rule, but in others, it becomes more intricate, with the rate of decay shifting abruptly at certain critical points. These shifts are not caused by the energy levels of the system alone, as one might expect, but by a subtle interference between the very short-range movements and the very long-range jumps. The act of measuring the system forces these two different types of motion to interact in a way that creates new, singular behaviors. This interference is what drives the system through different phases, creating a landscape of dynamics that is far richer than what is seen in standard, short-range quantum walks.

The study also revealed that the timing of the measurements plays a crucial role. If the detector checks the starting site too frequently, the walker can be frozen in place, effectively preventing it from ever leaving. This is a manifestation of the quantum Zeno effect, where constant observation halts evolution. However, the researchers showed that even with this effect, the underlying nature of the long-range hopping dictates the ultimate fate of the walker. By tuning the exponent that controls the decay of the hopping strength, one can control whether the walker is trapped, whether it wanders off, and how quickly the chances of finding it diminish. The results suggest that these different dynamical regimes are not just theoretical curiosities but could be observed in real physical systems, such as arrays of trapped ions or atoms in optical cavities, where the range of interactions can be precisely tuned.

Ultimately, this work provides a clear picture of how measurement and long-range connectivity shape the fate of a quantum particle. It shows that the boundary between returning home and wandering away is not a fixed property of the space the particle moves through, but a dynamic feature that emerges from the interplay between the particle's ability to jump far and the frequency with which it is watched. The researchers have identified the exact conditions under which a quantum walk becomes transient, and they have detailed the complex, non-smooth ways in which the probability of return decays in that state. These findings offer a new understanding of how quantum systems respond to observation, revealing that the simple act of checking for a return can trigger a cascade of changes that fundamentally alter the behavior of the system.

Technical Summary: Phase transitions in first-detection statistics of monitored long-range quantum walks

Problem Statement
The paper investigates the interplay between long-range hopping and projective measurements in one-dimensional quantum walks. While classical random walks and short-range quantum walks have well-established recurrence and transience properties (governed by Pólya's theorem and spectral dimensions, respectively), the behavior of quantum walks with power-law decaying hopping amplitudes ( $d^{-\alpha}$ ) under repeated stroboscopic monitoring remains unexplored. Specifically, the authors address how the long-range exponent $\alpha$ influences the First-Detection Return Probability (FDRP), $F_n$ , and whether the walk is recurrent (returning with unit probability) or transient (returning with probability $<1$ ).

Methodology
The study models a single quantum walker on a ring of $N$ sites with an initial state localized at site $|0\rangle$ . The system evolves under a Hamiltonian $\hat{H}_\alpha$ featuring long-range hopping where the amplitude decays as $d^{-\alpha}$ , normalized by a Kac factor $N_\alpha$ to ensure a well-defined thermodynamic limit. The dynamics are interspersed with projective measurements at stroboscopic times $t = n\tau$ to detect the walker's return to the origin.

The analysis relies on the quantum renewal equation, which links the first-detection return amplitude $\phi_n$ to the measurement-free Loschmidt amplitude $L_n = \langle 0 | e^{-i\hat{H}_\alpha n\tau} | 0 \rangle$ :
$\phi_n = L_n - \sum_{j=1}^{n-1} \phi_j L_{n-j}$
The authors solve this using generating functions (Z-transforms), where $\tilde{\phi}(z) = \tilde{L}(z) / (1 + \tilde{L}(z))$ . The asymptotic behavior of $F_n = |\phi_n|^2$ is determined by the non-analyticities (singularities) of $\tilde{L}(z)$ in the complex plane.

The Loschmidt amplitude is analyzed by examining the energy spectrum $E_k(\alpha)$ in the thermodynamic limit ( $N \to \infty$ ). The authors perform stationary phase approximations and contour integration, carefully treating the infrared ( $k=0$ ) and ultraviolet ( $k=\pi$ ) modes. They distinguish between the strong long-range regime ( $\alpha < 1$ ), where the spectrum is discrete, and the weak long-range regime ( $\alpha > 1$ ), where the spectrum becomes continuous.

Key Contributions and Results

Recurrence-to-Transience Phase Transition at $\alpha = 1$ :
The paper identifies a continuous phase transition in the total detection probability $P_\infty = \sum_{n \ge 1} F_n$ at the critical exponent $\alpha = 1$ .
- Recurrent Phase ( $\alpha \le 1$ ): The total detection probability is unity ( $P_\infty = 1$ ). In this regime, strong long-range hopping induces localization; the energy spectrum remains discrete (or effectively discrete in the limit), protecting the initial state from spreading. The walker is detected with certainty at the first attempt ( $F_n = \delta_{n,1}$ ).
- Transient Phase ( $\alpha > 1$ ): The total detection probability $P_\infty < 1$ . The spectrum becomes continuous, allowing the walker to spread. The system exhibits transient behavior.
Non-Monotonic Algebraic Decay of FDRP:
In the transient regime ( $\alpha > 1$ ), the FDRP decays algebraically as $F_n \sim n^{-\beta}$ . The decay exponent $\beta$ is a non-monotonic function of $\alpha$ , exhibiting non-analytic points at $\alpha = 5/3$ , $\alpha = 2$ , and $\alpha = 3$ .
- Regime $1 < \alpha \le 5/3$ : The decay is dominated by the ultraviolet mode ( $k=\pi$ ), yielding $F_n \sim n^{-3}$ . This matches the behavior of short-range quantum walks.
- Regime $5/3 \le \alpha < 2$ : The infrared mode ( $k=0$ ) dominates, but the interference with the ultraviolet mode modifies the scaling. The decay follows $F_n \sim n^{-2/(\alpha-1)}$ .
- Regime $2 < \alpha < 3$ : The decay is governed by the infrared mode with a modified exponent, $F_n \sim n^{-(4 - 2/(\alpha-1))}$ .
- Regime $\alpha \ge 3$ : The system effectively behaves like a short-range walk, with $F_n \sim n^{-3}$ . At $\alpha = 3$ , logarithmic corrections appear ( $F_n \sim n^{-3} \log n$ ).
Origin of Singularities:
The authors demonstrate that the non-analytic points in the decay exponent $\beta(\alpha)$ arise from the interference between infrared and ultraviolet energy modes induced by the projective measurements. Unlike unitary dynamics where infrared contributions are often subleading, the measurement back-action (via the renewal equation) creates a non-linear relationship where these modes compete. This competition leads to critical behavior intrinsic to the non-unitary dynamics, distinct from the spectral properties of the Hamiltonian alone.
Resonant Probing:
The study notes that for specific resonant probing times $\tau_{res}$ where the branch points of the generating functions coincide, the interference vanishes, and the system universally exhibits the short-range scaling $F_n \sim n^{-3}$ for all $\alpha > 1$ .

Significance and Claims
The paper claims to reveal a measurement-driven reorganization of the effective low-energy description in a single-particle sector. While measurement-induced phase transitions are typically studied in many-body systems via entanglement or purification, this work shows that such phenomena emerge in single-particle quantum walks through the tuning of the long-range exponent $\alpha$ .

The authors assert that the singularities in the FDRP decay are not solely determined by the low-energy spectrum but are a direct consequence of the interplay between coherent long-range spreading and stroboscopic projective measurements. They propose that these distinct dynamical regimes and critical exponents could be experimentally probed in atomic and molecular systems capable of tuning the long-range exponent, such as trapped ions, neutral atoms in photonic modes, and Rydberg atoms. The results provide a framework for understanding search efficiency and transport in monitored quantum networks with long-range interactions.

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