Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2609.10510

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

Sayan Roy, Shamik Gupta, Giovanna Morigi, Gabriele Perfetto

quant-phcond-mat.othercond-mat.stat-mechSubmitted Sep 9, 2026• Announced Sep 10, 2026• 0 citations
arXivHTMLPDF

Abstract

In a quantum walk, the first-detection return probability (FDRP) characterizes salient features, determining whether the quantum walk is transient or recurrent. We study the FDRP of quantum walks on a chain where the initial site is stroboscopically monitored by a detector and the walker performs long-range hopping between sites. We assume that the hopping strength decays with the distance $d$ as $d^{-α}$ and $α\geq 0$ and show that the power-law exponent $α$ critically determines the behavior of the FDRP. The value $α=1$ separates recurrent ($α<1$) from transient ($α>1$) quantum walks through a continuous phase transition in the total detection probability. For $α<1$, strong long-range hopping induces localization, resulting in unit total detection probability. Instead, for $α>1$ the long-range walk is transient and the return probability decays algebraically as a function of time as $t^{-β}$. The associated decay exponent $β$ features nonanalytic points as a function of $α$. Such singularities are not exclusively determined by the low-energy spectrum, but are caused by the interference between infrared and ultraviolet energy modes induced by projective measurements, signalling the emergence of critical behavior intrinsic to the non-unitary dynamics. These dynamics are solely controlled by tuning the long-range exponent $α$ and can thus be experimentally probed in atomic and molecular systems.

Comments
27 pages (7 main text + 20 supplemental material), 10 figures (3 figures + 7 supplemental figures)

Related papers

Spectral dimension controlling the decay of the quantum first-detection probability

Felix Thiel, David A. Kessler, Eli Barkai

We consider a quantum system that is initially localized at ${\mathbit{x}}_{\text{in}}$ and that is repeatedly projectively probed with a fixed period $\ensuremath{\tau}$ at position ${\mathbit{x}}_{\mathrm{d}}$. We ask for the probability ${F}_{n}$ that the system is detected at ${\mathbit{x}}_{\mathrm{d}}$ for the very first time, where $n$ is the number of detection attempts. We relate the asymptotic decay and oscillations of ${F}_{n}$ with the system's energy spectrum, which is assumed to be absolutely continuous. In particular, ${F}_{n}$ is determined by the Hamiltonian's measurement spectral density of states (MSDOS) $f(E)$ that is closely related to the density of energy states (DOS). We find that ${F}_{n}$ decays like a power law whose exponent is determined by the power-law exponent ${d}_{S}$ of $f(E)$ around its singularities ${E}^{*}$. Our findings are analogous to the classical first passage theory of random walks. In contrast to the classical case, the decay of ${F}_{n}$ is accompanied by oscillations with frequencies that are determined by the singularities ${E}^{*}$. This gives rise to critical detection periods ${\ensuremath{\tau}}_{c}$ at which the oscillations disappear. In the ordinary case ${d}_{S}$ can be identified with the spectral dimension associated with the DOS. Furthermore, the singularities ${E}^{*}$ are the van Hove singularities of the DOS in this case. We find that the asymptotic statistics of ${F}_{n}$ depend crucially on the initial and detection state and can be wildly different for out-of-the-ordinary states, which is in sharp contrast to the classical theory. The properties of the first-detection probabilities can alternatively be derived from the transition amplitudes. All our results are confirmed by numerical simulations of the tight-binding model, and of a free particle in continuous space both with a normal and with an anomalous dispersion relation. We provide explicit asymptotic formulas for the first-detection probability in these models.

arXiv:1802.05504•Physical Review A•Jun 6, 2018•18 citations

Running measurement protocol for the quantum first-detection problem

Dror Meidan, Eli Barkai, David A Kessler

Abstract The problem of the detection statistics of a quantum walker has received increasing interest. We investigate the effect of employing a moving detector, using a projective measurement approach with fixed sampling time , with the detector moving right before every detection attempt. For a tight-binding quantum walk on the line, the moving detector allows one to target a specific range of group velocities of the walker, qualitatively modifying the behavior of the quantum first-detection probabilities. We map the problem to that of a stationary detector with a modified unitary evolution operator and use established methods for the solution of that problem to study the first-detection statistics for a moving detector on a finite ring and on an infinite 1D lattice. On the line, the system exhibits a dynamical phase transition at a critical value of , from a state where the probability of detection decreases exponentially in time and the total detection probability is very small, to a state with power-law decay and a significantly higher total probability to detect the particle. The exponent describing the power-law decay of the detection probability at this critical is 10/3, as opposed to 3 for every larger . In addition, the moving detector strongly modifies the Zeno effect.

arXiv:1903.05907•Journal of Physics A Mathematical and Theoretical•Jul 17, 2019•9 citations

First Detected Arrival of a Quantum Walker on an Infinite Line

Felix Thiel, Eli Barkai, David A. Kessler

The first detection of a quantum particle on a graph is shown to depend sensitively on the distance ξ between the detector and initial location of the particle, and on the sampling time τ. Here, we use the recently introduced quantum renewal equation to investigate the statistics of first detection on an infinite line, using a tight-binding lattice Hamiltonian with nearest-neighbor hops. Universal features of the first detection probability are uncovered and simple limiting cases are analyzed. These include the large ξ limit, the small τ limit, and the power law decay with the attempt number of the detection probability over which quantum oscillations are superimposed. For large ξ the first detection probability assumes a scaling form and when the sampling time is equal to the inverse of the energy band width nonanalytical behaviors arise, accompanied by a transition in the statistics. The maximum total detection probability is found to occur for τ close to this transition point. When the initial location of the particle is far from the detection node we find that the total detection probability attains a finite value that is distance independent.

arXiv:1706.05168•Physical Review Letters•Jan 24, 2018•73 citations

Dynamical Quantum Phase Transitions in Spin Chains with Long-Range Interactions: Merging Different Concepts of Nonequilibrium Criticality

Bojan Žunkovič, Markus Heyl, Michael Knap +1 more

We theoretically study the dynamics of a transverse-field Ising chain with power-law decaying interactions characterized by an exponent α, which can be experimentally realized in ion traps. We focus on two classes of emergent dynamical critical phenomena following a quantum quench from a ferromagnetic initial state: The first one manifests in the time-averaged order parameter, which vanishes at a critical transverse field. We argue that such a transition occurs only for long-range interactions α≤2. The second class corresponds to the emergence of time-periodic singularities in the return probability to the ground-state manifold which is obtained for all values of α and agrees with the order parameter transition for α≤2. We characterize how the two classes of nonequilibrium criticality correspond to each other and give a physical interpretation based on the symmetry of the time-evolved quantum states.

arXiv:1609.08482•Physical Review Letters•Mar 27, 2018•275 citations

Entanglement transitions and multifractality in monitored free-fermions with random long-range hopping

Bo Fan, Antonio M. García-García

We study the entanglement dynamics of a one-dimensional chain of monitored non-interacting complex fermions with random power-law hopping characterized by a decay exponent $α$. For $α\lesssim 1$, in stark contrast with the case of hopping to nearest neighbors, the scaling of the entanglement entropy (EE) of the steady state with system size $L$ is faster than logarithmic for any monitoring or disorder strength and it tends towards a linear (volume-law) scaling for sufficiently small $α\lesssim 1/2$. For $α> 3/2$, the EE is in the area-law phase, namely, no scaling with $L$, for any monitoring strength. For $1 < α\lesssim 3/2$, we identify an $α$-dependent measurement-induced phase transition (MIPT) at a critical value of the monitoring strength separating the mentioned area-law and sub-volume-law phases. At this critical point, the EE scales logarithmically with system size, and the density-density correlation function, closely related to the EE, exhibits multifractal features. These results highlight the importance of superdiffusive classical hopping in the entanglement dynamic of quantum many-body systems and also help differentiate its role with respect to conventional sources of entanglement such as genuine quantum non-locality.

quant-pharXiv:2607.23871•Jul 28, 2026•0 citations

Quantum first detection of a quantum walker on a perturbed ring

Yajing Wang, Ruoyu Yin, Ling-Yu Dou +2 more

The problem of quantum first detection time has been extensively investigated in recent years. Employing the stroboscopic measurement protocol, we consider such a monitored quantum walk on sequentially or periodically perturbed rings and focus on the statistics of first detected return time, namely, the time it takes a particle to return to the initial state for the first time. Using time-independent perturbation theory, we obtain the general form of the eigenvalues and eigenvectors of the Hamiltonian. For the case of a sequentially perturbed ring system, we find steplike behaviors of ${\ensuremath{\sum}}_{n=1}^{\mathcal{N}}n{F}_{n}$ ($\ensuremath{\rightarrow}\ensuremath{\langle}n\ensuremath{\rangle}$ as $\mathcal{N}\ensuremath{\rightarrow}\ensuremath{\infty}$) when $\mathcal{N}$ increases, with two plateaus corresponding to integers, where ${F}_{n}$ is the first detected return probability at the $n\mathrm{th}$ detection attempt. Meanwhile, if the initial condition preserves the reflection symmetry, the mean return time is the same as the unperturbed system. For the periodically perturbed system, similar results can also appear in the case where the symmetry is preserved; however, the size of the ring, the interval between adjacent perturbations, and the initial position may change the mean return time in most cases. In addition, we find that the decay rate of the first detection probability ${F}_{n}$ decreases with the increase in perturbation amplitude. More profoundly, the symmetry-preserving setup of the initial conditions leads to the coincidence of ${F}_{n}$. The symmetry of the physical systems under investigation is deeply reflected in the quantum detection time statistics.

Physical Review Research•Mar 24, 2023•6 citations
In a quantum walk, the first-detection return probability (FDRP) characterizes salient features, determining whether the quantum walk is transient or recurrent. We study the FDRP of quantum walks on a chain where the initial site is stroboscopically monitored by a detector and the walker performs long-range hopping between sites. We assume that the hopping strength decays with the distance ddd as d−αd^{-α}d−α and α≥0α\geq 0α≥0 and show that the power-law exponent ααα critically determines the behavior of the FDRP. The value α=1α=1α=1 separates recurrent (α<1α<1α<1) from transient (α>1α>1α>1) quantum walks through a continuous phase transition in the total detection probability. For α<1α<1α<1, strong long-range hopping induces localization, resulting in unit total detection probability. Instead, for α>1α>1α>1 the long-range walk is transient and the return probability decays algebraically as a function of time as t−βt^{-β}t−β. The associated decay exponent βββ features nonanalytic points as a function of ααα. Such singularities are not exclusively determined by the low-energy spectrum, but are caused by the interference between infrared and ultraviolet energy modes induced by projective measurements, signalling the emergence of critical behavior intrinsic to the non-unitary dynamics. These dynamics are solely controlled by tuning the long-range exponent ααα and can thus be experimentally probed in atomic and molecular systems.
We consider a quantum system that is initially localized at ${\mathbit{x}}_{\text{in}}$ and that is repeatedly projectively probed with a fixed period $\ensuremath{\tau}$ at position ${\mathbit{x}}_{\mathrm{d}}$. We ask for the probability Fn{F}_{n}Fn​ that the system is detected at ${\mathbit{x}}_{\mathrm{d}}$ for the very first time, where nnn is the number of detection attempts. We relate the asymptotic decay and oscillations of Fn{F}_{n}Fn​ with the system's energy spectrum, which is assumed to be absolutely continuous. In particular, Fn{F}_{n}Fn​ is determined by the Hamiltonian's measurement spectral density of states (MSDOS) f(E)f(E)f(E) that is closely related to the density of energy states (DOS). We find that Fn{F}_{n}Fn​ decays like a power law whose exponent is determined by the power-law exponent dS{d}_{S}dS​ of f(E)f(E)f(E) around its singularities E∗{E}^{*}E∗. Our findings are analogous to the classical first passage theory of random walks. In contrast to the classical case, the decay of Fn{F}_{n}Fn​ is accompanied by oscillations with frequencies that are determined by the singularities E∗{E}^{*}E∗. This gives rise to critical detection periods ${\ensuremath{\tau}}_{c}$ at which the oscillations disappear. In the ordinary case dS{d}_{S}dS​ can be identified with the spectral dimension associated with the DOS. Furthermore, the singularities E∗{E}^{*}E∗ are the van Hove singularities of the DOS in this case. We find that the asymptotic statistics of Fn{F}_{n}Fn​ depend crucially on the initial and detection state and can be wildly different for out-of-the-ordinary states, which is in sharp contrast to the classical theory. The properties of the first-detection probabilities can alternatively be derived from the transition amplitudes. All our results are confirmed by numerical simulations of the tight-binding model, and of a free particle in continuous space both with a normal and with an anomalous dispersion relation. We provide explicit asymptotic formulas for the first-detection probability in these models.
Abstract The problem of the detection statistics of a quantum walker has received increasing interest. We investigate the effect of employing a moving detector, using a projective measurement approach with fixed sampling time , with the detector moving right before every detection attempt. For a tight-binding quantum walk on the line, the moving detector allows one to target a specific range of group velocities of the walker, qualitatively modifying the behavior of the quantum first-detection probabilities. We map the problem to that of a stationary detector with a modified unitary evolution operator and use established methods for the solution of that problem to study the first-detection statistics for a moving detector on a finite ring and on an infinite 1D lattice. On the line, the system exhibits a dynamical phase transition at a critical value of , from a state where the probability of detection decreases exponentially in time and the total detection probability is very small, to a state with power-law decay and a significantly higher total probability to detect the particle. The exponent describing the power-law decay of the detection probability at this critical is 10/3, as opposed to 3 for every larger . In addition, the moving detector strongly modifies the Zeno effect.
The first detection of a quantum particle on a graph is shown to depend sensitively on the distance ξ between the detector and initial location of the particle, and on the sampling time τ. Here, we use the recently introduced quantum renewal equation to investigate the statistics of first detection on an infinite line, using a tight-binding lattice Hamiltonian with nearest-neighbor hops. Universal features of the first detection probability are uncovered and simple limiting cases are analyzed. These include the large ξ limit, the small τ limit, and the power law decay with the attempt number of the detection probability over which quantum oscillations are superimposed. For large ξ the first detection probability assumes a scaling form and when the sampling time is equal to the inverse of the energy band width nonanalytical behaviors arise, accompanied by a transition in the statistics. The maximum total detection probability is found to occur for τ close to this transition point. When the initial location of the particle is far from the detection node we find that the total detection probability attains a finite value that is distance independent.
We theoretically study the dynamics of a transverse-field Ising chain with power-law decaying interactions characterized by an exponent α, which can be experimentally realized in ion traps. We focus on two classes of emergent dynamical critical phenomena following a quantum quench from a ferromagnetic initial state: The first one manifests in the time-averaged order parameter, which vanishes at a critical transverse field. We argue that such a transition occurs only for long-range interactions α≤2. The second class corresponds to the emergence of time-periodic singularities in the return probability to the ground-state manifold which is obtained for all values of α and agrees with the order parameter transition for α≤2. We characterize how the two classes of nonequilibrium criticality correspond to each other and give a physical interpretation based on the symmetry of the time-evolved quantum states.
We study the entanglement dynamics of a one-dimensional chain of monitored non-interacting complex fermions with random power-law hopping characterized by a decay exponent ααα. For α≲1α\lesssim 1α≲1, in stark contrast with the case of hopping to nearest neighbors, the scaling of the entanglement entropy (EE) of the steady state with system size LLL is faster than logarithmic for any monitoring or disorder strength and it tends towards a linear (volume-law) scaling for sufficiently small α≲1/2α\lesssim 1/2α≲1/2. For α>3/2α> 3/2α>3/2, the EE is in the area-law phase, namely, no scaling with LLL, for any monitoring strength. For 1<α≲3/21 < α\lesssim 3/21<α≲3/2, we identify an ααα-dependent measurement-induced phase transition (MIPT) at a critical value of the monitoring strength separating the mentioned area-law and sub-volume-law phases. At this critical point, the EE scales logarithmically with system size, and the density-density correlation function, closely related to the EE, exhibits multifractal features. These results highlight the importance of superdiffusive classical hopping in the entanglement dynamic of quantum many-body systems and also help differentiate its role with respect to conventional sources of entanglement such as genuine quantum non-locality.
The problem of quantum first detection time has been extensively investigated in recent years. Employing the stroboscopic measurement protocol, we consider such a monitored quantum walk on sequentially or periodically perturbed rings and focus on the statistics of first detected return time, namely, the time it takes a particle to return to the initial state for the first time. Using time-independent perturbation theory, we obtain the general form of the eigenvalues and eigenvectors of the Hamiltonian. For the case of a sequentially perturbed ring system, we find steplike behaviors of ${\ensuremath{\sum}}_{n=1}^{\mathcal{N}}n{F}_{n}$ ($\ensuremath{\rightarrow}\ensuremath{\langle}n\ensuremath{\rangle}$ as $\mathcal{N}\ensuremath{\rightarrow}\ensuremath{\infty}$) when N\mathcal{N}N increases, with two plateaus corresponding to integers, where Fn{F}_{n}Fn​ is the first detected return probability at the nthn\mathrm{th}nth detection attempt. Meanwhile, if the initial condition preserves the reflection symmetry, the mean return time is the same as the unperturbed system. For the periodically perturbed system, similar results can also appear in the case where the symmetry is preserved; however, the size of the ring, the interval between adjacent perturbations, and the initial position may change the mean return time in most cases. In addition, we find that the decay rate of the first detection probability Fn{F}_{n}Fn​ decreases with the increase in perturbation amplitude. More profoundly, the symmetry-preserving setup of the initial conditions leads to the coincidence of Fn{F}_{n}Fn​. The symmetry of the physical systems under investigation is deeply reflected in the quantum detection time statistics.