arXiv:1802.05504 · Physical Review A

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

Felix Thiel, David A. Kessler, Eli Barkai

Published Jun 6, 2018• 18 citations

Abstract

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.

DOI: 10.1103/physreva.97.062105

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Quantum random walk and tight-binding model subject to projective measurements at random times

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Dark states of quantum search cause imperfect detection

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First-detection time of a quantum state under random probing

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arXiv:2012.01763Physical Review AFeb 25, 202123 citations

arXiv:1802.05504 · Physical Review A

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

Felix Thiel, David A. Kessler, Eli Barkai

Published Jun 6, 2018• 18 citations

arXiv HTML PDF

Abstract

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}

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.

DOI: 10.1103/physreva.97.062105

Related papers

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

Sayan Roy, Shamik Gupta, Giovanna Morigi +1 more

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.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.

quant-pharXiv:2609.10510Sep 10, 20260 citations

Quantum walks: The first detected passage time problem

Harry Friedman, David A. Kessler, Eli Barkai

Even after decades of research, the problem of first passage time statistics for quantum dynamics remains a challenging topic of fundamental and practical importance. Using a projective measurement approach, with a sampling time τ, we obtain the statistics of first detection events for quantum dynamics on a lattice, with the detector located at the origin. A quantum renewal equation for a first detection wave function, in terms of which the first detection probability can be calculated, is derived. This formula gives the relation between first detection statistics and the solution of the corresponding Schrödinger equation in the absence of measurement. We illustrate our results with tight-binding quantum walk models. We examine a closed system, i.e., a ring, and reveal the intricate influence of the sampling time τ on the statistics of detection, discussing the quantum Zeno effect, half dark states, revivals, and optimal detection. The initial condition modifies the statistics of a quantum walk on a finite ring in surprising ways. In some cases, the average detection time is independent of the sampling time while in others the average exhibits multiple divergences as the sampling time is modified. For an unbounded one-dimensional quantum walk, the probability of first detection decays like (time)^{(-3)} with superimposed oscillations, with exceptional behavior when the sampling period τ times the tunneling rate γ is a multiple of π/2. The amplitude of the power-law decay is suppressed as τ→0 due to the Zeno effect. Our work, an extended version of our previously published paper, predicts rich physical behaviors compared with classical Brownian motion, for which the first passage probability density decays monotonically like (time)^{-3/2}, as elucidated by Schrödinger in 1915.Even after decades of research, the problem of first passage time statistics for quantum dynamics remains a challenging topic of fundamental and practical importance. Using a projective measurement approach, with a sampling time τ, we obtain the statistics of first detection events for quantum dynamics on a lattice, with the detector located at the origin. A quantum renewal equation for a first detection wave function, in terms of which the first detection probability can be calculated, is derived. This formula gives the relation between first detection statistics and the solution of the corresponding Schrödinger equation in the absence of measurement. We illustrate our results with tight-binding quantum walk models. We examine a closed system, i.e., a ring, and reveal the intricate influence of the sampling time τ on the statistics of detection, discussing the quantum Zeno effect, half dark states, revivals, and optimal detection. The initial condition modifies the statistics of a quantum walk on a finite ring in surprising ways. In some cases, the average detection time is independent of the sampling time while in others the average exhibits multiple divergences as the sampling time is modified. For an unbounded one-dimensional quantum walk, the probability of first detection decays like (time)^{(-3)} with superimposed oscillations, with exceptional behavior when the sampling period τ times the tunneling rate γ is a multiple of π/2. The amplitude of the power-law decay is suppressed as τ→0 due to the Zeno effect. Our work, an extended version of our previously published paper, predicts rich physical behaviors compared with classical Brownian motion, for which the first passage probability density decays monotonically like (time)^{-3/2}, as elucidated by Schrödinger in 1915.

arXiv:1611.05676Physical review. EMar 28, 201788 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.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.

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Quantum random walk and tight-binding model subject to projective measurements at random times

Debraj Das, Shamik Gupta

Abstract What happens when a quantum system undergoing unitary evolution in time is subject to repeated projective measurements to the initial state at random times? A question of general interest is: how does the survival probability S m , namely, the probability that an initial state survives even after m number of measurements, behave as a function of m ? We address these issues in the context of two paradigmatic quantum systems, one, the quantum random walk evolving in discrete time, and the other, the tight-binding model evolving in continuous time, with both defined on a one-dimensional periodic lattice with a finite number of sites N . For these two models, we present several numerical and analytical results that hint at the curious nature of quantum measurement dynamics. In particular, we unveil that when evolution after every projective measurement continues with the projected component of the instantaneous state, the average and the typical survival probability decay as an exponential in m for large m . By contrast, if the evolution continues with the leftover component, namely, what remains of the instantaneous state after a measurement has been performed, the survival probability exhibits two characteristic m values, namely, m 1 ⋆ ( N ) ∼ N and m 2 ⋆ ( N ) ∼ N δ with δ > 1. These scales are such that (i) for m large and satisfying m < m 1 ⋆ ( N ) , the decay of the survival probability is as m −2 , (ii) for m satisfying m 1 ⋆ ( N ) ≪ m < m 2 ⋆ ( N ) , the decay is as m −3/2 , while (iii) for m ≫ m 2 ⋆ ( N ) , the decay is as an exponential. The results for the dynamics with the leftover component, already known for the case of measurements at regular intervals, are being extended here to the case of measurements at random intervals. We find that our results hold independently of the choice of the distribution of times between successive measurements, as have been corroborated by our resuAbstract What happens when a quantum system undergoing unitary evolution in time is subject to repeated projective measurements to the initial state at random times? A question of general interest is: how does the survival probability S m , namely, the probability that an initial state survives even after m number of measurements, behave as a function of m ? We address these issues in the context of two paradigmatic quantum systems, one, the quantum random walk evolving in discrete time, and the other, the tight-binding model evolving in continuous time, with both defined on a one-dimensional periodic lattice with a finite number of sites N . For these two models, we present several numerical and analytical results that hint at the curious nature of quantum measurement dynamics. In particular, we unveil that when evolution after every projective measurement continues with the projected component of the instantaneous state, the average and the typical survival probability decay as an exponential in m for large m . By contrast, if the evolution continues with the leftover component, namely, what remains of the instantaneous state after a measurement has been performed, the survival probability exhibits two characteristic m values, namely, m 1 ⋆ ( N ) ∼ N and m 2 ⋆ ( N ) ∼ N δ with δ > 1. These scales are such that (i) for m large and satisfying m < m 1 ⋆ ( N ) , the decay of the survival probability is as m −2 , (ii) for m satisfying m 1 ⋆ ( N ) ≪ m < m 2 ⋆ ( N ) , the decay is as m −3/2 , while (iii) for m ≫ m 2 ⋆ ( N ) , the decay is as an exponential. The results for the dynamics with the leftover component, already known for the case of measurements at regular intervals, are being extended here to the case of measurements at random intervals. We find that our results hold independently of the choice of the distribution of times between successive measurements, as have been corroborated by our resu

arXiv:2105.00338Journal of Statistical Mechanics Theory and ExperimentMar 1, 202218 citations

Dark states of quantum search cause imperfect detection

Felix Thiel, Itay Mualem, Dror Meidan +2 more

We consider a quantum walk where a detector repeatedly probes the system with fixed rate 1/τ until the walker is detected.This is a quantum version of the first-passage problem.We focus on the total probability P det that the particle is eventually detected in some target state, for example, on a node r d on a graph, after an arbitrary number of detection attempts.Analyzing the dark and bright states for finite graphs and more generally for systems with a discrete spectrum, we provide an explicit formula for P det in terms of the energy eigenstates which is generically τ independent.We find that disorder in the underlying Hamiltonian renders perfect detection, P det = 1, and then expose the role of symmetry with respect to suboptimal detection.Specifically, we give a simple upper bound for P det that is controlled by the number of equivalent (with respect to the detection) states in the system.We also extend our results to infinite systems, for example, the detection probability of a quantum walk on a line, which is τ dependent and less than half, well below Polya's optimal detection for a classical random walk.We consider a quantum walk where a detector repeatedly probes the system with fixed rate 1/τ until the walker is detected.This is a quantum version of the first-passage problem.We focus on the total probability P det that the particle is eventually detected in some target state, for example, on a node r d on a graph, after an arbitrary number of detection attempts.Analyzing the dark and bright states for finite graphs and more generally for systems with a discrete spectrum, we provide an explicit formula for P det in terms of the energy eigenstates which is generically τ independent.We find that disorder in the underlying Hamiltonian renders perfect detection, P det = 1, and then expose the role of symmetry with respect to suboptimal detection.Specifically, we give a simple upper bound for P det that is controlled by the number of equivalent (with respect to the detection) states in the system.We also extend our results to infinite systems, for example, the detection probability of a quantum walk on a line, which is τ dependent and less than half, well below Polya's optimal detection for a classical random walk.

arXiv:1906.08112Physical Review ResearchOct 20, 202033 citations

First-detection time of a quantum state under random probing

David A. Kessler, Eli Barkai, K. Ziegler

We solve for the statistics of the first detection of a quantum system in a particular desired state, when the system is subject to a projective measurement at independent identically distributed random time intervals. We present formulas for the probability of detection in the $n\mathrm{th}$ attempt. We calculate as well the mean and mean square, both of the number of the first successful detection attempt and the time until first detection. We present explicit results for a particle initially localized at a site on a ring of size $L$ , probed at some arbitrary given site, in the case when the detection intervals are distributed exponentially. We prove that, for all interval distributions and finite-dimensional Hamiltonians, the mean detection time is equal to the mean attempt number times the mean time interval between attempts. We further prove that for the return problem when the initial and target state are identical, the total detection probability is unity and the mean attempts until detection is an integer, which is the size of the Hilbert space (symmetrized about the target state). We study an interpolation between the fixed time interval case to an exponential distribution of time intervals via the Gamma distribution with constant mean and varying width. The mean arrival time as a function of the mean interval changes qualitatively as we tune the interarrival time distribution from very narrow ($\ensuremath{\delta}$ peaked) to exponential, as resonances are wiped out by the randomness of the sampling.We solve for the statistics of the first detection of a quantum system in a particular desired state, when the system is subject to a projective measurement at independent identically distributed random time intervals. We present formulas for the probability of detection in the $n\mathrm{th}$ attempt. We calculate as well the mean and mean square, both of the number of the first successful detection attempt and the time until first detection. We present explicit results for a particle initially localized at a site on a ring of size $L$, probed at some arbitrary given site, in the case when the detection intervals are distributed exponentially. We prove that, for all interval distributions and finite-dimensional Hamiltonians, the mean detection time is equal to the mean attempt number times the mean time interval between attempts. We further prove that for the return problem when the initial and target state are identical, the total detection probability is unity and the mean attempts until detection is an integer, which is the size of the Hilbert space (symmetrized about the target state). We study an interpolation between the fixed time interval case to an exponential distribution of time intervals via the Gamma distribution with constant mean and varying width. The mean arrival time as a function of the mean interval changes qualitatively as we tune the interarrival time distribution from very narrow ($\ensuremath{\delta}$ peaked) to exponential, as resonances are wiped out by the randomness of the sampling.

arXiv:2012.01763Physical Review AFeb 25, 202123 citations

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}

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}

d

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.

n\mathrm{th}

attempt. We calculate as well the mean and mean square, both of the number of the first successful detection attempt and the time until first detection. We present explicit results for a particle initially localized at a site on a ring of size

L

, probed at some arbitrary given site, in the case when the detection intervals are distributed exponentially. We prove that, for all interval distributions and finite-dimensional Hamiltonians, the mean detection time is equal to the mean attempt number times the mean time interval between attempts. We further prove that for the return problem when the initial and target state are identical, the total detection probability is unity and the mean attempts until detection is an integer, which is the size of the Hilbert space (symmetrized about the target state). We study an interpolation between the fixed time interval case to an exponential distribution of time intervals via the Gamma distribution with constant mean and varying width. The mean arrival time as a function of the mean interval changes qualitatively as we tune the interarrival time distribution from very narrow ($\ensuremath{\delta}$ peaked) to exponential, as resonances are wiped out by the randomness of the sampling.