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

Physical Review A

Spectrum of an oscillator making a random walk in frequency

Robert M. Mazo, Christian Van den Broeck

Published Sep 1, 1986• 24 citations
Publisher

Abstract

The spectrum of a spectroscopically active particle which makes a random walk on a finite one-dimensional lattice, upon each of whose n sites the particle radiates or absorbs radiation with a different frequency, is discussed. The correlation function of the radiating dipole is found to decay asymptotically as a Gaussian times ${t}^{\mathrm{\ensuremath{-}}(n\mathrm{\ensuremath{-}}1)/2}$ in the case when the site frequencies are independent Gaussian random variables. Numerical simulations of the random walk confirm the asymptotic result. The difference between these results for a finite system and previous results for an infinite system are discussed.

DOI
10.1103/physreva.34.2364

Related papers

Dephasing by a continuous-time random walk process

Daniel M. Packwood, Yoshitaka Tanimura

Stochastic treatments of magnetic resonance spectroscopy and optical spectroscopy require evaluations of functions such as (exp(i ∫(0)(t) Q(s)ds)), where t is time, Q(s) is the value of a stochastic process at time s, and the angular brackets denote ensemble averaging. This paper gives an exact evaluation of these functions for the case where Q is a continuous-time random walk process. The continuous-time random walk describes an environment that undergoes slow steplike changes in time. It also has a well-defined Gaussian limit and so allows for non-Gaussian and Gaussian stochastic dynamics to be studied within a single framework. We apply the results to extract qubit-lattice interaction parameters from dephasing data of P-doped Si semiconductors (data collected elsewhere) and to calculate the two-dimensional spectrum of a three-level harmonic oscillator undergoing random frequency modulations.

arXiv:1205.0296•Physical Review E•Jul 26, 2012•4 citations

Spectral shifts produced by scattering on some model random media

Daniel F. V. James, E. Wolf

The effect of the statistical properties of fluctuating light sources on the spectrum of the emitted radiation has been the subject of intense research over the last few years. It was first shown that due to spatial correlations, the spectrum of light was not, in general, invariant on propagation, even in free space.1 It was also shown that these effects can lead to shifts of spectral lines, a result which is of interest particularly for astrophysics.2 In view of the intimate connection between radiation from localized sources and scattering from media of finite extent, similar spectral changes may also be produced by scattering. In this paper a model scatterer is introduced, whose dielectric response function is a random function of space and time and whose space-time correlation function is a Gaussian function in all its variables. Using the first-order Born approximation, such scattering media are shown to generate frequency shifts of spectral lines that imitate the Doppler effect in its main features. The shifts may be arbitrarily large. Some numerical simulations of such shifts are presented and compared with Doppler shifts.

Annual Meeting Optical Society of America•Jan 1, 1990•0 citations

Universal Regimes in Long-Time Asymptotic of Multilevel Quantum System Under Time-Dependent Perturbation

V. M. Akulin

In the framework of an exactly soluble model, one considers a typical problem of the interaction between radiation and matter: the dynamics of population in a multilevel quantum system subject to a time dependent perturbation. The algebraic structure of the model is taken richly enough, such that there exists a strong argument in favor of the fact that the behavior of the system in the asymptotic of long time has a universal character, which is system-independent and governed by the functional property of the time dependence exclusively. Functional properties of the excitation time dependence, resulting in the regimes of resonant excitation, random walks, and dynamic localization, are identified. Moreover, an intermediate regime between the random walks and the localization is identified for the polyharmonic excitation at frequencies given by the Liouville numbers.

Entropy•Jan 12, 2021•0 citations

The quantum Lévy walk

Manuel O Cáceres, Marco Nizama

We introduce the quantum Lévy walk to study transport and decoherence in a quantum random model. We have derived from second-order perturbation theory the quantum master equation for a Lévy-like particle that moves along a lattice through scale-free hopping while interacting with a thermal bath of oscillators. The general evolution of the quantum Lévy particle has been solved for different preparations of the system. We examine the evolution of the quantum purity, the localized correlation and the probability to be in a lattice site, all of them leading to important conclusions concerning quantum irreversibility and decoherence features. We prove that the quantum thermal mean-square displacement is finite under a constraint that is different when compared to the classical Weierstrass random walk. We prove that when the mean-square displacement is infinite the density of state has a complex null-set inside the Brillouin zone. We show the existence of a critical behavior in the continuous eigenenergy which is related to its non-differentiability and self-affine characteristics. In general, our approach allows us to study analytically quantum fluctuations and decoherence in a long-range hopping model.

arXiv:1112.3881•Journal of Physics A Mathematical and Theoretical•Oct 25, 2010•17 citations

Asymptotic behavior of decoherent and interacting quantum walks

André Ahlbrecht

In this thesis we study the propagation behavior of quantum particles which perform discrete-time quantum walks on spatial lattices.By definition, quantum walks are discrete-time evolutions such that a single time step couples only neighboring lattice sites.Additionally, we assume in this thesis that the dynamical law commutes with all translations of the spatial lattice, that is we consider translationally invariant quantum walks.The results presented in the following are separated into two parts: The first part is devoted to the study of single-particle quantum walks, and in the second part we investigate quantum walks of several particles.Single-particle quantum walks: Our aim is to study the asymptotic behavior of particles undergoing general translationally invariant quantum walks.Employing perturbation theory, we establish a direct method for computing the asymptotic distribution of the particle's position rescaled by an appropriate time-dependent factor.If the time evolution is implemented by a unitary operator commuting with all lattice translations and with empty pure point spectrum, we find linear scaling of the asymptotic position distribution.In this case, the asymptotic distribution of position divided by time is determined by the analog of the group velocity.If, on the other hand, the quantum walk is subject to decoherence, we expect that it behaves like a classical random walk.We provide explicit criteria under which this intuitive statement is true, that is, the asymptotic scaling of the particle's position distribution is that of a diffusion process.We find that the asymptotic distribution of position divided by the square root of time is a mixture of Gaussians in this case.However, we also provide examples of decoherent quantum walks which violate these criteria, and exhibit linear scaling of the asymptotic position distribution. Many-particle quantum walks:We investigate unitary quantum walks of two particles with collisional interactions.That is, the particles perform independent single-particle quantum walks if they are at distinct lattice sites, and an interaction takes place if both particles occupy the same lattice site.We prove that such quantum walks generically exhibit a molecular binding effect between the particles.More precisely, for a measurement of the particles' positions we have a bound on the probability to obtain a certain result which decays exponentially with the particles' distance.As a preparatory step for this analysis, we consider single-particle quantum walks with point defects, and provide a method to compute their eigenvalues in the spectral gap of the quantum walk without defect.We further generalize our model to arbitrary numbers of particles and study the behavior of local excitations in many-particle quantum walks.

•Jan 1, 2013•1 citation

The quantum Lévy walk

Manuel O. Cáceres, Marco Alfredo Nizama Mendoza

We introduce the quantum Levy walk to study transport and decoherence in a quantum random model. We have derived from second order perturbation theory the quantum master equation for a extit{Levy-like particle }that moves along a lattice through hopping scale-free while interacting with a thermal bath of oscillators. The general evolution of the quantum Levy particle has been solved for different preparations of the system. We examine the evolution of the quantum purity, the localized correlation, and the probability to be in a lattice site, all them leading to important conclusions concerning quantum irreversibility and decoherence features. We prove that the quantum thermal mean-square displacement is finite under a constraint that is different when compared to the classical Weierstrass random walk. We prove that when the mean-square displacement is infinite the density of state has a complex null-set inside the Brillouin zone. We show the existence of a critical behavior in the continuous eigenenergy which is related to its non-differentiability and self-affine characteristics. In general our approach allows to study analytically quantum fluctuations and decoherence in a long-range hopping model.

•Oct 1, 2010•0 citations
The spectrum of a spectroscopically active particle which makes a random walk on a finite one-dimensional lattice, upon each of whose n sites the particle radiates or absorbs radiation with a different frequency, is discussed. The correlation function of the radiating dipole is found to decay asymptotically as a Gaussian times t\ensuremath−(n\ensuremath−1)/2{t}^{\mathrm{\ensuremath{-}}(n\mathrm{\ensuremath{-}}1)/2}t\ensuremath−(n\ensuremath−1)/2 in the case when the site frequencies are independent Gaussian random variables. Numerical simulations of the random walk confirm the asymptotic result. The difference between these results for a finite system and previous results for an infinite system are discussed.
Stochastic treatments of magnetic resonance spectroscopy and optical spectroscopy require evaluations of functions such as (exp(i ∫(0)(t) Q(s)ds)), where t is time, Q(s) is the value of a stochastic process at time s, and the angular brackets denote ensemble averaging. This paper gives an exact evaluation of these functions for the case where Q is a continuous-time random walk process. The continuous-time random walk describes an environment that undergoes slow steplike changes in time. It also has a well-defined Gaussian limit and so allows for non-Gaussian and Gaussian stochastic dynamics to be studied within a single framework. We apply the results to extract qubit-lattice interaction parameters from dephasing data of P-doped Si semiconductors (data collected elsewhere) and to calculate the two-dimensional spectrum of a three-level harmonic oscillator undergoing random frequency modulations.
The effect of the statistical properties of fluctuating light sources on the spectrum of the emitted radiation has been the subject of intense research over the last few years. It was first shown that due to spatial correlations, the spectrum of light was not, in general, invariant on propagation, even in free space.1 It was also shown that these effects can lead to shifts of spectral lines, a result which is of interest particularly for astrophysics.2 In view of the intimate connection between radiation from localized sources and scattering from media of finite extent, similar spectral changes may also be produced by scattering. In this paper a model scatterer is introduced, whose dielectric response function is a random function of space and time and whose space-time correlation function is a Gaussian function in all its variables. Using the first-order Born approximation, such scattering media are shown to generate frequency shifts of spectral lines that imitate the Doppler effect in its main features. The shifts may be arbitrarily large. Some numerical simulations of such shifts are presented and compared with Doppler shifts.
In the framework of an exactly soluble model, one considers a typical problem of the interaction between radiation and matter: the dynamics of population in a multilevel quantum system subject to a time dependent perturbation. The algebraic structure of the model is taken richly enough, such that there exists a strong argument in favor of the fact that the behavior of the system in the asymptotic of long time has a universal character, which is system-independent and governed by the functional property of the time dependence exclusively. Functional properties of the excitation time dependence, resulting in the regimes of resonant excitation, random walks, and dynamic localization, are identified. Moreover, an intermediate regime between the random walks and the localization is identified for the polyharmonic excitation at frequencies given by the Liouville numbers.
We introduce the quantum Lévy walk to study transport and decoherence in a quantum random model. We have derived from second-order perturbation theory the quantum master equation for a Lévy-like particle that moves along a lattice through scale-free hopping while interacting with a thermal bath of oscillators. The general evolution of the quantum Lévy particle has been solved for different preparations of the system. We examine the evolution of the quantum purity, the localized correlation and the probability to be in a lattice site, all of them leading to important conclusions concerning quantum irreversibility and decoherence features. We prove that the quantum thermal mean-square displacement is finite under a constraint that is different when compared to the classical Weierstrass random walk. We prove that when the mean-square displacement is infinite the density of state has a complex null-set inside the Brillouin zone. We show the existence of a critical behavior in the continuous eigenenergy which is related to its non-differentiability and self-affine characteristics. In general, our approach allows us to study analytically quantum fluctuations and decoherence in a long-range hopping model.
In this thesis we study the propagation behavior of quantum particles which perform discrete-time quantum walks on spatial lattices.By definition, quantum walks are discrete-time evolutions such that a single time step couples only neighboring lattice sites.Additionally, we assume in this thesis that the dynamical law commutes with all translations of the spatial lattice, that is we consider translationally invariant quantum walks.The results presented in the following are separated into two parts: The first part is devoted to the study of single-particle quantum walks, and in the second part we investigate quantum walks of several particles.Single-particle quantum walks: Our aim is to study the asymptotic behavior of particles undergoing general translationally invariant quantum walks.Employing perturbation theory, we establish a direct method for computing the asymptotic distribution of the particle's position rescaled by an appropriate time-dependent factor.If the time evolution is implemented by a unitary operator commuting with all lattice translations and with empty pure point spectrum, we find linear scaling of the asymptotic position distribution.In this case, the asymptotic distribution of position divided by time is determined by the analog of the group velocity.If, on the other hand, the quantum walk is subject to decoherence, we expect that it behaves like a classical random walk.We provide explicit criteria under which this intuitive statement is true, that is, the asymptotic scaling of the particle's position distribution is that of a diffusion process.We find that the asymptotic distribution of position divided by the square root of time is a mixture of Gaussians in this case.However, we also provide examples of decoherent quantum walks which violate these criteria, and exhibit linear scaling of the asymptotic position distribution. Many-particle quantum walks:We investigate unitary quantum walks of two particles with collisional interactions.That is, the particles perform independent single-particle quantum walks if they are at distinct lattice sites, and an interaction takes place if both particles occupy the same lattice site.We prove that such quantum walks generically exhibit a molecular binding effect between the particles.More precisely, for a measurement of the particles' positions we have a bound on the probability to obtain a certain result which decays exponentially with the particles' distance.As a preparatory step for this analysis, we consider single-particle quantum walks with point defects, and provide a method to compute their eigenvalues in the spectral gap of the quantum walk without defect.We further generalize our model to arbitrary numbers of particles and study the behavior of local excitations in many-particle quantum walks.
We introduce the quantum Levy walk to study transport and decoherence in a quantum random model. We have derived from second order perturbation theory the quantum master equation for a extit{Levy-like particle }that moves along a lattice through hopping scale-free while interacting with a thermal bath of oscillators. The general evolution of the quantum Levy particle has been solved for different preparations of the system. We examine the evolution of the quantum purity, the localized correlation, and the probability to be in a lattice site, all them leading to important conclusions concerning quantum irreversibility and decoherence features. We prove that the quantum thermal mean-square displacement is finite under a constraint that is different when compared to the classical Weierstrass random walk. We prove that when the mean-square displacement is infinite the density of state has a complex null-set inside the Brillouin zone. We show the existence of a critical behavior in the continuous eigenenergy which is related to its non-differentiability and self-affine characteristics. In general our approach allows to study analytically quantum fluctuations and decoherence in a long-range hopping model.