Classical Stochastic Field Research Articles

Non-classical correlations of accelerated observers interacting with a classical stochastic noise beyond the single mode approximation

This paper investigates the behaviour of the quantum correlations for an accelerated two-qubit system during its interaction with a classical stochastic field, utilizing the Wigner function and concurrence. The non-classical behaviour is indicated by negative values of the Wigner function, while the degree of entanglement is demonstrated by the concurrence. To consider acceleration and interaction with a common or independent environment, we discard the single-mode approximation and utilize the Unruh construction of the quantum field mode in our analysis. Our results suggest that coherence suppression is caused by the acceleration effect, noise strength, and noise frequency. Through an examination of quantum concurrence, this study analyses the level of quantum entanglement that corresponds positively with negative values of the Wigner function. The findings indicate that coherence degradation in the initial system is reduced when observers interact independently with their environments, as opposed to interacting with a common environment. Additionally, the acceleration of both observers has an impact on coherence reduction. During system evolution, the Wigner function displays collapse and reappearance behaviour, while quantum entanglement undergoes local collapse and revival phenomena. Notably, common noise situations exhibit more rapid variation in both the Wigner function and entanglement compared to independent noise configurations.

Gravity, Quantum Fields and Quantum Information: Problems with Classical Channel and Stochastic Theories

In recent years an increasing number of papers have attempted to mimic or supplant quantum field theory in discussions of issues related to gravity by the tools and through the perspective of quantum information theory, often in the context of alternative quantum theories. In this article, we point out three common problems in such treatments. First, we show that the notion of interactions mediated by an information channel is not, in general, equivalent to the treatment of interactions by quantum field theory. When used to describe gravity, this notion may lead to inconsistencies with general relativity. Second, we point out that in general one cannot replace a quantum field by a classical stochastic field, or mock up the effects of quantum fluctuations by that of classical stochastic sources (noises), because in so doing important quantum features such as coherence and entanglement will be left out. Third, we explain how under specific conditions semi-classical and stochastic theories indeed can be formulated from their quantum origins and play a role at certain regimes of interest.

Intrinsic Entropy of Squeezed Quantum Fields and Nonequilibrium Quantum Dynamics of Cosmological Perturbations.

Density contrasts in the universe are governed by scalar cosmological perturbations which, when expressed in terms of gauge-invariant variables, contain a classical component from scalar metric perturbations and a quantum component from inflaton field fluctuations. It has long been known that the effect of cosmological expansion on a quantum field amounts to squeezing. Thus, the entropy of cosmological perturbations can be studied by treating them in the framework of squeezed quantum systems. Entropy of a free quantum field is a seemingly simple yet subtle issue. In this paper, different from previous treatments, we tackle this issue with a fully developed nonequilibrium quantum field theory formalism for such systems. We compute the covariance matrix elements of the parametric quantum field and solve for the evolution of the density matrix elements and the Wigner functions, and, from them, derive the von Neumann entropy. We then show explicitly why the entropy for the squeezed yet closed system is zero, but is proportional to the particle number produced upon coarse-graining out the correlation between the particle pairs. We also construct the bridge between our quantum field-theoretic results and those using the probability distribution of classical stochastic fields by earlier authors, preserving some important quantum properties, such as entanglement and coherence, of the quantum field.

Squeezing the Axion

We apply the squeezed state formalism to scalar field dark matter (e.g. axion) perturbations generated during inflation. As for the inflationary perturbations, the scalar field state becomes highly squeezed as modes exit the horizon. For as long as $H>m_\phi$ (with $H$ the Hubble rate and $m_\phi$ the scalar mass) the scalar field field does not interact during reheating, and we follow its evolution exactly as modes re-enter the horizon. We find that the quantum state remains squeezed after horizon re-entry during the hot big bang. This demonstrates a fact well-known in the theory of inflation: cosmological observables for scalar dark matter are accurately modelled by a classical stochastic field with a fixed phase. Our calculation covers all modes smaller than the present-day cosmic de Broglie wavelength. Larger scale modes mix gravitationally with the environment when $H<m_\phi$, and are thus expected to decohere.

Exact classical noise master equations: Applications and connections

All quantum systems are subject to noise and imperfections due to stray fields, inhomogeneities or drifting experimental controls. An understanding of the effects of noise and decoherence is critical to the progress towards fully functional quantum devices. In this perspective, we focus on noise in quantum systems which are modelled by a dynamic stochastic parameter in the Hamiltonian. We will outline exact evolution equations describing the ensemble average dynamics for a variety of common noise types and their connections. We will also highlight an approximate evolution equation valid in the weak noise limit for an arbitrary classical stochastic field. This framework should serve as a starting point for identifying signatures of differing noise types and optimisation of robust control protocols.

Noise spectroscopy of a quantum-classical environment with a diamond qubit

Knowing a quantum system's environment is critical for its practical use as a quantum device. Qubit sensors can reconstruct the noise spectral density of a classical bath, provided long enough coherence time. Here we present a protocol that can unravel the characteristics of a more complex environment, comprising both unknown coherently coupled quantum systems, and a larger quantum bath that can be modeled as a classical stochastic field. We exploit the rich environment of a Nitrogen-Vacancy center in diamond, tuning the environment behavior with a bias magnetic field, to experimentally demonstrate our method. We show how to reconstruct the noise spectral density even when limited by relatively short coherence times, and identify the local spin environment. Importantly, we demonstrate that the reconstructed model can have predictive power, describing the spin qubit dynamics under control sequences not used for noise spectroscopy, a feature critical for building robust quantum devices. At lower bias fields, where the effects of the quantum nature of the bath are more pronounced, we find that more than a single classical noise model are needed to properly describe the spin coherence under different controls, due to the back action of the qubit onto the bath.

Simulation of the hydrogen ground state in stochastic electrodynamics

Stochastic electrodynamics is a classical theory which assumes that the physical vacuum consists of classical stochastic fields with average energy in each mode, i.e., the zero-point Planck spectrum. While this classical theory explains many quantum phenomena related to harmonic oscillator problems, hard results on nonlinear systems are still lacking. In this work the hydrogen ground state is studied by numerically solving the Abraham–Lorentz equation in the dipole approximation. First the stochastic Gaussian field is represented by a sum over Gaussian frequency components, next the dynamics is solved numerically using OpenCL. The approach improves on work by Cole and Zou 2003 by treating the full problem and reaching longer simulation times. The results are compared with a conjecture for the ground state phase space density. Though short time results suggest a trend towards confirmation, in all attempted modellings the atom ionises at longer times.

Collapse and revival of quantum coherence for a harmonic oscillator interacting with a classical fluctuating environment

We address the dynamics of nonclassicality for a quantum system interacting with a noisy fluctuating environment described by a classical stochastic field. As a paradigmatic example, we consider a harmonic oscillator initially prepared in a maximally nonclassical state, e.g. a Fock number state or a Schroedinger cat-like state, and then coupled to either resonant or non-resonant external field. Stochastic modeling allows us to describe the decoherence dynamics without resorting to approximated quantum master equations, and to introduce non- Markovian effects in a controlled way. A detailed comparison among different nonclassicality criteria and a thorough analysis of the decoherence time reveal a rich phenomenology whose main features may be summarized as follows: i) classical memory effects increase the survival time of quantum coherence; ii) a detuning between the natural frequency of the system and the central frequency of the classical field induces revivals of quantum coherence.

Quantum probes for the spectral properties of a classical environment

We address the use of simple quantum probes for the spectral characterization of classical noisy environments. In our scheme a qubit interacts with a classical stochastic field describing environmental noise and is then measured after a given interaction time in order to estimate the characteristic parameters of the noise. In particular, we address estimation of the spectral parameters of two relevant kinds of non-Gaussian noise: random telegraph noise with Lorentzian spectrum and colored noise with $1/f^{\alpha}$ spectrum. We discuss the estimation precision achievable by quantum probes by analyzing in details the behavior of the quantum Fisher information (QFI) as a function of the noise and the other parameters. We found that population measurement on the qubit is optimal for noise estimation and also evaluate the optimal interaction times for the quantum probe, i.e. the values maximizing the QFI and the quantum signal-to-noise ratio. For random telegraph noise the QFI is inversely proportional to the square of the switching rate, meaning that the quantum signal-to-noise ratio is constant and thus the switching rate may be uniformly estimated with the same precision in its whole range of variation. For colored noise, the precision achievable in the estimation of ``color'', i.e. of the exponent $\alpha$, strongly depends on the structure of the environment, i.e. on the number of fluctuators describing the classical environment. For an environment modeled by a single random fluctuator estimation is more precise for pink noise, i.e. for $\alpha=1$, whereas by increasing the number of fluctuators, the QFI is has two local maxima, with the largest one drifting towards $\alpha=2$, i.e. brown noise.

Persistence of quantum information

There is an increasing interest in the role of macroscopic environments to our understanding of the basics of quantum theory. The knowledge of the implications of the quantum theory to other theories, especially to the statistical mechanics and the domain of validity has captivated scientists from the beginning of quantum description. In such a context, the presence of an environment is commonly thought as entanglement, decohering and mixing properties of quantum system. Generically, an environment is assumed to be a noisy reservoir or a heat bath. Whereas in common interpretation of statistical mechanics the heat bath is unspecified, in quantum systems a heat bath can also provide an indirect interaction between otherwise totally decoupled subsystems and consequently a means to entangle them \cite{cdkl,dvclp,bfp}. In simple example for the entanglement between two qubits due to the interaction with a common heat bath has been explicitly shown in \cite{b}. Whereas in that paper the bath is described by a collection of harmonic oscillators, it seems to be more reasonable to specify the bath by stochastic forces represented by stochastic fields. From a more general point of view we expect the bath should be better described in a stochastic manner and not by deterministic forces. In the present paper we consider a two level system (qubits) which are able to perform flip processes by a coupling to classical stochastic fields. Thus we bridge the gap between quantum and classical probability theory. This problem is related to many other questions of quantum optics and quantum electronics where quantum statistical aspects arising from the intrinsic quantum character of the system while the possible time-dependence of system parameters may be interpreted as the influence of classical thermal fluctuations.

Quantum two-state dynamics driven by stationary non-Markovian discrete noise: Exact results

We consider the problem of stochastic averaging of a quantum two-state dynamics driven by non-Markovian, discrete noises of the continuous time random walk type (multistate renewal processes). The emphasis is put on the proper averaging over the stationary noise realizations corresponding, e.g., to a stationary environment. A two-state non-Markovian process with an arbitrary non-exponential distribution of residence times (RTDs) in its states with a finite mean residence time provides a paradigm. For the case of a two-state quantum relaxation caused by such a classical stochastic field we obtain the explicit exact, analytical expression for the averaged Laplace-transformed relaxation dynamics. In the limit of Markovian noise (implying an exponential RTD), all previously known results are recovered. We exemplify new more general results for the case of non-Markovian noise with a biexponential RTD. The averaged, real-time relaxation dynamics is obtained in this case by numerically exact solving of a resulting algebraic polynomial problem. Moreover, the case of manifest non-Markovian noise with an infinite range of temporal autocorrelation (which in principle is not accessible to any kind of perturbative treatment) is studied, both analytically (asymptotic long-time dynamics) and numerically (by a precise numerical inversion of the Laplace-transformed averaged quantum relaxation).

Quantum systems subject to the action of classical stochastic fields

In this paper we investigate the averaged dynamics of quantum systems under the influence of classical stochastic fields. This influence is modeled using a stochastic Hamiltonian evolution for the system-density matrix. From the averaged dynamics, a general characterization of the short-time decoherence behavior is obtained. General applicable short-time perturbation expansions for the input-output fidelity and its generalizations for mixed states are developed. The master equations of the systems that can be worked out without a perturbative expansion, i.e., any system subject to dispersive noise and the quantum harmonic oscillator subject to amplitude noise, are extensively analyzed. In both cases all non-Markovian features are worked out in an exact way. We apply these cases to the study of heating of trapped ions.

Phase transition of the dissipative one-dimensional Ising model.

The thermodynamic properties of a one-dimensional Ising model in a classical stochastic field of Gaussian type or, equivalently, in an infinite reservoir of harmonic oscillators, is investigated. A finite-temperature phase transition is found, if there is dissipation. This model is applied to a real one-dimensional magnetic system and semiquantitative agreement is obtained. \textcopyright{} 1996 The American Physical Society.

QUANTUM FIELD THEORY OF A DISSIPATIVE SYSTEM

Quantum dissipative scalar and fermionic fields are studied here from classical stochastic fields. These classical stochastic fields are the outcome of the relativistic generalization of Nelson's stochastic mechanics based on a new microlocal geometry. Results show that the dissipation is the external classical phenomena whereas quantum nature comes from within (microlocal structure).

Entropy of the gravitational field.

We derive a formula for the nonequilibrium entropy of a classical stochastic field in terms of correlation functions of this field. The formalism is then applied to define the entropy of gravitational perturbations (both gravitational waves and density fluctuations). We calculate this entropy in a specific cosmological model (the inflationary Universe) and find that on scales of interest in cosmology the entropy in both density perturbations and gravitational waves exceeds the entropy of statistical fluctuations of the microwave background. The nonequilibrium entropy discussed here is a measure of loss of information about the system. We discuss the origin of the entropy in our cosmological models and compare the definition of entropy in terms of correlation functions with the microcanonical definition in quantum statistical mechanics.

Entropy of a classical stochastic field and cosmological perturbations.

We propose a general definition of nonequilibrium entropy of a classical stochastic field. As an example of particular interest in cosmology we apply this definition to compute the entropy of density perturbations in an inflationary Universe. On the scales of structures in the Universe, the entropy of density perturbations dominates over the statistical fluctuations of the entropy of cosmic microwave photons, indicating the relevance of the entropy of density fluctuations for structure formation.