Quantum Brownian Motion Research Articles

Quarkonium polarization in medium from open quantum systems and chromomagnetic correlators

We study the spin-dependent in-medium dynamics of quarkonia by using the potential nonrelativistic QCD (pNRQCD) and the open quantum system framework. We consider the pNRQCD Lagrangian valid up to the order rM0=r and r0M=1M in the double power counting. By considering the Markovian condition and applying the Wigner transformation upon the diagonal spin components of the quarkonium density matrix with the semiclassical expansion, we systematically derive the Boltzmann transport equation for quarkonia with polarization dependence in the quantum optical limit. Unlike the spin-independent collision terms governed by certain chromoelectric field correlators, new gauge invariant correlators of chromomagnetic fields determine the recombination and dissociation terms with polarization dependence at the order we are working. We also derive a Lindblad equation describing the in-medium transitions between spin-singlet and spin-triplet heavy quark-antiquark pairs in the quantum Brownian motion limit. The Lindblad equation is governed by new transport coefficients defined in terms of the chromomagnetic field correlators. Our formalism is generic and valid for both weakly coupled and strongly coupled quark gluon plasmas. It can be further applied to study spin alignment of vector quarkonia in heavy ion collisions. Published by the American Physical Society 2024

Strong-coupling quantum thermodynamics of quantum Brownian motion based on the exact solution of its reduced density matrix

Strong-coupling quantum thermodynamics of quantum Brownian motion based on the exact solution of its reduced density matrix

Independent-oscillator model and the quantum Langevin equation for an oscillator: a review

This review provides a brief and quick introduction to the quantum Langevin equation for an oscillator, while focusing on the steady-state thermodynamic aspects. A derivation of the quantum Langevin equation is carefully outlined based on the microscopic model of the heat bath as a collection of a large number of independent quantum oscillators, the so-called independent-oscillator model. This is followed by a discussion on the relevant ‘weak-coupling’ limit. In the steady state, we analyze the quantum counterpart of energy equipartition theorem which has generated a considerable amount of interest in recent literature. The free energy, entropy, specific heat, and third law of thermodynamics are discussed for one-dimensional quantum Brownian motion in a harmonic well. Following this, we explore some aspects of dissipative diamagnetism in the context of quantum Brownian oscillators, emphasizing upon the role of confining potentials and also upon the environment-induced classical-quantum crossover. We discuss situations where the system-bath coupling is via the momentum variables by focusing on a gauge-invariant model of momentum-momentum coupling in the presence of a vector potential; for this problem, we derive the quantum Langevin equation and discuss quantum thermodynamic functions. Finally, the topic of fluctuation theorems is discussed (albeit, briefly) in the context of classical and quantum cyclotron motion of a particle coupled to a heat bath.

Quantum tunnelling and thermally driven transitions in a double-well potential at finite temperature

We explore dissipative quantum tunnelling, a phenomenon central to various physical and chemical processes, utilizing a model based on a double-well potential. This paper aims to bridge gaps in understanding the crossover from thermal activation to quantum tunnelling, a domain still shrouded in mystery despite extensive research. By numerically investigating a model derived from Caldeira–Leggett’s work on quantum Brownian motion, examining both Lindblad and stochastic Schrödinger dynamics, we offer new insights into the transition states in the crossover region. Contrary to a common belief that temperature strongly dampens all quantum effects, our findings reveal that under certain conditions temperature instead alters the nature of tunnelling from a deterministic and periodic process to a stochastic yet still very quantum phenomenon. This underscores the profound influence of quantum effects on transition rates and the critical role of temperature in modulating tunnelling behaviours. Additionally, we introduce a new model for quantum Brownian motion that takes Lindblad form and is formulated as a modification of the widely known model found in Breuer and Petruccione. In our approach, we remove the zero-temperature singularity resulting in a better description of low-temperature quantum Brownian motion near a potential minima. Despite these advancements, we recognize persistent challenges in accurately simulating the dynamics at extremely low temperatures for arbitrary potentials, particularly those that cannot be closely approximated locally by a quadratic function.

Quantum Brownian motion induced by a scalar field in Einstein’s universe

The Brownian motion of a point particle induced by quantum vacuum fluctuations of a massless real scalar field in Einstein’s universe is studied. By assuming the small displacement condition, the dispersion in the momentum and position of a point particle coupled to the massless scalar field are obtained. As a consequence of the homogeneity and isotropy properties of the Einstein’s Universe, we find that all components of these physical observables are identical. We also examine divergent behaviors associated with the physical momentum and position dispersions, which we attribute to the IR1×\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{1}\ imes $$\\end{document}S3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{3}$$\\end{document} compact topology of Einstein’s universe. Finally, based on the small displacement condition assumed, we analyze the limit of validity of our investigation.

Page-curve-like entanglement dynamics in open quantum systems

The entanglement entropy of a black hole and that of its Hawking radiation are expected to follow the so-called Page curve: After an increase in line with Hawking’s calculation, it is expected to decrease back to zero once the black hole has fully evaporated, as demanded by unitarity. Recently, a simple system-plus-bath model has been proposed which shows a similar behavior. Here, we make a general argument as to why such a Page-curve-like entanglement dynamics should be expected to hold generally for system-plus-bath models at small coupling and low temperatures, when the system is initialized in a pure state far from equilibrium. The interaction with the bath will then generate entanglement entropy, but it eventually has to decrease to the value prescribed by the corresponding mean-force Gibbs state. Under those conditions, it is close to the system ground state. We illustrate this on two paradigmatic open-quantum-system models, the exactly solvable harmonic quantum Brownian motion and the spin-boson model, which we study numerically. In the first example we find that the intermediate entropy of an initially localized impurity is higher for more localized initial states. In the second example, for an impurity initialized in the excited state, the Page time—when the entropy reaches its maximum—occurs when the excitation has half decayed. Published by the American Physical Society 2024

Heat distribution in quantum Brownian motion.

We investigate the heat statistics in a relaxation process of quantum Brownian motion described by the Caldeira-Leggett model. By employing the normal mode transformation and the phase-space formulation approach, we can analyze the quantum heat distribution within an exactly dynamical framework beyond the traditional paradigm of Born-Markovian and weak-coupling approximations. It is revealed that the exchange fluctuation theorem for quantum heat generally breaks down in the strongly non-Markovian regime. Our results may improve the understanding about the nonequilibrium thermodynamics of open quantum systems when the usual Markovian treatment is no longer appropriate.

Anisotropic motion of an electric dipole in a photon gas near a flat conducting boundary

The quantum Brownian motion of a single neutral particle with nonzero electric dipole moment placed in a photon gas at fixed temperature and close to a conducting wall is here examined. The interaction of the particle with the photon field leads to quantum dispersions of its linear and angular momenta, whose magnitudes depend on the temperature, distance to the wall, and also on the dipole moment characteristics. It is shown that for typical experimental parameters the amount of energy held by the dipole rotation is expressively larger than the one related to the center of mass translation. Furthermore, the particle kinetic energy in presence of a thermal bath can decrease if the wall is added to the system, representing a novel quantum cooling effect where the work done by the quantum vacuum extracts energy from the particle. Finally, possible observable consequences are discussed.

Chromoelectric field correlator for quarkonium transport in the strongly coupled mathcal{N} = 4 Yang-Mills plasma from AdS/CFT

Previous studies have shown that a gauge-invariant correlation function of two chromoelectric fields connected by a straight timelike adjoint Wilson line encodes crucial information about quark-gluon plasma (QGP) that determines the dynamics of small-sized quarkonium in the medium. Motivated by the successes of holographic calculations to describe strongly coupled QGP, we calculate the analog gauge-invariant correlation function in strongly coupled mathcal{N} = 4 supersymmetric Yang-Mills theory at finite temperature by using the AdS/CFT correspondence. Our results indicate that the transition processes between bound and unbound quarkonium states are suppressed in strongly coupled plasmas, and moreover, the leading contributions to these transition processes vanish in both the quantum Brownian motion and quantum optical limits of open quantum system approaches to quarkonia.

R. Fürth’s 1933 paper “On certain relations between classical statistics and quantum mechanics” [“Über einige Beziehungen zwischen klassischer Statistik und Quantenmechanik”, Zeitschrift für Physik, 81 143–162

We present a translation of the 1933 paper by R. Fürth in which a profound analogy between quantum fluctuations and Brownian motion is pointed out. Fürth highlights the existence of uncertainty relations involving the variance of a statistically conserved quantity of a non-equilibrium thermodynamic indicator and the variance of the corresponding current velocity. The phenomenon is entirely classical and traces back to the effect of a fluctuating environment on a measured system. In some sense, Fürth’s paper also opened the way to the stochastic methods of quantization developed almost 30 years later by Edward Nelson and others.

Memory effects displayed in the evolution of continuous variable system

We analyze non-Markovian memory effects displayed by the quantum Brownian motion modelled as a quantum harmonic oscillator coupled to a bath consisting of harmonic oscillators. In particular, the evolution of fidelity, relative entropy and quantum entanglement (measured by negativity) is considered for a family of Gaussian states (one and two mode squeezed vacuum and three mode basset-hound states). It is shown how to enhance detection of memory effects by adding an additional (ancillary) mode such that only a single mode (a system) is subjected to the evolution. It is observed that even a small amount of squeezing allows detection of non-Markovian memory effects.

Quantum Brownian motion induced by an inhomogeneous tridimensional space and a S1× R3 topological space-time

In this paper we investigate the Quantum Brownian motion of a point particle induced by quantum vacuum fluctuations of a massless scalar field in (3 + 1)-dimensional Minkowski spacetime with distinct conditions (Dirichlet, Neumann, mixed and quasiperiodic). The modes of the field are confined and compactified to a finite length region, which consequently provides a natural measure scale for the system. Useful expressions for the Wightman function have been obtained, which allow us to calculate analytical expressions for the velocity dispersion in all condition cases considered. We also obtain expressions for the velocity dispersion in the short and late time regimes. Finally, we exhibit some graphs in order to show the behavior of the velocity dispersions, discussing important divergencies that are present in our results.

Ancillary Gaussian modes activate the potential to witness non-Markovianity

We study how the number of employed modes impacts the ability to witness non-Markovian evolutions via correlation backflows in continuous-variable quantum dynamics. We first prove the existence of non-Markovian Gaussian evolutions that do not show any revivals in the correlations between the mode evolving through the dynamics and a single ancillary mode. We then demonstrate how this scenario radically changes when two ancillary modes are considered. Indeed, we show that the same evolutions can show correlation backflows along a specific bipartition when three-mode states are employed, and where only one mode is subjected to the evolution. These results can be interpreted as a form of activation phenomenon in non-Markovianity detection and are proven for two types of correlations, entanglement and steering, and two classes of Gaussian evolutions, a classical noise model and the quantum Brownian motion model.

Realizing the Quantum Relative Entropy of Two Noisy States using the Hudson-Parthasarathy Equations

The idea of noisy states can be derived through a quantum relative entropy over a given time period and construct the average value of X at time based on the system variables. A random Hermitian matrix is used to represent the quantum system observables with BATH states. The Hudson-Parthasarathy (HP equation) context for stochastic processes allows us to simulate quantum relative entropy using quantum Brownian motion. The Sudarshan-Lindblad's density evolution matrix equation was already derivable in generalized form in my previous work. This paper's goal is to illustrate how the HP equation may be used to estimate the density matrix for noise in a perturbed quantum system of a stochastic process. The last stage involves using MATLAB to estimate and simulate a random density matrix and measure the quantum average T_r (ρ(t)X) at various times. These formulas would be helpful in determining how sensitive the evolving/evolved states are to changes in the Hamiltonian of the noise operators in a sensitivity/robustness study of quantum systems.

Decoherence factor as a convolution: an interplay between a Gaussian and an exponential coherence loss

We identify and investigate the origin and nature of the transition between Gaussian and exponential forms of decoherence: the decoherence factor (that controls the time dependence of the off-diagonal terms of the density matrix expressed in the pointer basis representation) is the convolution of the Fourier transforms of the spectral density and of the overlap (between the eigenstates the environment with and without couplings to the system). Spectral density alone tends to lead to the (approximately) Gaussian decay of coherence while the overlap alone results in a (largely) exponential decay. We show that these two contributions combine as a convolution, their relative importance controlled by the strength of the system-environment coupling. The resulting decoherence factor in the strong and weak coupling limits leads to predominantly Gaussian or exponential decay, respectively, as is demonstrated with two paradigmatic examples of decoherence—a spin-bath model and the quantum Brownian motion.

Long-time Tails in Quantum Brownian Motion of a charged particle in a magnetic field

We analyse the long-time tails of a charged quantum Brownian particle in a harmonic potential in the presence of a magnetic field using the Quantum Langevin Equation as a starting point. We analyse the long-time tails in the position-autocorrelation function, position–velocity correlation function and velocity-autocorrelation function. We study these correlations for a Brownian particle coupled to Ohmic and Drude baths, via position coordinate coupling. At finite temperatures we notice a crossover from a power-law to an exponentially decaying behaviour around the thermal time scale ħkBT. We analyse how the appearance of the cyclotron frequency in our study of a charged quantum Brownian particle affects the behaviour of the long time tails and contrast it with the case of a neutral quantum Brownian particle.

Piecewise ensemble averaging stochastic Liouville equations for simulating non-Markovian quantum dynamics

Here we present a novel stochastic Liouville equation with piecewisely correlated noises, in which the inter-piece correlation is rigorously incorporated by a convolution integral involving functional derivatives. Due to the feature of piecewise correlation, we can perform piecewise ensemble average and serve the average of the preceding interval as the initial condition of the subsequent propagation. This strategy avoids the long-time stochastic average and the statistical errors are saturated at long times. By doing so, we circumvent the intrinsic difficulty of the stochastic simulations caused by the fast increase in the variance of the quantum Brownian motion. Therefore, as demonstrated by the numerical examples, the proposed method enables us to simulate the long-time quantum dissipative dynamics with long memories in the non-perturbative regime.

Quantum simulation of nonequilibrium dynamics and thermalization in the Schwinger model

We present simulations of nonequilibrium dynamics of quantum field theories on digital quantum computers. As a representative example, we consider the Schwinger model, a ($1+1$)-dimensional U(1) gauge theory, coupled through a Yukawa-type interaction to a thermal environment described by a scalar field theory. We use the Hamiltonian formulation of the Schwinger model discretized on a spatial lattice. With the thermal scalar fields traced out, the Schwinger model can be treated as an open quantum system and its real-time dynamics are governed by a Lindblad equation in the Markovian limit. The interaction with the environment ultimately drives the system to thermal equilibrium. In the quantum Brownian motion limit, the Lindblad equation is related to a field theoretical Caldeira-Leggett equation. By using the Stinespring dilation theorem with ancillary qubits, we perform studies of both the nonequilibrium dynamics and the preparation of a thermal state in the Schwinger model using IBM's simulator and quantum devices. The real-time dynamics of field theories as open quantum systems and the thermal state preparation studied here are relevant for a variety of applications in nuclear and particle physics, quantum information and cosmology.

Exact master equation for generalized quantum Brownian motion with momentum-dependent system-environment couplings

In this paper, we generalize the quantum Brownian motion to include momentum-dependent system-environment couplings. The conventional QBM model corresponds to the spacial case $W_k = V_k$. The generalized QBM is more complicated but the generalization is necessary. This is because the particle transition and the pair production between the system and the environment represent two very different physical processes, and usually cannot have the same coupling strengths. Thus, the conventional QBM model, which is well-defined at classical level, is hardly realized in real quantum physical world. We discuss the physical realizations of the generalized QBM in different physical systems, and derive its exact master equation for both the initial decoupled states and initial correlated states. The Hu-Paz-Zhang master equation of the conventional QBM model is reproduced as a special case. We find that the renormalized Brownian particle Hamiltonian after traced out all the environmental states induced naturally a momentum-dependent potential, which also shows the necessity of including the momentum-dependent coupling in the QBM Hamiltonian. In the Hu-Paz-Zhang master equation, such a renormalized potential is misplaced so that the correct renormalization Hamiltonian has not been found. With the exact master equation for both the initial decoupled and and initial correlated states, the issues about the initial jolt which is a long-stand problem in the Hu-Paz-Zhang master equation is also re-examined. We find that the so-called "initial jolt", which has been thought to be an artificial effect due to the use of the initial decoupled system-environment states, has nothing do to with the initial decoupled state. The new exact master equation for the generalized QBM also has the potential applications to photonics quantum computing.

Quantum Brownian motion of a charged oscillator in a magnetic field coupled to a heat bath through momentum variables

We study the Quantum Brownian motion of a charged particle moving in a harmonic potential in the presence of an uniform external magnetic field and linearly coupled to an Ohmic bath through momentum variables. We analyse the growth of the mean square displacement of the particle in the classical high temperature domain and in the quantum low temperature domain dominated by zero point fluctuations. We also analyse the Position Response Function and the long time tails of various correlation functions. We notice some distinctive features, different from the usual case of a charged quantum Brownian particle in a magnetic field and linearly coupled to an Ohmic bath via position variables.