Quantum Brownian Motion Research Articles

Path integral approach to heat in quantum thermodynamics

We study the heat statistics of a quantum Brownian motion described by the Caldeira-Leggett model. By using the path integral approach, we introduce a concept of the quantum heat functional along every pair of Feynman paths. This approach has the advantage of improving our understanding about heat in quantum systems. First, we demonstrate the microscopic reversibility of the system by connecting the heat functional to the forward and time-reversed probabilities. Second, we analytically prove the quantum-classical correspondence of the heat functional and their statistics, which allows us to obtain better intuitions about the difference between classical and quantum heat.

Testing collapse models by a thermometer

Collapse models postulate that space is filled with a collapse noise field, inducing quantum Brownian motions which are dominant during the measurement, thus causing collapse of the wave function. An important manifestation of collapse noise field, if any, is thermal energy generation, thus disturbing the temperature profile of a system. The experimental investigation of collapse-driven heating effect has provided, so far, the most promising test of collapse models against standard quantum theory. In this paper, we calculate the collapse-driven heat generation for a three-dimensional multi-atomic Bravais lattice, by solving stochastic Heisenberg equations. We perform our calculation for the mass-proportional Continuous Spontaneous Localization collapse model with non-white noise. We obtain the temperature distribution of a sphere under stationary-state and adiabatic surface conditions. However, the exact quantification of effect highly depends on the value of cutoff in the collapse noise spectrum.

Quantum Brownian motion in a magnetic field: Transition from monotonic to oscillatory behaviour

We investigate the Brownian motion of a charged particle in a magnetic field. We study this in the classical and quantum domains. In both domains, we observe a qualitatively interesting transition of the mean square displacement from a monotonic to a damped oscillatory behaviour as one increases the strength of the magnetic field. We notice that these features of the mean square displacement are robust and remain essentially the same for an Ohmic dissipation model and a single relaxation time model for the memory kernel. The predictions stemming from our analysis can be tested against experiments in trapped cold ions.

Non-Markovian quantum Brownian motion in one dimension in electric fields

Quantum Brownian motion is the random motion of quantum particles suspended in a field (or an effective field) resulting from their collision with fast-moving modes in the field. It provides us with a fundamental model to understand various physical features concerning open systems in chemistry, condensed-matter physics, biophysics, and optomechanics. In this paper, without either the Born-Markovian or rotating-wave approximation, we derive a master equation for a charged-Brownian particle in one dimension coupled with a thermal reservoir in electric fields. The effect of the reservoir and the electric fields is manifested as time-dependent coefficients and coherent terms, respectively, in the master equation. The two-photon correlation between the Brownian particle and the reservoir can induce nontrivial squeezing dynamics to the particle. We derive a current equation including the source from the driving fields, transient current from the system flowing into the environment, and the two-photon current caused by the non-rotating-wave term. The presented results then are compared with that given by the rotating-wave approximation in the weak-coupling limit, and these results are extended to a more general quantum network involving an arbitrary number of coupled-Brownian particles. The presented formalism might open a way to better understand exactly the non-Markovian quantum network.

Strong Coupling Corrections in Quantum Thermodynamics.

Quantum systems strongly coupled to many-body systems equilibrate to the reduced state of a global thermal state, deviating from the local thermal state of the system as it occurs in the weak-coupling limit. Taking this insight as a starting point, we study the thermodynamics of systems strongly coupled to thermal baths. First, we provide strong-coupling corrections to the second law applicable to general systems in three of its different readings: As a statement of maximal extractable work, on heat dissipation, and bound to the Carnot efficiency. These corrections become relevant for small quantum systems and vanish in first order in the interaction strength. We then move to the question of power of heat engines, obtaining a bound on the power enhancement due to strong coupling. Our results are exemplified on the paradigmatic non-Markovian quantum Brownian motion.

Quantum Brownian motion under generalized position measurements: a converse Zeno scenario

We study the quantum Brownian motion of a harmonic oscillator undergoing a sequence of generalized position measurements. Our exact analytical results capture the interplay of the measurement backaction and dissipation. Here we demonstrate that no freeze-in Zeno effect occurs upon increasing the monitoring frequency. A similar behavior is also found in the presence of generalized momentum measurements.

Belavkin filter for mixture of quadrature and photon counting process with some control techniques

The Belavkin filter for the H-P Schrodinger equation is derived when the measurement process consists of a mixture of quantum Brownian motions and conservation/Poisson process. Higher-order powers of the measurement noise differentials appear in the Belavkin dynamics. For simulation, we use a second-order truncation. Control of the Belavkin filtered state by infinitesimal unitary operators is achieved in order to reduce the noise effects in the Belavkin filter equation. This is carried out along the lines of Luc Bouten. Various optimization criteria for control are described like state tracking and Lindblad noise removal.

Quantum thermodynamics from the nonequilibrium dynamics of open systems: Energy, heat capacity, and the third law.

In a series of papers, we intend to take the perspective of open quantum systems and examine from their nonequilibrium dynamics the conditions when the physical quantities, their relations, and the laws of thermodynamics become well defined and viable for quantum many-body systems. We first describe how an open-system nonequilibrium dynamics (ONEq) approach is different from the closed combined system+ environment in a global thermal state (CGTs) setup. Only after the open system equilibrates will it be amenable to conventional thermodynamics descriptions, thus quantum thermodynamics (QTD) comes at the end rather than assumed in the beginning. The linkage between the two comes from the reduced density matrix of ONEq in that stage having the same form as that of the system in the CGTs. We see the open-system approach having the advantage of dealing with nonequilibrium processes as many experiments in the near future will call for. Because it spells out the conditions of QTD's existence, it can also aid us in addressing the basic issues in quantum thermodynamics from first principles in a systematic way. We then study one broad class of open quantum systems where the full nonequilibrium dynamics can be solved exactly, that of the quantum Brownian motion of N strongly coupled harmonic oscillators, interacting strongly with a scalar-field environment. In this paper, we focus on the internal energy, heat capacity, and the third law. We show for this class of physical models, amongst other findings, the extensive property of the internal energy, the positivity of the heat capacity, and the validity of the third law from the perspective of the behavior of the heat capacity toward zero temperature. These conclusions obtained from exact solutions and quantitative analysis clearly disprove claims of negative specific heat in such systems and dispel allegations that in such systems the validity of the third law of thermodynamics relies on quantum entanglement. They are conceptually and factually unrelated issues. Entropy and entanglement will be the main theme of our second paper on this subject matter.

Invariant eigen-operator calculated vibration mode of lattice in the case of absorbing an atom onto crystal surface

The influence of diffusion and defects of crystal surface on the crystal vibration mode are an important and basic subject in surface physics research. The frequency of lattice vibration corresponds to the energy band of the system. Since the vibrations of the atoms in the crystal lattice are not isolated from each other, and the crystal lattice is periodic, thereby forming a lattice wave in the crystal. The lattice wave represents that all the atoms in the crystal vibrate at an identical frequency, which is often called a vibration mode. The lattice chain model has been studied as the vibrating mode of phonon and the energy-band in solid state physics. The vibrating modes of the lattice chain model have been analyzed with the Newton equation and the Born-von-Karman boundary condition in the literaure. In general, it is difficult to solve this problem due to the complex nonlinear characteristic of the interactions between the matter particles and the environment. Noting the complicacy in directly diagonalizing quantum Hamiltonian operator of a long chain, we introduce the invariant eigenoperator method (IEO) for deriving the energy gap of a given crystal lattice without solving its eigenstates in the Heisenberg picture. The Heisenberg equation is as important as the Schrödinger equation. However, it has been seldom used for directly deriving the energy-gap in previous studies. Following the Heisenberg's original idea that most observable physical quantity in quantum mechanics is energy spectrum, Hong-yi Fan, one of the authors of the present paper, developed the IEO method. This method provides a natural result of combining both the Schrödinger operator and the Heisenberg equation. Using the IEO method, we study the vibration modes of crystal lattice, which are affected by absorbing an atom with mass m0, which is different from the mass of atom in the crystal. Moreover, the attractive potential constantβ0 of the lattice surface differs from the inner constantβ. With the help of invariant eigen-operator method, we deduce the vibration mode ω=√(2β(1-cosh α))/ħm, where α=ln[-(mβ0+m0(-2β+β0)+√β0√-4 mm0β+(m+m0)2β0)/2m0β]. Our numerical results show that vibration mode ω depends not only on the absorption potential and the mass of the absorbed atom, but also on the mass of the lattice atom and the inner potential. In general, by discussing the vibration modes via some numerical solutions or approximate methods, we show the relations between the system vibration modes with different parameters which describe the environment influences. These results can deepen our understanding of quantum Brownian motion and demonstrate the applicability of the IEO method.

Heisenberg-Langevin versus quantum master equation

The quantum master equation is an important tool in the study of quantum open systems. It is often derived under a set of approximations, chief among them the Born (factorization) and Markov (neglect of memory effects) approximations. In this article we study the paradigmatic model of quantum Brownian motion of an harmonic oscillator coupled to a bath of oscillators with a Drude-Ohmic spectral density. We obtain analytically the \emph{exact} solution of the Heisenberg-Langevin equations, with which we study correlation functions in the asymptotic stationary state. We compare the \emph{exact} correlation functions to those obtained in the asymptotic long time limit with the quantum master equation in the Born approximation \emph{with and without} the Markov approximation. In the latter case we implement a systematic derivative expansion that yields the \emph{exact} asymptotic limit under the factorization approximation \emph{only}. We find discrepancies that could be significant when the bandwidth of the bath $\Lambda$ is much larger than the typical scales of the system. We study the \emph{exact} interaction energy as a \emph{proxy} for the correlations missed by the Born approximation and find that its dependence on $\Lambda$ is similar to the \emph{discrepancy} between the exact solution and that of the quantum master equation in the Born approximation. We quantify the regime of validity of the quantum master equation in the Born approximation with or without the Markov approximation in terms of the system's relaxation rate $\gamma$, its \emph{unrenormalized} natural frequency $\Omega$ and $\Lambda$: $\gamma/\Omega \ll 1$ and \emph{also} $\gamma \Lambda/\Omega^2 \ll 1$. The reliability of the Born approximation is discussed within the context of recent experimental settings and more general environments.

Phase diagrams for quantum Brownian motion on two-dimensional Bravais lattices

We study quantum Brownian motion (QBM) models for a particle in a dissipative environment coupled to a periodic potential. We review QBM for a particle in a one-dimensional periodic potential and extend the study to that for a particle in two-dimensional (2D) periodic potentials of four Bravais lattice types: square, rectangular, triangular (hexagonal), and centered rectangular. We perform perturbative renormalization group analyses to derive the zero temperature flow diagrams and phase boundaries for a particle in these potentials, and observe localization behavior dependent on the anisotropy of the lattice parameters.

On the Small Mass Limit of Quantum Brownian Motion with Inhomogeneous Damping and Diffusion

We study the small mass limit (or: the Smoluchowski-Kramers limit) of a class of quantum Brownian motions with inhomogeneous damping and diffusion. For Ohmic bath spectral density with a Lorentz-Drude cutoff, we derive the Heisenberg-Langevin equations for the particle's observables using a quantum stochastic calculus approach. We set the mass of the particle to equal $m = m_{0} \epsilon$, the reduced Planck constant to equal $\hbar = \epsilon$ and the cutoff frequency to equal $\Lambda = E_{\Lambda}/\epsilon$, where $m_0$ and $E_{\Lambda}$ are positive constants, so that the particle's de Broglie wavelength and the largest energy scale of the bath are fixed as $\epsilon \to 0$. We study the limit as $\epsilon \to 0$ of the rescaled model and derive a limiting equation for the (slow) particle's position variable. We find that the limiting equation contains several drift correction terms, the quantum noise-induced drifts, including terms of purely quantum nature, with no classical counterparts.

Quantum Brownian Motion at Strong Dissipation Probed by Superconducting Tunnel Junctions.

We have investigated the phase dynamics of a superconducting tunnel junction at ultralow temperatures in the presence of high damping, where the interaction with environmental degrees of freedom represents the leading energy scale. In this regime, theory predicts the dynamics to follow a generalization of the classical Smoluchowski description, the quantum Smoluchowski equation, thus, exhibiting overdamped quantum Brownian motion characteristics. For this purpose, we have performed current-biased measurements on the small-capacitance Josephson junction of a scanning tunneling microscope placed in a low impedance environment at milli-Kelvin temperatures. We can describe our experimental findings with high accuracy by using a quantum phase diffusion model based on the quantum Smoluchowski equation. In this way we experimentally demonstrate that overdamped quantum systems follow quasiclassical dynamics with significant quantum effects as the leading corrections.

Geometric quantum speed limits: a case for Wigner phase space

The quantum speed limit is a fundamental upper bound on the speed of quantum evolution. However, the actual mathematical expression of this fundamental limit depends on the choice of a measure of distinguishability of quantum states. We show that quantum speed limits are qualitatively governed by the Schatten-p-norm of the generator of quantum dynamics. Since computing Schatten-p-norms can be mathematically involved, we then develop an alternative approach in Wigner phase space. We find that the quantum speed limit in Wigner space is fully equivalent to expressions in density operator space, but that the new bound is significantly easier to compute. Our results are illustrated for the parametric harmonic oscillator and for quantum Brownian motion.

Nonexponential Quantum Decay under Environmental Decoherence.

A system prepared in an unstable quantum state generally decays following an exponential law, as environmental decoherence is expected to prevent the decay products from recombining to reconstruct the initial state. Here we show the existence of deviations from exponential decay in open quantum systems under very general conditions. Our results are illustrated with the exact dynamics under quantum Brownian motion and suggest an explanation of recent experimental observations.

Bose polaron as an instance of quantum Brownian motion

We study the dynamics of a quantum impurity immersed in a Bose-Einstein condensate as an open quantum system in the framework of the quantum Brownian motion model. We derive a generalized Langevin equation for the position of the impurity. The Langevin equation is an integrodifferential equation that contains a memory kernel and is driven by a colored noise. These result from considering the environment as given by the degrees of freedom of the quantum gas, and thus depend on its parameters, e.g. interaction strength between the bosons, temperature, etc. We study the role of the memory on the dynamics of the impurity. When the impurity is untrapped, we find that it exhibits a super-diffusive behavior at long times. We find that back-flow in energy between the environment and the impurity occurs during evolution. When the particle is trapped, we calculate the variance of the position and momentum to determine how they compare with the Heisenberg limit. One important result of this paper is that we find position squeezing for the trapped impurity at long times. We determine the regime of validity of our model and the parameters in which these effects can be observed in realistic experiments.

Quantum analysis and thermodynamic operator relations in stochastic energetics

We introduce a model of the quantum Brownian motion coupled to a classical neat bath by using the operator differential proposed in the quantum analysis. We then define the heat operator by adapting the idea of the stochastic energetics. The introduced operator satisfies the relations which are analogous to the first and second laws of thermodynamics.

Momentum coupling in non-Markovian quantum Brownian motion

We consider a model of non-Markovian Quantum Brownian motion that consists of an harmonic oscillator bilinearly coupled to a thermal bath, both via its position and momentum operators. We derive the master equation for such a model and we solve the equations of motion for a generic Gaussian system state. We then investigate the resulting evolution of the first and second moments for both an Ohmic and a super-Ohmic spectral density. In particular, we show that, irrespective of the specific form of the spectral density, the coupling with the momentum enhances the dissipation experienced by the system, accelerating its relaxation to the equilibrium, as well as modifying the asymptotic state of the dynamics. Eventually, we characterize explicitly the non-Markovianity of the evolution, using a general criterion which relies on the positivity of the master equation coefficients.

Decoherence, discord, and the quantum master equation for cosmological perturbations

We examine environmental decoherence of cosmological perturbations in order to study the quantum-to-classical transition and the impact of noise on entanglement during inflation. Given an explicit interaction between the system and environment, we derive a quantum master equation for the reduced density matrix of perturbations, drawing parallels with quantum Brownian motion, where we see the emergence of fluctuation and dissipation terms. Although the master equation is not in Lindblad form, we see how typical solutions exhibit positivity on super-horizon scales, leading to a physically meaningful density matrix. This allows us to write down a Langevin equation with stochastic noise for the classical trajectories which emerge from the quantum system on super-horizon scales. In particular, we find that environmental decoherence increases in strength as modes exit the horizon, with the growth driven essentially by white noise coming from local contribution to environmental correlations. Finally, we use our master equation to quantify the strength of quantum correlations as captured by discord. We show that environmental interactions have a tendency to decrease the size of the discord and that these effects are determined by the relative strength of the expansion rate and interaction rate of the environment. We interpret this in terms of the competing effects of particle creation versus environmental fluctuations, which tend to increase and decrease the discord respectively.

Adjoint master equation for quantum Brownian motion

Quantum brownian motion is a fundamental model for a proper understanding of open quantum systems in different contexts such as chemistry, condensed matter physics, bio-physics and opto- mechamics. In this paper we propose a novel approach to describe this model. We provide an exact and analytic equation for the time evolution of the operators, and we show that the corresponding equation for the states is equivalent to well-known results in literature. The dynamics is expressed in terms of the spectral density, regardless the strength of the coupling between the system and the bath. Our allows to compute the time evolution of physically relevant quantities in a much easier way than previous formulations allow to. An example is explicitly studied.