Bath Of Oscillators Research Articles

Generalized quantum Fokker-Planck, diffusion, and Smoluchowski equations with true probability distribution functions

Traditionally, quantum Brownian motion is described by Fokker-Planck or diffusion equations in terms of quasiprobability distribution functions, e.g., Wigner functions. These often become singular or negative in the full quantum regime. In this paper a simple approach to non-Markovian theory of quantum Brownian motion using true probability distribution functions is presented. Based on an initial coherent state representation of the bath oscillators and an equilibrium canonical distribution of the quantum mechanical mean values of their coordinates and momenta, we derive a generalized quantum Langevin equation in c numbers and show that the latter is amenable to a theoretical analysis in terms of the classical theory of non-Markovian dynamics. The corresponding Fokker-Planck, diffusion, and Smoluchowski equations are the exact quantum analogs of their classical counterparts. The present work is independent of path integral techniques. The theory as developed here is a natural extension of its classical version and is valid for arbitrary temperature and friction (the Smoluchowski equation being considered in the overdamped limit).

Vibrational energy pooling in CO on NaCl(100): Methods

Vibrational states as high as n=15 have been experimentally observed in CO molecules adsorbed in a monolayer on the NaCl(100) surface after pumping the n=0→1 vibrational transition with a short (5 μs) infrared laser pulse. These high states become populated from successive single vibrational quantum exchanges between CO molecules on the surface, CO(m)+CO(n)→CO(m−1)+CO(n+1), mediated by dipole–dipole interactions and driven by the anharmonicity of the CO bond vibration. The rates for all of the possible channels of vibrational energy flow in the CO/NaCl(100) system, exchange, relaxation, and fluorescence, were calculated using perturbation theory for a model in which the CO bond vibration is treated as a Morse oscillator and is coupled to a bath of harmonic oscillators with a Debye density of states representing the underlying NaCl substrate. These rates form a Master equation that governs the overall vibrational population dynamics of CO molecules in the monolayer, and was solved using kinetic Monte Carlo (KMC) techniques. Time-dependent vibrational population distributions, Pn(t), representing the probability of finding a CO molecule in the monolayer in vibrational state n at time t, were obtained from the KMC simulations. The results are in good accord with experiment. The maximum achievable excitation is found to be limited by a crossover in the vibrational relaxation and excitation transfer rates with increasing quantum number.

Aharonov-Bohm ring with fluctuating flux

We consider a non-interacting system of electrons on a clean one-channel Aharonov-Bohm ring which is threaded by a fluctuating magnetic flux. The flux derives from a Caldeira-Leggett bath of harmonic oscillators. We address the influence of the bath on the following properties: one- and two-particle Green's functions, dephasing, persistent current and visibility of the Aharonov-Bohm effect in cotunneling transport through the ring. For the bath spectra considered here (including Nyquist noise of an external coil), we find no dephasing in the linear transport regime at zero temperature. PACS numbers: 73.23.-b, 73.23.Hk, 73.23.Ra, 03.65.Yz

Fourth-order quantum master equation and its Markovian bath limit

Fourth-order quantum master equations (FQMEs) are derived in both time nonlocal and local forms for a general system Hamiltonian, with new detailed expressions for the fourth-order kernel, where the bath correlation functions are explicitly decoupled from the system superoperators. Further simplifications can be made for the model of linearly coupled harmonic oscillator bath. Consideration of the high temperature Ohmic bath limit leads to a general Markovian FQME with compact forms of time independent superoperators. Two examples of this equation are then considered. For the system of a quantum particle in a continuous potential field, the equation reduces to a known form of the quantum Fokker–Planck equation, except for a fourth-order potential renormalization term that can be neglected only in the weak system-bath interaction regime. For a two-level system with off-diagonal coupling to the bath, fourth-order corrections do not alter the relaxation characteristics of the second-order equation and introduce additional coherence terms in the equations for the off-diagonal elements.

Approach to quantum Kramers’ equation and barrier crossing dynamics

We have presented a simple approach to quantum theory of Brownian motion and barrier crossing dynamics. Based on an initial coherent state representation of bath oscillators and an equilibrium canonical distribution of quantum-mechanical mean values of their co-ordinates and momenta we have derived a c number generalized quantum Langevin equation. The approach allows us to implement the method of classical non-Markovian Brownian motion to realize an exact generalized non-Markovian quantum Kramers' equation. The equation is valid for arbitrary temperature and friction. We have solved this equation in the spatial diffusion-limited regime to derive quantum Kramers' rate of barrier crossing and analyze its variation as a function of the temperature and friction. While almost all the earlier theories rest on quasiprobability distribution functions (e.g., Wigner function) and path integral methods, the present work is based on true probability distribution functions and is independent of path integral techniques. The theory is a natural extension of the classical theory to quantum domain and provides a unified description of thermally activated processes and tunneling.

Iterative path integral formulation of equilibrium correlation functions for quantum dissipative systems

We present an iterative path integral algorithm for computing multitime correlation functions of a quantum system coupled to a dissipative bath of harmonic oscillators. By splitting the Boltzmann operator into two parts and reordering the propagators in the expression for canonical correlation functions, we are able to transform the evolution time contour into a symmetric one so that a forward propagation and a backward one are specified. Because the memory induced by the bath through the Feynman–Vernon influence functional decays rapidly in the complex time plane, long-time correlations are negligible. Taking advantage of this fact, we show that the correlation function can be obtained via an iterative procedure. The method is used to calculate three-time correlation functions of a dissipative two-level system.

Decoherence and dissipation during a quantum XOR gate operation

The dynamics of a generic quantum XOR gate operation involving two interacting qubits being coupled to a bath of quantum harmonic oscillators is explored. By use of the formally exact quasiadiabatic-propagator path-integral methodology we study the time-resolved evolution of this interacting and decohering two-qubit system in presence of time-dependent external fields. The quality of the XOR gate operation is monitored by evaluating the four characteristic gate quantifiers: fidelity, purity, the quantum degree, and the entanglement capability of the gate. Two different types of errors for the XOR operation have been modeled, i.e., (i) bit-flip errors and (ii) phase errors. The various quantifiers are systematically investigated vs the strength of the interqubit coupling and vs both, the environmental temperature and the (Ohmic-like) bath-interaction strength. Our main findings are that these four gate quantifiers depend only very weakly on temperature, but are extremely sensitive to the bath-interaction strength. Interestingly enough, however, we find that the XOR gate operation deteriorates only weakly upon decreasing the interqubit coupling strength. This generic case study yields lower bounds on the quality of realistic XOR gate operations.

Nonequilibrium generalization of Förster–Dexter theory for excitation energy transfer

Förster–Dexter theory for excitation energy transfer (EET) is generalized for the account of short time nonequilibrium kinetics due to the nonstationary bath relaxation. The final rate expression is presented as a spectral overlap between the time dependent stimulated emission and the stationary absorption profiles, which allows experimental determination of the time dependent rate. For a harmonic oscillator bath model, an explicit rate expression is derived and model calculations are performed in order to examine the dependence of the nonequilibrium kinetics on the excitation–bath coupling strength and the temperature. Relevance of the present theory with recent experimental findings and possible future theoretical directions are discussed.

Cavity QED Deutsch quantum computer

The two-atom correlation scheme originally proposed by Davidovich, Brune, Raimond, and Haroche for measuring the decoherence of a mesoscopic superposition of coherent states of a QED cavity field is shown to be equivalent to a quantum computer solving Deutsch's problem. Using the existing analysis of decoherence in the Master equation formalism, and other important losses in this system, the final probability for obtaining the correct result for the computation is found in terms of the time period between atom traversals, the number of photons in the cavity, and the precision of the atomic velocity. The error due to decoherence in this system amounts to a phase error, and in the Master equation approach is a linear effect at small time scales. By explicitly considering the dynamics of the decoherence process when the system is coupled to a bath of oscillators with finite mode cutoff the error due to decoherence is found to decrease significantly and becomes a quadratic effect at short-time scales.

Fokker–Planck and Langevin Equations from the Forward–Backward Path Integral

Starting from a forward-backward path integral of a point particle in a bath of harmonic oscillators, we derive the Fokker–Planck and Langevin equations with and without inertia. Special emphasis is placed upon the correct operator order in the time evolution operator. The crucial step is the evaluation of a Jacobian with a retarded time derivative by analytic regularization.

Mechanisms of decoherence at low temperatures

We briefly review the oscillator and spin bath models of quantum environments, which can be used to describe the low-energy dynamics of open quantum systems. We then use them to discuss both the mechanisms causing decoherence at low T, and the dynamics of this decoherence. This is done first for a central two-level system coupled to these environments – the results can be applied to the dynamics of quantum nanomagnets and superconducting SQUIDs, where large-scale tunnelling of magnetisation and flux take place. Decoherence in these systems is caused principally by coupling to electrons and nuclear spins – the spin bath couplings are particularly dangerous at low T. We may also generalise these models to discuss quantum measurements in which the measured system, the measuring apparatus, and the environment are treated quantum mechanically. The results can be used to calculate the dynamics of coupled SQUIDs and/or nanomagnets, in which one acts as a measuring apparatus and the other exhibits large-scale quantum superpositions. The same model can be used to describe coupled qubits.

A uniqueness-theorem for “linear” thermal baths

We consider systems in contact with a “linear” thermal bath, modeled by an additive thermal noise and an additive dissipation term which depends linearly on the system velocity. It is shown that the dissipation term and the bath temperature uniquely fix all statistical properties of the noise, without referring to any microscopic details of the bath. While the fluctuation dissipation theorem fixes only the second moment (correlation) of the noise, our present theorem extends to all moments. As a consequence, any linear thermal bath can be imitated by a harmonic oscillator bath model and the noise statistics is always Gaussian.

Semiclassical description of diffraction and its quenching by the forward–backward version of the initial value representation

It is shown that the forward–backward (FB) version of the semiclassical (SC) initial value representation (IVR) is able to describe quantum interference/coherence (i.e., diffraction) of particles transmitted by a two-slit potential. (In contrast, the linearized approximation to the SC-IVR, which leads to the classical Wigner model, is unable to do so.) FB-IVR calculations are also used to describe the (partial) quenching of this interference structure (i.e., “de-coherence”) when the two-slit potential is coupled to a bath of harmonic oscillators.

Radiating dipoles in photonic crystals

The radiation dynamics of a dipole antenna embedded in a photonic crystal are modeled by an initially excited harmonic oscillator coupled to a non-Markovian bath of harmonic oscillators representing the colored electromagnetic vacuum within the crystal. Realistic coupling constants based on the natural modes of the photonic crystal, i.e., Bloch waves and their associated dispersion relation, are derived. For simple model systems, well-known results such as decay times and emission spectra are reproduced. This approach enables direct incorporation of realistic band structure computations into studies of radiative emission from atoms and molecules within photonic crystals. We therefore provide a predictive and interpretative tool for experiments in both the microwave and optical regimes.

A time-dependent discrete variable representation method

We have developed a novel discrete variable representation (DVR) method where not only the amplitudes of the wave function at the DVR grid points can change but also the positions of these grid points can move as a function of time. Since the Gauss–Hermite basis set is used as the primitive basis functions (PBF) to construct the DVR basis set, the method appears as a semiclassical one with a small number of PBF but converges very fast to the quantum with an increasing PBF. We have investigated the dynamics of a reaction coordinate with or without coupling to a heat bath of harmonic oscillators to demonstrate the validity of the proposed method. The excellent agreement of the calculated tunneling probabilities with numbers obtained by traditional quantum grid method (FFT) and the fast computability of the present method compared to the latter are remarkable.

DEVIATION OF COHERENT STATE CAUSED BY DISSIPATION

The time evolution of a coherent state was studied in the dissipative system, a harmonic oscillator coupling with a bath of harmonic oscillators with Ohmic spectral density. We define a deviation of uncertainty relation versus the squeezed coherent state as [Formula: see text]. It is found that for the case of η ≪ ω0, namely, the weak dissipation, Δ oscillates with a small amplitude. The system is in a squeezed coherent state essentially and the dissipation only leads to a small deviation. For both strong (η ≫ ω0) and critical (η ~ ω0) dissipations, Δ is divergent with respect to t and the coherence of state is destroyed.

Molecular Dynamics with Nonadiabatic Transitions: A Comparison of Methods

Surface hopping (SH) and density matrix evolution (DME) methods which simulate the dynamics of quantum systems embedded in a classical environments are compared with exact quantum-dynamical calculations. These methods are applied to study the inelastic collisions of a classical particle with a five-level quantum harmonic oscillator. One-dimensional, two-state models representing electronic transitions are also treated. In addition, the methods are applied to the dynamics of a proton in a bistable potential bilinearly coupled to the bath of classical harmonic oscillators. Vibrational spectra calculated by both methods compare well with each other. The SH results are, in general, closer to the results of a full quantum treatment than the corresponding DME values. The DME method breaks down in the case of extended coupling with reflection at low energies.

Influence of population decay on the short time signal in four-wave mixing experiments

Oscillatory dephasing is a characteristic of non-Markovian relaxation phenomena when the coherence losses are essentially due to pure dephasing processes. In this paper, we wish to consider the case where the phase relaxation is essentially due to population decay. To this end, we calculate the transient four-wave mixing signal in the short pulse limit from a three-level sytem nondiagonally coupled to a bath of harmonic oscillators. We will show that no oscillatory dephasing is observed in the non-Markovian regime, only a nonexponential decay. Therefore, oscillatory dephasing appears to be a signature of pure dephasing processes.

Exactly solvable quantum model for electrochemical electron-transfer reactions.

We consider electron exchange between a metal electrode and a solvated reactant coupled to a harmonic oscillator bath. In the wide-band approximation the time development of the occupation probability for the reactant orbital can be calculated explicitly. From the behavior at long times we derive an expression for the reaction rate that is valid for all strengths of the electronic interaction between the metal and the reactant. The rate constant is related to the scattering matrix for electron exchange between a metal substrate and a scanning tunneling microscope via an electroactive adsorbate.

Fokker–Planck Equations as Scaling Limits of Reversible Quantum Systems

We consider a quantum particle moving in a harmonic exterior potential and linearly coupled to a heat bath of quantum oscillators. Caldeira and Leggett derived the Fokker–Planck equation with friction for the Wigner distribution of the particle in the large-temperature limit; however, their (nonrigorous) derivation was not free of criticism, especially since the limiting equation is not of Lindblad form. In this paper we recover the correct form of their result in a rigorous way. We also point out that the source of the diffusion is physically restrictive under this scaling. We investigate the model at a fixed temperature and in the large-time limit, where the origin of the diffusion is a cumulative effect of many resonant collisions. We obtain a heat equation with a friction term for the radial process in phase space and we prove the Einstein relation in this case.