Quantum Fluctuation-dissipation Theorem Research Articles

Unification of Thermal and Quantum Noises in Gravitational-Wave Detectors.

Contemporary gravitational-wave detectors are fundamentally limited by thermal noise-due to dissipation in the mechanical elements of the test mass-and quantum noise-from the vacuum fluctuations of the optical field used to probe the test-mass position. Two other fundamental noises can in principle also limit sensitivity: test-mass quantization noise due to the zero-point fluctuation of its mechanical modes and thermal excitation of the optical field. We use the quantum fluctuation-dissipation theorem to unify all four noises. This unified picture shows precisely when test-mass quantization noise and optical thermal noise can be ignored.

Quantum bounds and fluctuation-dissipation relations

In recent years, there has been intense attention on the constraints posed by quantum mechanics on the dynamics of the correlation at low temperatures, triggered by the postulation and derivation of quantum bounds on the transport coefficients or on the chaos rate. However, the physical meaning and the mechanism enforcing such bounds is still an open question. Here, we discuss the quantum fluctuation-dissipation theorem (the KMS conditions) as the principle underlying bounds on correlation time scales. By restating the problem in a replicated space, we show that the quantum bound to chaos is a direct consequence of the KMS condition, as applied to a particular pair of two-time correlation and response functions. Encouraged by this, we describe how quantum fluctuation-dissipation relations act in general as a blurring of the time-dependence of correlations, which can imply bounds on their decay rates. Thinking in terms of fluctuation-dissipation opens a direct connection between bounds and other thermodynamic properties.

Quantum fluctuation-dissipation theorem far from equilibrium

Fluctuations associated with relaxations in far-from-equilibrium regime is of fundamental interest for a large variety of systems within broad scales. Recent advances in techniques such as spectroscopy have generated the possibility for measuring the fluctuations of the mesoscopic systems in connection to the relaxation processes when driving the underlying quantum systems far from equilibrium. We present a general nonequilibrium Fluctuation-Dissipation Theorem (FDT) for quantum Markovian processes where the detailed-balance condition is violated. Apart from the fluctuations, the relaxation involves extra correlation that is governed by the quantum curl flux emerged in the far-from-equilibrium regime. Such a contribution vanishes for the thermal equilibrium, so that the conventional FDT is recovered. We finally apply the nonequilibrium FDT to the molecular junctions, elaborating the detailed-balance-breaking effects on the optical transmission spectrum. Our results have the advantage of and exceed the scope of the fluctuation-dissipation relation in the perturbative and near equilibrium regimes, and is of broad interest for the study of quantum thermodynamics.ation in the perturbative and near equilibrium regimes, and is of broad interest for the study of quantum thermodynamics.

Nuclear Quantum Effects in Liquid Water at Near Classical Computational Cost Using the Adaptive Quantum Thermal Bath.

We demonstrate the accuracy and efficiency of a recently introduced approach to account for nuclear quantum effects (NQEs) in molecular simulations: the adaptive quantum thermal bath (adQTB). In this method, zero-point energy is introduced through a generalized Langevin thermostat designed to precisely enforce the quantum fluctuation-dissipation theorem. We propose a refined adQTB algorithm with improved accuracy and report adQTB simulations of liquid water. Through extensive comparison with reference path integral calculations, we demonstrate that it provides excellent accuracy for a broad range of structural and thermodynamic observables as well as infrared vibrational spectra. The adQTB has a computational cost comparable to that of classical molecular dynamics, enabling simulations of up to millions of degrees of freedom.

The Quantum Coherence Induced by Geometric Curvature of Gauge Field in Non-equilibrium Quantum Dynamics

The study of non-equilibrium quantum dynamics receives attentions recently. However, the nature and effects of nonequilibirumness such as detailed-balance breaking and the relationship to the underlying intrinsic geometry is still unclear. In this study, we uncover that a gauge field will be induced by the nonequilibriumness in the coherence representation and furthermore its internal geometrical curvature is directly related to the degree of detailed balance breaking. The nonequilibriumness of the quantum system induces an intrinsic geometric curvature which can enhance the quantum coherence, leading to the possibility of the space time origin for the non-local quantum correlations or the possibility of curved space time emergence from nonequilibrium quantum dynamics. We also uncovered that the internal curvature of the gauge field provides a bridge to connect the generalized quantum fluctuation dissipation theorem to the fluctuation theorem and time irreversibility of the quantum dynamics. The quantum time irreversibility is due to the path dependent factor along any particular path in an internal curved space, which is analogous to the Wilson lines (or Wilson loops) in Abelian gauge theory. We also found that the steady state quantum coherence disappears when the non-trivial internal curvature vanishes for the quantum system coupled with environments. When the curvature is relatively small indicating weak detailed balance breaking, the coherence increases as curvature increases. The internal curvature can provide a general and direct quantitative measure of the detail-balance breaking for any quantum/classical non-equilibrium systems, even without knowing the underlying steady state distribution or the steady state flux. Using an example of two harmonic oscillators coupled to two environments with different temperatures, we explicitly show the dependence of the internal curvature and quantum coherence.

Axiomatic construction of quantum Langevin equations

A phenomenological construction of quantum Langevin equations, based on the physical criteria of (i) the canonical equal-time commutators, (ii) the Kubo formula, (iii) the virial theorem and (iv) the quantum fluctuation–dissipation theorem is presented. The case of a single harmonic oscillator coupled to a large external bath is analysed in detail. This allows to distinguish a Markovian semi-classical approach, due to Bedeaux and Mazur, from a non-Markovian full quantum approach, due to to Ford, Kac and Mazur. The quantum-fluctuation–dissipation theorem is seen to be incompatible with a Markovian dynamics. Possible applications to the quantum spherical model are discussed.

The Fluctuation-Dissipation Theorem as a Diagnosis and Cure for Zero-Point Energy Leakage in Quantum Thermal Bath Simulations.

Quantum thermal bath (QTB) simulations reproduce statistical nuclear quantum effects via a Langevin equation with a colored random force. Although this approach has proven efficient for a variety of chemical and condensed-matter problems, the QTB, as many other semiclassical methods, suffers from zero-point energy leakage (ZPEL). The absence of a reliable criterion to quantify the ZPEL without resorting to demanding comparisons with path integral-based calculations has so far hindered the use of the QTB for the simulation of real systems. In this work, we establish a quantitative connection between ZPEL in the QTB framework and deviations from the quantum fluctuation-dissipation theorem (FDT) that can be monitored along the simulation. This provides a rigorous general criterion to detect and quantify the ZPEL without any a priori knowledge of the system under study. We then use this criterion to build an adaptive QTB method that strictly enforces the quantum FDT at all frequencies via an on-the-fly, spectrally resolved fine-tuning of the system-bath coupling coefficients. The validity of the adaptive approach is first demonstrated on a simple two-oscillator model. It is then applied to two more realistic problems: the description of the vibrational properties of a model aluminum crystal at low temperature and the simulation of the liquid-solid phase transition in a 13-atom neon cluster. In both systems, the standard QTB results are strongly altered by the ZPEL, which can be essentially eliminated using the adaptive approach.

Nuclear quantum effects in molecular dynamics simulations

To take into account nuclear quantum effects on the dynamics of atoms, the path integral molecular dynamics (PIMD) method used since 1980s is based on the formalism developed by R. P. Feynman. However, the huge computation time required for the PIMD reduces its range of applicability. Another drawback is the requirement of additional techniques to access time correlation functions (ring polymer MD or centroid MD). We developed an alternative technique based on a quantum thermal bath (QTB) which reduces the computation time by a factor of ∼20. The QTB approach consists in a classical Langevin dynamics in which the white noise random force is replaced by a Gaussian random force having the power spectral density given by the quantum fluctuation-dissipation theorem. The method has yielded satisfactory results for weakly anharmonic systems: the quantum harmonic oscillator, the heat capacity of a MgO crystal, and isotope effects in 7LiH and 7LiD. Unfortunately, the QTB is subject to the problem of zero-point energy leakage (ZPEL) in highly anharmonic systems, which is inherent in the use of classical mechanics. Indeed, a part of the energy of the high-frequency modes is transferred to the low-frequency modes leading to a wrong energy distribution. We have shown that in order to reduce or even eliminate ZPEL, it is sufficient to increase the value of the frictional coefficient. Another way to solve the ZPEL problem is to combine the QTB and PIMD techniques. It requires the modification of the power spectral density of the random force within the QTB. This combination can also be seen as a way to speed up the PIMD.

Lattice-assisted proton hopping in oxides at low temperatures

Stimulated diffusion of protons in oxides such as ABO3 crystals and rutile TiO2 is discussed in the context of quantum Brownian motion. A self-consistent lattice-assisted proton hopping (LAPH) model is developed by going from white noise (characteristic of the standard stochastic theory of superionic conduction) to colored noise in the Markovian limit. This model differs from the commonly used ion jump models in that the hydrogen diffusion rate prefactor is identified as a quantity proportional to the frequency of phonon assistance. Application of the quantum fluctuation–dissipation theorem suggests that the dynamic activation energy for diffusion is a function of a bath-mode frequency. The LAPH model can predict enhanced rates of barrier jumping at room temperature compared to thermally activated proton diffusion. This indicates that low-temperature solid oxide devices are potential candidates for use in hydrogen energy research. The LAPH model offers a valid explanation for the mechanism of high protonic mobility recently observed for TiO2 in a picosecond transient pump-probe experiment. This unexpected dominant lattice relaxation channel must be considered as a new classical-like (but low-temperature) proton transfer mechanism. For vibration-assisted protonic jumps to occur at low temperature, the phonon assistance must be classified as a low-frequency vibration specific to each lattice.

QUANTUM FLUCTUATION DISSIPATION THEOREM REVISITED: REMARKS AND CONTRADICTIONS

The quantum fluctuation dissipation theorem (QFDT) in the Callen–Welton [ Phys. Rev.83 (1951) 34] form is critically revisited. We show that the role of the system eigenvalues is in general not correctly accounted for by the accepted form of the QFDT. As a consequence, a series of quantum results claimed in the literature, like the presence of zero point fluctuations, the violation of the quantum regression hypothesis, the non-white spectrum of the Langevin force, etc. emerge as a consequence of an incorrect application of the theorem. In this context the case of the single harmonic oscillator is illustrated as a typical example where the accepted form of the QFDT is proven to fail.

Second-Order Langevin Equation in Quantized Hamilton Dynamics

We derive a semi-classical Langevin equation that, in conjuction with a quantum fluctuation–dissipation theorem, can be used to run canonical dynamics and to determine thermal averages in the second-order quantized Hamilton dynamics (QHD-2). In contrast to our earlier work [J. Chem. Phys. 122 (2005) 234109], the bath is no longer treated classically, and the friction is represented at the second-order. As a result, the quantum fluctuation variables are kept close to their optimal values, similarly to the recently developed phase-space canonical averaging for QHD-2 [J. Chem. Phys. 126 (2007) 204108]. The theory is exact for the quantum harmonic oscillator and the free particle, and gives good agreement with the quantum answer for a variety of quartic potentials.

Quantum equations for vibrational dynamics on metal surfaces

A first principles treatment of the vibrational dynamics of molecular chemisorbates on metal surfaces is presented. It is shown that the mean field quantum evolution of the vibrational position operator is determined by a quantum Langevin equation with an electronic friction. In the mean field limit, the quantum noise and friction are related by the quantum fluctuation-dissipation theorem. The classical limit of this model is shown to agree with previously proposed models. A criterion is presented to describe the validity of the weak-coupling approximation and equations of motion for the dynamics in the presence of strong nonadiabatic coupling to electron-hole pairs are presented.

Comment on “Dynamic range of nanotube- and nanowire-based electromechanical systems” [Appl. Phys. Lett. 86, 223105 (2005)

We investigate the role of quantum effects (e.g. zero-point energy fluctuations) in the physics of nanotube- and nanowire-based electromechanical sensors as discussed in a recent article [Postma et al., Appl. Phys. Lett. 86, 223105 (2005)]. Employing the quantum fluctuation-dissipation theorem we find that these effects pose additional limits on the dynamic range of nanomechanical resonators.

Fundamental aspects of quantum Brownian motion

With this work we elaborate on the physics of quantum noise in thermal equilibrium and in stationary nonequilibrium. Starting out from the celebrated quantum fluctuation-dissipation theorem we discuss some important consequences that must hold for open, dissipative quantum systems in thermal equilibrium. The issue of quantum dissipation is exemplified with the fundamental problem of a damped harmonic quantum oscillator. The role of quantum fluctuations is discussed in the context of both, the nonlinear generalized quantum Langevin equation and the path integral approach. We discuss the consequences of the time-reversal symmetry for an open dissipative quantum dynamics and, furthermore, point to a series of subtleties and possible pitfalls. The path integral methodology is applied to the decay of metastable states assisted by quantum Brownian noise.

Quantum collective creep: A quasiclassical Langevin equation approach

The dynamics of an elastic medium driven through a random medium by a small applied force is investigated in the low-temperature limit where quantum fluctuations dominate. The motion proceeds via tunneling of segments of the manifold through barriers whose size grows with decreasing driving force f. At zero temperature and in the limit of small drive, the average velocity has the form $v\ensuremath{\propto}\mathrm{exp}[\ensuremath{-}\mathrm{const}\mathrm{}/{\ensuremath{\Elzxh}}^{\ensuremath{\alpha}}{f}^{\ensuremath{\mu}}].$ For strongly dissipative dynamics, there is a wide range of forces where the dissipation dominates and the velocity--force characteristics takes the form $v\ensuremath{\propto}\mathrm{exp}[\ensuremath{-}S(f)/\ensuremath{\Elzxh}],$ with $S(f)\ensuremath{\propto}{1/f}^{(d+2\ensuremath{\zeta})/(2\ensuremath{-}\ensuremath{\zeta})}$ the action for a typical tunneling event, the force dependence being determined by the roughness exponent \ensuremath{\zeta} of the d-dimensional manifold. This result agrees with the one obtained via simple scaling considerations. Surprisingly, for asymptotically low forces or for the case when the massive dynamics is dominant, the resulting quantum creep law is not of the usual form with a rate proportional to $\mathrm{exp}[\ensuremath{-}S(f)/\ensuremath{\Elzxh}];$ rather we find $v\ensuremath{\propto}\mathrm{exp}{\ensuremath{-}[S(f)/\ensuremath{\Elzxh}{]}^{2}}$ corresponding to \ensuremath{\alpha}=2 and $\ensuremath{\mu}=2(d+2\ensuremath{\zeta}\ensuremath{-}1)/(2\ensuremath{-}\ensuremath{\zeta}),$ with \ensuremath{\mu}/2 the naive scaling exponent for massive dynamics. Our analysis is based on the quasiclassical Langevin approximation with a noise obeying the quantum fluctuation-dissipation theorem. The many space and time scales involved in the dynamics are treated via a functional renormalization group analysis related to that used previously to treat the classical dynamics of such systems. Various potential difficulties with these approaches to the multi-scale dynamics---both classical and quantum---are raised and questions about the validity of the results are discussed.

Quantum fluctuation-dissipation theorem: a time-domain formulation

A time-domain formulation of the equilibrium quantum fluctuation–dissipation theorem (FDT) in the whole range of temperatures is presented. In the classical limit, the FDT establishes a proportionality relation between the dissipative part of the linear response function and the derivative of the corresponding equilibrium correlation function. At zero temperature, the FDT takes the form of Hilbert transform relations between the dissipative part of the response function and the corresponding symmetrized equilibrium correlation function, and thus allows to establish a connection with the analytic signal theory. The harmonic oscillator is treated as a basic example. The time-domain formulation of the relation between dissipation and fluctuations is especially valuable in out-of-equilibrium dynamics. We show on the specific example of the free particle quantum Brownian motion that it is the case in the discussion of the aging phenomena.

Aging effects in free quantum Brownian motion

The two-time correlation function C xx ( t, t′) of the displacement x( t)− x( t 0) of a free quantum Brownian particle with respect to its position at a given time t 0 is calculated analytically in the framework of the Caldeira and Leggett ohmic dissipation model. As a result, at any temperature T, C xx(t,t′) exhibits aging, i.e. it depends explicitly on both times t and t′ and not only on the time difference τ= t− t′, even in the limit of large age t′ (t 0⩽t′⩽t) , in contrast with a dynamic variable in equilibrium such as the particle velocity. The equilibrium quantum fluctuation-dissipation theorem (QFDT) has to be modified in order to relate the response function χ xx ( t, t′) to C xx ( t, t′), since this latter quantity takes into account even those fluctuations of the displacement which take place during the waiting time t w = t′− t 0. We describe the deviation from QFDT in terms of an effective inverse temperature β eff ( τ, t w ). The behaviour of this quantity as a function of τ for given values of T and t w is analysed. In the classical limit it is shown that β eff ( τ, t w )= βD( τ)/[ D( τ)+ D( t w )], where D( t) denotes the time-dependent diffusion coefficient.

Real-time nonequilibrium dynamics of quantum glassy systems

We develop a systematic analytic approach to aging effects in quantum disordered systems in contact with an environment. Within the closed-time path-integral formalism we include dissipation by coupling the system to a set of independent harmonic oscillators that mimic a quantum thermal bath. After integrating over the bath variables and averaging over disorder we obtain an effective action that determines the real-time dynamics of the system. The classical limit yields the Martin-Siggia-Rose generating functional associated to a colored noise. We apply this general formalism to a prototype model related to the $p$ spin-glass. We show that the model has a dynamic phase transition separating the paramagnetic from the spin-glass phase and that quantum fluctuations depress the transition temperature until a quantum critical point is reached. We show that the dynamics in the paramagnetic phase is stationary but presents an interesting crossover from a region controlled by the classical critical point to another one controlled by the quantum critical point. The most characteristic property of the dynamics in a glassy phase, namely aging, survives the quantum fluctuations. In the sub-critical region the quantum fluctuation-dissipation theorem is modified in a way that is consistent with the notion of effective temperatures introduced for the classical case. We discuss these results in connection with recent experiments in dipolar quantum spin-glasses and the relevance of the effective temperatures with respect to the understanding of the low temperature dynamics.

Quantum Aging in Mean-Field Models

We study the real-time dynamics of quantum models with long-range interactions coupled to a heat bath within the closed-time path-integral formalism. We show that quantum fluctuations depress the transition temperature. In the subcritical region there are two asymptotic time regimes with (i) stationary and (ii) slow aging dynamics. We extend the quantum fluctuation-dissipation theorem to the nonequilibrium case in a consistent way with the notion of an effective temperature that drives the system in the aging regime. The classical results are recovered for $\ensuremath{\Elzxh}\ensuremath{\rightarrow}0$.

Radiating electron: Fluctuations without dissipation in the equation of motion

The quantum fluctuation-dissipation theorem is generally manifested in the equation of motion through the appearance of a dissipative (frictional) term as well as a fluctuating force. Here it is shown that the equation of motion for a free electron interacting with the radiation field, when expressed in terms of the renormalized (observed) mass, displays fluctuations without dissipation. Despite this apparent contradiction, the equation is consistent with the fluctuation-dissipation theorem.