Intermolecular Forces and Generalized Response Functions in Liouville Space

The quasi-equilibrium behaviour of isolated nuclear spin systems in full and reduced Liouville spaces is discussed. We focus in particular on the reduced Liouville spaces used in the low-order correlations in Liouville space (LCL) simulation method, a restricted-spin-space approach to efficiently modelling the dynamics of large networks of strongly coupled spins. General numerical methods for the calculation of quasi-equilibrium expectation values of observables in Liouville space are presented. In particular, we treat the cases of a time-independent Hamiltonian, a time-periodic Hamiltonian (with and without stroboscopic sampling) and powder averaging. These quasi-equilibrium calculation methods are applied to the example case of spin diffusion in solid-state nuclear magnetic resonance. We show that there are marked differences between the quasi-equilibrium behaviour of spin systems in the full and reduced spaces. These differences are particularly interesting in the time-periodic-Hamiltonian case, where simulations carried out in the reduced space demonstrate ergodic behaviour even for small spins systems (as few as five homonuclei). The implications of this ergodic property on the success of the LCL method in modelling the dynamics of spin diffusion in magic-angle spinning experiments of powders is discussed.

The critical dynamics of charge-density waves (CDW's) just above the threshold for steady-state sliding motion are studied via a renormalization-group expansion in 4-\ensuremath{\epsilon} dimensions. This necessitates the simultaneous treatment of the long-wavelength low-frequency dynamics as well as the singular behavior associated with the jumps of local regions during which the local velocities become large. Scaling forms are obtained for response and correlation functions in the critical region, and exponent identities are derived. To leading order in \ensuremath{\epsilon}, the exponent \ensuremath{\zeta} of the CDW current ${\mathit{I}}_{\mathrm{CDW}}$\ensuremath{\sim}(V-${\mathit{V}}_{\mathit{T}}$${)}^{\mathrm{\ensuremath{\zeta}}}$ just above the threshold voltage ${\mathit{V}}_{\mathit{T}}$ is found to be \ensuremath{\zeta}=1-\ensuremath{\epsilon}/6+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$), in reasonable agreement with recent numerical simulations in two and three dimensions. The spatial correlations in the local velocities of the CDW decay exponentially with a correlation length, which diverges with an exponent \ensuremath{\nu} exactly equal to 1/2. Nevertheless, finite-size scaling of many quantities will be determined by a different exponent ${\ensuremath{\nu}}_{\mathrm{FS}}$=1/2+\ensuremath{\epsilon}/8+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$)=2/d+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$). The other exponents are related to \ensuremath{\zeta}, \ensuremath{\nu}, and ${\ensuremath{\nu}}_{\mathrm{FS}}$. Simple forms of long-wavelength low-frequency correlation and response functions in the moving phase are obtained to all orders in perturbation theory.

The dynamics of liquids is discussed with the help of time-dependent correlation functions. They are related to response functions by the fluctuation-dissipation theorem. This theorem enables to relate experimentally measured inelastic scattering data to the Fourier-transformed correlation and response functions. The Laplace transform of the correlation functions can be represented as a continuous fraction with suitable residual terms (memory functions). The projection formalism of Mori and Zwanzig is introduced.KeywordsCorrelation FunctionMemory FunctionRelaxation FunctionLiouville OperatorGeneralize Langevin EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Thermofield dynamics (TFD) arose out of efforts to define operators at finite temperatures in the theory of superconductivity.1 Since then it has developed into a powerful formalism for dealing with quantum field theories at finite temperatures2-4 and has found numerous applications in non-equilibrium phenomena5 and in quantum optics.6-15 The central idea in TFD is to represent a density operator as a vector in an extended Hilbert space. The same idea, in fact, also forms the basis of another formalism known as the Liouville space representation.16-19 The difference between the two formalisms is as follows. Consider a density operator ρ on a Hilbert space ℌ. Any operator on ℌ and, in particular, the density operator ρ can be expanded in terms of the operators |M > constitutes a complete orthonormal basis in ℌ. In TFD the operators |M > . In Liouville space representation,16-19 on the other hand, |M >< N| are regarded as the basis vectors in the Hilbert space ℌ of linear operators on ℌ. In what follows, we shall confine ourselves to the TFD representation and consider its application to quantum optics. In particular, we shall highlight its usefulness for exact or approximate solution of master equations encountered in quantum optics.

Dynamics in condensed phase for systems involving phase functions obeying Gaussian statistics

We study the mode-coupling approximation for the Kardar-Parisi-Zhang equation in the strong-coupling regime. By constructing an ansatz consistent with the asymptotic forms of the correlation and response functions we determine the upper critical dimension d(c) = 4 and the expansion z = 2-(d-4)/4+O((4-d)2) around dc. We find the exact z = 3/2 value in d = 1, and estimate the values z approximately 1.62, z approximately 1.78 in d = 2, 3. The result dc = 4 and the expansion around dc are very robust and can be derived just from a mild assumption on the relative scale on which the response and correlation functions vary as z approaches 2.

The excitation of a molecule in a medium, by absorption of a light pulse, is described in terms of density operators and the Liouville–von Neumann equation for the whole system. This is divided into a primary region containing the molecule and its close neighbors to which it may be bonded or strongly interacting, plus a secondary region including the remaining medium. A time-dependent self-consistent field procedure factors the density operator into primary and secondary components, each of which interacts with the applied light field. Direct absorption and indirect absorption are described with superoperators in a Liouville space formalism. Equations for the response of the primary region are developed assuming a stochastic medium. The treatment allows for dissipation due to collective electronic or vibrational excitations in the medium. Applications to photodesorption of molecules from metal surfaces are briefly discussed. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 367–375, 2000

Neural network quantum states such as ansatz wave functions have shown a great deal of promise for finding the ground state of spin models. Recently, work has focused on extending this idea to mixed states for simulating the dynamics of open systems. Most approaches so far have used a purification ansatz where a copy of the system Hilbert space is added, which when traced out gives the correct density matrix. Here we instead present an extension of the restricted Boltzmann machine which directly represents the density matrix in Liouville space. This allows the compact representation of states which appear in mean-field theory. We benchmark our approach on two different versions of the dissipative transverse-field Ising model, which show our ansatz is able to compete with other state-of-the-art approaches. Published by the American Physical Society 2024

A generalized response function is presented that can describe several of the practically important transient-response features of MOS systems. It allows for deviations from strict logarithmic time dependencies yet is mathematically tractable in performing linear response analysis. Fits of the generic response function to experimental data are discussed, including the short-term recovery due to hole transport, the long-term recovery due to trapped hole annealing, and the long-term, time-dependent buildup of interface traps. Analytic results for the convolution integral of linear response theory are derived for a square irradiation pulse, and some simple applications are discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Superconductivity exhibits elegant macroscopic quantum coherence in such a way that the many-body physics can be understood in a one-body way, described by the superconducting phase, and its quantum conjugate variable, the charge. When two superconductors are connected to each other through a tunnel junction, the charge tunneling can be controlled by the phase difference φ, leading to many interesting phenomena. For decades, because of the robustness of phase coherence in large junctions, a simple classical approach by modeling the system by a damped pendulum to the problem is successful and overwhelmed.(Tinkham 1996) However, as the sub-micro fabrication techniques had emerged in the 1990’s, ultra small junctions were found exhibiting stronger quantum fluctuations in phase due to charging effect, which cannot be overlooked. For exploring the novel phenomena in the opposite limit, people have made devices with robust charging effect. These phenomena can be well understood by treating the charge tunneling as a noncoherent perturbation to the quantum states described by charge. However, in the case when the Josephson energy and charging energy are competing, neither approach gives a satisfactory description. One of the attempts is to include the coherent nature in the charge tunneling processes by introducing a phase correlation function in time, which quantifies the robustness of the phase coherence.(Ingold & Nazarov 1991) The correlation function, which has been studied in many other fields, has a universal relation to the dissipation of the system, called fluctuation-dissipation theorem. Taken in this sense, dissipation is an important controlling parameter in the phase coherence robustness. If the environment impedance of the junction is much smaller than the quantum resistance (for Cooper pairs) RQ=h/4e2, the phase fluctuation is strongly damped, leading to pronounced phase coherence.(Devoret, Esteve et al. 1990) Indeed, why the classical model is so successful? The pioneer work done by Cadeira and Leggett(Caldeira & Leggett 1983) has pointed out that the environment impedance plays an important role. Their work stimulated many studies in dissipation-driven phase transitions in various systems including Josephson junctions.(Leggett, Chakravarty et al. 1987; Schon & Zaikin 1990) In this article, we will focus on the responses of Josephson junctions under microwave irradiation. Theoretically the phenomenon can be explained by the phase dynamics under an ac driving. In section 2, we will start from a classical picture by considering a Langevin

We present a diagrammatic formalism for the time correlation functions of density fluctuations for an excluded volume lattice gas on a simple d-dimensional hypercubic lattice. We consider a multicomponent system in which particles of different species can have different transition rates. Our theoretical approach uses a Hilbert space formalism for the time dependent dynamical variables of a stochastic process that satisfies the detailed balance condition. We construct a Liouville matrix consistent with the dynamics of the model to calculate both the equation of motion for multipoint densities in configuration space and the interactions in the diagrammatic theory. A Boley basis of fluctuation vectors for the Hilbert space is used to develop two formally exact diagrammatic series for the time correlation functions. These theoretical techniques are generalizations of methods previously used for spin systems and atomic liquids, and they are generalizable to more complex lattice models of liquids such as a lattice gas with attractive interactions or polymer models. We use our formalism to construct approximate kinetic theories for the van Hove correlation and self-correlation function. The most simple approximation is the mean field approximation, which is exact for the van Hove correlation function of a one component system but an approximation for the self-correlation function. We use our first diagrammatic series to derive a two site multiple scattering approximation that gives a simple analytic expression for the spatial Fourier transform of the self-correlation function. We employ our second diagrammatic series to derive a simple mode coupling type approximation that provides a system of equations that can be solved for the self-correlation function.

The calculation of dynamic response functions is expected to be an early application benefiting from rapidly developing quantum hardware resources. The ability to calculate real-time quantities of strongly correlated quantum systems is one of the most exciting applications that can easily reach beyond the capabilities of traditional classical hardware. Response functions of fermionic systems at moderate momenta and energies corresponding roughly to the Fermi energy of the system are a potential early application because the relevant operators are nearly local, and the energies can be resolved in moderately short real time, reducing the spatial resolution and gate depth required. This is particularly the case in quasielastic electron and neutrino scattering from nuclei, a topic of great interest in the nuclear and particle physics communities and directly related to experiments designed to probe neutrino properties. In this work we use current quantum hardware and error mitigation protocols to calculate response functions for a highly simplified nuclear model through calculations of a 2-point real time correlation function for a modified Fermi-Hubbard model in two dimensions with three distinguishable nucleons on four lattice sites

It is known that a uniformly accelerating detector gets excited in the inertial vacuum of a field, as if it were immersed in a thermal bath. We study the effect that the choice of the quantum state has on the response of the detector. The ``correlation function'' 〈\ensuremath{\Psi}\ensuremath{\Vert}\ensuremath{\varphi}(x)\ensuremath{\varphi}(x')\ensuremath{\Vert}\ensuremath{\Psi}〉 is computed in a Fock state, in the coherent state, and in the thermal state. It is then shown that the response function is a sum of two terms---the first being the same as the response in the vacuum state and a second term which is state dependent. The limit of the response function as \ensuremath{\Elzxh}\ensuremath{\rightarrow}0 is discussed and the response of the inertial and accelerated detectors compared in this limit. The response in the large-${n}_{\mathrm{k}}$ limit of the Fock state is compared with the response in the coherent state. We conclude that the thermal response of an accelerated detector in the inertial vacuum has interesting similarities to its response in other states.

This paper deals with an application of Monte Carlo Markov Chain. This Monte Carlo sampling technique was first introduced by Metropolis et al. to provide estimates of the statistical mechanical Boltzmann averages of parameters of a system of interacting particles. To illustrate an application of this method we have chosen the square spin-flip Ising model, i.e., a regular array of sites with each site occupied by a spin, which is allowed only two orientations. On such a model we have considered estimates of internal energy, magnetization, response functions and relaxation time, at the critical temperature and for different lattice sites. The problem of the accuracy of estimates is discussed in details and it has been shown how the moving block bootstrap can be applied successfully to assess standard errors and to construe confidence intervals for the computed averages. It can been seen also that, when correlation functions have a long-time behaviour, the moving block bootstrap works better than other methods based on subseries values.

The Markov approximation significantly simplifies formulations for the statistical moments of a wave propagating in a random medium. For the phase fluctuations, the Markov approximation is expected to be valid if the propagation range is much greater than the scale of largest inhomogeneities in a medium. In the atmospheric boundary layer, this scale can be several hundred meters, indicating that the Markov approximation might be inapplicable for relatively small ranges. In this paper, using geometrical acoustics, the correlation function and variance of the phase fluctuations of a plane sound wave are calculated without the Markov approximation and compared to previous results based on this approximation. The mean sound field and the spatial mutual coherence function (MCF) are also analyzed by expressing them in terms of the phase fluctuations. It is shown that for ranges smaller than the scale of largest inhomogeneities, the variance of the phase fluctuations is significantly smaller than that found with the Markov approximation. For large ranges, the relative difference between the two results tends to zero, while the absolute difference remains constant and can be much greater than unity. For the MCF, the Markov approximation is valid for both small and large ranges.