Anharmonic Research Articles

Perturbation theory in a microcanonical ensemble

The microcanonical ensemble is a natural starting point of statistical mechanics. However, when it comes to perturbation theory in statistical mechanics, mainly the canonical and grand canonical ensembles have been used. In this article we show how the microcanonical ensemble can be directly used to carry out perturbation theory for both non-interacting and interacting systems. We obtain the first non-trivial order answers for the specific heat of anharmonic oscillators and for the virial expansion in real gases. They are in exact agreement with the results obtained from the canonical ensemble. In addition, we show how crossover functions for the specific heat of anharmonic oscillators can be constructed using a microcanonical ensemble and also how the subsequent terms of the virial expansion can be obtained. We find that if we consider quantum free particles in a one-dimensional box of extension L, then both the ensembles give an unusual result for the first correction to the specific heat in the high temperature limit.

Solving anharmonic oscillator with null states: Hamiltonian bootstrap and Dyson-Schwinger equations

Solving anharmonic oscillator with null states: Hamiltonian bootstrap and Dyson-Schwinger equations

Extension of the normal shock wave relations for calorically imperfect gases

An extension to the normal shock relations for a thermally perfect, calorically imperfect gas, modelling the vibrational excitation with an anharmonic oscillator model and including the influence of electronic modes, is derived and studied. Such additional considerations constitute an extension to the work achieved in the past, which modelled the caloric imperfections with a harmonic oscillator for vibrational energy and did not consider the effect of electronic energy. Additionally, the newly derived expressions provide physical insights into the limitations of experimentation for replicating flight conditions, which is demonstrated through providing solutions at different upstream temperatures. The results are compared with direct simulation Monte Carlo simulations for nitrogen and air, with the extent of the caloric imperfection of the gas showing excellent agreement. For low upstream temperatures, the extended relations are found to be in good agreement with the original normal shock wave expressions, but the results diverge for higher upstream temperatures that would be more representative of real flows. The results show that the new expressions depart from ideal gas theory for Mach numbers in excess of 4.9 at wind-tunnel conditions and for any Mach number above 3.0 at flight conditions. It is also shown that the traditional harmonic oscillator model and the anharmonic oscillator model begin to diverge at Mach number 3.0 for molecular oxygen gas and at Mach number 5.0 for an air mixture at flight conditions.

$$L^p$$-$$L^q$$ Boundedness of Fourier Multipliers Associated with the Anharmonic Oscillator

$$L^p$$-$$L^q$$ Boundedness of Fourier Multipliers Associated with the Anharmonic Oscillator

On the One‐Dimensional Transition State Theory and the Relation between Statistical and Deterministic Oscillation Frequencies of Anharmonic Energy Wells

AbstractThe transition state theory allows the development of approximated models useful to study the non‐equilibrium evolution of systems undergoing transformations between two states (e.g., chemical reactions). In a simplified 1D setting, the characteristic rate constants are typically written in terms of a temperature‐dependent characteristic oscillation frequency , describing the exploration of the phase space. As a particular case, this statistical oscillation frequency can be defined for an arbitrary convex potential energy well. This value is compared here with the deterministic oscillation frequency of the corresponding anharmonic oscillator. It is proved that there is a universal relationship between statistical and deterministic frequencies, which is the same for classical and relativistic mechanics. The independence of this relationship from the adopted physical laws gives it an interesting thermodynamic and pedagogical meaning. Several examples clarify the meaning of this relationship from both physical and mathematical viewpoints.

Quantum infinite square well as a limiting case of higher order anharmonic oscillator

We present a new type of continuous anharmonic potential and analyze its higher order asymptotic limit. The potential display infinite square well-like behavior when anharmonic order N goes to infinity. Consequently, wave functions and eigenvalues of ground and first excited states of anharmonic oscillators also asymptotically approach to those of infinite square well.

Non-Wiener Dynamics of an Ensemble of Identical Anharmonic Oscillators

Non-Wiener Dynamics of an Ensemble of Identical Anharmonic Oscillators

The non-Wiener dynamics of an ensemble of identical anharmonic oscillators

The model of an ensemble of anharmonic oscillators in the field of a common broadband photon-free thermostat is reduced to a model of an ensemble of identical N-level particles in terms of the algebraic perturbation theory. In one region the model describes the Dicke superadiance, and in the other one—oscillating dependence of the collective radiation intensity on the number of the ensemble oscillator. The collective radiation proves to be suppressed with a certain number of oscillators.

The effects of the variable mass on the electronic and nonlinear optical properties of octic anharmonic oscillators

The effects of the variable mass on the electronic and nonlinear optical properties of octic anharmonic oscillators

Reassignment of the Photoelectron Spectrum of Methylketene Using a Hybrid Model of Harmonic and Anharmonic Oscillators to Compute Franck-Condon Factors.

We constructed a hybrid model of harmonic and anharmonic oscillators to compute Franck-Condon factors and interpret the photoelectron spectrum of methylketene. The equilibrium structures of methylketene and its cation were optimized, and then, the harmonic and anharmonic vibrational frequencies were computed using the B3LYP, PBE0, APFD, and ωB97XD approaches of the density functional theory. The photoelectron spectrum of methylketene was simulated by computing the Franck-Condon factors with both the harmonic and hybrid models. The adiabatic ionization energy of methylketene was computed by using the CCSD(T) approach extrapolating to the complete basis set limit. The simulated photoelectron spectra are consistent with those from the experiment for both the harmonic and hybrid models. However, the error in band positions is reduced by using the hybrid model. The computed adiabatic ionization energies of methylketene are in agreement with the experiment, with the smallest error being 0.017 eV. Our interpretation based on the theoretical spectrum led to the reassignment of the experimental photoelectron spectrum of methylketene.

Thermal rectification in mass-asymmetric one-dimensional anharmonic oscillator lattices with and without a ballistic spacer

In this work we perform a systematic analysis of various structural parameters that have influence on the thermal rectification effect, i.e. asymmetrical heat flow, and the negative differential thermal resistance —reduction of the heat flux as the applied thermal bias is increased— present in a one-dimensional, segmented mass-graded system consisting of a coupled nearest-neighbor harmonic oscillator lattice (ballistic spacer) and two diffusive leads (modeled by a substrate potential) attached to the lattice at both boundaries. At variance with previous works, we consider the size of the spacer as smaller than that of the leads. Also considered is the case where the leads are connected along the whole length of the oscillator lattice; that is, in the absence of the ballistic spacer. Upon variation of the system’s parameters it was determined that the performance of the device, as quantified by the spectral properties, is largely enhanced in the absence of the ballistic spacer for the small system-size limit herein considered.

Simulations and theory of power spectral density functions for time-dependent and anharmonic Langevin oscillators.

Simulations and theory are presented for the power spectral density functions (PSDs) of particles in time-dependent and anharmonic potentials, including the effects of a thermal environment leading to damping and fluctuating forces. We investigate three one-dimensional perturbations to the harmonic oscillator of which two are time-dependent changes in the natural frequency of the oscillator, while the other is a time-independent extension of the quadratic potential to include a quartic term. We investigate the effect of these perturbations on two PSDs of the motion that are used in experiments on trapped nano-oscillators. We also derive and numerically test the PSDs for the motion of a spherical nanoparticle in a Paul trap. We found that the simple harmonic Langevin oscillator's PSDs are good approximations for the x and y coordinates' PSDs for small values of the parameter q of the Mathieu equation, but the difference can be more than a factor of two as "q" increases. We also numerically showed that the presence of a permanent electric dipole on the nanosphere can significantly affect the PSDs in the x and y coordinates.

Bootstrapping the gap in quantum spin systems

In this work we report on a new bootstrap method for quantum mechanical problems that closely mirrors the setup from conformal field theory (CFT). We use the equations of motion to develop an analogue of the conformal block expansion for matrix elements and impose crossing symmetry in order to place bounds on their values. The method can be applied to any quantum mechanical system with a local Hamiltonian, and we test it on an anharmonic oscillator model as well as the (1 + 1)-dimensional transverse field Ising model (TFIM). For the anharmonic oscillator model we show that a small number of crossing equations provides an accurate solution to the spectrum and matrix elements. For the TFIM we show that the Hamiltonian equations of motion, translational invariance and global symmetry selection rules imposes a rigorous bound on the gap and the matrix elements of TFIM in the thermodynamic limit. The bound improves as we consider larger systems of crossing equations, ruling out more finite-volume solutions. Our method provides a way to probe the low energy spectrum of an infinite lattice from the Hamiltonian rigorously and without approximation.

Continuation of the Stieltjes series to the large regime by finite-part integration

We devise a prescription to use a novel convergent expansion in the strong-asymptotic regime for the Stieltjes integral and its generalizations (Galapon EA. 2017 Proc. R. Soc A 473 , 20160567) to sum the associated divergent series of Stieltjes across all asymptotic regimes. The novel expansion makes use of the divergent negative-power moments, which we treated as Hadamard’s finite part integrals. The result allowed us to compute the ground state energy of the quartic, sextic anharmonic oscillators as well as the P T symmetric cubic oscillator, and the funnel potential across all perturbation regimes from a single expansion that is built from the divergent weak-coupling perturbation series and incorporates the known leading-order strong-coupling behaviour of the spectra.

Polynomial solutions of generalized quartic anharmonic oscillators

This paper deals with the partial solution of the energy eigenvalue problem for generalized symmetric quartic oscillators. Algebraization of the problem is achieved by expressing the Schrödinger operator in terms of the generators of a nilpotent group, which we call the quartic group. Energy eigenvalues are then seen to depend on the values of the two Casimir operators of the group. This dependence exhibits a scaling law which follows from the scaling properties of the group generators. Demanding that the potential gives rise to polynomial solutions in a particular Lie algebra element puts constraints on the four potential parameters, leaving only two of them free. For potentials satisfying such constraints, at least one of the energy eigenvalues and the corresponding eigenfunctions can be obtained in closed analytic form by pure algebraic means. With our approach, we extend the class of quasi-exactly solvable quartic oscillators which have been obtained in the literature by means of the more common textrm{sl}(2,{mathbb {R}}) algebraization. Finally, we show how solutions of the generalized quartic oscillator problem give rise to solutions for a charged particle moving in particular non-constant electromagnetic fields.

Partition functions of non-Lagrangian theories from the holomorphic anomaly

The computation of the partition function in certain quantum field theories, such as those of the Argyres-Douglas or Minahan-Nemeschansky type, is problematic due to the lack of a Lagrangian description. In this paper, we use the holomorphic anomaly equation to derive the gravitational corrections to the prepotential of such theories at rank one by deforming them from the conformal point. In the conformal limit, we find a general formula for the partition function as a sum of hypergeometric functions. We show explicit results for the round sphere and the Nekrasov-Shatashvili phases of the Ω background. The first case is relevant for the derivation of extremal correlators in flat space, whereas the second one has interesting applications for the study of anharmonic oscillators.

Entire hypergeometric approximants for the ground state energy perturbation series of the quartic, sextic and octic anharmonic oscillators

In resumming a divergent series, one is frequently confronted with situations where predictions from traditional resummation techniques might not accurate when considering strong-coupling values. Besides, in literature, we barely can find a resummation algorithm for which the convergence is faster for strongly-divergent ( Gevrey−k , k>1) series than a divergent one ( Gevrey−1 ). In this work, we use appropriate hypergeometric functions to approximate a divergent series ( n! large order growth factor) and strongly-divergent series ( (2n)! and (3n)! large order growth factors). We found that the convergence of the predicted non-perturbative parameters to their exact values is faster for the challenging Gevrey−3 than the Gevrey−2 series which in turn is faster than the divergent Gevrey−1 series. To explain such interesting feature, we suggest a type of analytic continuation where the parametrized divergent hypergeometric series is represented as a finite sum over entire hypergeometric functions. The algorithm is applied to sum the weak-coupling series of the ground state energy of the quartic (divergent), sextic (Gevrey−2) and Octic (Gevrey−3) anharmonic oscillators. Accurate results are obtained for the ground state energy as well as the predicted non-perturbative parameters. While traditional resummation techniques fail to give reliable results for large coupling values, our algorithm gives the expected limit as the coupling goes to infinity.

High-performance Kerr quantum battery

We propose and investigate the performance of a hybrid quantum battery, the so-called Kerr quantum battery, which consists of two interacting quantum oscillators, i.e., the charger is a harmonic oscillator and the battery is an anharmonic oscillator involving the Kerr nonlinearity. Such a setup creates nonuniform spacing between energy levels of the quantum oscillator that increases with the energy level. We find that the Kerr quantum battery can store more energy than the qubit battery and reaches maximum stored energy faster than the harmonic oscillator battery. In particular, the average charging power of the Kerr quantum battery is larger than the qubit battery. Furthermore, most of the stored energy in the Kerr quantum battery can be extracted for work. All of the properties of the Kerr quantum battery are controlled by the strength of nonlinearity, in which the enhancement of the nonlinearity transforms the battery from a harmonic oscillator to a qubit.

Quasi-classical simulations of resonance Raman spectra based on path integral linearization.

Based on a linearization approximation coupled with path integral formalism, we propose a method derived from the propagation of quasi-classical trajectories to simulate resonance Raman spectra. This method is based on ground state sampling followed by an ensemble of trajectories on the mean surface between the ground and excited states. The method was tested on three models and compared to a quantum mechanics solution based on a sum-over-states approach: harmonic and anharmonic oscillators and the HOCl molecule (hypochlorous acid). The method proposed is able to correctly characterize resonance Raman scattering and enhancement, including the description of overtones and combination bands. The absorption spectrum is obtained at the same time, and the vibrational fine structure can be reproduced for long excited state relaxation times. The method can also be applied to dissociating excited states (as is the case for HOCl).

Bayesian Estimation of Experimental Parameters in Stochastic Inertial Systems: Theory, Simulations, and Experiments with Objects Levitated in Vacuum

High-quality nanomechanical oscillators can sensitively probe force, mass, or displacement in experiments bridging the gap between the classical and quantum domain. Dynamics of these stochastic systems is inherently determined by the interplay between acting external forces, viscous dissipation, and random driving by the thermal environment. The relevance of inertia then dictates that both position and momentum must, in principle, be known to fully describe the system, which makes its quantitative experimental characterization rather challenging. We introduce a general method of Bayesian inference of the force field and environmental parameters in stochastic inertial systems that operates solely on the time series of recorded noisy positions of the system. The method is first validated on simulated trajectories of model stochastic harmonic and anharmonic oscillators with damping. Subsequently, the method is applied to experimental trajectories of particles levitating in tailored optical fields and used to characterize the dynamics of particle motion in a nonlinear Duffing potential, a static or time-dependent double-well potential, and a nonconservative force field. The presented inference procedure does not make any simplifying assumptions about the nature or symmetry of the acting force field and provides robust results with trajectories 2 orders of magnitude shorter than those typically required by alternative inference schemes. In addition to being a powerful tool for quantitative data analysis, it can also guide experimentalists in choosing appropriate sampling frequency (at least 20 measured points per single characteristic period) and length of the measured trajectories (at least 10 periods) to estimate the force field and environmental characteristics with a desired accuracy and precision.