Quantum Anharmonic Oscillator Research Articles

Derivation of Miller’s rule for the nonlinear optical susceptibility of a quantum anharmonic oscillator

Miller’s rule is an empirical relation between the nonlinear and linear optical coefficients that applies to a large class of materials but has only been rigorously derived for the classical Lorentz model with a weak anharmonic perturbation. In this work, we extend the proof and present a detailed derivation of Miller’s rule for an equivalent quantum-mechanical anharmonic oscillator. For this purpose, the classical concept of velocity-dependent damping inherent to the Lorentz model is replaced by an adiabatic switch-on of the external electric field, which allows a unified treatment of the classical and quantum-mechanical systems using identical potentials and fields. Although the dynamics of the resulting charge oscillations, and hence the induced polarizations, deviate due to the finite zero-point motion in the quantum-mechanical framework, we find that Miller’s rule is nevertheless identical in both cases up to terms of first order in the anharmonicity. With a view to practical applications, especially in the context of ab initio calculations for the optical response where adiabatically switched-on fields are widely assumed, we demonstrate that a correct treatment of finite broadening parameters is essential to avoid spurious errors that may falsely suggest a violation of Miller’s rule, and we illustrate this point by means of a numerical example.

Squeezing and bunching of photons of coherent light interacting with two quantum anharmonic oscillators coupled through the angular momentum

Squeezing and bunching of photons of coherent light interacting with two quantum anharmonic oscillators coupled through the angular momentum

Description of CuInP<sub>2</sub>S<sub>6</sub> ferrielectrics in a mixed Ising model

The appearance of spontaneous polarization in CuInP2S6 ferrielectrics is related to the second order Jahn-Teller effect for copper cations located in a double-well local potential, the stereoactivity of indium cations located in a three-well local potential, as well as the valence fluctuations of phosphorus cations. The paraelectric to ferrielectric phase transition is primarily determined by the coupling of indium cations with their surroundings. This transition can be analyzed using the mixed Ising model with spins s = 1/2 and S = 1. The spectrum of pseudospin fluctuations at different temperatures was calculated using a mean-field approach for a set of quantum anharmonic oscillators. The results were then compared with Raman spectroscopy data for CuInP2S6 crystal. The analysis indicates that the lattice anharmonicity below 150 K, is mainly determined by the indium sublattice, leading to the coexistence of the glassy state and ferrielectric phase. Above 150 K, the anharmonicity of the copper sublattice activates the ionic conductivity and results in the existence of a long-ranged fluctuated cluster of spontaneous polarization in a temperature interval of the paraelectric phase above TC.

Ordering/displacive ferrielectricity in 2D CuInP2S6

Using DFT-based molecular dynamics simulation of Cu+ cations flipping and In3+ cations displacive dynamics, we clarify the dipole ordering of CuInP2S6 ferrielectrics through the second order Jahn-Teller effect which determines the double-well potential for copper cations as well as three-well potential for indium cations inside the structural layers. The temperature dependence of the spatial distribution of Cu+ and In3+ cations is also described within the quantum anharmonic oscillator model. In3+ cations play a decisive role in the character of the polar ordering: at ambient or for positive pressures, the first order transition occurs; with the rise of negative compression, the ferrielectric transition evolves to the second order. The peculiarities of spontaneous polarization switching are related to the contributions of copper and indium cationic sublattices.

Involvement of cationic sublattices in multiple polarization states in CuInP2S6 ferrielectrics

The contribution of copper and indium cationic sublattices on polarization hysteresis loops in CuInP2S6 van der Waals ferrielectrics has been studied within the quantum anharmonic oscillator model that considers the ordering dynamics of Cu+ cations in the double-well local potential and displacive dynamics of In3+ cations in the three-well shaped potential. The multiple polarization states are manifested as single, double, and triple P(E) hysteresis loops. The triple loops originate from a field-induced ferrielectric to ferroelectric phase transition within a certain range of the phase diagram with continuous increase in the spontaneous polarization below the second-order transition from the paraelectric phase to the ferrielectric state.

Investigating Single Quantum Anharmonic Oscillator with Perturbation Theory

Investigating Single Quantum Anharmonic Oscillator with Perturbation Theory

Approach to Equilibrium of Statistical Systems: Classical Particles and Quantum Fields Off-Equilibrium

Non-equilibrium evolution at absolute temperature T and approach to equilibrium of statistical systems in long-time (t) approximations, using both hierarchies and functional integrals, are reviewed. A classical non-relativistic particle in one spatial dimension, subject to a potential and a heat bath (hb), is described by the non-equilibrium reversible Liouville distribution (W) and equation, with a suitable initial condition. The Boltzmann equilibrium distribution Weq generates orthogonal (Hermite) polynomials Hn in momenta. Suitable moments Wn of W (using the Hn’s) yield a non-equilibrium three-term hierarchy (different from the standard Bogoliubov–Born–Green–Kirkwood–Yvon one), solved through operator continued fractions. After a long-t approximation, the Wn’s yield irreversibly approach to equilibrium. The approach is extended (without hb) to: (i) a non-equilibrium system of N classical non-relativistic particles interacting through repulsive short range potentials and (ii) a classical ϕ4 field theory (without hb). The extension to one non-relativistic quantum particle (with hb) employs the non-equilibrium Wigner function (WQ): difficulties related to non-positivity of WQ are bypassed so as to formulate approximately approach to equilibrium. A non-equilibrium quantum anharmonic oscillator is analyzed differently, through functional integral methods. The latter allows an extension to relativistic quantum ϕ4 field theory (a meson gas off-equilibrium, without hb), facing ultraviolet divergences and renormalization. Genuine simplifications of quantum ϕ4 theory at high T and large distances and long t occur; then, through a new argument for the field-theoretic case, the theory can be approximated by a classical ϕ4 one, yielding an approach to equilibrium.

Towards learning optimized kernels for complex Langevin

We present a novel strategy aimed at restoring correct convergence in complex Langevin simulations. The central idea is to incorporate system-specific prior knowledge into the simulations, in order to circumvent the NP-hard sign problem. In order to do so, we modify complex Langevin using kernels and propose the use of modern auto-differentiation methods to learn optimal kernel values. The optimization process is guided by functionals encoding relevant prior information, such as symmetries or Euclidean correlator data. Our approach recovers correct convergence in the non-interacting theory on the Schwinger-Keldysh contour for any real-time extent. For the strongly coupled quantum anharmonic oscillator we achieve correct convergence up to three-times the real-time extent of the previous benchmark study. An appendix sheds light on the fact that for correct convergence not only the absence of boundary terms, but in addition the correct Fokker-Plank spectrum is crucial.

Simulating the Dynamics of Stock Price via PSO-Assisted Quantum Anharmonic Oscillator Model: Case of Jakarta Composite Index

The Jakarta Composite Index (JCI) stock price dynamics have been modeled by a quantum anharmonic oscillator and the PSO (Particle Swarm Optimization) Algorithm. Some of the constants that affect the probability density of return are the ability of the market makers to control the market (γ), the behaviour of contrarians and the trend followers to the price return (c), and the investor behaviour towards perceived volatility (k). The simulation results have produced the slightest error of the JCI at 8.36% for the opportunity density and 3.6% for the stock price returns. Forward prediction for the next three months using the exponential smoothing method resulted in a 17.77% error in the opportunity density of the stock price return and a 10.6% error in the stock price return. Based on those results, it can be concluded that the stock price dynamics can be modelled using an anharmonic quantum oscillator where the value of liquidity and volatility in the previous period affects the investor and the stock's price return in the next period.

Monte Carlo simulations for quantum harmonic and anharmonic oscillators within epistemically restricted phase-space representation

We investigate Monte Carlo simulations within the epistemically-restricted phase-space (ERPS) representation of quantum mechanics for harmonic and anharmonic oscillators. Compared to the variational Monte Carlo (VMC) simulations in the conventional representation, the calculation time in the ER phase-space representation is quite competitive. We also numerically show that the distribution function of the global random variable on the order of Planck’s constant in the ER phase-space representation does not significantly affect the energy accuracy, yet a particular choice of the distribution function may give a better calculation time. This work gives an alternative approach to the Monte Carlo simulations beyond those in the conventional representation of quantum mechanics.

First and Second-Order Energy Eigenvalues of One-Dimensional Quantum Harmonic and Anharmonic Oscillator with Linear, Quadratic, Cubic and Polynomial Perturbation Potential

This work is aimed at obtaining the energy eigenvalues for one-dimensional quantum harmonic and anharmonic oscillators perturbed by linear, quadratic, cubic and polynomial potentials. To obtain the solutions of the energy eigenvalues, we employed the time-independent perturbation theory to calculate the first and the second-order energy correction, which we used to obtain the complete generalised energy eigenvalues of the quantum harmonic oscillators with linear, quadratic, cubic and polynomial perturbation potential of the same unperturbed Hamiltonian (H).

Vacuum energy and renormalization of the field-independent term

Due to its construction, the nonperturbative renormalizationgroup (RG) evolution of the constant, field-independent term (which is constantwith respect to field variations but depends on the RG scale k) requiresspecial care within the Functional Renormalization Group (FRG) approach.In several instances, the constant term of the potential has nophysical meaning. However, there are special cases where it receives importantapplications. In low dimensions (d = 1), in a quantum mechanical model, thisterm is associated with the ground-state energy of the anharmonic oscillator.In higher dimensions (d = 4), it is identical to the Λ termof the Einstein equationsand it plays a role in cosmic inflation. Thus, instatistical field theory, in flat space, the constant term could be associatedwith the free energy, while in curved space, it could be naturally associatedwith the cosmological constant. It is known that one has to use a subtractionmethod for the quantum anharmonic oscillator in d = 1 to remove the k 2term that appears in the RGflow in its high-energy (UV) limit in order to recover thecorrect results for the ground-state energy. The subtraction is needed becausethe Gaussian fixed point is missing in the RG flow once the constant term isincluded. However, if the Gaussian fixed point is there, no furthersubtraction is required. Here, we propose a subtraction method for k 4 andk 2 terms of the UV scaling of the RG equations for d = 4 dimensions if theGaussian fixed point is missing in the RG flow with the constant term. Finally,comments on the application of our results to cosmological models are provided.

Quantum Anharmonic Oscillators: A Truncated Matrix Approach

This study aims at implementing a truncated matrix approach based on harmonic oscillator eigenfunctions to calculate energy eigenvalues of anharmonic oscillators containing quadratic, quartic, sextic, octic, and decic anharmonicities. The accuracy of the matrix method is also tested. Using this method, the wave functions of the anharmonic oscillators were written as a linear combination of some finite number of harmonic oscillator basis states. Results showed that calculation with 100 basis states generated accurate energies of oscillators with relatively small coupling constants, with computation time less than 1 minute. Including more basis states could result in more correct digits. For instance, using 300 harmonic oscillator basis states in a simple Mathematica code in about 8 minutes, highly accurate energies of the oscillators were obtained for relatively small coupling constants, with up to 15 correct digits. Reasonable accuracy was also found for much larger coupling constants with at least three correct digits for some low lying energies of the oscillators reported in this study. Some of our results contained more correct digits than other results reported in the literature.

Энергетические характеристики системы квантовых ангармонических осцилляторов

Энергетические характеристики системы квантовых ангармонических осцилляторов

Stable solvers for real-time Complex Langevin

This study explores the potential of modern implicit solvers for stochastic partial differential equations in the simulation of real-time complex Langevin dynamics. Not only do these methods offer asymptotic stability, rendering the issue of runaway solution moot, but they also allow us to simulate at comparatively large Langevin time steps, leading to lower computational cost. We compare different ways of regularizing the underlying path integral and estimate the errors introduced due to the finite Langevin time steps. Based on that insight, we implement benchmark (non-)thermal simulations of the quantum anharmonic oscillator on the canonical Schwinger-Keldysh contour of short real-time extent.

Laser-Driven Anharmonic Oscillator: Ground-State Dissociation of the Helium Hydride Molecular Ion by Midinfrared Pulses.

The vibrational motion of molecules represents a fundamental example of an anharmonic oscillator. Using a prototype molecular system, HeH^{+}, we demonstrate that appropriate laser pulses make it possible to drive the nuclear motion in the anharmonic potential of the electronic ground state, increasing its energy above the potential barrier and facilitating dissociation by purely vibrational excitation. We find excellent agreement between the frequency-dependent response of the helium hydride molecular cation to both classical and quantum mechanical simulations, thus removing any ambiguities through electronic excitation. Our results provide access to the rich dynamics of anharmonic quantum oscillator systems and pave the way to state-selective control schemes in ground-state chemistry by the adequate choice of the laser parameters.

Anharmonic oscillator: a solution

It is shown that for the one-dimensional quantum anharmonic oscillator with potential V(x) = x 2 + g 2 x 4 the perturbation theory (PT) in powers of g 2 (weak coupling regime) and the semiclassical expansion in powers of ℏ for energies coincide. It is related to the fact that the dynamics in x-space and in (gx)-space corresponds to the same energy spectrum with effective coupling constant ℏg 2. Two equations, which govern the dynamics in those two spaces, the Riccati–Bloch (RB) and the generalized Bloch (GB) equations, respectively, are derived. The PT in g 2 for the logarithmic derivative of wave function leads to PT (with polynomial in x coefficients) for the RB equation and to the true semiclassical expansion in powers of ℏ for the GB equation, which corresponds to a loop expansion for the density matrix in the path integral formalism. A two-parametric interpolation of these two expansions leads to a uniform approximation of the wavefunction in x-space with unprecedented accuracy ∼10−6 locally and unprecedented accuracy ∼10−9–10−10 in energy for any g 2 ⩾ 0. A generalization to the radial quartic oscillator is briefly discussed.

A Model Effective Mass Quantum Anharmonic Oscillator and Its Thermodynamic Characterization

Using a model anharmonic oscillator with asymptotically decreasing effective mass to study the effect of compositional grading on the quantum mechanical properties of a semiconductor heterostructure, we determine the exact bound states and spectral values of the system. Furthermore, we show that ordering ambiguity only brings about a spectral shift on the quantum anharmonic oscillator with spatially varying effective mass. A study of thermodynamic properties of the system reveals a resonance condition dependent on the magnitude of the anharmonicity parameter. This resonance condition is seen to set a critical value on the said parameter beyond which a complex valued entropy which is discussed, emerges.

Energy Eigenvalues of Quantum Anharmonic Oscillators: Exact Expression for the Pure λx2m Oscillators and a Simple Expression for x2 + λx2m Oscillators

We propose an exact expression for the energy eigenvalues of the pure λx2m quantum anharmonic oscillators. The only information needed for the purpose is K0(m,n), which is the first term of strong-coupling expansion for oscillator with potential λx2m for the nth excited state. We then propose simple expression for the eigenvalues of the x2 + λx2m oscillators which reproduce both the ground state and excited states energies of these oscillators with a fairly good accuracy. The present formula is found to better reproduce the energy eigenvalues than those calculated by earlier authors.

Numerical simulation of quantum anharmonic oscillator, embedded within an infinite square well potential, by matrix methods using Gnumeric spreadsheet

The quantum anharmonic oscillator has been solved numerically using matrix diagonalization technique. The interaction potential consisting of quadratic () and quartic (λx4) terms is embedded within an infinite square well potential of appropriate width, ‘a’ and its sine eigen functions are used as basis functions ‘N’ for the employed matrix method. The energy eigen values for the resultant Hamiltonian are solved in a free open source software (FOSS), Gnumeric, a simple worksheet environment. The numerical parameters ‘a’ and ‘N’ are optimized to converge to the expected energies for harmonic oscillator and those for anharmonic oscillator from perturbation theory for small values of physical parameter, ‘λ’. The pure quartic oscillator is studied for both small and large values of λ and validated with results obtained from other numerical techniques. The breakdown of perturbation approximation for large values of λ is also shown.