A Simple Path Integration For the Time Dependent Oscillator

Path integral approach to driven quantum harmonic oscillator using Markov chain Monte Carlo methods

We have simulated the ground states of quantum harmonic oscillators driven either by constant forces of different magnitudes or time-dependent driving forces. The expectation values of position for various combinations of mass, natural angular frequency, and the coupling constant $λ$ were calculated for both driving modes. For constant forcing, coherent states were obtained. The results for both forcing scenarios match the theoretically expected values almost exactly. For the simulations, the Metropolis algorithm was implemented on a discrete time lattice to evaluate the imaginary time path in…

Path Integration for the Plane Pendulum with Finite Amplitude

Exact path integration for the one dimensional potential $V=b^2\cos 2q$ which describes the finite amplitude pendulum is presented.

Tailoring the correlation and anti-correlation behavior of path-entangled photons in Glauber-Fock oscillator lattices

We demonstrate that single-photon as well as biphoton revivals are possible in a new class of dynamic optical systems-the so-called Glauber-Fock oscillator lattices. In these arrays, both Bloch-like oscillations and dynamic delocalization can occur which can be described in closed form. The bunching and anti-bunching response of path-entangled photons can be pre-engineered in such coupled optical arrangements and the possibility of emulating Fermionic behavior in this family of lattices is also considered. We elucidate these effects via pertinent examples and we discuss the prospect of experim…

Generalizations to Feynman's Path Integration Methods in One Dimension

This paper reviews and generalizes Feynman's path integration methods which use time slicing with straight line segments and Fourier sine series. The generalizations are done from variational calculus considerations and in one dimension for simplicity in demonstrating concepts.

Path Integrals and Their Application to Dissipative Quantum Systems

Introduction Path Integrals - Introduction - Propagator - Free Particle - Path Integral Representation of Quantum Mechanics - Particle on a Ring - Particle in a Box - Driven Harmonic Oscillator - Semiclassical Approximation - Imaginary Time Path Integral Dissipative Systems - Introduction - Environment as Collection of Harmonic Oscillators - Effective Action Damped Harmonic Oscillator - Partition Function - Ground State Energy and Density of States - Position Autocorrelation Function

Stochastic Path Integral Analysis of the Continuously Monitored Quantum Harmonic Oscillator

We consider the evolution of a quantum simple harmonic oscillator in a general Gaussian state under simultaneous time-continuous weak position and momentum measurements. We deduce the stochastic evolution equations for position and momentum expectation values and the covariance matrix elements from the system's characteristic function. By generalizing the Chantasri-Dressel-Jordan (CDJ) formalism (Chantasri et al.~2013 and 2015) to this continuous variable system, we construct its stochastic Hamiltonian and action. Action extremization gives us the equations for the most-likely readout paths an…

Path Integration on Darboux Spaces

In this paper the Feynman path integral technique is applied to two-dimensional spaces of non-constant curvature: these spaces are called Darboux spaces $\DI$--$\DIV$. We start each consideration in terms of the metric and then analyze the quantum theory in the separable coordinate systems. The path integral in each case is formulated and then solved in the majority of cases, the exceptions being quartic oscillators where no closed solution is known. The required ingredients are the path integral solutions of the linear potential, the harmonic oscillator, the radial harmonic oscillator, the mo…

Extended Feynman Formula for the Harmonic Oscillator by the Discrete Time Method

We calculate the Feynman formula for the harmonic oscillator beyond and at caustics by the discrete formulation of path integral. The extension has been made by some authors, however, it is not obtained by the method which we consider the most reliable regularization of path integral. It is shown that this method leads to the result with, especially at caustics, more rigorous derivation than previous.

用于描述谐振子及其他一维问题本征态的路径分布

Path distributions for describing eigenstates of the harmonic oscillator and other 1-dimensional problems

本文通过重新表示波函数的积分表达式为路径端点间运动的特征动量形式，利用驻相分析推广方法得到精确描述谐振子及其他一维问题本征态的路径分布，且该分布对所有传播时间有效。

A physical basis for the phase in Feynman path integration

In the path integral formulation of quantum mechanics, the phase factor Exp[iS(x[t])] is associated with every path x[t]. Summing this factor over all paths yields Feynman's propagator as a sum-over-paths. In the original formulation, the complex phase was a mathematical device invoked to extract wave behaviour in a particle framework. In this paper we show that the phase itself can have a physical origin in time reversal, and that the propagator can be drawn by a single deterministic path.

Nonclassical phase-space trajectories for the damped harmonic quantum oscillator

The phase-space path-integral approach to the damped harmonic oscillator is analyzed beyond the Markovian approximation. It is found that pairs of nonclassical trajectories contribute to the path-integral representation of the Wigner propagating function. Due to the linearity of the problem, the sum coordinate of a pair still satisfies the classical equation of motion. Furthermore, it is shown that the broadening of the Wigner propagating function of the damped oscillator arises due to the time-nonlocal interaction mediated by the heat bath.

Path Integration on a Quantum Computer

We study path integration on a quantum computer that performs quantum summation. We assume that the measure of path integration is Gaussian, with the eigenvalues of its covariance operator of order j^{-k} with k>1. For the Wiener measure occurring in many applications we have k=2. We want to compute an $\e$-approximation to path integrals whose integrands are at least Lipschitz. We prove: 1. Path integration on a quantum computer is tractable. 2. Path integration on a quantum computer can be solved roughly $\e^{-1}$ times faster than on a classical computer using randomization, and exponential…

On the path integration for the potential barrier $V_{0}\cosh ^{-2}(ωx)$

The propagator associated to the potential barrier $V=V_{0}\cosh ^{-2}(ωx)$ is obtained by solving path integrals. The method of delta functionals based on canonical and other transformations is used to reduce the path integral for this potential into a path integral for the Morse potential problem. The dimensional extension technique is seen to be essential for performing the multiple integral representation of the propagator. The correctly normalized scattering wave functions and the scattering function are derived. To test the method employed, the free particle and the $δ-$function barrier …

Cryogenic sapphire oscillator with exceptionally high long-term frequency stability

We report on the development of a sapphire cryogenic microwave resonator oscillator long-term fractional frequency stability of 2x10^-17Sqrt[τ] for integration times τ>10^3 s and negative drift of about 2.2x10^-15/day. The short-term frequency instability of the oscillator is highly reproducible and also state-of-the-art: 5.6x10^-16 for an integration time of τ~ 20 s.

Quantum correlation in degenerate optical parametric oscillators with mutual injections

We theoretically and numerically study the quantum dynamics of two degenerate optical parametric oscillators with mutual injections. The cavity mode in the optical coupling path between the two oscillator facets is explicitly considered. Stochastic equations for the oscillators and mutual injection path based on the positive $P$ representation are derived. The system of two gradually pumped oscillators with out-of-phase mutual injections is simulated, and its quantum state is investigated. When the incoherent loss of the oscillators other than the mutual injections is small, the squeezed quadr…

Scheme to measure squeezing and phase properties of a harmonic oscillator

We propose a simple scheme to measure squeezing and phase properties of a harmonic oscillator. We treat in particular the case of a the field, but the scheme may be easily realized in ion traps. It is based on integral transforms of measured atomic properties as atoms exit a cavity. We show that by measuring atomic polarizations it is possible, after a given integration, to measure several properties of the field.

Conditional generation of path-entangled optical NOON states

We propose a measurement protocol to generate path-entangled NOON states conditionally from two pulsed type II optical parametric oscillators. We calculate the fidelity of the produced states and the success probability of the protocol. The trigger detectors are assumed to have finite dead time, and for short pulse trigger fields they are modeled as on/off detectors with finite efficiency. Continuous-wave operation of the parametric oscillators is also considered.

谐振子路径积分蒙特卡洛热力学能与哈密顿能量的解析评估

Analytical evaluations of the Path Integral Monte Carlo thermodynamic and Hamiltonian energies for the harmonic oscillator

本文利用谐振子的普适离散虚时传播子解析评估热力学能与哈密顿能量，证明哈密顿能量可优化至十二阶并在三个珠子内收敛，同时揭示二阶原始近似传播子计算热力学能时收敛极慢。

Relaxation of two coupled quantum oscillators to quasi-equilibrium states based on path integrals

The paper addresses the problem of relaxation of open quantum systems. Using the path integral methods we found an analytical expression for time-dependent density matrix of two coupled quantum oscillators interacting with different baths of oscillators. The expression for density matrix was found in the linear regime with respect to the coupling constant between selected oscillators. Time-dependent spatial variances and covariance were investigated analytically and numerically. It was shown that asymptotic variances in the long-time limit are always in accordance with the fluctuation dissipat…

Path Integration on Hermitian Hyperbolic Space

In this paper the path integral technique is applied to the quantum motion on the Hermitian hyperbolic space HH(2). The Schrödinger equation on this space separates in 12 coordinate systems which are closely related to the coordinate systems on the two-dimensional hyperboloid. For six coordinate systems out of the twelve it is possible to find a path integral solution.

The Berry Phase for Simple Harmonic Oscillators

We evaluate the Berry phase for a "missing" family of the square integrable wavefunctions for the linear harmonic oscillator, which cannot be derived by the separation of variables (in a natural way). Instead, it is obtained by the action of the maximal kinematical invariance group on the standard solutions. A simple closed formula for the phase (in terms of elementary functions) is found by integration with the help of a computer algebra system.

实时路径积分的高效评估

Efficient evaluation of real-time path integrals

针对实时路径积分高维振荡导致的数值困难，提出将高维积分分解为低维振荡积分，结合Picard-Lefschetz或程函近似与快速傅里叶变换，高效评估量子力学、量子场论及量子引力中的路径积分。

On the equivalence of the Pais-Uhlenbeck oscillator model and two non-Hermitian Harmonic Oscillators

A system of two independent Bosonic Harmonic Oscillators is converted into the respective fourth-order derivative Pais-Uhlenbeck oscillator model. The conversion procedure displays transparently how the quantization of the fourth-order derivative Pais-Uhlenbeck oscillator has to be performed in order not to suffer from the divergence problems of the vacuum state and path integrals as conjectured most recently by P. D. Mannheim in his article ``Determining the normalization of the quantum field theory vacuum, with implications for quantum gravity" [arXiv:2301.13029 [hep-th]]. In order to make t…

通过有限路径积分极限获得收敛的微扰级数：应用于各向异性振子的强耦合能量

Convergent perturbative series via finite path integral limits: application to energy at strong coupling of the anharmonic oscillator

通过有限路径积分极限获得收敛的微扰级数，应用于各向异性振子强耦合能量计算，结果在强耦合下与精确值一致。

Path Integration in Conical Space

Quantum mechanics in conical space is studied by the path integral method. It is shown that the curvature effect gives rise to an effective potential in the radial path integral. It is further shown that the radial path integral in conical space can be reduced to a form identical with that in flat space when the discrete angular momentum of each partial wave is replaced by a specific non-integral angular momentum. The effective potential is found proportional to the squared mean curvature of the conical surface embedded in Euclidean space. The path integral calculation is compatible with the S…

Path integration in the field of a topological defect: the case of dispiration

The motion of a particle in the field of dispiration (due to a wedge disclination and a screw dislocation) is studied by path integration. By gauging $SO(2) \otimes T(1)$, first, we derive the metric, curvature, and torsion of the medium of dispiration. Then we carry out explicitly path integration for the propagator of a particle moving in the non-Euclidean medium under the influence of a scalar potential and a vector potential. We obtain also the winding number representation of the propagator by taking the non-trivial topological structure of the medium into account. We extract the energy s…

On the numerical evaluation of real-time path integrals: Double exponential integration and the Maslov correction

Ooura's double exponential integration formula for Fourier transforms is applied to the oscillatory integrals occuring in the path-integral description of real-time Quantum Mechanics. Due to an inherent, implicit regularization multi-dimensional Gauss-Fresnel integrals are obtained numerically with high precision but modest number of function calls. In addition, the Maslov correction for the harmonic oscillator is evaluated numerically with an increasing number of time slices in the path integral thereby clearly demonstrating that the real-time propagator acquires an additional phase $ - π/2 $…

The Wigner distribution function for the $\mathfrak{su}(2)$ finite oscillator and Dyck paths

Recently, a new definition for a Wigner distribution function for a one-dimensional finite quantum system, in which the position and momentum operators have a finite (multiplicity-free) spectrum, was developed. This distribution function is defined on discrete phase-space (a finite square grid), and can thus be referred to as the Wigner matrix. In the current paper, we compute this Wigner matrix (or rather, the pre-Wigner matrix, which is related to the Wigner matrix by a simple matrix multiplication) for the case of the $\mathfrak{su}(2)$ finite oscillator. The first expression for the matrix…

Euclidean Path Integral and Higher-Derivative Theories

We consider the Euclidean path integral approach to higher-derivative theories proposed by Hawking and Hertog (Phys. Rev. D65 (2002), 103515). The Pais-Uhlenbeck oscillator is studied in some detail. The operator algebra is reconstructed and the structure of the space of states revealed. It is shown that the quantum theory results from quantizing the classical complex dynamics in which the original dynamics is consistently immersed. The field-theoretical counterpart of Pais-Uhlenbeck oscillator is also considered.

淬火量子倒置振子的 Schwinger-Keldysh 路径积分形式体系

Schwinger-Keldysh path integral formalism for a Quenched Quantum Inverted Oscillator

本文利用 Schwinger-Keldysh 形式体系和不变算符方法研究淬火量子倒置振子，导出其生成函数与 OTOC，并据此分析不同时间尺度行为及混沌特征。

Quantum Rényi entropy by optimal thermodynamic integration paths

Despite being a well-established operational approach to quantify entanglement, Rényi entropy calculations have been plagued by their computational complexity. We introduce here a theoretical framework based on an optimal thermodynamic integration scheme, where the Rényi entropy can be efficiently evaluated using regularizing paths. This approach avoids slowly convergent fluctuating contributions and leads to low-variance estimates. In this way, large system sizes and high levels of entanglement in model or first-principles Hamiltonians are within our reach. We demonstrate it in the one-dimens…

四次振荡器的路径积分：配分函数的精确解析公式

Path integral for the quartic oscillator: An accurate analytic formula for the partition function

本文通过路径积分和最小灵敏度原理，为四次振荡器配分函数推导出解析公式，精确再现了自由能、极限行为和微扰级数发散等关键特征。