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IEEE Biennial Congress of Argentina (ARGENCON)

Spurious Photon States at the Quantum Harmonic Oscillator with Glauber-Bessel Hamiltonian

Huber Nieto–Chaupis

Published Sep 18, 2024• 0 citations
Publisher

Abstract

Commonly, coherent states can be represented in terms of an exact expansion of Fock's states and a Gaussian distribution. This paper explores the harmonic quantum oscillator through integer-order Bessel functions expected to replace Gaussian profiles. As consequence of having opted Bessel polynomials, the annihilation and creator operators acquire a new mathematical form. This led to construct a kind of Glauber-Bessel Hamiltonian that reproduces standard quantum oscillator energies as well as spurious terms that are interpreted as an inherent noise. In praxis, this novel Hamiltonian can be employed to model cases in which there is evidence of spurious pulses in an experimental scenario of photons correlation without any formalism of perturbation theory.

DOI
10.1109/argencon62399.2024.10735844

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Commonly, coherent states can be represented in terms of an exact expansion of Fock's states and a Gaussian distribution. This paper explores the harmonic quantum oscillator through integer-order Bessel functions expected to replace Gaussian profiles. As consequence of having opted Bessel polynomials, the annihilation and creator operators acquire a new mathematical form. This led to construct a kind of Glauber-Bessel Hamiltonian that reproduces standard quantum oscillator energies as well as spurious terms that are interpreted as an inherent noise. In praxis, this novel Hamiltonian can be employed to model cases in which there is evidence of spurious pulses in an experimental scenario of photons correlation without any formalism of perturbation theory.
Here a special case of perturbation in quantum harmonic oscillator is studied. Here we assume the perturbed potential to be a Harmonic Oscillator that has been shifted in the position space.We construct the new creation and annihilation operators for the new Hamiltonian to find out its energy eigenstates. What is interesting about the solution of this system is that we find out that the ground state of this system is a coherent state of the ground state of the original harmonic oscillator. We know that the Hamiltonian of electromagnetic field is similar to that of the harmonic oscillator where the electric field and magnetic field are expressed in terms of time evolution of annihilation and creation operators. According to the quantum electromagnetic field theory, photon states are expressed as the coherent states of the ground states of the unperturbed harmonic oscillator.In the previous research works,it has been established how coherent states provide a better picture of Electric field compared to the energy eigenstates. This paper mainly deals with why the coherent states provide a more complete picture of electric and magnetic field theory compared to just energy eigenstates. Here we find how our perturbation act as a coherent state generator and adds a new term in the Hamiltonian. The Perturbation is thought to arise from the only available interaction i.e. observation. The perturbation also explains how observation leads to coherence of photon states. We will find out how the new perturbation affects the time evolution of electromagnetic field. This also explains how the classical electromagnetic field theory takes into account the factor of observation, due to which the classical laws are in perfect agreement with the quantum laws.
The main goal of this paper is to present an alternative method to construct new kinds of nonlinear coherent states. To do this, we first establish a class of hypergeometric type of generalized displacement operators, 1 F r ([0], [0, 1, ..., r − 1], za †), act on the vacuum state of the harmonic oscillator and generate normalized quantum states of the Fock space which admit a resolution of the identity through a positive definite measure on the complex plane. Furthermore, realization of the compact form of these states, as functions of the position coordinate x for r = 2, leads to a generating function of the Hermite polynomials in terms of the modified Bessel function. Finally, studying some statistical characters reveals that they have indeed non-classical features such as squeezing, an anti-bunching effect and sub-Poissonian statistics, too.
The squeezed harmonic oscillator Hamiltonian H = ωa†a + αa2 + α*a†2 is analysed, where a† and a are harmonic oscillator creation and annihilation operators, ω is real, and α is a time independent constant. For the case that ω2 − 4|α|2 > 0 it is known that the Hamiltonian possesses real and positive eigenvalues. The exact eigenstates are constructed by using a Bogoliubov transformation to express them in terms of the algebra and states of the harmonic oscillator. Known properties of these states are reviewed and new results are derived. Modified coherent states are constructed from the exact eigenstates and shown to yield a unit projection operator for the original harmonic oscillator Hilbert space via generalized Gaussian integrals. Transition elements between harmonic oscillator coherent states are then evaluated exactly by using the properties of the modified coherent state projection operator.
The elementary quantum theory of quasi-classical states for mechanical harmonic oscillators is applied to the radiation oscillators describing light, and thus used to define quasi-classical states of light. These states are called “coherent”, because the expectation value of any normally ordered product of creation and annihilation operators factorizes for them. A theoretical discussion of possible sources for coherent states is followed by experimental evidence for their existence. The mathematical properties of coherent states, e.g., nonorthogonality and overcompleteness, are explained in order to use them to represent states and operators. These representations provide a suitable replacement for the classical phase space description of light, which is forbidden by the uncertainty principle. The quantum phase space approach leads to the definition of the Wigner distribution, the Q-function, the P-function, and Gaussian mixed states.
Abstract In this chapter we collect a number of key elements of quantum optics relevant for quantum information theory. We begin by discussing the quantization of the electromagnetic field in terms of harmonic oscillators with ladder operators and the corresponding Fock space. Based on this description we introduce Glauber’s coherent states and examine their properties as proper quantum analogues of the classical harmonic oscillator. We then study the description of states of light in a quantum-mechanical phase space by employing quasi-probability distributions such as the Wigner function. With these tools at hand we then focus on the family of Gaussian states and their compact description in terms of covariance matrices and vectors of first moments. We review the formalism of Gaussian operations, symplectic operations, the Williamson normal form, and the Bloch-Messiah decomposition. Finally, we take a look at entanglement of Gaussian states.
In this paper is demonstrated that quantum ef-fects can be derived in a straightforward manner from the classical radiation intensity equation. To accomplish this, it is employed the equation done by Hartemann and Kerman inside the framework of Nonlinear Compton backscattering. Thus, through mathematical procedures involving the usage of integer-order Bessel functions, it is derived expressions that are in fully agreement to the quantum mechanics formulation of coherence as the one given by Glauber by which a Gaussian profile is representing the coherent states of light. The resulting classical field of emission is interpreted from the spectrum consisting of classical radiated energy against Doppler frequency. It has yielded peaks that are been interpreted ad emitted photons.