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The forming mechanism of spontaneous emission noise flux radiated from hydrogen-like atoms by means of vibrational Hamiltonian

J. Jahanpanah

Published Mar 1, 2021• 2 citations
Publisher

Abstract

The main goal of this article is to investigate the origin of spontaneous emission radiation by applying a new method. In this method, the vibrational Hamiltonian of hydrogen-like atoms (HLAs) is first constructed in accordance with the Hamiltonian of a simple harmonic oscillator. The Hamiltonian is then used to derive the vibrational motion equations of HLAs in a Heisenberg picture. The Langevin equation will ultimately be formed by adding the dissipative term and fluctuating (Langevin) force according to the fluctuation–dissipation theorem. The solution of the Langevin equation provides the exact data on the fluctuations that occur for the electron position during its rather fast vibrational oscillations. On the other hand, the fluctuations in the position of the oscillating electron, in turn, lead to the fluctuations in its potential and kinetic energies so that the appearance of the potential and kinetic noise fluxes is vulnerable. It is finally demonstrated that the potential and kinetic noise fluxes commonly act as an internal source of producing the external noise flux emitted from HLAs in the form of spontaneous emission with a Lorentzian profile. In contrast with all previous procedures, no ambient effect has been involved to describe the forming mechanism of spontaneous emission for the first time.

DOI
10.1063/5.0036017

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Calculating the noise flux of spontaneous emission radiated from Hydrogen-like atoms by means of vibrational Hamiltonian and ignoring from ambient effects

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A theoretical model is introduced for constructing the vibrational Hamiltonian of Hydrogen-like atoms (HLA). The Hamiltonian is then used to derive the vibrational motion equations of HLA in Heisenberg picture. The Langevin equation will ultimately be formed after adding the dissipative term and fluctuating (Langevin) force according to the fluctuation-dissipation theorem (FDT). The positional noise flux is then defined as the correlation function of fluctuations that happens for the electron position during its rather fast vibrational oscillations ( ). On the other hand, the positional fluctuations led to the fluctuations in the potential and kinetic energies of oscillating electron so that the appearance of the potential and kinetic noise fluxes is vulnerable. The positional, potential, and kinetic noise fluxes of oscillating electron will be determined by solving the Langevin equation in frequency domain. It is finally demonstrated that the potential and kinetic noise fluxes commonly act as an internal source for producing the external noise flux emitted from HLA in the form of spontaneous emission with a Lorentzian profile. In contrast with all previous procedures, no ambient effect has been involved to describe the forming mechanism of spontaneous emission for the first time.

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Quantum theory of damping – Heisenberg–Langevin approach

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In the previous chapter, we developed the equation of motion for a system as it evolved under the influence of an unobserved (reservoir) system. We used the density matrix approach and worked in the interaction picture. In this chapter, we consider the same problem of the system-reservoir interaction using a quantum operator approach. We again eliminate the reservoir variables. The resulting equations for the system operators include, in addition to the damping terms, the noise operators which produce fluctuations. These equations have the form of classical Langevin equations, which describe, for example, the Brownian motion of a particle suspended in a liquid. The Heisenberg–Langevin approach discussed in this chapter is particularly suitable for the calculation of two-time correlation functions of the system operator as is, for example, required for the determination of the natural linewidth of a laser. We first consider the damping of the harmonic oscillator by an interaction with a reservoir consisting of many other simple harmonic oscillators. This system describes, for example, the damping of a single-mode field inside a cavity with lossy mirrors. The reservoir, in this case, consists of a large number of phonon-like modes in the mirrors. We also consider the decay of the field due to its interaction with an atomic reservoir. An interesting application of the theory of the system–reservoir interaction is the evolution of an atom inside a damped cavity. It is shown that the spontaneous transition rate of the atom can be substantially enhanced if it is placed in a resonant cavity.

Cambridge University Press eBooks•Sep 4, 1997•2 citations
The main goal of this article is to investigate the origin of spontaneous emission radiation by applying a new method. In this method, the vibrational Hamiltonian of hydrogen-like atoms (HLAs) is first constructed in accordance with the Hamiltonian of a simple harmonic oscillator. The Hamiltonian is then used to derive the vibrational motion equations of HLAs in a Heisenberg picture. The Langevin equation will ultimately be formed by adding the dissipative term and fluctuating (Langevin) force according to the fluctuation–dissipation theorem. The solution of the Langevin equation provides the exact data on the fluctuations that occur for the electron position during its rather fast vibrational oscillations. On the other hand, the fluctuations in the position of the oscillating electron, in turn, lead to the fluctuations in its potential and kinetic energies so that the appearance of the potential and kinetic noise fluxes is vulnerable. It is finally demonstrated that the potential and kinetic noise fluxes commonly act as an internal source of producing the external noise flux emitted from HLAs in the form of spontaneous emission with a Lorentzian profile. In contrast with all previous procedures, no ambient effect has been involved to describe the forming mechanism of spontaneous emission for the first time.
A theoretical model is introduced for constructing the vibrational Hamiltonian of Hydrogen-like atoms (HLA). The Hamiltonian is then used to derive the vibrational motion equations of HLA in Heisenberg picture. The Langevin equation will ultimately be formed after adding the dissipative term and fluctuating (Langevin) force according to the fluctuation-dissipation theorem (FDT). The positional noise flux is then defined as the correlation function of fluctuations that happens for the electron position during its rather fast vibrational oscillations ( ). On the other hand, the positional fluctuations led to the fluctuations in the potential and kinetic energies of oscillating electron so that the appearance of the potential and kinetic noise fluxes is vulnerable. The positional, potential, and kinetic noise fluxes of oscillating electron will be determined by solving the Langevin equation in frequency domain. It is finally demonstrated that the potential and kinetic noise fluxes commonly act as an internal source for producing the external noise flux emitted from HLA in the form of spontaneous emission with a Lorentzian profile. In contrast with all previous procedures, no ambient effect has been involved to describe the forming mechanism of spontaneous emission for the first time.
The relativistic properties of Hydrogen-like atoms (HLAs) are here investigated in the Heisenberg picture for the first time. The relativistic vibrational Hamiltonian (RVH) is first defined as a power series of harmonic oscillator Hamiltonian by using the relativistic energy eigenvalue . By applying the first-order RVH (proportional to ) to the Heisenberg equation, a pair of coupled equations is turned out for the relativistic motion of the electron’s position and linear momentum. A simple comparison of the first-order relativistic and nonrelativistic equations reveals this reality that the natural (fundamental) frequency of HLA (like entropy) is slowly raised by increasing the atomic number from . The second-order RVH (proportional to ) has then been implemented to determine an exact expression for the electron relativistic frequency in the different atomic energy levels. In general, the physical role of RVH is fundamental because it not only specifies the temporal relativistic variations of position, velocity, and linear momentum of the oscillating electron, but also identifies the corresponding relativistic potential, kinetic, and mechanical energies. The results will finally be testified by demonstrating energy conservation.
The relativistic behavior of Hydrogen-like atoms (HLAs) is investigated in Heisenberg picture for the first time. The relativistic vibrational Hamiltonian (RVH) is first defined as a power series of harmonic oscillator Hamiltonian by using the relativistic energy eigenvalue . By applying the first-order RVH (proportional to ) to Heisenberg equation, a pair of coupled equations is turned out for the motion of electron position and its relativistic linear momentum. A simple comparison of the first-order relativistic and nonrelativistic equations reveals this reality that the natural (fundamental) frequency of electron oscillation (like entropy) is slowly raised by increasing the atomic number. The second-order RVH (proportional to ) have then been implemented to determine an exact expression for the electron relativistic frequency in the different atomic energy levels. In general, the physical role of RVH is fundamental because it not only specifies the temporal relativistic variations of position, velocity, and linear momentum of oscillating electron, but also identifies the corresponding relativistic potential, kinetic, and mechanical energies. The results will finally be testified by demonstrating the energy conservation.
We present a formal quantum-mechanical treatment of the damping and fluctuations of a generic radiator atom due to collisions with foreign gas atoms. By working in the Heisenberg picture, the fluctuating noise sources are identified with the damping. Under the same approximations as the conventional line-shape broadening theory in the impact (Markov) approximation, the fluctuation-dissipation theorem is satisfied in the form of the conservation of an equal-time commutator for the atomic operators. The implications of this type of approach for the quantum theory of light propagation in an active medium is addressed. Simple cases for two-level and three-level \ensuremath{\Lambda}-type systems are given explicitly.
The recent first-principles derivation of Langevin-Bloch equations of motion for near-resonant atoms embedded in a linear dielectric host leads to a clear, direct physical reinterpretation of fundamental physical principles as applied to one of the most basic problems of quantum mechanics: the renormalization of the spontaneous emission rate of an atom in a dielectric host. Here we show the role of the Lorentz local field, the fluctuation-dissipation theorem, the wave number and wave vector, the density of states, Fermi's golden rule, the electromagnetic energy density, canonical quantization, and the classical constitutive relations in this context.
In the previous chapter, we developed the equation of motion for a system as it evolved under the influence of an unobserved (reservoir) system. We used the density matrix approach and worked in the interaction picture. In this chapter, we consider the same problem of the system-reservoir interaction using a quantum operator approach. We again eliminate the reservoir variables. The resulting equations for the system operators include, in addition to the damping terms, the noise operators which produce fluctuations. These equations have the form of classical Langevin equations, which describe, for example, the Brownian motion of a particle suspended in a liquid. The Heisenberg–Langevin approach discussed in this chapter is particularly suitable for the calculation of two-time correlation functions of the system operator as is, for example, required for the determination of the natural linewidth of a laser. We first consider the damping of the harmonic oscillator by an interaction with a reservoir consisting of many other simple harmonic oscillators. This system describes, for example, the damping of a single-mode field inside a cavity with lossy mirrors. The reservoir, in this case, consists of a large number of phonon-like modes in the mirrors. We also consider the decay of the field due to its interaction with an atomic reservoir. An interesting application of the theory of the system–reservoir interaction is the evolution of an atom inside a damped cavity. It is shown that the spontaneous transition rate of the atom can be substantially enhanced if it is placed in a resonant cavity.