The entangled photons components are found to be created in the lossless nanocavity with resonance mode. The smallness of Γ ( 0 ≤ Γ<< g- coupling constant for electro-dipolar interaction) was revealed playing the crucial part in their production. It’s known that Г determines limits (ω<sub>c</sub> ± Γ) of photon frequency deflection from the mode frequency ω<sub>c</sub>, when photon passes through empty cavity. When Γ = 0 and ω<sub>a</sub>=ω<sub>c</sub> , the Hamiltonian is time independent and has two eigenstates with eigenvalies (ω<sub>a</sub> ± g). Each state is superposition of the upper and lower atomic states, taken with signs plus and minus respectively. These states are stationary and form a time-depended superposition. Matrix elements of the interaction Hamiltonian, taken between that superposition and atomic unperturbed states, contain two anti-phases components of entangled photons. Since Γ = 0, their emission out of cavity is forbidden so they interfere, producing beatings of the resonance mode by sin (g•t). When 0 < Γ<< g those beatings become quasi-stationary, and with probability proportional to Γ/4g they go out through the partly transparent mirror and disintegrate into two photons, each of them taking its own spectral place outside the cavity. This process is illustrated by 3D-plots in the (ω, t)-space.
The exact theory of generating the "entangled photons" by excited motionless two level atom in one-dimensional
high finesse nanocavity with a single resonance linearly polarized mode, decaying at the rate Γ, is presented. We
have investigated the evolution of resonance emission out of the macromolecule-like system "nanocavity with
resonance mode and excited atom" in area 0≤ Γ≤ 0.2g , where g is a coupling constant for electro-dipolar
interaction. We have revealed that the source of arising of the entangled photons at the outlet of nanocavity is
disintegration of metastable interference superpositional field structure, ejected through the partly transparent
mirror out of the cavity. This structure is produced by the self-consistent AC Stark effect, created by electrical
fields of rotating atomic dipole and resonance mode. The field splits atomic levels and radiation transitions
between them, producing in the nanocavity pairs of anti-phase (ω<sub>a</sub> ± g)- photons. Since the rate of mode
damping Γ << g these photons form in the cavity metastable interference superposional structure consisting of
resonance mode carring wave with amplitude modulated with frequency 2g. The profiles of (ω<sub>a</sub> ± g )-
components have identical form exp(-t Γ)t Γ, with average lifetime in the cavity being estimated as 4ln Γ/Γ.
This work is a continuation of papers presented at the Optics and Photonics Symposium 2009 and is devoted to the case of the high finesse nanocavity with the average photon escaping rate Γ = η c/R << g -coupling constant. The case is of special interest as possible pretender for a qubit in quantum computers. The probability distribution to find photon in the (ωk , t )-space, investigated in the interval 0≤Γ/4g <<1, has triplet structure with very low central peak and satellites at frequencies ≈ ( ω<sub>a</sub> ± g). The latter emerge as result of emission from two upper atomic split sublevels. The peak is produced by ensuing reabsorptions of satellites by atom through its upper sublevels. Oscillating as t<sup>2</sup>•cos(gt) and decaying fast, the peak is accompanied with the simultaneously arising satellites, When the peak disappears the satellites become stable. The stability is quenched with continuum of final states. The profile of structure consisting of two identical components has the time-dependence t • exp(−Γt/4) and the width of satellites is by order less than the distance between them. These components with frequencies (ω<sub>a</sub> ± g ) have the average photon energies equal to 1/2(ω<sub>a</sub> ± g ) where factor "1/2 "accounts for normalization condition. The satellites amplitudes reach maximum at Γ/4g = 0.05. The profile of the structure has the form Γt • exp (−Γt/4) with maximum attained for Γ/4g =0,05 and average photon cavity life-time proportional to 4lnΓ/Γ. We name the structure "entangled photon."
Nonperturbative theoretical analysis of the temporal evolution of a spontaneous photon with atomic frequency ωa, emitted by a motionless two-level atom in a one-dimensional high-finesse nanocavity into a single resonance decaying mode, is presented. The explicit solution of the Schrödinger equation was found in an interaction picture with use of the Green functions technique. It has been assumed that emission leaks out of the empty cavity by exponential law at rate Γ, which is a function of coupling constant g, distance between the mirrors, penetrability coefficient of the left mirror, and the velocity of light. The stationary superpositional co-phased structure of two photons with the same profiles and average frequencies 1/2(ωa ± g), quenched with continuum of final photonics states, has been revealed. The profile of this structure has been found to have the form Γt exp(−Γt/4) with maximum attained for Γ/4g = 0.05 and average photon cavity lifetime equal to 4lnΓ/ Γ.
The time-dependent spectral properties and nonlinear dynamics of a spontaneous photon emitted by two level atom, trapped in damped nanocavity and coupled to a single resonance mode, have been investigated. The results have been obtained with the help of new nonperturbative approach by solving exactly the Schrödinger equation. The theory accounts exactly and simultaneously processes of emission, reabsorptions and leakage of photon. The solution was found with the Green functions formalism, supplemented with the novel algorithm in operating causal singular functions and fundamentals of the theory of quasi-stationary systems. The obtained distributions and emission dynamics are presented with plots as functions of photon frequency detuning and time for various value Γ/4g. For Γ/4g <1 the spectrum is a triplet inside the cavity and doublet outside it. In this case the total emission probability is described by decaying oscillations. For Γ/4g ≥ 1 and Γ/4g >> 1 the spectrum consists of a single central line decaying exponentially with profile depending on value Γ/4g.
The nonperturbative theory of the cooperative spontaneous emission from a two level atom trapped in one-dimensional
damped nanocavity with a single resonance mode is presented. The time-dependent spectral properties
and nonlinear dynamics of a separate photon emission by the macroolecular-like system "excited atom coupled to a
resonance decaying mode" have been analyzed. The investigation has been carried out by solving the Schrödinger
equation in the interaction picture with the help of the Green functions method in the Heitler-Ma's form . The
formalism was supplemented with the novel algorithm in operating causal singular functions and with fundamentals
of the theory of quantum quasi-stationary systems. The proposed theory accounts automatically of both
reabsorptions of emitted photon and its simultaneous escaping out of the cavity. Solutions of the wave equation
were found without using intermediate virtual states and series expansions. In accordance with the theory of quasistationary
systems the field of mode decaying exponentially in the empty nanocavity was represented with the
Lorenz-shaped packet of stationary photonic states (quasi-modes). The electro-dipolar interaction between the atom
and the mode field was adopted to be switched on suddenly. The expressions and plots of emission spectral
densities probabilities together with photon emission probability dynamics as functions of time for various ratios
Γ/4g. For Γ/4g<1 the transient emission spectrum reveals the presence of two symmetrical side-bands and the
central peak interconnected with each other in the area of interaction with the atom. Since the central component
oscillates, decaying simultaneously in time at two rates infinity Γ/2 and ~ Γ/4, in the area of interaction the emission
is a triplet with satellites oscillating in the interaction area and being stable outside of it. So the final spectrum is a
doublet outside of nanocavity. On the contrary for Γ/4g ≥1
the spectrum is a singlet, and the emission occurs in
exponentially decaying way.
The nonperturbative theory of the cooperative spontaneous emission from a two level atom trapped in one-dimentional
damped nanocavity with a single resonance mode is presented. The time-dependent spectral properties and nonlinear
dynamics of a separate photon emission by the micro-molecular-like system "excited atom coupled to a resonance
decaying mode" have been analyzed. The investigation has been carried out by solving the Schrödinger equation in the
interaction picture with the help of the Green functions method in the Heitler-Ma's form. The formalism was
supplemented with the novel algorithm in operating causal singular functions and with fundamentals of the theory of
quasi-stationary systems. The proposed theory accounts automatically of both reabsorptions of emitted photon and its
simultaneous escaping out of the cavity. Solutions of the wave equation were found without using intermediate virtual
states and series expansions. In accordance with the theory of quasi-stationary systems the field of mode decaying
exponentially in the empty nanocavity was represented with the Lorenz-shaped packet of stationary photonic states
(quasi-modes). The electro-dipolar interaction between the atom and the mode field was adopted to be switched on
suddenly. The expressions and plots of emission probabilities spectral densities together with photon emission
probability dynamics as functions of time for various ratios Γ/4g of photon escaping rate Γ and coupling constant g are presented. For Γ/4g <1 the transient emission spectrum reveals the presence of two symmetrical side-bands and of
the central peak, the latter decaying in time at the rate ∝ Γ/2 so that the final spectrum is a doublet. In this case the
photon emission probability is described by decaying oscillations. On the contrary for Γ/4g ≥ 1 the spectrum is a singlet and the emission occurs in exponentially decaying ways.
The impact of a two level ultracold ion vibrations on its spontaneous emission in a single resonance mode microcavity has been analyzed nonperturbatively. The 'recoiless' emission of photon by the moving ion has been revealed. The resonance exchange with energy through the AC Stark- sublevels of the 'ion + cavity mode'-subsystem has been pointed out as one of effective mechanism enhancing greatly energy transform from spontaneous radiation to vibrations.
The fundamentals of nonperturbative theory of spontaneous emission of two level atom coupled to single-mode resonance field in 1D damped cavity are outlined. The theory predicts striking dependence of the emission spectral line shape on the ratio (Gamma) <SUB>c</SUB>/g of cavity passive passband to the coupling constant g. When (Gamma) <SUB>c</SUB> equals 4g, the line shape transforms from the `two peaks'--form into singlet, thus allowing to find g.
The time-dependent spontaneous emission spectral line shape of two level motionless atom placed in common antinode of two symmetrically detuned lossless modes has been investigated nonperturbatively. The line shape and its temporal evolution have been found to be a function of modes detuning and natural line width of isolated atom. The spectrum varies between singlet and triplet taking on the well-known Lorentz-shape form when detuning and natural line width become much larger than the coupling constant of interaction of the atom with modes.
Cavity-controlled spontaneous emission spectrum of two level atom coupled to two lossless symmetrically detuned modes has been found rigorously to be a function of the system's energy eigenvalues and displays triplet, doublet or singlet form depending on detuning, natural linewidth and coupling constant.
The Green function formalism together with the new algorithm in operating with -functions has been used to obtain the 3-peaked spectrum of spontaneous emission of an atom in a weakly damped cavity. The single peaked spectrum and exponential decay law with short (1/Tc) and long (lc/g2) decay times have been shown to exist in the strongly damped cavity.