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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Arising of entangled photon in the high finesse nanocavity

Vladislav Cheltsov, Anton V. Cheltsov

Published Sep 8, 2011• 0 citations
Publisher

Abstract

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 ≈ ( ωa ± 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 t2•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 (ωa ± g ) have the average photon energies equal to 1/2(ωa ± 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."

DOI
10.1117/12.892294

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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 ≈ ( ωa ± 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 t2•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 (ωa ± g ) have the average photon energies equal to 1/2(ωa ± 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 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 (ωa ± 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 (ωa ± g )- components have identical form exp(-t Γ)t Γ, with average lifetime in the cavity being estimated as 4ln Γ/Γ.
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 (ωc ± Γ) of photon frequency deflection from the mode frequency ωc, when photon passes through empty cavity. When Γ = 0 and ωa=ωc , the Hamiltonian is time independent and has two eigenstates with eigenvalies (ωa ± 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.
Of fundamental importance in quantum optics has been the investigation of the radiative coupling of a collection of atoms to the field of a resonant cavity1. Within this context we are conducting a research program to investigate the interaction of optically prepared two-state atoms with the fields of high finesse optical cavities. With regard to our specific observations, we have employed a sodium atomic beam apparatus to make direct spectroscopic measurements of the normal-mode splitting for the coupled oscillator system formed by the collective atomic polarization of N two-state atoms and a single mode of a resonator of finesse of approximately 2x10⁴. At the transition wavelength of 589 nm (³, S_(1/2), F = 2, m_F = 2→3 P_(3/2), m_F = 3 and for one of our spherical mirror standing-wave cavities, the relevant parameters for the system are as follows: transverse atomic decay rate γ_⊥ = 2πX5X10⁶/S, cavity damping rate K/γ⊥= 0.18, and single-atom coupling coefficient g/γ_⊥ = 0.2.
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.