Cooperative Spontaneous Emission from an Initially Fully Excited System of Three Identical Two‐level Atoms

Abstract

A b s t r a c t . Using Lehmberg's master equation and taking into account dipole-dipole near-field interaction the time-dependent intensity pattern, intensity -and angular correlations of the cooperative spontaneous radiation from an initially fully excited system of three identical two-level atoms arc calculated. The intensity pattern developes lobes in different directions a t different times in such a wsy that contrary to the two-atom system the spatial distribution of the total number of the emittcd photons differs from that of the single-atom radiation pattern. I n particular three atoms placed in propcr distances on a straight line emit preferentially their energy along this line. At the beginning of the emission process the probability to find two photons simultaneously in the same (or geometrical equivalent) direction is always highest but in the coursc of time, contrary to the twoatom ease, it may well happen that the probability of finding two photons in quite different directions becomes more favoured. Kooperative spontane Emission cines anflinglich vollstlindig angeregten Systems von drci gleichen Zwei-Niveau-AtomenI n h a l tsiibersicht. Es wcrdcn der richtungsabhangigc Intcnsitats-Zeit-Verlauf sowie die Intensitats-und Winkelkorrelationen der spontanen Strahlung eincs kooperativ emittierenden, anfanglich vollstandig angeregten Systems von drei gleichen Zwei-Niveau-Atomen mit Hilfe der Lehmbergschen Mastergleichung untcr Beriicksichtigung dcr Dipol-Dipol-Nahfeldwechselwirkung bereehnet. Die Richtcharakteristik der Intcnsitat zeigt zu verschiedenen Zeiten Keulen in untersehiedlichen Richtungen derart, daB im Gegensatz zum Zwei-Atomfall die Richtungsverteilurig der insgesanit emittierten Photonen von der Ein-Atomrichtcharakteristilr abweicht.Insbesondere strahlen drei, in geeignetem Abstsnd auf ciner Geraden angeorclnete Atome vorzugswcisc in Richtung der Vcrbindongslinie. Zri Beginn des Ausstrahlungsprozesses ist die Wahrseheinlichlieit, zwei Photonen gleichzcitig in dorselben (hzw. gcomctrisch Iquivalenten) Richtung zu finclen, immer am groBten. Im Verlaaf der Zoit kann cs jcdoch im Gegensatz zum Zwei-Atomsystem dazu kommen, daB die simultitnc Emission zwcicr Photonen in vijllig verschiedene Richtnngcn bevorzugt ist.

Kinetics and Optical Properties of the Strongly Driven Gas Medium of Interacting Atoms

“…Moreover, as it is seen from the structure of the model Hamiltonian, quantum recoil effects are not taken into account. On a theoretical basis quantum recoil effects, as described in the recent work [45], can be eliminated because they require that atomic motion on the timescale of spontaneous emission be small compared to the time taken to move a wavelength distance, as otherwise, the Rabi frequency cannot be considered a constant (see the works [34,35,45]). The quite high sample temperatures allow us to impose such limitations on the model.…”

“…(33), (34), (35), (36), we obtain the following system of linear equations for the unknown coefficients A ± n , B ± n , C ± n : The special function integral cosine has the following properties used in this paper:…”

“…Substituting the expressions (37), (38), (39) into the system of Eqs. (33), (34), (35), (36), we obtain the system of linear equations for the unknown coefficients A ± n , B ± n , C ± n (see Appendix). Comparing the coefficients at the same exponents on the right and left sides of the above equations, we find that:…”

“…For comparison, the prior theoretical work [45] describes the evolution of two dipole-dipole interacting atoms in a vacuum with only one atom being initially excited; [36] describes two two-level atoms independently interacting with local thermal or squeezed reservoirs, this time taking into account the possibility of their initial simultaneous excitation, but neglecting dipole-dipole interactions. Works [34,35] follow the approximations of [32,33], just adding an additional state, corresponding to two simultaneously excited atoms, to the model. Here they examine possible two-photon jumps, that differ from our description of "jumps" (we model the dipole-dipole interaction by an operator keeping appropriate pairs of atomic transition operators that are neglected in the approach in [32]: see more detailed explanation in the next section "Model").…”

“…The certain amount of techniques for deriving the master equation is in use today, but most of them are limited to the case of the non-interacting atoms (see, for instance, [6,[25][26][27][28][29][30][31]) or frozen positions for each atom (see, for example, works [32][33][34][35][36]). A review of the relatively early developments in the theory of the effects of binary collisions on the spectral profiles is presented in [37] with respect to the simplified master equations for one-particle density matrix elements including some phenomenological items taking into account atomic transitions at the short-range collisions.…”