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Physical Review A

Transition from incoherence to coherence in the spontaneous emission of extended systems

C. T. Lee

Published Apr 1, 1976• 12 citations
Publisher

Abstract

Using the results of a previous paper, we study the evolution of a completely excited system of $N$ two-level atoms, distributed in a circular cylinder with dimensions much larger than the wavelength, interacting with all modes of radiation field. We emphasize the distinction between $r$-conserving ($r\mathrm{C}$) and $r$-nonconserving ($r\mathrm{NC}$) processes ($r$ is the so-called cooperation number). Considering the number of photons emitted as the discrete independent variable, this evolution is described by a partial difference equation. Numerical solutions of this equation show the transition from $r\mathrm{NC}$ dominance at the beginning to $r\mathrm{C}$ dominance later. This is also a transition from incoherent to coherent emission of radiation.

DOI
10.1103/physreva.13.1657

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Using the results of a previous paper, we study the evolution of a completely excited system of NNN two-level atoms, distributed in a circular cylinder with dimensions much larger than the wavelength, interacting with all modes of radiation field. We emphasize the distinction between rrr-conserving (rCr\mathrm{C}rC) and rrr-nonconserving (rNCr\mathrm{NC}rNC) processes (rrr is the so-called cooperation number). Considering the number of photons emitted as the discrete independent variable, this evolution is described by a partial difference equation. Numerical solutions of this equation show the transition from rNCr\mathrm{NC}rNC dominance at the beginning to rCr\mathrm{C}rC dominance later. This is also a transition from incoherent to coherent emission of radiation.
The evolution of a completely excited system of N two-level atoms, distributed over a large region and interacting with all modes of radiation field, is studied. The distinction between r-conserving (RC) and r-nonconserving (RNC) processes is emphasized. Considering the number of photons emitted as the discrete independent variable, the evolution is described by a partial difference equation. Numerical solution of this equation shows the transition from RNC dominance at the beginning to RC dominance later. This is also a transition from incoherent to coherent emission of radiation.
We introduce a straightforward quantum-electrodynamic approach to the problem of superradiant spontaneous emission from a system of two-level atoms, and also discuss two classical superradiant systems. The same physical concepts underlie our treatment of both the quantum and classical cases. Explicit expressions are found which describe the time evolution and directional character of the radiated power. Our approach applies to an arbitrary number of atoms which are confined to a region with linear dimensions which may be large compared with the mean wavelength of the emitted radiation. The radiative decay half-life of a large many-atom system is found to be much shorter than the natural lifetime of a single atom, but not as short as the half-life of a small atomic system with the same number of atoms. The far-field radiation pattern appropriate to the case of a circular cylinder of emitters which have been excited by a plane-wave pulse is considered in detail. Polar plots are presented of the radiation patterns produced when the atoms are excited by a plane-wave pulse. Exciting pulses traveling in directions both parallel and at an angle of 10\ifmmode^\circ\else\textdegree\fi{} to the axis of the cylinder are considered. A discussion and generalization of Dicke's "coherence-brightening" criterion is given.
Coherent spontaneous emission from a system of NNN two-level atoms interacting with a quantized radiation field is treated for the case in which Dicke's "cooperation number" rrr is macroscopically large. It is shown how to modify the quasiclassical approach to this problem to incorporate quantum effects that are lost in the self-consistent-field approximation. The statistics of the emitted radiation are found to vary markedly with the initial state of the system of atoms. The photon statistics tend to that typical of blackbody radiation when the initial state of the atomic system is that which would result from incoherent pumping ($m\ensuremath{\sim}r$). When, on the other hand, the atoms are initially in a superradiant state ($m\ensuremath{\ll}r$), the emitted radiation may be represented approximately by a coherent state.
A system of NNN identical two-level atoms, distributed over an extended region, completely excited initially, and interacting with all modes of radiation field is considered. A rate equation is proposed to describe the evolution of such a system and the transition from initial nondirectional emission to eventual directional emission is reinvestigated. An analytical expression is obtained for the critical point of this transition.
Fully-excited two-level atoms separated by less than the transition wavelength cooperatively emit light in a short burst, a phenomenon called superradiance by R. Dicke in 1954. The burst is characterized by a maximum intensity scaling with the square of the number of atoms NNN and a temporal width reduced by NNN compared to the single atom spontaneous decay time. Both effects are usually attributed to a synchronization of the electric dipole moments of the atoms occurring during the process of light emission. Contrary to this explanation, it was recently shown by use of a quantum path description that the peak intensity results from the quantum correlations among the atoms when occupying symmetric Dicke states. Here we investigate from this perspective the temporal evolution of the ensemble, starting in the small sample limit, i.e., when the atoms have mutual separations much smaller than the transition wavelength λλλ and pass down the ladder of symmetric Dicke states. In addition, we explore the temporal evolution for the case of non-interacting atoms with mutual separations much larger than λλλ. We show that in this case a similar superradiant burst of the emitted radiation is observed if the quantum correlations of the atoms are generated by conditional photon measurements retaining the atomic ensemble within or close to the symmetric subspace.
We investigate the emergence of cooperative radiation phenomena in ensembles of two-level atoms coupled to a lossy resonant cavity beyond the Markovian and mean-field approximations. By deriving a complete analytical solution for the two-emitter case and employing a numerically exact method for larger ensembles, we characterize the full transition from Markovian to non-Markovian collective dynamics for systems of up to 10310^3103 emitters. Our results reveal three distinct regimes: a Markovian phase exhibiting the standard superradiant burst, a non-Markovian phase featuring spontaneous superabsorption of the emitted field, and a critical regime marked by pulsed collective emission. We show that the critical spectral width separating these behaviors increases monotonically with the number of emitters, demonstrating that environmental memory effects can be enhanced by cooperativity. Finally, we find that the superradiant scaling of the peak intensity progressively degrades with increasing system size, approaching a subquadratic law in the limit of a perfect cavity. In this regime, spontaneous superabsorption emerges as a distinct manifestation of non-Markovian cooperativity.