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Physical Review A

Superradiant Level Shift and Its Possible Detection in a Transient Optical Experiment

Hans Morawitz

Published Mar 1, 1973• 33 citations
Publisher

Abstract

The superradiant level shift of a system of $N$ two-level molecules, interacting with the radiation field, is calculated for pencil-like geometry. Introducing the self-energy ${\ensuremath{\Sigma}}^{(N)}(E)$ of the molecular systems due to its radiative interaction, we first discuss the trivial ($N=1$) and familiar ($N=2$) cases in more detail. The analytic properties of the self-energy ${\ensuremath{\Sigma}}^{(N)}(E)$ are used to derive dispersion relations for its real and imaginary parts ${\ensuremath{\Delta}}^{(N)}(E)$, ${\ensuremath{\Gamma}}^{(N)}(E)$. We then generalize to the case of $N\ensuremath{\gg}1$, employing a self-consistency argument advanced by Arecchi and Courtens. The modified Dicke expression for the cooperative spontaneous decay rate is then used in the dispersion relation for the shift ${\ensuremath{\Delta}}^{(N)}$ and a value of \ensuremath{\sim}6 MHz calculated corresponding to typical molecular-gas parameters. The physical measurability of the shift is discussed and a Brewer-Shoemaker experiment in a molecular gas is proposed for the observation of the shift.

DOI
10.1103/physreva.7.1148

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The superradiant level shift of a system of NNN two-level molecules, interacting with the radiation field, is calculated for pencil-like geometry. Introducing the self-energy \ensuremathΣ(N)(E){\ensuremath{\Sigma}}^{(N)}(E)\ensuremathΣ(N)(E) of the molecular systems due to its radiative interaction, we first discuss the trivial (N=1N=1N=1) and familiar (N=2N=2N=2) cases in more detail. The analytic properties of the self-energy \ensuremathΣ(N)(E){\ensuremath{\Sigma}}^{(N)}(E)\ensuremathΣ(N)(E) are used to derive dispersion relations for its real and imaginary parts \ensuremathΔ(N)(E){\ensuremath{\Delta}}^{(N)}(E)\ensuremathΔ(N)(E), \ensuremathΓ(N)(E){\ensuremath{\Gamma}}^{(N)}(E)\ensuremathΓ(N)(E). We then generalize to the case of N\ensuremath≫1N\ensuremath{\gg}1N\ensuremath≫1, employing a self-consistency argument advanced by Arecchi and Courtens. The modified Dicke expression for the cooperative spontaneous decay rate is then used in the dispersion relation for the shift \ensuremathΔ(N){\ensuremath{\Delta}}^{(N)}\ensuremathΔ(N) and a value of \ensuremath{\sim}6 MHz calculated corresponding to typical molecular-gas parameters. The physical measurability of the shift is discussed and a Brewer-Shoemaker experiment in a molecular gas is proposed for the observation of the shift.
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.
Finding a comprehensive and general description of the collective Lamb shift and cooperative broadening in a radiatively interacting system is a long-standing open question. Both energy levels and linewidth of individual atoms are modified by the exchange of real and virtual photons making up the dipole-dipole interaction. We introduce a method to theoretically study weakly-driven, low-excited ensembles of two-level atoms, and obtain an analytic description of the collective Lamb shift and linewidth via a self-consistent formalism including infinite order of correlations which stem from only two-body interactions. We predict the dependency of these quantities, as measurables, on system parameters: the number density of the ensemble, the detuning of an external probe field, and the geometry of the sample.
Superradiant decay is accompanied by two kinds of collective lineshifts, an induced shift and the spontaneous "collective Lamb shift." Both form as sum of dipole-dipole interaction-induced level shifts between atoms in the system. We have developed a procedure to obtain numerical results on this model that self-consistently incorporates the shifts. The induced shift displays large non-zero values early in the system evolution. In addition, its effect on the superradiant system is studied: there is only a very small dephasing effect on the decay rate. While the induced shift is largely absent in not-too strongly driven systems, this parameter region might provide a good experimental regime for measuring the collective Lamb shift. These results can have important consequences for highly sensitive systems, such as quantum information science or atomic clocks.
In order to study the cooperative emission of radiation from an extended system of NNN two-level atoms, we use the master-equation approach within the Born and Markov approximations. All the properties of the emission of radiation can be deduced from the diagonalization of the coupling matrix $\ensuremath{\Gamma}$, where an element ${\ensuremath{\Gamma}}_{\mathrm{jk}}$ gives the coupling between the atoms labeled jjj and kkk via the electromagnetic field. We choose a pencil-shaped volume and study analytically the coupling matrix in the limit for small and large Fresnel numbers. For Fresnel numbers of order unity numerical methods are used. Thus we prove that, for the pencil-shaped geometry, any elaborated model must take into account the interaction of the atoms with the infinity of electromagnetic field modes lying inside two small solid angles about the axis of the largest linear dimension, which we label the zzz axis. In other words, the intensity of radiation emitted by any atom of the system depends on its position with respect to the others and this "propagation effect" cannot be neglected when interpreting superradiance experiments. For small Fresnel numbers the zzz dependence predominates, in the opposite case the (x,yx, yx,y) dependence does, while for Fresnel numbers of order unity the dependence of all three components must be taken into account. Preliminary calculations are given for which only the zzz dependence is considered; for Fresnel numbers of magnitude of order unity or smaller, the effect of propagation along the zzz axis displays large delays for the appearance of superradiance with respect to the time at which the superradiant pulse is maximum when the "propagation effects" are neglected (one-mode model).
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.
We discuss the cooperative decay of initial atomic excitation for a pencil-shaped active volume filled with two-level atoms. As long as the length of the sample is much smaller than a certain maximal cooperation length, the atom-field interaction producing the superradiant pulse can be treated in terms of the simplest possible laser model (single mode). The basic laser master equation turns out to be exactly solvable if specified for the superradiance limit which is characterized by two conditions: (i) The photons escape from the low-QQQ cavity so fast that they cannot feed themselves back into atomic excitation to any appreciable amount (this no-feedback condition is equivalent to the above-mentioned requirement for the length of the sample). (ii) The incoherent atomic decay due to natural relaxation is so slow that the individual atomic dipoles do not dephase before engaging themselves cooperatively in the interaction with the electromagnetic field. The present paper presents the derivation and general discussion of the equations describing the statistical properties of atoms and field in a superradiant pulse. Analytical and numerical solutions will be presented in a subsequent paper.