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Physical Review Letters

Two-Ion Interference: Bragg Scattering and Superradiance

Richard G. Brewer

Published Dec 23, 1996• 14 citations
Publisher

Abstract

Bragg scattering and superradiance, always considered independently, are examined jointly for the case of two trapped ions. The two phenomena are seen to be different representations of the same basic two-ion interference resulting from a common coherent optical preparation. Solutions of a master equation for two four-level atoms show that fringes (spatial interference) and a variation of the spontaneous emission lifetime with ion-ion distance (temporal interference) can occur either simultaneously or separately. Unlike a classical calculation, not all scatterings result in interference, a quantum effect not considered in interpreting recent experiments.

DOI
10.1103/physrevlett.77.5153

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Bragg scattering and superradiance, always considered independently, are examined jointly for the case of two trapped ions. The two phenomena are seen to be different representations of the same basic two-ion interference resulting from a common coherent optical preparation. Solutions of a master equation for two four-level atoms show that fringes (spatial interference) and a variation of the spontaneous emission lifetime with ion-ion distance (temporal interference) can occur either simultaneously or separately. Unlike a classical calculation, not all scatterings result in interference, a quantum effect not considered in interpreting recent experiments.
A superradiance theory is developed for two identical hydrogenic ions (four states each) in a microtrap, as in recent experiments. The ions oscillate (micromotion) due to the trap's radio frequency (rf) electric quadrupole field. One signature of superradiance is a deviation of the two-ion average upper state decay rate \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{} from the one-ion value \ensuremath{\gamma}. A master equation is derived, giving a fractional change in the upper state lifetime \ensuremath{\gamma}/\ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}-1=sinkR/2kR [J02{\mathit{J}}_{0}^{2}J02​(z)-2J12{\mathit{J}}_{1}^{2}J12​(z)+...]cos\ensuremath{\Phi}, where k=2\ensuremath{\pi}/\ensuremath{\lambda}, \ensuremath{\lambda} is the emission wavelength, R is the ion-ion distance, Jn{\mathit{J}}_{\mathit{n}}Jn​(z) is a Bessel function of integer order n, and z=ka, a is the ion amplitude of motion, and \ensuremath{\Phi} is the two-ion relative phase due to the preparation. In the Lamb-Dicke regime, a, J02{\mathit{J}}_{0}^{2}J02​(z)\ensuremath{\approxeq}1 and thus superradiance is not influenced significantly by ion motion. This damped sinusoid is diluted by the factor of 1/2 due to destructive interference. Superradiance vanishes in the absence of coherent preparation, e.g., with inversion, as indicated by the time evolution of the two-ion dipole correlation function. Fringes and a beat at the rf are predicted in forward scattering, the most elementary form of optical free induction decay.
We report the first observation of a microscopically resolved interaction between two trapped atoms. This is also the first observation of two-ion superradiance and subradiance, the most elementary example of Dicke's theory. Its signature is the variation of the spontaneous emission decay rate of a laser-cooled two-ion crystal, +1.5+1.5%+1.5 and $\ensuremath{-}1.2%$, as the ion-ion distance is changed from 1380 to 1540 nm, in agreement with a no-free-parameter theory. The experiment uses an ion microtrap (80 \ensuremath{\mu}m radius) with planar electrodes and a new few-atom technique for measuring lifetimes.
We report the first observation of interference in the light scattered from two trapped atoms ({sup 198}Hg{sup +} ions localized in a linear Paul trap). The visibility of the interference fringes can be explained in the frame-work of Bragg scattering by a harmonic crystal, but with important differences compared to the case of a large crystal. Comparison of the experimental data with theory shows that the interference pattern offers another method to determine ion temperatures and separations. Furthermore, by exploiting the atom`s internal structure we have found a way to obtain {open_quotes}which path{close_quotes} information without invoking the position-momentum uncertainty relation. If the light scattered by the atoms is detected in a polarization-sensitive way, then it is possible to selectively demonstrate either the particle-nature or the wave-nature of the scattered photons.
Dicke superradiance, i.e., the enhanced spontaneous emission of coherent radiation, is often attributed to radiation emitted by synchronized dipoles coherently oscillating in phase. At the same time, Dicke derived superradiance assuming atoms in entangled Dicke states which do not display any dipole moment. To shed light on this apparent paradox, we study the intensity distribution arising from two identical two-level atoms prepared either in an entangled Dicke state or in a separable atomic state with a nonvanishing dipole moment. We find that the two configurations produce similar far-field intensity patterns, although they stem from fundamentally distinct types of coherence: while in the second case the atoms display coherence among the individual particles, leading to Young-type interference like that known from classical dipoles, atoms in Dicke states possess collective quantum coherence stemming from entanglement and quantum correlations of the state, leading to an interference pattern resulting from enhanced spontaneous emission. This demonstrates that the radiation generated by synchronized dipoles and Dicke superradiance---even if they do produce the same interference pattern---are fundamentally distinct phenomena and have to be interpreted in different ways.
While the spontaneous emission from independent emitters provides spatially uncorrelated photons-a typical manifestation of quantum randomness, the interference of the coherent scattering leads to a well-defined intensity pattern-a feature described by linear optics. We here demonstrate experimentally how the interplay between the two mechanisms in large systems of quantum emitters leads to spatial variations of photon correlations. The implementation with trapped ion crystals in free space allows us to observe the anticorrelation between photon rates and variance of the photon number distributions in chains of up to 18 ions. For smaller crystals of four ions, the transition from antibunching to bunching and super-Poissonian statistics of the scattered light is reported. For higher numbers of scatterers, the photon statistics still display a strong deviation from the fully incoherent scattering case. Our results illustrate how the interference of coherent scattering, combined with spontaneous emission, provides a control mechanism for the light statistics.
Second-order interference effects in the spontaneous emission by two identical two-level atoms, separated a distance apart and both initially excited, are considered. On the basis of the analysis in our previous paper published in 2004, within fourth-order perturbation, detailed analysis of the second-order quantum interference in the emitted field is made. The interference arises both from single-photon states and two-photon states owing to interferences between different decaying sequences. By detailed analysis we have found some cancellations between single-photon and two-photon interference terms, and an interpretation of this is given. The contributions from the single- and the two-photon states as well as the total interference are analysed numerically for various interatomic distances and times. The contribution from a two-photon state is mostly larger than that from a single-photon state except when the atomic electric dipole is large, in which case the two contributions can be comparable. It is shown that the emission patterns from the two interacting atoms provide more information on atom–atom correlations than the decay time evolution of the atomic states population does. The number of interference fringes and the distance between them depend on the interatomic distance in a similar fashion to Young's interference fringes.