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Journal of Physics B Atomic Molecular and Optical Physics

Two-photon polarization Fourier spectroscopy of metastable atomic hydrogen

Alan J. Duncan, Z. A. Sheikh, H. J. Beyer, Hans Kleinpoppen

Published Mar 14, 1997• 6 citations
Publisher

Abstract

A novel Fourier-transform spectroscopic method using two-photon polarization to determine the spectral distribution of the two photons emitted in the spontaneous decay of metastable atomic hydrogen is described. The method uses birefringent retardation plates and takes advantage of the subtle interplay between the spectral properties and the entangled polarization properties of the radiation emitted in the decay. Assuming the validity of the theoretical spectral distribution, it is shown that the experimental results agree well with theory. On the other hand, success in solving the inverse problem of determining the spectral distribution from the experimental results is limited by the small number of experimental points. However, making reasonable assumptions it is deduced that the observed spectrum is characterized by a broadband signal of width and centre angular frequency in good agreement with the predictions of and , respectively, obtained from the theoretical spectral distribution modified to take account of the absorption of the two-photon radiation in air. The values of 1.5 fs for the coherence time and 440 nm for the coherence length for single photons of the two-photon pair which are obtained from the measured bandwidth imply that, in the ideal case, these values are determined by the essentially zero lifetime of the virtual intermediate state of the decay process rather than the long lifetime of the metastable state which, it is suggested, determines the coherence time and coherence length appropriate to certain types of fourth-order interference experiments.

DOI
10.1088/0953-4075/30/5/025

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A novel Fourier-transform spectroscopic method using two-photon polarization to determine the spectral distribution of the two photons emitted in the spontaneous decay of metastable atomic hydrogen is described. The method uses birefringent retardation plates and takes advantage of the subtle interplay between the spectral properties and the entangled polarization properties of the radiation emitted in the decay. Assuming the validity of the theoretical spectral distribution, it is shown that the experimental results agree well with theory. On the other hand, success in solving the inverse problem of determining the spectral distribution from the experimental results is limited by the small number of experimental points. However, making reasonable assumptions it is deduced that the observed spectrum is characterized by a broadband signal of width and centre angular frequency in good agreement with the predictions of and , respectively, obtained from the theoretical spectral distribution modified to take account of the absorption of the two-photon radiation in air. The values of 1.5 fs for the coherence time and 440 nm for the coherence length for single photons of the two-photon pair which are obtained from the measured bandwidth imply that, in the ideal case, these values are determined by the essentially zero lifetime of the virtual intermediate state of the decay process rather than the long lifetime of the metastable state which, it is suggested, determines the coherence time and coherence length appropriate to certain types of fourth-order interference experiments.
The Stirling two-photon source based upon the decay of metastable atomic hydrogen has been applied to study angular and polarization correlations and the photon coherence length, and to an experimental confirmation of the Breit-Teller hypothesis. Selecting opposite directions for the detection of the two photons in coincidence, polarization correlations were measured with both linear and circular polarizers inserted into the photon paths. The results of these Einstein-Podolski-Rosen (EPR) type experiments favour quantum mechanics over local realistic theories in common with all other published studies. By inserting an additional linear polarizer in one detection arm it was, further-more, possible to test for the first time (and exclude) a new class of local realistic theories. The coherence length of the photons in one detection arm was measured by the introduction (in that arm) of retardation ("multiwave") plates oriented with their fast axes at 45° with respect to the linear polarization which in turn is determined by the linear polarizer in the other detection arm. If the retardation is just sufficient to prevent the two orthogonal components emerging from the multiwave plate from interfering as a result of their path difference, the polarization of the coincidence signal is destroyed. The coherence length of 350 nm found in this way is extremely short, only approximately 1.5 times the median wavelength of 243 nm of the two-photon decay. The corresponding coherence time t coh = l coh / c is approximately 10 −15 s which confirms that it is not determined by the long lifetime of the 2S state (~ 1/8 s) but by the virtual states which mediate the two-photon decay. Given measurements at enough values of retardation this method can, in principle, be used to determine the spectral distribution of the two-photon decay. The current measurements also confirm the Breit-Teller hypothesis that the two-photon decay is unaffected by fine structure and hyperfine structure interactions.
The Paper first summarizes fundamental aspects and results of the quantum electrodynamical theory of the two-photon radiationfromthe decay of the metastable 2 2 S 1/2 atomic hydrogen state. After a brief description of the second improved Stirling two-photon coincidence experiment polarization correlations of the two-photon decay are described in which both two or three linear polarizers are applied in order to test predictions of such correlations based upon quantum mechanics and local realistic theories (i.e., Einstein-Podolsky-Rosen type experiments). It is particularly noticeable that the three-polarizer coincidence measurement provided the largest difference (about 40%) between the Bell limits of local realistic theories and quantum mechanics so far. Apart from confirming in addition the correlations of right-right and left-left circularly polarized two-photon correlations a new type of coherence analysis of the two-photon radiation has been carried out experimentally and theoretically. A result of it is the measured coherence time of τ coh = 1.2 · 10 −15 s and coherence length of l coh = c · τ coh = 350 nm of the two-photon emission. By applying a theoretical model of the two-photon radiation linked to cascade transitions the coherence length can be estimated to l coh ≈ 100 nm in agreement by order of magnitude with the experimental data.
We report a novel method of determining the coherence length of photons produced in the two-photon decay of metastable atomic deuterium by observing the depolarization of the photons in what is essentially a true single-photon interference experiment. In the Stirling source two photons propagating in the ±z directions are detected in coincidence. The degree of polarization of the radiation emerging from a multiwave plate placed on one side of the source as a function of the optical path difference 6 introduced by the multiwave plate between the orthogonally polarized components is determined by measuring the Stokes' parameters. The results confirm the dependence of the depolarizing effect of the multiwave plate on 6, agree with the quantum mechanical predictions and allow a measurement to be made of the coherence length of single photons of the two-photon pair. \nWe also report the results t>f an experiment which, for the first time, demonstrates the action of an achromatic half-wave plate on the polarization state of the two-photon radiation emitted by atomic deuterium in the metastable 2Sv, state The results agree with the quantum mechanical predictions and confirm the hypothesis of Breit and Teller that the fine and hyperfine interaction play no role in the emission process.
The two-photon decay mode of hydrogenic atoms from an arbitrary state (n1{n}_{1}n1​, l1{l}_{1}l1​, m1{m}_{1}m1​) to an arbitrary state (n2{n}_{2}n2​, l2{l}_{2}l2​, m2{m}_{2}m2​) is studied within the framework of nonrelativistic quantum mechanics. In the dipole approximation, these decay rates, which involve infinite summation over intermediate states, are derived exactly via a general second-order matrix element obtained by Kelsey and Macek and an implicit technique introduced by Dalgarno and Lewis. The results are expressed in terms of hypergeometric functions. For transitions ${n}_{1}s\ensuremath{\rightarrow}{n}_{2}s$, our results reduce to those of Klarsfeld whose starting point is the Coulomb Green's function. For transitions to the ground state, an alternative expression involving a simple one-dimensional integral is presented. The decay rate of the 2s2s2s metastable state of atomic hydrogen is calculated as an illustration of the method. The result, $\frac{1}{\ensuremath{\tau}}=8.2284$ ${\mathrm{sec}}^{\ensuremath{-}1}$, agrees with Klarsfeld. For transitions of ${n}_{1}s\ensuremath{\rightarrow}1s$ and ${n}_{1}d\ensuremath{\rightarrow}1s ({n}_{1}\ensuremath{\ge}3)$, the transition rates exhibit interesting and unexpected structures. In particular, "zeros" are found in the two-photon emission spectrum indicating that two-photon emission is not possible at certain frequencies. Physically, these "zeros" are the result of destructive interference between the radiating dipole terms associated with the sum over intermediate states. In addition to the emission spectrum the expected coincidence signal between two detectors monitoring the two photons simultaneously emitted during a two-photon transition is calculated as a function of the angle between the detectors. The angular distribution for the ${n}_{1}d\ensuremath{\rightarrow}1s$ transitions is shown to be significantly different from the ${n}_{1}s\ensuremath{\rightarrow}1s$ transitions. Finally, a possible experiment is suggested to test the results presented in this paper.