Photon pairs with tailor-made entanglement obtained from the two-photon decay of atomic hydrogen

Abstract

From the work by W. Perrie et al. ͓Phys. Rev. Lett. 54, 1790 ͑1985͔͒, it is known that the photon pairs that are emitted in the 2s 1/2 → 1s 1/2 ͑two-photon͒ decay of atomic hydrogen are quantum mechanically correlated, i.e., entangled. However, less information is available about the degree of polarization entanglement between the two photons if an arbitrary geometry is considered for collecting the photons. In this paper, we study the effect of the decay geometry on the degree of polarization entanglement between the two emitted photons. Results are shown for the 2s 1/2 → 1s 1/2 and 3d 5/2 → 1s 1/2 two-photon transitions of atomic hydrogen. The outlined theory is general and can be applied also to heavier elements. To demonstrate the influence of relativistic and multipole effects, results are also shown for the 3d 5/2 → 1s 1/2 transition of hydrogenlike uranium.

Polarization correlation in the two-photon decay of atomic hydrogen: nonlocality versus entanglement

“…As pointed out in reference [26] for the 2s 1/2 → 1s 1/2 decay of hydrogen-like ions, this typical behaviour of the concurrence (and the Bell violation for a pure final state of the photon pair) can be understood as a geometrical effect due to the conservation of angular momentum. Note that Figure 2 is obtained for the special geometry of a two-photon measurement in which the emission of the 'first' photon is fixed along the direction of the quantization axis: θ 1 = 0…”

“…In the past, this formalism has been applied widely in order to explore not only the total decay rates [5] but also the energy as well as angular distributions [7,24,25] and even the correlation in the polarization state of the photons [26]. Below, we therefore restrict ourselves to a rather short account of the basic expressions, just enough for introducing the quantum measures for the nonlocality and nonseparability of the state of the photon pair.…”

“…Having defined the 'geometry' in the last subsection, the two-photon density matrix for an atomic transition from the initial state |n i κ i to the final state |n f κ f is given within the helicity representation by [26,30]: (1) where k 1,2 denotes the wave vector and λ 1,2 = ±1 the helicity of the photons, that is the spin projection onto their direction of propagation. In the helicity representation, the two spin projections λ = ±1 correspond to the right and left circular polarization states |σ ± of a photon.…”

Photon polarization in the two-photon decay of heavy hydrogen-like ions

“…For many years, these investigations have dealt not only with the total and energy-differential transition rates but also with the angular correlations between the emitted photons. Today, much of the present interest is focused on the polarization properties of the photons whose quantum correlations may reveal unique information about the electronic structure of the ions as well as about the entanglement in such systems [13][14][15]. Up to the present, however, most investigations on the polarization of the correlated photons have been carried out for the (twophoton) decay of neutral hydrogen or low-Z ions.…”

“…Most naturally, such an analysis can be performed by means of the density matrix that is associated to the spin-polarization of the emitted photons. Since the application of the density matrix formalism to the two-photon decay has been discussed at several places elsewhere [15,21], here we just display the helicity representation of the (two-photon) spin density matrix in its final form:…”