Physical Review A

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

Thomas Radtke, A. Surzhykov, S. Fritzsche

Published Feb 25, 2008• 17 citations

Abstract

From the work by W. Perrie et al. [Phys. Rev. Lett. 54, 1790 (1985)], it is known that the photon pairs that are emitted in the $2{s}_{1∕2}\ensuremath{\rightarrow}1{s}_{1∕2}$ (two-photon) 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 $2{s}_{1∕2}\ensuremath{\rightarrow}1{s}_{1∕2}$ and $3{d}_{5∕2}\ensuremath{\rightarrow}1{s}_{1∕2}$ 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 $3{d}_{5∕2}\ensuremath{\rightarrow}1{s}_{1∕2}$ transition of hydrogenlike uranium.

DOI: 10.1103/physreva.77.022507

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

Thomas Radtke, A. Surzhykov, S. Fritzsche

The European Physical Journal DJul 11, 20085 citations

Angular and polarization correlations in two--photon decay of hyperfine 2s states in hydrogenlike ions

F. Fratini

arXiv:1203.6601Mar 29, 20120 citations

Parametrization of the angular correlation and degree of linear polarization in two-photon decays of hydrogenlike ions

Pedro Manuel Duarte Gonçalves Amaro, F. Fratini, S. Fritzsche +3 more

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Angular and polarization analysis for two-photon decay of 2s hyperfine states of hydrogenlike uranium

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

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Quantum correlations in the two-photon decay of few-electron ions

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Physical Review A

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

Thomas Radtke, A. Surzhykov, S. Fritzsche

Published Feb 25, 2008• 17 citations

Publisher

Abstract

From the work by W. Perrie et al. [Phys. Rev. Lett. 54, 1790 (1985)], it is known that the photon pairs that are emitted in the

2{s}_{1∕2}\ensuremath{\rightarrow}1{s}_{1∕2}

(two-photon) 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

2{s}_{1∕2}\ensuremath{\rightarrow}1{s}_{1∕2}

and

3{d}_{5∕2}\ensuremath{\rightarrow}1{s}_{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

3{d}_{5∕2}\ensuremath{\rightarrow}1{s}_{1∕2}

transition of hydrogenlike uranium.

DOI: 10.1103/physreva.77.022507

Related papers

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

Thomas Radtke, A. Surzhykov, S. Fritzsche

From the work by Perrie et al. [Phys. Rev. Lett. 54, 1790 (1985)], photon pairs from the 2s 1/2 → 1s 1/2 (two-photon) decay of atomic hydrogen are known to be quantum mechanically correlated. In these experiments, the polarization states of the photons emitted in back-to-back geometry were shown to violate the Bell inequality as a qualitative sign of nonlocality and entanglement. In the present contribution, we analyze how these nonlocal quantum correlations, as given by the violation of the Bell inequality, differ from the concurrence as a true entanglement measure. Results are shown for both quantifiers in dependence of the decay geometry and the initial polarization of the atoms for the 2s 1/2 → 1s 1/2 and 3d 5/2 → 1s 1/2 two-photon decay of atomic hydrogen. These results display the difference between nonlocality and entanglement and, hence, may stimulate further experiments on nonlocal quantum correlations in atomic systems.From the work by Perrie et al. [Phys. Rev. Lett. 54, 1790 (1985)], photon pairs from the 2s 1/2 → 1s 1/2 (two-photon) decay of atomic hydrogen are known to be quantum mechanically correlated. In these experiments, the polarization states of the photons emitted in back-to-back geometry were shown to violate the Bell inequality as a qualitative sign of nonlocality and entanglement. In the present contribution, we analyze how these nonlocal quantum correlations, as given by the violation of the Bell inequality, differ from the concurrence as a true entanglement measure. Results are shown for both quantifiers in dependence of the decay geometry and the initial polarization of the atoms for the 2s 1/2 → 1s 1/2 and 3d 5/2 → 1s 1/2 two-photon decay of atomic hydrogen. These results display the difference between nonlocality and entanglement and, hence, may stimulate further experiments on nonlocal quantum correlations in atomic systems.

The European Physical Journal DJul 11, 20085 citations

Angular and polarization correlations in two--photon decay of hyperfine 2s states in hydrogenlike ions

F. Fratini

The angular and polarization properties of the photon pair emitted in the two--photon decay of $2s$ hyperfine states in hydrogenlike ions are investigated within the relativistic Dirac framework and second order perturbation theory. The studied transitions are $2s_{1/2}\,(F=1,0)\to1s_{1/2}\,(F=1,0)$ in Hydrogen atom and $2s_{1/2}\,(F=4,3)\to1s_{1/2}\,(F=4,3)$ in hydrogenlike Uranium ion. Two different emission patterns are found: For non-spin-flip transitions, the angular correlation ( $i.e.$ the angular distribution of the emitted photons) is of the type $\sim 1+\cos^2θ$ , while the linear polarizations of the emitted photons are approximately parallel one another; For spin-flip transitions, the angular correlation is of the type $\sim 1-1/3\cos^2θ$ , while the linear polarizations of the emitted photons are approximately orthogonal one another.The angular and polarization properties of the photon pair emitted in the two--photon decay of $2s$ hyperfine states in hydrogenlike ions are investigated within the relativistic Dirac framework and second order perturbation theory. The studied transitions are $2s_{1/2}\,(F=1,0)\to1s_{1/2}\,(F=1,0)$ in Hydrogen atom and $2s_{1/2}\,(F=4,3)\to1s_{1/2}\,(F=4,3)$ in hydrogenlike Uranium ion. Two different emission patterns are found: For non-spin-flip transitions, the angular correlation ($i.e.$ the angular distribution of the emitted photons) is of the type $\sim 1+\cos^2θ$, while the linear polarizations of the emitted photons are approximately parallel one another; For spin-flip transitions, the angular correlation is of the type $\sim 1-1/3\cos^2θ$, while the linear polarizations of the emitted photons are approximately orthogonal one another.

arXiv:1203.6601Mar 29, 20120 citations

Parametrization of the angular correlation and degree of linear polarization in two-photon decays of hydrogenlike ions

Pedro Manuel Duarte Gonçalves Amaro, F. Fratini, S. Fritzsche +3 more

The spontaneous two-photon emission in hydrogenlike ions is investigated within the framework of second-order perturbation theory and Dirac's equation. Special attention is paid to the angular correlation of the emitted photons as well as to the degree of linear polarization of one of the two photons, if the second is just observed under arbitrary angles. Expressions for the angular correlation and the degree of linear polarization are expanded in powers of cosine functions of the two-photon opening angle, whose coefficients depend on the atomic number and the energy sharing of the emitted photons. The effects of including higher (electric and magnetic) multipoles upon the emitted photon pairs beyond the electric-dipole approximation are also discussed. Calculations of the coefficients are performed for the transitions $2{s}_{1/2}\ensuremath{\rightarrow}1{s}_{1/2}$, $3{d}_{3/2}\ensuremath{\rightarrow}1{s}_{1/2}$, and $3{d}_{5/2}\ensuremath{\rightarrow}1{s}_{1/2}$, along the entire hydrogen isoelectronic sequence ($1\ensuremath{\le}Z\ensuremath{\le}100$).The spontaneous two-photon emission in hydrogenlike ions is investigated within the framework of second-order perturbation theory and Dirac's equation. Special attention is paid to the angular correlation of the emitted photons as well as to the degree of linear polarization of one of the two photons, if the second is just observed under arbitrary angles. Expressions for the angular correlation and the degree of linear polarization are expanded in powers of cosine functions of the two-photon opening angle, whose coefficients depend on the atomic number and the energy sharing of the emitted photons. The effects of including higher (electric and magnetic) multipoles upon the emitted photon pairs beyond the electric-dipole approximation are also discussed. Calculations of the coefficients are performed for the transitions $2{s}_{1/2}\ensuremath{\rightarrow}1{s}_{1/2}$, $3{d}_{3/2}\ensuremath{\rightarrow}1{s}_{1/2}$, and $3{d}_{5/2}\ensuremath{\rightarrow}1{s}_{1/2}$, along the entire hydrogen isoelectronic sequence ($1\ensuremath{\le}Z\ensuremath{\le}100$).

arXiv:1208.3169Physical Review AOct 16, 201215 citations

Angular and polarization analysis for two-photon decay of 2s hyperfine states of hydrogenlike uranium

L. Safari, Pedro Manuel Duarte Gonçalves Amaro, José Paulo Santos +1 more

The amplitude of two-photon transitions between hyperfine states in hydrogenlike ions is derived based on the relativistic Dirac equation and second-order perturbation theory. We study angular and linear polarization properties of the photon pair emitted in the decay of $2s$ states, where spin-flip and non-spin-flip transitions are highlighted. We pay particular attention to hydrogenlike uranium, since it is an ideal candidate for investigating relativistic and high-multipole effects, such as spin-flip transitions. Two types of emission patterns are identified: (i) non-spin-flip transitions are found to be characterized by an angular distribution of the type $W(\ensuremath{\theta})\ensuremath{\sim}1+{cos}^{2}\ensuremath{\theta}$ while the polarizations of the emitted photons are parallel; and (ii) spin-flip transitions have somewhat smaller decay rates and are found to be characterized by an angular distribution of the type $W(\ensuremath{\theta})\ensuremath{\sim}1\ensuremath{-}1/3{cos}^{2}\ensuremath{\theta}$ while the polarizations of the emitted photons are orthogonal, where $\ensuremath{\theta}$ is the angle between photons directions. Deviations due to nondipole and relativistic contributions are evaluated for both types of transitions. This work is the first step toward exploring the effect of the nucleus over the angular and polarization properties of the photon pairs emitted by two-photon transitions.The amplitude of two-photon transitions between hyperfine states in hydrogenlike ions is derived based on the relativistic Dirac equation and second-order perturbation theory. We study angular and linear polarization properties of the photon pair emitted in the decay of $2s$ states, where spin-flip and non-spin-flip transitions are highlighted. We pay particular attention to hydrogenlike uranium, since it is an ideal candidate for investigating relativistic and high-multipole effects, such as spin-flip transitions. Two types of emission patterns are identified: (i) non-spin-flip transitions are found to be characterized by an angular distribution of the type $W(\ensuremath{\theta})\ensuremath{\sim}1+{cos}^{2}\ensuremath{\theta}$ while the polarizations of the emitted photons are parallel; and (ii) spin-flip transitions have somewhat smaller decay rates and are found to be characterized by an angular distribution of the type $W(\ensuremath{\theta})\ensuremath{\sim}1\ensuremath{-}1/3{cos}^{2}\ensuremath{\theta}$ while the polarizations of the emitted photons are orthogonal, where $\ensuremath{\theta}$ is the angle between photons directions. Deviations due to nondipole and relativistic contributions are evaluated for both types of transitions. This work is the first step toward exploring the effect of the nucleus over the angular and polarization properties of the photon pairs emitted by two-photon transitions.

arXiv:1407.4063Physical Review AJul 29, 20149 citations

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

A. Surzhykov, Thomas Radtke, P. Indelicato +1 more

We have applied the density matrix and second-order perturbation theory in order to re-analyze the two-photon decay of hydrogen-like ions for the polarization of emitted light. Special attention is paid to the linear polarization of one of the photons, while the spin state of the second photon is supposed to be unobserved. For such an “angle-polarization” correlation of the two photons we investigate the contributions that arise from non electric-dipole terms in the expansion of the electron-photon interaction. Detailed calculations are performed for the 2s 1/2 →1s 1/2 and 3d 5/2 →1s 1/2 transitions in neutral hydrogen as well as in hydrogen-like uranium.We have applied the density matrix and second-order perturbation theory in order to re-analyze the two-photon decay of hydrogen-like ions for the polarization of emitted light. Special attention is paid to the linear polarization of one of the photons, while the spin state of the second photon is supposed to be unobserved. For such an “angle-polarization” correlation of the two photons we investigate the contributions that arise from non electric-dipole terms in the expansion of the electron-photon interaction. Detailed calculations are performed for the 2s 1/2 →1s 1/2 and 3d 5/2 →1s 1/2 transitions in neutral hydrogen as well as in hydrogen-like uranium.

The European Physical Journal Special TopicsMar 1, 20098 citations

Quantum correlations in the two-photon decay of few-electron ions

F. Fratini, Malte C. Tichy, Thorsten Jahrsetz +3 more

A theoretical study of the polarization entanglement of two photons emitted in the decay of metastable ionic states is performed within the framework of density-matrix theory and second-order perturbative approach. Particular attention is paid to relativistic and nondipole effects that become important for medium- and high- $Z$ ions. To analyze these effects, the degree of entanglement is evaluated both in the dipole approximation and within rigorous relativistic theory. Detailed calculations are performed for the two-photon $2{s}_{1/2}\ensuremath{\rightarrow}1{s}_{1/2}$ transition in hydrogenlike ions as well as for the $1{s}_{1/2} 2{s}_{1/2} {}^{1}{S}_{0}\ensuremath{\rightarrow}1{s}_{1/2}^{2} {}^{1}{S}_{0}$, $1{s}_{1/2} 2{s}_{1/2} {}^{3}{S}_{1}\ensuremath{\rightarrow}1{s}_{1/2}^{2} {}^{1}{S}_{0}$, and $1{s}_{1/2} 2{p}_{1/2} {}^{3}{P}_{0}\ensuremath{\rightarrow}1{s}_{1/2}^{2} {}^{1}{S}_{0}$ transitions in heliumlike ions.A theoretical study of the polarization entanglement of two photons emitted in the decay of metastable ionic states is performed within the framework of density-matrix theory and second-order perturbative approach. Particular attention is paid to relativistic and nondipole effects that become important for medium- and high-$Z$ ions. To analyze these effects, the degree of entanglement is evaluated both in the dipole approximation and within rigorous relativistic theory. Detailed calculations are performed for the two-photon $2{s}_{1/2}\ensuremath{\rightarrow}1{s}_{1/2}$ transition in hydrogenlike ions as well as for the $1{s}_{1/2} 2{s}_{1/2} {}^{1}{S}_{0}\ensuremath{\rightarrow}1{s}_{1/2}^{2} {}^{1}{S}_{0}$, $1{s}_{1/2} 2{s}_{1/2} {}^{3}{S}_{1}\ensuremath{\rightarrow}1{s}_{1/2}^{2} {}^{1}{S}_{0}$, and $1{s}_{1/2} 2{p}_{1/2} {}^{3}{P}_{0}\ensuremath{\rightarrow}1{s}_{1/2}^{2} {}^{1}{S}_{0}$ transitions in heliumlike ions.

arXiv:1011.5816Physical Review AMar 21, 201116 citations

From the work by W. Perrie et al. [Phys. Rev. Lett. 54, 1790 (1985)], it is known that the photon pairs that are emitted in the

2{s}_{1∕2}\ensuremath{\rightarrow}1{s}_{1∕2}

2{s}_{1∕2}\ensuremath{\rightarrow}1{s}_{1∕2}

and

3{d}_{5∕2}\ensuremath{\rightarrow}1{s}_{1∕2}

3{d}_{5∕2}\ensuremath{\rightarrow}1{s}_{1∕2}

transition of hydrogenlike uranium.

The angular and polarization properties of the photon pair emitted in the two--photon decay of

2s

hyperfine states in hydrogenlike ions are investigated within the relativistic Dirac framework and second order perturbation theory. The studied transitions are

2s_{1/2}\,(F=1,0)\to1s_{1/2}\,(F=1,0)

in Hydrogen atom and

2s_{1/2}\,(F=4,3)\to1s_{1/2}\,(F=4,3)

in hydrogenlike Uranium ion. Two different emission patterns are found: For non-spin-flip transitions, the angular correlation (

i.e.

the angular distribution of the emitted photons) is of the type

\sim 1+\cos^2θ

, while the linear polarizations of the emitted photons are approximately parallel one another; For spin-flip transitions, the angular correlation is of the type

\sim 1-1/3\cos^2θ

, while the linear polarizations of the emitted photons are approximately orthogonal one another.

2s

states, where spin-flip and non-spin-flip transitions are highlighted. We pay particular attention to hydrogenlike uranium, since it is an ideal candidate for investigating relativistic and high-multipole effects, such as spin-flip transitions. Two types of emission patterns are identified: (i) non-spin-flip transitions are found to be characterized by an angular distribution of the type $W(\ensuremath{\theta})\ensuremath{\sim}1+{cos}^{2}\ensuremath{\theta}$ while the polarizations of the emitted photons are parallel; and (ii) spin-flip transitions have somewhat smaller decay rates and are found to be characterized by an angular distribution of the type $W(\ensuremath{\theta})\ensuremath{\sim}1\ensuremath{-}1/3{cos}^{2}\ensuremath{\theta}$ while the polarizations of the emitted photons are orthogonal, where $\ensuremath{\theta}$ is the angle between photons directions. Deviations due to nondipole and relativistic contributions are evaluated for both types of transitions. This work is the first step toward exploring the effect of the nucleus over the angular and polarization properties of the photon pairs emitted by two-photon transitions.

Z

ions. To analyze these effects, the degree of entanglement is evaluated both in the dipole approximation and within rigorous relativistic theory. Detailed calculations are performed for the two-photon $2{s}_{1/2}\ensuremath{\rightarrow}1{s}_{1/2}$ transition in hydrogenlike ions as well as for the $1{s}_{1/2} 2{s}_{1/2} {}^{1}{S}_{0}\ensuremath{\rightarrow}1{s}_{1/2}^{2} {}^{1}{S}_{0}$, $1{s}_{1/2} 2{s}_{1/2} {}^{3}{S}_{1}\ensuremath{\rightarrow}1{s}_{1/2}^{2} {}^{1}{S}_{0}$, and $1{s}_{1/2} 2{p}_{1/2} {}^{3}{P}_{0}\ensuremath{\rightarrow}1{s}_{1/2}^{2} {}^{1}{S}_{0}$ transitions in heliumlike ions.