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Physical Review A

Optical angular momentum: Multipole transitions and photonics

Davıd L. Andrews

Published Mar 15, 2010• 36 citations

Abstract

The premise that multipolar decay should produce photons uniquely imprinted with a measurably corresponding angular momentum is shown in general to be untrue. To assume a one-to-one correlation between the transition multipoles involved in source decay and detector excitation is to impose a generally unsupportable one-to-one correlation between the multipolar form of emission transition and a multipolar character for the detected field. It is specifically proven impossible to determine without ambiguity, by use of any conventional detector, and for any photon emitted through the nondipolar decay of an atomic excited state, a unique multipolar character for the transition associated with its generation. Consistent with the angular quantum uncertainty principle, removal of a detector from the immediate vicinity of the source produces a decreasing angular uncertainty in photon propagation direction, reflected in an increasing range of integer values for the measured angular momentum. In such a context it follows that when the decay of an electronic excited state occurs by an electric quadrupolar transition, for example, any assumption that the radiation so produced is conveyed in the form of ``quadrupole photons'' is experimentally unverifiable. The results of the general proof based on irreducible tensor analysis invite experimental verification, and they signify certain limitations on quantum optical data transmission.

DOI: 10.1103/physreva.81.033825

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The premise that multipolar decay should produce photons uniquely imprinted with a measurably corresponding angular momentum is shown in general to be untrue. To assume a one-to-one correlation between the transition multipoles involved in source decay and detector excitation is to impose a generally unsupportable one-to-one correlation between the multipolar form of emission transition and a multipolar character for the detected field. It is specifically proven impossible to determine without ambiguity, by use of any conventional detector, and for any photon emitted through the nondipolar decay of an atomic excited state, a unique multipolar character for the transition associated with its generation. Consistent with the angular quantum uncertainty principle, removal of a detector from the immediate vicinity of the source produces a decreasing angular uncertainty in photon propagation direction, reflected in an increasing range of integer values for the measured angular momentum. In such a context it follows that when the decay of an electronic excited state occurs by an electric quadrupolar transition, for example, any assumption that the radiation so produced is conveyed in the form of ``quadrupole photons'' is experimentally unverifiable. The results of the general proof based on irreducible tensor analysis invite experimental verification, and they signify certain limitations on quantum optical data transmission.

General expressions for the dependence of the polarization of the radiation emitted in an atomic electronic transition upon the angle of emission are derived on the basis of a vector-spherical-harmonic/Bessel-function description of the photon. The concept of a photon of a specific multipolarity, i.e. quadrupole, octupole, etc, and identifiable as such, is found to be unviable. The angular distribution and dependence upon distance from the source of the transverse and parallel components of the electric and magnetic fields arising from E1, E2, M1 and M2 atomic electronic emissions are elucidated.

Schemes for the communication and registration of optical angular momentum depend on the fidelity of transmission between optical system components. It is known that electron spin can be faithfully relayed between exciton states in quantum dots; it has also been shown by several theoretical and experimental studies that the use of beams conveying orbital angular momentum can significantly extend the density and efficiency of such information transfer. However, it remains unclear to what extent the operation of such a concept at the single photon level is practicable—especially where this involves optical propagation through a material system, in which forward scattering events can intervene. The possibility of transmitting and decoding angular momentum over nanoscale distances itself raises other important issues associated with near-field interrogation. This paper provides a framework to address these and related issues. A quantum electrodynamical representation is constructed and used to pursue the consequences of individual photons, from a Laguerre–Gaussian beam, undergoing single and multiple scattering events in the course of propagation. In this context, issues concerning orbital angular momentum conservation, and its possible compromise, are tackled by identifying the relevant components of the electromagnetic scattering and coupling tensors, using an irreducible Cartesian basis. The physical interpretation broadly supports the fidelity of quantum information transmission, but it also identifies potential limitations of principle.

Recent experiments involving the interaction of optical vortices with atoms in quadrupole transitions have shown it to be accompanied by the exchange of orbital angular momentum (OAM) between the electronic states of the atom and the optical vortex field. Earlier work, both theoretical and experimental, had ruled out the transfer of a vortex OAM to the electronic degrees of freedom in an electric dipole atomic transition and it has been confirmed that the lowest multipolar order involving an OAM transfer to the electronic motion is indeed the electric quadrupole. Hitherto, the quadrupole transition involving optical vortices has not been quantified and we thus set out to evaluate the absorption rate accompanied by an OAM transfer with reference to the $6^{2}S_{1/2}\ensuremath{\rightarrow}5^{2}D_{5/2}$ in Cs when cesium atoms are subject to the field of a linearly polarized optical vortex. Our results assuming typical experimentally accessible parameters indicate that the absorption rate for moderate light intensities is lower than the quadrupole spontaneous emission rate but should still be within the measurement capabilities of modern spectroscopic techniques.

We present an experimental demonstration of a form of the angular uncertainty principle for single photons. Producing light from type I down-conversion, we use spatial light modulators to perform measurements on signal and idler photons. By measuring states in the angle and orbital angular momentum basis, we demonstrate the uncertainty relation of Franke-Arnold et al (2004 New J. Phys. 6 103). We consider two manifestations of the uncertainty relation. In the first we herald the presence of a photon by detection of its paired partner and demonstrate the uncertainty relation on this single photon. In the second, we perform orbital angular momentum measurements on one photon and angular measurements on its correlated partner exploring, in this way, the uncertainty relation through non-local measurements.

The resonant interaction of optical radiation with atomic vapors can excite the atoms into anisotropic states with long lifetimes. The conventional description of these states uses an expansion in a basis of multipole moments, which are represented by irreducible tensor operators. We demonstrate an experimental method that permits the excitation of these atomic multipole moments in specific, experimentally controllable spatial orientations. Complementary optical methods permit the observation of multipole moments of arbitrary order and a direct measurement of their spatial orientation. We demonstrate these procedures in the ground state of atomic sodium, using optical pumping to polarize the angular-momentum substates. We show how the experimental results can be used to separate signal contributions that originate from different multipole moments.