Stephen M. Barnett

Free-space information transfer using light beams carrying orbital angular momentum

We demonstrate the transfer of information encoded as orbital angular momentum (OAM) states of a light beam. The transmitter and receiver units are based on spatial light modulators, which prepare or measure a laser beam in one of eight pure OAM states. We show that the information encoded in this way is resistant to eavesdropping in the sense that any attempt to sample the beam away from its axis will be subject to an angular restriction and a lateral offset, both of which result in inherent uncertainty in the measurement. This gives an experimental insight into the effects of aperturing and misalignment of the beam on the OAM measurement and demonstrates the uncertainty relationship for OAM.

Unitary Phase Operator in Quantum Mechanics

The difficulties in formulating a natural and simple operator description of the phase of a quantum oscillator or single-mode electromagnetic field have been known for some time. We present a unitary phase operator whose eigenstates are well-defined phase states and whose properties coincide with those normally associated with a phase. The corresponding phase eigenvalues form only a dense subset of the real numbers. A natural extension to the definition of a time-measurement operator yields a corresponding countable infinity of eigenvalues.

On the Hermitian Optical Phase Operator

Abstract It has long been believed that no Hermitian optical phase operator exists. However, such an operator can be constructed from the phase states. We demonstrate that its properties are precisely in accord with the results of semiclassical and phenomenological approaches when such approximate methods are valid. We find that the number-phase commutator differs from that originally postulated by Dirac. This difference allows the consistent use of the commutator for inherently quantum states. It also leads to the correct periodic phase behaviour of the Poisson bracket in the classical regime.

Quantum correlations in optical angle-orbital angular momentum variables

Entanglement in a Twist The strong correlations observed in quantum mechanically entangled particles, such as photons, offer potential for secure communication and quantum information processing. Leach et al. (p. 662) now show such strong quantum correlations between the complementary variables—angular position and orbital angular momentum—of two photons created during the parametric down-conversion process in a nonlinear crystal. This demonstration of entanglement in an angular basis establishes that angles are genuine quantum observables and can therefore be considered a resource for quantum information processing, capable of secure, high-dimension, key distribution. Strong quantum correlations are induced between the angular position and angular momentum of two photons. Entanglement of the properties of two separated particles constitutes a fundamental signature of quantum mechanics and is a key resource for quantum information science. We demonstrate strong Einstein, Podolsky, and Rosen correlations between the angular position and orbital angular momentum of two photons created by the nonlinear optical process of spontaneous parametric down-conversion. The discrete nature of orbital angular momentum and the continuous but periodic nature of angular position give rise to a special sort of entanglement between these two variables. The resulting correlations are found to be an order of magnitude stronger than those allowed by the uncertainty principle for independent (nonentangled) particles. Our results suggest that angular position and orbital angular momentum may find important applications in quantum information science.

Orbital angular momentum and nonparaxial light beams

Abstract The simple relationship between total angular momentum and energy and the seemingly natural separation of the angular momentum into spin and orbital components in the paraxial approximation, are investigated for a general nonparaxial form of monochromatic beam with near cylindrical symmetry.

Detection of a Spinning Object Using Light’s Orbital Angular Momentum

Doppler Effect with a Twist The Doppler shift is a familiar and well-understood effect in acoustics. Radar guns use the same effect to determine the speed of moving vehicles. Applied to a rotating object side-on, however, a linear Doppler effect would register no movement. Using twisted light, whereby photons are imprinted with a given amount of optical angular momentum, Lavery et al. (p. 537; see the Perspective by Marrucci) detected rotation with an analogous angular Doppler shift, which may be useful for remote sensing and observational astronomy. Orbital angular momentum modes of light can be used to detect rotation. [Also see Perspective by Marrucci] The linear Doppler shift is widely used to infer the velocity of approaching objects, but this shift does not detect rotation. By analyzing the orbital angular momentum of the light scattered from a spinning object, we observed a frequency shift proportional to product of the rotation frequency of the object and the orbital angular momentum of the light. This rotational frequency shift was still present when the angular momentum vector was parallel to the observation direction. The multiplicative enhancement of the frequency shift may have applications for the remote detection of rotating bodies in both terrestrial and astronomical settings.

Optical angular-momentum flux

We introduce the angular-momentum flux as the natural description of the angular momentum carried by light. We present four main results: (i) angular-momentum flux is the flow of angular momentum across a surface and, in conjunction with the more familiar angular-momentum density, expresses the conservation of angular momentum. (ii) The angular-momentum flux for a light beam about its axis (or propagation direction) can be separated into spin and orbital parts. This separation is gauge invariant and does not rely on the paraxial approximation. (iii) Angular-momentum flux can describe the propagation of angular momentum in other geometries, but the identification of spin and orbital parts is then more problematic. We calculate the flux for a component of angular momentum that is perpendicular to the axis of a light beam and for the field associated with an electric dipole. (iv) The theory can be extended to quantum electrodynamics.

The enigma of optical momentum in a medium

It is 100 years since Minkowski and Abraham first gave rival expressions for the momentum of light in a material medium. At the single-photon level, these correspond, respectively, either to multiplying or dividing the free-space value () by the refractive index (n). The debate that this work started has continued till the present day, punctuated by the occasional publication of ‘decisive’ experimental demonstrations supporting one or other of these values. We review the compelling arguments made in support of the Minkowski and Abraham forms and are led to the conclusion that both momenta are correct. We explain why two distinct momenta are needed to describe light in a medium and why each appears as the natural, and experimentally observed, momentum in appropriate situations.

Phase in quantum optics

Diracs prescription for quantisation does not lead to a unique phase operator for the electromagnetic field. The authors consider the commonly employed phase operators due to Susskind and Glogower (1964) and their extension to unitary exponential phase operators. However, they find that phase measuring experiments respond to a different operator. They discuss the form of the measured phase operator and its properties.

Uncertainty principle for angular position and angular momentum

The uncertainty principle places fundamental limits on the accuracy with which we are able to measure the values of different physical quantities (Heisenberg 1949 The Physical Principles of the Quantum Theory (New York: Dover); Robertson 1929 Phys. Rev. 34 127). This has profound effects not only on the microscopic but also on the macroscopic level of physical systems. The most familiar form of the uncertainty principle relates the uncertainties in position and linear momentum. Other manifestations include those relating uncertainty in energy to uncertainty in time duration, phase of an electromagnetic field to photon number and angular position to angular momentum (Vaccaro and Pegg 1990 J. Mod. Opt. 37 17; Barnett and Pegg 1990 Phys. Rev. A 41 3427). In this paper, we report the first observation of the last of these uncertainty relations and derive the associated states that satisfy the equality in the uncertainty relation. We confirm the form of these states by detailed measurement of the angular momentum of a light beam after passage through an appropriate angular aperture. The angular uncertainty principle applies to all physical systems and is particularly important for systems with cylindrical symmetry.