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Journal of Optics A Pure and Applied Optics

Quantum cores of optical phase singularities

Michael Berry, Mark R. Dennis

Published Apr 20, 2004• 54 citations
Publisher

Abstract

The nodal line singularities (optical vortices) of classical scalar optics are smoothed in quantum optics, because of spontaneous emission into unoccupied modes. The radius of the 'quantum core' surrounding each classical singularity is proportional to ??. A trapped excited atom, steered into a nodal line of the classical field, is a possible detector for the effect. Analogous phenomena are anticipated for other waves, for example sound, where the silence at a nodal line is disturbed by pressure fluctuations of the fluid molecules.

DOI
10.1088/1464-4258/6/5/006

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Journal of Modern Optics•Jul 23, 2008•17 citations
The nodal line singularities (optical vortices) of classical scalar optics are smoothed in quantum optics, because of spontaneous emission into unoccupied modes. The radius of the 'quantum core' surrounding each classical singularity is proportional to ??. A trapped excited atom, steered into a nodal line of the classical field, is a possible detector for the effect. Analogous phenomena are anticipated for other waves, for example sound, where the silence at a nodal line is disturbed by pressure fluctuations of the fluid molecules.
Optical vortices are points of zero intensity in a two dimensional, classical optical field. As first discussed by Berry and Dennis [Berry, M. V. and Dennis, M. R., "Quantum cores of optical phase singularities" Journal of Optics A 6, S178-S180 (2004)] these singularities are replaced by 'quantum cores' in a deeper level of description. In a fully quantized theory of optical fields an excited atom trapped at the singularity can emit light spontaneously and hence soften the perfect zero of an optical vortex. More recently Barnett [Barnett, S. M., "On the quantum cores of a optical vortex," Journal of Modern Optics 55, 2279-2292 (2008)] presented a more realistic analysis of quantum cores which accounts for the effect of the trapping potential on the transition dynamics inside a vortex core. Here, we revisit the scenario of emission near an optical vortex in the realistic setting of Barnett.
The nodal structure of the Schr\"odinger wave function is used to describe an NNN-particle system in a stationary quantum state. In terms of nodal lines (quantized flux lines) together with their circulation numbers, two fields are defined: the microscopic, singular nodal flux field n (r) and the macroscopic, regular nodal flux density field N (r). Their definition arises in a natural way from a quantum-mechanical vector potential defined by the gradient of the multivalued phase function associated with the pattern of quantized flux lines. The flux of the fields n (r) and N (r) is quantized and shown to be proportional to the circulation of the velocity in the system. The unit circulation for a system of bosons with a particle spin s=0s=0s=0 and mass m0{m}_{0}m0​ is equal to hm0\frac{h}{{m}_{0}}m0​h​. In the case of charged particles, the field (cq)N(r)(\frac{c}{q})\mathbf{N}(\mathbf{r})(qc​)N(r) resembles Jehle's formulation of the magnetic field of a lepton as a superposition of quantized flux lines. In our case, however, the presence of quantized flux lines, forming closed loops follows from the mere existence of a current density and does not have to be assumed. By a simple argument, using permutation symmetry, it is shown that a flux quantum of the size hc2q\frac{\mathrm{hc}}{2q}2qhc​ is possible.
Spinor Bose–Einstein condensates (BECs) and singular optical systems have both recently served as sandboxes to create and study analogs of phenomena from other fields of physics that are otherwise difficult to create and control experimentally. Interfacing singular optics and spinor BECs allows us to take advantage of and build on the foundations of singular optics to create and describe complex spin textures in BECs that serve as analogs of other systems. Here, the complete BEC wavefunctions are precisely engineered via a two-photon Raman interaction to contain π-symmetric (lemon, star) or 2π-symmetric (saddle, spiral) C-point singularities. The optical Raman beams are singular optical beams that contain these singularities and transfer them to the condensate, thereby creating vector-vortex spin textures—the spinor counterparts to scalar vortices—in pseudo-spin-1/2 BECs. With a version of atom-optic polarimetry, we can measure the Stokes parameters of the atomic cloud and characterize the singularities by the patterns present in their ellipse fields or by the C-point index. In the low density limit, these spin textures are analogs of optical vector vortices and should have dynamics driven by a matter-wave Gouy phase. With precise tuning of Raman beam parameters, we can create full Bloch BECs that contain every possible superposition between two states in the atomic cloud. Full Bloch BECs are similar to topologically stable magnetic skyrmions such as those created in thin metal films and nanowires, which may prove useful for atom-spintronics and topological quantum processes.
Conceptual problems in quantum optics are investigated by a study of the physical significance of various types of description of the field of a single mode. The descriptions are examined through the effect of the field on matter. Both undamped and damped modes are considered, the descriptions being those of various degrees of precision in classical and quantum-mechanical formalisms. The matter consists of a large number of "molecules," and the interaction is studied by means of low-order perturbation theory. Formal expressions are obtained for the energy interchange between field and molecules in terms of the zeroth-order (uncoupled) operators, with the effect of the commutation properties of the field variables explicitly separated from that of the excitation of the field, in second order. The significance of taking expectation values of the operator results is discussed, and it is concluded that this procedure is justified only when the experiment under consideration involves an averaging process; thus, the molecular variables should be averaged, but not necessarily those of the field. Conditions under which the averaging process is justified are considered, and it is reasoned that, when such a process is not justified, a statistical description of the field can result only in a statistical description of the interaction process. Results of various field descriptions are compared, and it is shown that for induced emission and for spontaneous emission from coherently oscillating molecules, classical and quantum-mechanical fields are equivalent. However, for a self-consistent theory of spontaneous emission from molecules that are not oscillating coherently, the quantum-mechanical commutation properties of the field are essential if the molecules are treated quantum-mecahnically, and the field must be treated classically if the molecules are treated classically; the type of description used for the excitation of the field is inconsequential. Photoelectric emission is examined, and, within the approximation framework used, shown to be accurately described by a classical treatment of the field. It is pointed out that there does not necessarily exist a connection between "photon statistics," that is, the statistics associated with an ensemble of fields, and photocurrent fluctuations in one or more detectors coupled to the same field.
An optical vortex is a line around which the phase increases by an integer multiple of 2π. It follows that the phase on the line itself is undefined and hence the field must have zero amplitude there. Berry and Dennis have suggested that this line of darkness is smoothed by a ‘quantum core’ with a radius proportional to ħ1/2 and have illustrated this idea by considering the competition between stimulated and spontaneous emission by an excited atom placed in the vicinity of the vortex. We show here that a similar phenomenon may be seen in absorption when the quantum state of motion of the absorbing atom is taken into consideration. There is, however, an underlying quantum singularity in which the absorption events for an atom centred on the vortex core can take place only if accompanied by a transfer of angular momentum to the atomic motion. The nature of this singularity relies on the evolution of an entangled state between the electronic and motional degrees of freedom of the trapped atom. We comment briefly on the effects of field quantisation on this quantum core of the optical vortex.