Shimony–Wolf states and hidden coherences in classical light

Abstract

The classical theory of polarisation coherence is briefly summarised and then extended. The extension is motivated by the recognition that the traditional theory of two-point coherence provides only what we identify as ‘diagonal’ correlation functions and their associated two-point coherence matrices. It is pointed out that a wider focus is possible when taking account of the three-sector vector space underlying all two-point coherences in classical optics. This reveals the possibility of observing a new type of ‘off-diagonal’ correlations that arise when the correlation functions under investigation are associated with points in two distinct vector spaces, pairs of points that are not analogous to the pairs of space points or time points that underlie traditional measures of spatial and temporal coherence. Quantum theory has experience with correlations engaging such ‘cross-sector’ coherences, for example in tests of Bell inequalities, and the quantum formulations are shown to be easily adopted by classical theory without incorporating quantum features in the optical signals. The familiar theory of classical coherence that is associated with the pioneering work of Emil Wolf is extended in conformance with three criteria advanced by Abner Shimony to obtain formulas for correlation functions and for the Bell measure $\mathcal{S}$ of coherence. Values of $\mathcal{S}$ greater than the standard upper limit $\mathcal{S}=2$ are predicted for certain classical Shimony–Wolf fields, indicating strong cross-sector coherence, but only when standard measures of coherence such as degree of polarisation $\mathcal{P}$ are minimised. Experimental results confirming the predictions for cross-sector coherence are exhibited.

Keywords:

• classical coherence theory of light
• correlation vector space
• cross-sector correlation
• entanglement in classical light

Acknowledgements

It is a great pleasure to acknowledge discussions over many years with Emil Wolf, to thank many colleagues for frank and helpful debates, and to mention Xiao-Feng Qian and Bethany Little particularly for their creative theoretical and experimental contributions that allowed the data shown in Fig. 2 to be obtained.

Notes

No potential conflict of interest was reported by the author.

1 Of course one may say that the field simply jumps from one fully polarised state to another so rapidly that no specific polarisation is able to be recorded. But in regard to detection, this is exactly the same as non-deterministic, and a theoretical description of such a random process has to be effectively stochastic.

2 There are well-known measures of coherence that won’t be treated here. One is correlations of higher order, of intensities or other combinations of more than two fields. The quantum theory of coherence based on photon counting raises entirely different considerations [8,9]. We will not enter this domain but recognise its existence by making sure that experimental examinations of classical results to be described below are not dependent on photon-counting detection techniques.

3 The issue of “locality" often promotes a confusing discussion of tests of Bell inequalities. We accept Shimony’s conclusion that what matters in such tests is the independence of the sectors of the joint vector space under examination, as in our polarisation and temporal function spaces. This is logically distinct from the locational separation of detectors that is employed in many experiments on photon pairs to ensure independence (for example, see [43–47]).

Additional information

Funding

Partial financial support has been provided by National Science Foundation [grant number NSF PHY-0855701] and [grant number NSF PHY-1203931].