Light

The Degenerate Case in Parametric Down Conversion

Part II: Phase Difference and Polarisation

Caroline H Thompson

26:5:99

Introduction

In recent "quantum entanglement" experiments, a special configuration of PDC is often used (see, for example, Kwiat et al, Phys Rev Lett 75, 4337-4341 (1995)). The crystal is aligned so as to produce overlapping cones of output, one polarised vertically, the other horizontally. The parts used as "EPR photon pairs" are the small regions in which the cones overlap. The QT story is that in these regions we get either vertical or horizontal polarisation, with the two simultaneous "down-converted photons" having orthogonal polarisation. My own story is that both components are present in every output.

Moreover, the components tend to be of equal amplitude, and whatever the phase relationship between vertical and horizontal is for one output it is identical (or, possibly, always inverse) for the other. This feature is caused by what I am becoming convinced is a fact, that the phases of the outputs in the degenerate case (i.e. when the outputs are exactly half the frequency of the pump laser) are determined by those of the pump, except that there are two possibilities. Each will tend to happen equally often, as there is no reason for one to be preferred. The actual choice will depend on random factors, and can be considered to be due to the local "zero point field". Since both vertical and horizontal components are locked, in some sense, to the pump, the phase difference between them at the point of initiation must be either 0 or o .

Now let us see how this might explain the result of the 1998 Innsbruck experiment by Gregor Weihs et al (quant-ph/9810080 and now published in Physical Review Letters 81, 5039 (1998)). The ingredients comprise a source of the kind discussed, a few "wave plates", adjustable "modulators" (acting in the same way as a wave plate only with the relative velocities of the vertical and horizontal components not fixed but able to be modulated by adjusting the voltage) and a Wollaston prism polariser. In the view of the initiators of the experiment, there are some other critical components - random number generators that are used to switch the modulators between two settings - but these play no part in the logic I am discussing.

Thus the apparatus generates pairs of signals, provides the means to alter the phase difference between the two components of each, then analyses the result using wollaston prisms. Let us study how this analysis is done.

Fig. 1: The resultant of equal-amplitude vertical and horizontal components of light, for four special values of the phase difference.

On each side of the experiment, beams that were initially identical (but may now have different phase differences) impinge on a Wollaston prism, set at 45 degrees. It is straightforward to work out mathematically, using ordinary classical optics, what will happen, but possibly Fig. 1 will help. Because the prism is at 45 degrees to both components, it is the projection of the fields that count. The oscillations of the fields are phase-related, so we get interference. The net result for the special phase differences of Fig. 1 is shown in Table 1, for a prism alligned so that linear polarisation in the +45ºdirection exits from the "+" output. The mathematics shows that, as in other kinds of interference, so long as Malus' Law holds the intensities vary sinusoidally as the phase difference varies, allowing deduction of the intermediate values.

	Output intensity
Phase Difference	+	-
0	1	0
p /2	1/2	1/2
p	0	1
3p /2	1/2	1/2

Table 1: Output intensities from a Wollaston prism, for input comprising equal vertical and horizontal components, each of e/m amplitude 1/Ö2.

It is at once clear how the output seems to support the QT photon hypothesis, as it looks as if the output, at least in two special cases, all exits by one port. There is something wrong, though. Where the table says 1/2 it means that the intensity is 1/2, and this does not necessarily mean that the probability of detection is 1/2. The intensities vary sinusoidally, but the probabilities vary according to rules that depend on the operating characteristics of the detectors. The theorists have lived for a long time now under the impression that they were dealing with photons, and have not, therefore, tended to test for linearity. My current hypothesis on the experiment, using information from email correspondence as well as the published paper, is that the detectors are not likely to be linear. The nonlinearity is likely to result in nondetection of a high proportion of weak signals, so that, in particular, probabilities that one might expect to be 1/2 are in fact less than this.

Thus the combined effect of phase difference adjustment and a Wollaston prism at 45 degrees is, mathematically, logically, just the same as the effect of using a polariser set at an angle determined by the experimenter. He determines the "angle" by his choice of phase plates and by the modulator setting. There is another factor involved, though. This is the exact frequency of the pulse. Over this he has no direct control. He is able to control frequency only to within a "band-width". From the behaviour of the system, I would deduce that this band-width is such that there is a difference of a few complete cycles between the phase shift for the shortest and that for the longest, for a given setting of the modulator. The light (as in glass, say) travels at different speeds at different wavelengths, and the variation is not, it appears, exactly the same for the two components. My hypothesis is that each pair of pulses produced by the source has exactly the same frequency - exactly half the current pump frequency. This is the true "hidden variable" of the experiment, causing phase differences that are identical for the members of a pair but vary randomly as the pump varies.

Taking this resultant phase difference as the hidden variable rather than the frequency, we find we have the typical EPR setup, for which the standard realist analysis (see my paper at quant-ph/9903066) predicts a sinusoidal variation in the coincidence rates, with minimum of 1/8, maximum of 3/8. The fact that the actual curves have minima almost zero is explained by the above-mentioned nonlinearity of the detectors.

Or it is possible that we are seeing the beginnings of another effect: the fact that, for any fixed frequency, the outputs will fall into two "phase sets", as mentioned earlier. If the dispersion of the pump is great, it will obliterate this effect, and it appears from correspondence that in the actual experiment this is the case. If, however, great effort is devoted to ensuring that the frequency spread of the detected signals is very narrow, and the signal is not sent over great distances along fibre cables (risking scaling up of any phase difference), it might be possible to produce output with different statistical properties. Instead of the hidden variable taking on all values with equal probability, it might take on just the two, 0 and o .

I have explored the consequences of this in a draft paper, Weihs.tex, available on request. The paper also covers the maths of the Wollaston prism, and is intended for posting in the quant-ph archive and possible publication in Phys Rev A or elsewhere. It also, incidentally, hints at other phenomena - "induced coherence" (see Zou, X Y, Wang, L J and Mandel, L, Physical Review Letters 67, 318 (1991)) in particular, in which nature seems to be following my model.