The
Degenerate Case in Parametric Down Conversion
Part
I: Frequency Variations
Caroline
H Thompson
Web site: http://www.aber.ac.uk/~cat
15:2:99
Background
Parametric Down Conversion (PDC) is the name given to the production of a pair of supposedly quantum-entangled "photons" by a "non-linear crystal" when illuminated by a suitable laser. Quantum theory (QT) predicts that the pair will have "conjugate" frequencies, the sum of their energies equalling that of the pump "photon" that induced them. For pairs of widely differing frequencies, it does seem likely on energy-conservation grounds that the frequencies will be negatively correlated. Energy conservation gives z 1 + z 2 = z 0, where z 1 and z 2 are the output frequencies ("signal" and "idler" respectively) and z 0 the pump frequency. There is little room for doubt. If z 1 is high, z 2 must be low, and vice versa.
Many interesting optical experiments involve, however, the special "degenerate" case, in which z 1 and z 2 are (almost?) equal, both being z 0/2. I have for a few years now had strong doubts as to whether the same negative correlation is to be expected in this case. There are small frequency variations, making up the observed "band width", but I can see no a priori reason for supposing that the two pulses differ in any way, other than polarisation. The observed frequency spread could arise mainly from variations in the pump frequency, which generally has quite a wide band width. The variations in output frequency could be perfectly positively correlated, with the frequency varying precisely in parallel for the two beams. The assumption of negative correlation — that they are "conjugate" — could be just a carry-over from the theory of the non-degenerate case. I have yet to see evidence that it happens.
I believe that the physics behind the generation of degenerate pairs may well force them to be identical in frequency, and that this frequency can be of considerably narrower bandwidth than would appear from observation. (Remember that we cannot observe the frequency spread within individual pulses, only in ensembles of them. See also Appendix A) But my beliefs in this area are not currently in question. I wish to put to you a few facts that suggest that the empirical evidence favours my hypothesis and, incidentally, requires no "non-local" quantum effects for its explanation.
The 1997 and 1998 Geneva Bell Test
Experiments
The experiment that first raised my doubts was that of Wolfgang Tittel et al. in Geneva in 1987, which is reported as demonstrating non-local Bell test violations over a distance of 10 k (quant-ph archive (http://xxx.lanl.gov) ref 9707042; revised 12:6:98 and Phys Rev A 57, 3229). It seemed to me that one could achieve the observed high-visibility coincidence curve much more easily with positive correlations than with negative ones. With negative ones, the interference fringes would come in different places in the two locations and so any variation in the pump frequency would ruin the result. Yet the pump bandwidth was quite wide. Using the usual quantum theory reasoning, about indistinguishability of paths etc., this simple experimental fact would not have been obvious. No sign of ordinary first order interference can be detected, and the theory assumes — from a classical point of view quite unjustifiably — that it follows that there is none.
I had some correspondence in 1987 with Nicolas Gisin, a member of the Geneva team, and later with Tittel. I also discussed the matter briefly with Gisin at a conference. He agreed that he had not tested for the negative correlation, and said he would look into it.
In 1998, another experiment was reported (quant-ph archive (http://xxx.lanl.gov) ref 9809025), and this showed signs that they had at least considered the problem. But my notes report:
I've got his new papers, and am (a) pleased to find a definite hook on which to hang my "proof" that the detection loophole is in action: they mention (page 8, near bottom) that R+- + R++ is not constant! QED! But (b) I'm currently stuck as to an explanation of their graph (fig 3) showing variations as you vary the phase differences simultaneously in the same and opposite directions at the two stations. They also show the result varying one at a time. Why on earth should the frequencies for BEL and BNX differ?? Wavelengths are the same. I can't find any mention of an asymmetry that could explain this.
As I understand it, it is this graph in Fig. 3 that is supposed to reassure us that the frequencies are in fact negatively correlated, but I still have not found out just what the graph means. I have asked and had no reply.
The experiment used temperature to control the path-length differences in the interferometers. They could vary the temperature gradually, it seems, but perhaps not keep it steady at designated values. Hence it may not have been possible for them to make the direct comparisons that were needed. These would have involved keeping the temperature at one station constant and observing the change in coincidence rates as that of the other is changed very very slightly, so that the direct interference fringes, if they had been visible, would have been moved by less than a wavelength. The trick would be to start with temperatures that give the maximum coincidence rate. We then change one of the temperatures very slightly — we increase it, say. Now if we have negative correlations, we shall get back to our maximum visibility most easily by decreasing the temperature at the other station; if positive, by increasing it, to keep it in line with the first station. (Practical note: interpretation will have to take into account the possibility that we do not have true "rotational invariance": frequency spreads may not be wide enough to ensure that all phase differences are equally likely.)
New Evidence: "Cancellation of
Chromatic Dispersion"
Thus the question of whether or not we have negative frequency correlations in EPR experiments apparently remains unresolved, but I have recently realised that a new theory has been applied. Is this new theory valid? Does the experimental evidence really confirm it? It seems to me that it had no reason to be invented other than to explain a rather extraordinary fact that had emerged in experiments such as Tittel's.
For in these experiments, the time-resolution had been, it seems, better than they could reasonably have hoped for, given their data on band widths (see Appendix A). Even without any "subtraction of accidentals" (see my papers, quant-ph/9711044, for example), the coincidence patterns were extraordinarily good. Tittel et al. interpreted this as meaning that the "photons" were behaving as if "chromatic dispersion" — the spreading out of frequencies in the course of transmission, in this case along 10 k or so of fibre-optic cable — hardly happened at all. (My own interpretation is that the frequencies were very much sharper than theory predicted, and in any case identical for the two stations. There was no reason for dispersion of this kind to be present to weaken the correlations.)
Gisin discusses the matter in quant-ph 9901043. The reader is strongly advised to consult the relevant section (pp 4-5). The argument concerns the idea that the "chromatic dispersion" (the spreading of frequencies in glass, with red going faster than blue) at the centre frequency, 1310 nm, happens to vanish for the particular fibres used. Because the law governing dispersion is quadratic, deviations upwards and downwards by the same frequency shift will experience the same delay, to first approximation. Hence our supposedly negatively correlated pulses will have the same delay, and no effect of the dispersion will show up on coincidence counts. The difference in detection times will be the same for all pairs, so long as the deviations in frequency are small enough for the approximation to be valid. As he says, "this phenomenon, called 2-photon chromatic dispersion cancellation, is essential for long distance Bell experiments using optical fibres".
I have now read one of the references that he gives on the subject — Steinberg, A M, Kwiat, P G and Chiao, R Y, "Dispersion cancellation in a measurement of the single-photon propagation velocity in glass", PRL 68, 2421 (1992). I believe I have found a glaring error in the interpretation. I shall try and explain the problem, but obviously it would be best if the reader had the paper available as well.
An experiment is described that is designed to investigate velocity of propagation and chromatic dispersion of light in glass, and the effect on coincidences between signal and idler outputs from a pumped nonlinear crystal. The two outputs from a nonlinear crystal are used as inputs, from opposite sides, to a beamsplitter. The two beams emerging from the beamsplitter are sent to detectors (D1 and D2). A block of glass is inserted along one path, and a "trombone prism" can be adjusted to produce a delay along the other so as to equalise the times along both routes.
It is argued that the glass spreads ("chirps") the light so that the red component arrives first then the blue, with the spread of frequencies of bandwidth 60 fsec. Quantum theory predicts that photons of opposite colours must be detected by the two detectors, i.e. when there is a coincidence, it must have been either red in D1 and blue in D2 or blue in D1 and red in D2. Steinberg et al. proceed to show by an example that this will ensure that the interval between the two detections is the same, to first order, whether the two photons were both transmitted or both reflected at the beamsplitter. (According to QT, because of the supposed indivisibility of photons, it is only in these two cases that we can get coincidences.)
But if you check, it turns out that in their example, with blue detected by D1 and red by D2, the interval is the same for both the possibilities (i.e. for both transmitted or both reflected). D2 registers before D1. But it seems to me that they never looked at the other case, with red at D2 and blue at D1. For this case, D2 registers after D1. So though they may be of the same magnitude, the time differences are opposite in sign, and hence contribute to a wider, shallower, dip in the coincidence curve. They have failed, in fact, to account for "dispersion cancellation", and for the observed narrow dip.
They do give "a more formal derivation" of the effect, which assumes the coincidence rate to be given by "Glauber's correlation function", with details to be published in Phys Rev A. The information given in the PRL paper suggests that the formal method may have the same inadequacies as the example, looking at one choice of colour pairs only. The whole experiment in fact (going along for the moment with the QT idea that we always detect conjugate colours) can produce coincidences of type:
D1 blue; D2 red; RR
D1 blue; D2 red; TT
D1 red; D2 blue; RR
D1 red; D2 blue; TT
where R = reflection and T = transmission at the beamsplitter.
In the actual experiment, all possibilities will occur. If all time-differences were of the same sign, they could be made approximately zero by adjusting the "trombone" delay, but if half are of one sign and half the other, there will be an irreducible broadening of the time interval. This is not observed.
Conclusion
I am far from happy with the whole idea of "cancellation" of chromatic dispersion, and if this does not happen then the idea of negative frequency correlations in long-distance EPR experiments is not viable. The existing theory seems far from satisfactory — indeed, as far as I can tell, we are offered two quite distinct theories by Gisin and Steinberg, so they cannot both be right. Steinberg's version seems to me to be intrinsically flawed, and Gisin's somewhat strains the credibility. The alternative — that the photons are not "conjugate" but identical, and that each pair is almost dispersion-free — is so simple that surely it should be investigated? The physics of this alternative is well known: it is just classical optics, with perhaps a minor modification to allow for the existence of pulses and the possibility that they can be both short and almost dispersion-free.
I intend in Part II to cover other properties of degenerate PDC outputs: their phase and polarisation relationships. It is hoped to show that a complete local causal explanation emerges for all optical experiments employing such light. The explanation is broadly compatible with the ideas of Stochastic Electrodynamics (SED) (see http://homepages.tesco.net/~trevor.marshall). SED covers general characteristics, averaged over time, but I feel that a more detailed theory is required for the experiments concerned here. Can I call my ideas so far a "theory"? What I believe I may have found is the beginnings of a consistent (local realist) description of the phenomena. This, if not itself a theory, could form a sound basis for one.
Appendix A: Coherence Length and Band
Width
Both QT and classical theory agree on a formula that predicts that energy variations multiplied by time is in some sense likely to be constant. In the QT case, this is a form of Heisenberg Uncertainty Principle. In the classical case, it arises from Fourier theory. If you have a short pulse of light and take its Fourier transform, you will find that a wide spread of frequencies is involved. Conversely, if a certain light source produces light of many frequencies, one expects the "coherence length" to be very short.
It is assumed in both QT and classical theories that pulse length and coherence length are much the same thing. When dealing with classical light sources, the distinction may be unimportant. The difference becomes crucial, though, when dealing with laser sources. These produce light with coherence lengths enormously longer than any envisaged in classical theory. They can also produce very short pulses. Fourier theory would suggest that these short pulses must have wide frequency spreads, but I believe that experiments such as those of Tittel et al. achieve their dramatic correlations as a result of even short pulses being, in reality, sometimes of high coherence — almost dispersion-free. The kind of effect mentioned above, in which frequency variations are from one pair to the next and not within pairs, may be quite common.
One author who has come to the same conclusion independently and had the courage to publish is Paul Wesley, in his book "Classical Quantum Theory", (Benjamin Wesley, 1996, available from author at Weiherdammstrasse 24, 78176 Blumberg, Germany).
In Tittel's 1997 experiment, they state that the band width for the down-converted photons was about 90 nm FWHM (Full Width at Half Maximum of frequency spectrum). They find that the envelope of their "interferogram" suggests a single-photon coherence length around 10.2 l m, which is, they imply, just as expected from theory. In a sense they may be right. These figures may well be correct. The estimated coherence length may be a reasonable one for individual pulses. They would argue that this is because the applied theory, "chromatic dispersion cancellation", combined with their bandwidth estimate, is correct. I suspect, though, that what we are seeing is the simple empirical fact of the clear coincidence pattern, and ad hoc theory contrived to explain it.
This Appendix has not proved as clear-cut as I expected! I admit to confusion and the need for further work.
Appendix B: Glass can Improve Coherence!
The Steinberg et al. experiment shows a very interesting feature that they do not remark on. Possibly it is an illusion, and the graphs that seem to show it (Fig. 3 (a) and (b)) are not as comparable as it would appear. Taking the graphs at face value, though, it seems that the presence of the glass reduces the dip dramatically, showing that, in some sense, the degree of coherence is increased when the glass is present. And indeed, in a sense it may well be, if we treat the matter classically. The spread (chirped) signal will be of lower amplitude than the unspread one, but it may well be almost dispersion-free at any given point along its length.
Perhaps this phenomenon is already well known in applications?