Department of Chemistry

Saint John's University

College of Saint Benedict

The schematic diagram below shows a Mach-Zehnder interferometer for photons. When the experiment is run so that there is only one photon in the apparatus at any time, the photon is always detected at D_{2}and never at D_{1}.(1,2,3) The qualitative explanation is that there are two paths to each detector and, therefore, the probability amplitudes for these paths may interfer constructively or destructively. For detector D_{2}the probability amplitudes for the two paths interfer constructively, while for detector D_{1}they interfer destructively. A quantitative quantum mechanical analysis of this striking phenomenon is outlined below. The photon leaves the source, S, traveling in the y-direction. Whether the photon takes the upper or lower path it interacts with a beam splitter, a mirror, and another beam splitter before reaching the detectors. \BS\ (S)- -> - -\- - - - - - - -\M+----> y-direction |\ |\ | | | | | | | | | v | | | | x-direction \|BS\|M\- - - - - - - -\- - - ->D\ |\_{2}BS= Beam splitter |D= Detector |M= Mirror vS= SourceD_{1}Orthonormal basis states:(1x2 vectors) Photon moving in the x-direction: |x> = |1| < x | = (1 0) < x | x > = 1 |0| Photon moving in the y-direction: |y> = |0| < y | = (0 1) < y | y > = 1 |1| < y | x > = < x | y > = 0Operators:(2x2 matrices) Operator for photon interaction with the mirror:M= |0 1| |1 0| Operator for photon interaction with the beam splitter:BS= |T iR| |iR T| T and R are the transmission and reflection amplitudes. For the half- silvered mirrors used in this example they are: T = R = 1/2^{1/2}= .707Operations:After interacting with a beam splitter a photon is in a linear superposition of |x> and |y> in which the components are 90 degrees out of phase.BS|x> = [|x> + i|y>]/2^{1/2}BS|y> = [i|x> + |y>]/2^{1/2}BS M BS|y> = i|y> Interaction with the mirror merely changes the direction of the photon.M|x> = |y>M|y> = |x>Matrix elements:< x |M| x > = 0 < y |M| x > = 1 < x |M| y > = 1 < y |M| y > = 0 < x |BS| x > = < y |BS| y > = 1/2^{1/2}< y |BS| x > = < x |BS| y > = i/2^{1/2}Dirac brackets are read from right to left. In Dirac's notation < x |M| y > is the amplitude that a photon initially moving in the y- direction will be moving in the x-direction after interacting with the mirror. |< x |M| y >|^{2}is the probability that a photon initially moving in the y-direction will be moving in the x-direction after interacting with the mirror. |< y |BS| y >|^{2}is the probability that a photon initially moving in the y-direction will be found moving in the y-direction after interacting with the beam splitter. (A) For the photon to be detected at D_{1}it must be in the state |x> after interacting with two beam splitters and a mirror in the configuration shown above. The probability that a photon will be detected at D_{1}: < x |BS M BS| y > = 0 thus |< x |BS M BS|y>|^{2}= 0 (B) For the photon to be detected at D_{2}it must be in the state |y> after interacting with two beam splitters and a mirror in the configuration shown above. The probability that a photon will be detected at D_{2}: < y |BS M BS| y > = i thus |< y |BS M BS| y >|^{2}= 1 It is also instructive to use Dirac's notation to examine upper and lower paths. (A') < D_{1}| y > = < D_{1}| y >_{upper}+ < D_{1}| Y >_{lower}= < x |BS| x >< x |M| y >< y |BS| y > + < x |BS| y >< y |M| x >< x |BS| y > = {1/2^{1/2})*1*(1/2^{1/2}) + (i/2^{1/2})*1*(i/2^{1/2}) = 1/2 - 1/2 = 0 This shows that upper and lower paths have the photon arriving 180 degrees out of phase. Thus the photon suffers destructive interference at D_{1}. (B') < D_{2}| y > = < D_{2}| y >_{upper}+ < D_{2}| Y >_{lower}= < y |BS| x >< x |M| y >< y |BS| y > + < y |BS| y >< y |M| x >< x |BS| y > = (i/2^{1/2})*1*(1/2^{1/2}) + (1/2^{1/2})*1*(i/2^{1/2}) = i/2 + i/2 = i Thus, |< D_{2}| y >|^{2}= 1 This calculation shows that the upper and lower paths have the photon arriving in phase at D_{2}. If either path (upper or lower) is blocked the interference no longer occurrs and the photon reaches D_{1}25% of the time and D_{2}25%. Of course, 50% of the time it is absorbed by the blocker.Lower path blocked:Probability photon reaches D_{1}: |< x |BS| x >< x |M| y >< y |BS| y >|^{2}= 1/4 Probability photon reaches D_{2}: |< y |BS| x >< x |M| y >< y |BS| y >|^{2}= 1/4Upper path blocked:Probability photon reaches D_{1}: |< x |BS| y >< y |M| x >< x |BS| y >|^{2}= 1/4 Probability photon reaches D_{2}: |< y |BS| y >< y |M| x >< x |BS| y >|^{2}= 1/4References:

- P. Grangier, G. Roger, and A. Aspect, "Experimental Evidence for Photon Anticorrelation Effects on a Beam Splitter: A New Light on Single Photon Interferences," Europhys. Lett. 1, 173-179 (1986).
- V. Scarani and A. Suarez, "Introducing Quantum Mechanics: One-particle Interferences," Am. J. Phys. 66, 718-721 (1998).
- Kwiat, P, Weinfurter, H., and Zeilinger, A, "Quantum Seeing in the Dark," Sci. Amer. Nov. 1996, pp 72-78.

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