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 D2 and
    never at D1.(1,2,3)

    The quantum mechanical analysis of this striking phenomenon is outlined
    below.  There are two paths (upper and lower) to each detector, and they 
    both contain a beam splitter, a mirror, and another beam splitter before 
    the detectors are reached. At the beam splitters the the probability 
    amplitude for transmission is 1/21/2, while for reflection it is i/21/2. 
    The origin of the 90o phase difference between transmission and reflection 
    is found in the principle of energy conservation as is shown in the appendix.
 
                     Upper Path
			    
		\ BS              \
      (S)- -> - -\- - - - T - - - -\ M            
		 |\                |\              
		 |                 |               
		 |                 |
                 R                 |
                 |                 |                            
		 |                 |                    
		\|             BS \|
	       M \- - - - - - - - -\- - - -> D2
		  \                |\                 BS = Beam splitter (50/50)
		     Lower Path    |                  D  = Detector
                 		   |                  M  = Mirror
                                   v                  S  = Source
				   D1		   

    Because there are two paths to each detector the probability amplitudes for these 
    paths may interfer constructively or destructively when added. For detector D2 
    the probability amplitudes for the two paths interfer constructively, while for 
    detector D1 they interfer destructively. 

    For example, the probability for the photon being detected at D2 is calculated 
    as follows:


    (1)  P(D2) = |< D2 | S >|2 = |< D2 | T >< T | S > + < D2 | R >< R | S >|2      

                              = |(i/21/2)(1/21/2) + (1/21/2)(i/21/2)|2 = 1


    The probability that the photon will be detected at D1 is:


    (2)  P(D1) = |< D1 | S >|2 = |< D1 | T >< T | S > + < D1 | R >< R | S >|2      

                              = |(1/21/2)(1/21/2) + (i/21/2)(i/21/2)|2 = 0


    Appendix:

    Suppose there is no phase difference between transmission and reflection.  
    Then the probability amplitudes for transmission and reflection are both 1/21/2.
    Under these circumstances equations (1) and (2) become


    (1')  P(D2) = |(1/21/2)(1/21/2) + (1/21/2)(1/21/2)|2 = 1

    (2')  P(D1) = |(1/21/2)(1/21/2) + (1/21/2)(1/21/2)|2 = 1


    This result violates the principle of conservation of energy because the 
    original photon has a probability of 1 of being detected at D1 and also 
    a probability of 1 of being detected at D2. In other words, the number 
    of photons has doubled. Thus, there must be a phase difference between 
    transmission and reflection, and a 90o phase difference, as shown above, 
    conserves energy.

    References:

    
 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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