Using Dirac Notation to Analyze Single Particle Interference

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

    The quantum mechanical analysis of this striking phenomenon is outlined
    below.  The photon leaves the source, S, and whether it takes the upper 
    or lower path it interacts with a beam splitter, a mirror, and another 
    beam splitter before reaching the detectors. At the beam splitters there  
    is a 50% chance that the photon will be transmitted and a 50% chance that 
    it will be reflected.
                     Upper Path
		\ BS              \
      (S)- -> - -\- - - - T - - - -\ M            
		 |\                |\              
		 |                 |               
		 |                 |
                 R                 |
                 |                 |                            
		 |                 |                    
		\|             BS \|
	       M \- - - - - - - - -\- - - -> D2
		  \                |\                 BS = Beam splitter (50/50)
		     Lower Path    |                  D  = Detector
                 		   |                  M  = Mirror
                                   v                  S  = Source

    After the first beam splitter the photon is in an even linear superposition 
    of being transmitted and reflected. Reflection involves a 90o (p/2) 
    phase change which is represented by exp(ip/2) = i, where i = (-1)1/2. (See 
    the appendix for a simple justification of the 90o phase difference 
    between transmission and reflection.) Thus the state after the first beam splitter 
    is given by equation (1).

    (1)        |y> = [|T> + i|R>]/21/2    

    Now |T> and |R> will be written in terms of |D1> and |D2> the states they 
    evolve to at detection. |T> reaches |D1> by transmission and |D2> by 

    (2)        |T> = [|D1> + i|D2>]/21/2

    |R> reaches |D1> by reflection and |D2> by transmission.

    (3)        |R> = [i|D1> + |D2>]/21/2

    Equations (2) and (3) are substituted into equation (1).

    (4)        |y> = [|D1> + i|D2> + i2|D1> + i|D2>]/2

    It is clear (i2 = -1) that the first and third terms cancel (the amplitudes
    are 180o out of phase), so that we end up with a final state given
    by equation 5.

    (5)        |y> = i|D2>

    The probability of an event is the square of the absolute magnitude of the
    probability amplitude.

    (6)        P(D2) = |i|2 = 1

    Thus this analysis is in agreement with the experimental outcome that no
    photons are ever detected at D1. 


    Suppose there is no phase difference between transmission and reflection.  
    Then equations (1), (2), and (3) become

    (1')        |y> = [|T> + |R>]/21/2    
    (2')        |T> = [|D1> + |D2>]/21/2

    (3')        |R> = [|D1> + |D2>]/21/2

    Substitution of equations (2') and (3') into equation (1') yields

    (4')        |y> = |D1> + |D2>

    Thus, the detection probabilities at the two detectors are:
    (5')        P(D1) = 1  and  P(D2) = 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.

  1. 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).

  2. V. Scarani and A. Suarez, "Introducing Quantum Mechanics: One-particle Interferences," Am. J. Phys. 66, 718-721 (1998).

  3. Kwiat, P, Weinfurter, H., and Zeilinger, A, "Quantum Seeing in the Dark," Sci. Amer. Nov. 1996, pp 72-78.

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