Using Dirac Notation to Analyze Single Particle Interference 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 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. After the first beam splitter the photon is in an even linear superposition of being transmitted and reflected. Reflection involves a 90 o (π/2) phase change which is represented by exp(iπ/2) = i, where i = (-1)1/2. (See the appendix for a simple justification of the 90 o phase difference between transmission and reflection.) Thus the state after the first beam is given by equation 257.1. \[ | \psi = \left( \frac{[ | T> + i|R ]}{2} \right) ^{ \frac{1}{2}} \nonumber \] Now |T> and |R> will be written in terms of |D 1 > and |D 2 > the states they evolve to at detection. |T> reaches |D 1 > by transmission and |D 2 > by reflection. \[ |T> = \left( \frac{[|D1> + i|D2>]}{2} \right) ^{ \frac{1}{2}} \nonumber \] |R> reaches |D 1 > by reflection and |D 2 > by transmission. \[ |R> = \left( \frac{[i|D1> + |D2>]}{2} \right) ^{ \frac{1}{2}} \nonumber \] Equations 257.2 and 257.3 are substituted into equation 257.1. \[ | \psi > = \frac{[|D1> + i|D2> + i2|D1> + i|D2>]}{2} \nonumber \] It is clear (i 2 = -1) that the first and third terms cancel (the amplitudes are 180 o out of phase), so that we end up with a final state given by equation 257.5. \[ | \psi> = i|D_2> \nonumber \] The probability of an event is the square of the absolute magnitude of the probability amplitude. \[ P(D_2) = |i|^2 = 1 \nonumber \] Thus this analysis is in agreement with the experimental outcome that no photons are ever detected at D 1 . Appendix : Suppose there is no phase difference between transmission and reflection. Then equations 257.1, 257.2, and 257.3 become \[ | \psi = \left( \frac{[|T> + |R>]}{2} \right) ^{ \frac{1}{2}} \nonumber \] \[ | T> = \left( \frac{[|D1> + |D2>]}{2} \right) ^{ \frac{1}{2}} \nonumber \] \[ | \psi = \left( \frac{[|D_1> + |D_2> |D_1> + D_2>]}{2} \right) ^{ \frac{1}{2}} \nonumber \] Substitution of equations 257.8 and 257.9 into equation 257.7 yields \[ | \psi > = |D_1> + |D_2> \nonumber \] Thus, the detection probabilities at the two detectors are: \[ P(D_1) = 1 and P(D_2) = 1 \nonumber \] This result violates the principle of conservation of energy because the original photon has a probability of 1 of being detected at D 1 and also a probability of 1 of being detected at D 2 . In other words, the number of photons has doubled. Thus, there must be a phase difference between transmission and reflection, and a 90 o phase difference, as shown above, conserves energy. References : 1. P. Grangier, G. Roger, and A. Aspect, "Experimental Evidence for Photon Anticorrelation Effects on a Beam Splitter: A New Light on Single 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.