This time the setup is a Mach−Zehnder interferometer, which consists of two beam splitters (S_{1} and S_{2}), two mirrors (M_{1} and M_{2}), and two photodetectors (D_{1} and D_{2}) arranged as in Fig. 1.4.1. A particular twist of the experiment we are about to discuss^{[1]} is the possible presence of a “bomb” — a photodetector so sensitive that it will explode if it absorbs a single photon. (For simplicity’s sake we make the usual assumption that all detectors including the bomb are 100% efficient.)

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Imagine, to begin with, that neither S_{2} nor the bomb is present. A beam of photons (a.k.a. a light beam) enters S_{1} from the left. Classically described, two beams emerge, each with half the intensity of the incoming beam. Described in quantum-mechanical terms, each incoming photon has a 50% chance of being detected by D_{1} (indicating that the photon was reflected upward by S_{1}) and an equal chance of being detected by D_{2} (indicating that the photon went horizontally through S_{1}).

If S_{2} (but as yet no bomb) is present, Rule B applies. Here is what we need to know about the amplitudes associated with the alternatives (reflection by M_{1} or reflection by M_{2}): they are equal *except *that each reflection causes a phase shift of 90°. In other words, it rotates the amplitude anticlockwise by 90°. This is equivalent to multiplying it by ** i** = √(−1). (The magnitude of the phase shift depends on the materials used. For the sake of convenience we imagine using materials for which it equals 90°.)

Each of the alternatives leading to D_{1} involves two reflections, so the corresponding amplitudes are equal (*i*^{2}A, say). The probability of detection by D_{1} is therefore given by

p_{B,1} = |*i*^{2}A + *i*^{2}A|^{2} = 4|A|^{2}.

The alternative leading to D_{2} via M_{1} involves three reflections, so the corresponding amplitude equals *i*^{3}A, while the alternative leading to D_{2} via M_{2} involves a single reflection, the corresponding amplitude thus being ** i**A. Since the two amplitudes differ by a factor

*i*^{2}= −1, the probability of detection by D

_{2}is

p_{B,2} = |** i**A +

*i*^{3}A|

^{2}= |

**A −**

*i***A|**

*i*^{2}= 0.

Finally, if both S_{2} and the bomb are present, the alternative taken by the photon is measured. If the bomb explodes, this indicates that the photon went via M_{1}, and if it does not explode, this indicates that the photon went via M_{2}. If it went via M_{2}, either photodetector responds with probability ^{1}⁄_{2}. Thus:

- If the bomb is
*absent*, D_{1}“clicks”*every time*(in 100% of all cases), whereas D_{2}*never*“clicks”. - If the bomb is
*present*, it explodes half of the time (in 50% of all cases); and if it doesn’t explode, D_{1}and D_{2}are*equally likely*to respond (each “clicks” in 25% of all cases).

Now suppose that the bomb is present. Is it possible, using the present setup, to ascertain the presence of the bomb *without* setting it off? Stop to think about this before you proceed.

The answer is affirmative, albeit only in 25% of the tests. If the bomb explodes, which happens in 50% of the tests, we have failed. If the bomb is present and D_{1} responds, which happens in 25% of the tests, we have learned nothing, for D_{1} also responds if the bomb is absent. But if D_{2} responds, which happens in the remaining 25% of the tests, we have succeeded, for D_{2} would not have responded if the bomb had been absent.

When a version of this experiment was demonstrated at a science fair in Groningen, the Netherlands, in 1995, the reactions of non-physicists differed markedly from those of physicists.^{[2]} Everyone was perplexed, for the detection of the photon by D_{2} seems to have contradictory implications:

- The bomb was present.
- The photon never came near the bomb.

If the photon never came near the bomb, how was it possible to learn that the bomb was present? While most ordinary folks thought that some physicist will eventually solve this puzzle, the physicists themselves were decidedly less hopeful that a satisfactory explanation will be found.

1. [↑] Elitzur, A.C. and Vaidman, L. (1993). Quantum mechanical interaction-free measurements, *Foundations of Physics* 23, 987–997.

2. [↑] du Marchie van Voorthuysen, E.H. (1996). Realization of an interaction-free measurement of the presence of an object in a light beam, *American Journal of Physics* 4 (12), 1504–1507.