5 Bomb testing

This time the setup is a Mach−Zehnder inter­fer­om­eter, which con­sists of two beam split­ters (S1 and S2), two mir­rors (M1 and M2), and two pho­tode­tec­tors (D1 and D2) 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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Figure 1.5.1 The “Bomb testing” exper­i­ment of Elitzur and Vaidman

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Imagine, to begin with, that nei­ther S2 nor the bomb is present. A beam of pho­tons (a.k.a. a light beam) enters S1 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 D1 (indi­cating that the photon was reflected upward by S1) and an equal chance of being detected by D2 (indi­cating that the photon went hor­i­zon­tally through S1).

If S2 (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 M1 or reflec­tion by M2): 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 D1 involves two reflec­tions, so the cor­re­sponding ampli­tudes are equal (i2A, say). The prob­a­bility of detec­tion by D1 is there­fore given by

pB,1 = |i2A + i2A|2 = 4|A|2.

The alter­na­tive leading to D2 via M1 involves three reflec­tions, so the cor­re­sponding ampli­tude equals i3A, while the alter­na­tive leading to D2 via M2 involves a single reflec­tion, the cor­re­sponding ampli­tude thus being iA. Since the two ampli­tudes differ by a factor i2 = −1, the prob­a­bility of detec­tion by D2 is

pB,2 = |iA + i3A|2 = |iA − iA|2 = 0.

Finally, if both S2 and the bomb are present, the alter­na­tive taken by the photon is mea­sured. If the bomb explodes, this indi­cates that the photon went via M1, and if it does not explode, this indi­cates that the photon went via M2. If it went via M2, either pho­tode­tector responds with prob­a­bility 12. Thus:

  • If the bomb is absent, D1 “clicks” every time (in 100% of all cases), whereas D2 never “clicks”.
  • If the bomb is present, it explodes half of the time (in 50% of all cases); and if it doesn’t explode, D1 and D2 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 D1 responds, which hap­pens in 25% of the tests, we have learned nothing, for D1 also responds if the bomb is absent. But if D2 responds, which hap­pens in the remaining 25% of the tests, we have suc­ceeded, for D2 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 D2 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.

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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.