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Then, a detector located in the “down” port after the last beam splitter will click. Following our previous analysis, if there is an object in the upper arm of the first interferometer, the amplitude α2 is suppressed and then the intensity of the light going downwards at the end of the first interferometer is given by I˜2 = |α1 |2 = cos4 π 2n . 80) Hence, if there are objects in the upper half of each interferometer, after passing m beam splitters, the intensity downwards will be I˜m = cos2m π 2n .

Thus, we have a method of detecting the object without interacting with it. However, the method has at most a 50% efficiency. In the second, more efficient scheme, proposed by Kwiat et al. [27], instead of a single Mach−Zehnder interferometer, a series of n such interferometers is used, the amplitude reflectivity r of each beam splitter being equal to cos(π/2n), and the amplitude transmissivity t = sin(π/2n). The scheme is illustrated in Fig. 15. As we show, this arrangement has the property that a photon entering at the lower part of the interferometer and passing through all n interferometers will be transferred into the upper part of the interferometer.

There are two contributions to the state of the photon. The first one is from the pass Br1 → Md1 → Br2 and the second one is from the pass Bt1 → Mu1 → Bt2 , where Bri and Bti mean reflection and transmission at the ith beam splitter, respectively. The first beam has been reflected twice at beam splitters B1 and B2 and once at the mirror Md1 , three reflections giving a phase factor i3 , and the amplitude α1 = −i cos2 π 2n . 76) The second beam has been transmitted twice at the beam splitters and reflected once at the mirror Mu1 , giving the total transmission coefficient multiplied by the factor i as α2 = i sin2 π 2n .

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