8 The ESW experiment

Rule A, you will recall, applies “if the inter­me­diate mea­sure­ments are made (or if it is pos­sible to find out what their out­comes would have been if they had been made),” and Rule B applies “if the inter­me­diate mea­sure­ments are not made (and if it is impos­sible to find out what their out­comes would have been if they had been made).” The two-​​slit exper­i­ment we will dis­cuss in this sec­tion demon­strates the ratio­nale behind the cryptic clauses in parentheses.

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ESW setup

Figure 1.8.1 Setup of the ESW experiment

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In this exper­i­ment[1,2] atoms are used instead of elec­trons. All atoms are of the same type — Cesium-​​133, say — and all start out in the same excited state. Placed in front of the slits are two ini­tially sep­a­rate microwave res­o­nance cav­i­ties, each tuned to the energy dif­fer­ence ΔE between this excited state and the atoms’ ground state and thus capable of “holding” pho­tons “car­rying” the energy ΔE. The design of each cavity more­over ensures that the prob­a­bility with which an atom is found to emerge from it in the ground state equals unity — pro­vided, of course, that the appro­priate mea­sure­ment is made.

In what sense can such a cavity hold a photon? A rea­son­ably safe inter­pre­ta­tion is that if a (100% effi­cient) pho­tode­tector were inserted into the cavity, it would detect (and absorb) a photon. And in what sense does a photon carry the energy ΔE? A rea­son­ably safe inter­pre­ta­tion is that the pho­tode­tector would absorb (and gain) this energy.

The two res­o­nance cav­i­ties are sep­a­rated from each other by a pair of electro-​​optical shutters,which remain closed for now. Atoms are launched, one at a time, with nothing to pre­dict the par­tic­ular cavity through which any given atom will pass. (Before an atom is launched, the photon left behind by the pre­vious atom is absorbed and thus “removed” from the cavity.) Each atom leaves a mark on the screen. How will the marks be distributed?

Focus on a single atom, after it has hit the screen but before the photon is removed. This is a sit­u­a­tion in which it is pos­sible to find out what the out­come of an inter­me­diate mea­sure­ment would have been if it had been made. The inter­me­diate mea­sure­ment, had it been made, would have deter­mined the slit taken by the atom. The reason why we can find out what its out­come would have been is the fol­lowing strict cor­re­la­tion between the out­come of this mea­sure­ment and the cavity con­taining the photon: if the atom were found to emerge from the left slit, the prob­a­bility of absorbing the photon in the left cavity would be 1, and if the atom were found to emerge from the right slit, the prob­a­bility of absorbing the photon in the right cavity would be 1. Hence if the photon is detected in the left cavity, a mea­sure­ment of the slit taken by the atom would have indi­cated the left slit, and if the photon is detected in the right cavity, a mea­sure­ment of the slit taken by the atom would have indi­cated the right slit. Thus Rule A applies.

Let us color the marks: those made by atoms that left a photon in the left cavity green, and those made by atoms that left a photon in the right cavity red. The dotted curve in Fig. 1.8.2 gives the dis­tri­b­u­tion of the green marks, the dashed curve that of the red marks. The solid curve is the sum of the two dis­tri­b­u­tions. The green marks are dis­trib­uted as we expect from atoms that went through the left slit, and the red marks are dis­trib­uted as we expect from atoms that went through the right slit. (Com­pare Fig. 1.8.2 with Fig. 1.4.2.)

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Plot A

Figure 1.4.2

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Between the shut­ters there is a (100% effi­cient) pho­to­sensor. If the shut­ters are opened before the photon is absorbed, quantum mechanics pre­dicts that the sensor will absorb the energy ΔE with prob­a­bility 12. Since we now have a single cavity instead of two, infor­ma­tion about the slit taken by the photon is no longer avail­able. (It has become cus­tomary to say that the infor­ma­tion has been “erased.” What has actu­ally been erased, how­ever, is merely the pos­si­bility of obtaining the infor­ma­tion.) Does this mean that Rule B now applies? If this exper­i­ment is done with suf­fi­ciently many atoms, will the overall dis­tri­b­u­tion of marks exhibit inter­fer­ence fringes?

The answer has to be neg­a­tive, for the mea­sure­ment involving the photon is made after the atom has hit the screen. The deci­sion about which mea­sure­ment to per­form — to deter­mine the cavity that held the photon or to deter­mine the behavior of the pho­to­sensor upon opening the shut­ters — comes too late to affect the overall dis­tri­b­u­tion of marks.

But we now have another way of col­oring the marks: yellow if the pho­to­sensor responds, blue if it fails to respond. Quantum mechanics pre­dicts that the yellow marks will exhibit the same inter­fer­ence pat­tern as elec­trons in a two-​​slit exper­i­ment under the con­di­tions stip­u­lated by Rule B (Fig. 1.4.3). Because the overall dis­tri­b­u­tion of marks is the same in both ver­sions of the exper­i­ment, the blue marks will exhibit the com­ple­men­tary inter­fer­ence pat­tern, having maxima where the other has minima, and vice versa.

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Plot B

Figure 1.9.1 Dis­tri­b­u­tion of marks if the exper­i­ment is done with open shut­ters. The dotted curve gives the dis­tri­b­u­tion of yellow marks, the dashed curve that of blue marks. The solid curve — the sum of the two dis­tri­b­u­tions — is the same as in Fig. 1.8.2.

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All “yellow” atoms and all “blue” atoms have some­thing in common, but as their respec­tive behav­iors lack clas­sical coun­ter­parts, we have no ready name for it. We may say that the “yellow” atoms went through the slits in phase} while the “blue” atoms went through the slits out of phase. We use these phrases for the fol­lowing reason. If there is to be a max­imum at the center of the screen, the phases of the ampli­tudes asso­ci­ated with the alter­na­tives “through L” and “through R” must differ by an even mul­tiple of 180° — the alter­na­tives must be “in phase.” And if there is to be a min­imum instead, the ampli­tudes must differ by an odd mul­tiple of 180° — the alter­na­tives must be “out of phase.” (As you will remember, the phase of the ampli­tude asso­ci­ated with a particle’s prop­a­ga­tion from point A to point B is pro­por­tional to the dis­tance between A and B.)

The “green” atoms, we noted, behave like atoms that went through L, while the “red” atoms behave like atoms that went through R. Like­wise, the “yellow” atoms behave like atoms that went through the slits in phase (inas­much as they dis­play the cor­re­sponding inter­fer­ence pat­tern, while the “blue” atoms behave like atoms that went through the slits out of phase. Cannot we con­clude from this that the “green” atoms actu­ally went through L, that the “red” atoms actu­ally went through R, that the “yellow” atoms actu­ally went through the slits in phase, and that the “blue” atoms actu­ally went through the slits out of phase? After all, if it looks like a duck, swims like a duck, and quacks like a duck, then it prob­ably is a duck. The problem with this con­clu­sion is that it seems to imply the pos­si­bility of influ­encing the past!

If the exper­i­menters deter­mine the cavity that held the photon, they learn through which slit the cor­re­sponding atom went. If they open the shut­ters and observe whether or not the sensor responds, they learn how the atom went through the slits — in phase or out of phase. They cannot make the atom go through L or through R, yet by doing the former exper­i­ment, they can make sure that it went through a single slit — either L or R. Nor can they make an atom go through the slits in phase or out of phase, yet by doing the latter exper­i­ment they can make sure that it when through both slits. (Fur­ther dis­cus­sion of the meaning of “both” can be found here.) And since they can choose between the two exper­i­ments after the atom has made its mark on the screen, they can, by their choice, con­tribute to deter­mine the atom’s past behavior.

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Logical flow chart for the ESW experiment

Figure 1.9.2 A log­ical flow chart for the ESW experiment

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1. [↑] Englert, B.G., Scully, M.O., and Walther, H. (1994). The duality in matter and light, Sci­en­tific Amer­ican 271 (6), 56–61.

2. [↑] Scully, M.O., Englert, B.G., and Walther, H. (1991). Quantum optical tests of com­ple­men­tarity, Nature 351 (6322), 111–116.