Understanding the quantum Cheshire Cat experiment of Denkmayr et al. W.M. Stuckey* Department of Physics, Elizabethtown College, Elizabethtown, PA 17022 Michael Silberstein Department of Philosophy, Elizabethtown College, Elizabethtown, PA 17022 Department of Philosophy, University of Maryland, College Park, MD 20742 Timothy McDevitt Department of Mathematics, Elizabethtown College, Elizabethtown, PA 17022 Abstract The authors of a July 2014 paper in Nature Communications made the spectacular claim that their interferometer experiment instantiated the spatial separation of neutrons and their spins. This is akin to the Cheshire Cat being in one place while its grin is in another and they call it the quantum Cheshire Cat experiment. Herein we provide the pedagogic calculation and comparison of the intensities used and observed in their experiment. These simple quantum calculations of intensity, which are not provided in their paper, and the resulting comparison of theory to the experimental outcomes render the experiment easy to understand. The amplitude squared at any point in the device gives the neutrons per second (intensity) at that location. When the magnetic field B z is introduced to the interferometer s lower (upper) path it turns some of the S x + > ( S x >) amplitude there into S x > ( S x + >) amplitude, i.e., it creates (removes) some neutrons per second associated with spin S x > at that location. The weak values presented by the authors in support of their claim are nothing more than expansion coefficients in the weak field approximation to these intensities. Thus, the weak values do not in any way contradict what is readily apparent with the magnetic field and neutron spins per our calculation of intensity. While the quantum mechanical theory is in excellent agreement with the experimental outcomes, it is clearly not the case that the magnetic field in the lower path is operating on the spin of the neutrons in the upper path, or vice-versa. Thus, given this analysis, there is absolutely no reason to conclude that neutrons and their spins are spatially separated in this experiment.
I. Introduction Using a neutron interferometer, Denkmayr et al. claim (1) to have instantiated, for the first time, the quantum Cheshire Cat experiment. In a quantum Cheshire Cat experiment (2), a particle is spatially separated from one of its properties, just as the Cheshire Cat can be spatially separated from its grin in the Lewis Carroll story Alice s Adventures in Wonderland (3). This is quite a spectacular claim, so naturally the publication of Denkmayr et al. s paper was received with much fanfare. Of course, any such spectacular claim should also be received with exceptional scrutiny. Luckily, the calculations required to understand the experimental results in this case are accessible to the undergraduate physics student. It is perhaps precisely because these calculations are pedagogic that they are omitted in Denkmayr et al. s paper. That is unfortunate because, as will become clear when we provide these missing calculations, they allow one to understand Denkmayr et al. s experimental results quite clearly. While this detailed scrutiny reveals a beautiful marriage of experiment with simple quantum mechanics theory, it does not support their spectacular claim. In their own words, Denkmayr et al. claim, Figures 3 and 4 already clearly demonstrate the effect predicted by refs 23 25: an absorber with high transmissivity has on average no significant effect on the measurement outcome if it is placed in path I. It is only effective if it is placed in path II. In contrast to that, a small magnetic field has on average a significant effect only in path I, while it has none in path II. Therefore, any probe system that interacts with the Cheshire Cat system weakly enough will on average be affected as if the neutron and its spin are spatially separated. In a paper that challenges this claim in the same fashion as we do here (although, without the details supplied here), Corrêa et al. write (4), This is what they mean when they say that the experimental results suggest that the system behaves as if the neutrons go through one beam path, while their magnetic moment travels along the other. But as we have seen here, the results can be explained as simple quantum interference, with no separation between the neutron and its spin. As we will show, there is a bit more to the experimental outcomes than quantum interference, but Corrêa et al. do identify the primary problem with Denkmayr et al. s assertion that neutrons have been separated from their spins. The amplitude squared at any point in the device gives the neutrons per second (intensity) at that location. When the magnetic field B z is introduced to the
lower path (I) it turns some of the S x + > amplitude there into S x > amplitude, i.e., it creates some neutrons per second associated with spin S x > at that location. The S x > amplitude from the lower path combines with the S x > amplitude from the upper path to contribute to the S x > amplitude at the detector O and thus, the neutrons per second at O. [Only S x > amplitude exists at O, because a S x > spin selector precedes detector O.] This also creates interference at O between the S x > contributions from the upper and lower paths when a phase difference is introduced between paths. Placing the magnetic field in the upper path (II) has the opposite effect, since it turns some of the S x > amplitude into S x + > amplitude in the upper path, thereby decreasing the S x > amplitude at O. Since there is no S x > contribution from the lower path, there is no interference at O in this case. The weak values presented by the authors in support of their claim are nothing more than expansion coefficients in the weak field approximation to these intensities. Thus, the weak values do not in any way contradict what is readily apparent with the magnetic field and neutron spins per our calculation of intensity. It is clearly not the case that the magnetic field in the lower path is operating on the spin of the neutrons in upper path, or vice-versa. Indeed, since the amplitude squared at any location gives the neutrons per second at that location, and the amplitude carries the spin designations S x > and/or S x + > at each location, and the spin designations are never independent of the neutronsper-second aspect of the amplitude, it is not even the case that neutrons behave as if they are separated from their spins in this experiment. Thus, given this analysis, there is absolutely no reason to conclude that neutrons and their spins are spatially separated in this experiment. We begin with a simple explanation of the experiment proper. II. The Experiment The experiment and its results are summarized in Denkmayr et al. s Figures 3 and 4 below. To understand the essential elements of the experiment, you need to know that spin rotators create S x + > (or + > for short) on path I and S x > (or > for short) on path II (by brown boxes in Figure 3) just after the neutrons pass through the first beam splitter (entering from the left in Figure 3). Path I is the lower path and path II is the upper path. The two detectors are O (labeled by I o in yellow boxes of Figures 3 & 4) and H (labeled by I H in yellow box of Figure 4). A > spin selector immediately precedes the detector O (red box labeled SA) while the entire signal is sampled at H. Thus, when a partial (weak) absorber (brown bar in Figure 3) is placed in path I it diminishes the + > amplitude contributing to the amplitude going to the spin selector.
But, the > spin selector deletes that effect on the amplitude going to O, so there is no change in the intensity at O. However, when the partial absorber is placed in path II it diminishes the > amplitude contributing to the amplitude going to the spin selector, so this decrease in the amplitude reaches O giving rise to a slight decrease in the intensity at O. The experimenters therefore conclude that the neutrons reaching O are taking path II, i.e., a minimally disturbing measurement will find the Cat in the upper beam path. This part of the experiment is straightforward and requires no detailed analysis. It is the second part of the experiment, i.e., the introduction of a magnetic field, that yields the controversial part of the conclusion, i.e., while its grin will be found in the lower one. Their take-home message from Figure 4 is that we see an effect in the intensity at O when a small (weak) magnetic field B z is introduced on path I, but not on path II. Thus, they conclude that the grin (neutron s spin) is taking path I while the Cat (neutron) is taking path II. However, we will show that the results in Figure 4 are easy to understand without any hint that the property and particle are spatially separated. We do that by simply providing the details of the derivations of the intensities at O and H when the magnetic field B z is introduced on path I and on path II, i.e., their Eqs (13) (16). These calculations do not appear in their paper (as is customary for such pedagogic calculations), but these missing details reveal quite clearly that the results of Figure 4 do not entail the spatial separation of the neutrons and their spins. III. Calculations As with any interferometer experiment, we can compute the amplitudes at O and H by collecting contributions along each path then combining them through the last beam splitter. The intensities are then obtained by squaring the amplitudes (Born s rule). We start with B z in the upper path (II). In the upper path, after the first beam splitter and spin rotator, our amplitude is, since a 50-50 beam splitter introduces a factor of and refection introduces a factor of i. Thus, after the mirror (middle black bar in the Figures) we have. The magnetic field effect is given by the unitary operator / on the amplitude at that point, as shown in their Eq (8). Accordingly, the magnetic field effect is
/ 2 2 2 2 since. Thus, we have going into the phase shifter. The phase shifter contributes a factor of / and last beam splitter contributes another in route to O, so the upper path contributes / to the amplitude just past the second beam splitter in route to O. For the lower path (I) the amplitude is after the first beam splitter and spin rotator. After the mirror we have and finally after the phase shifter and last beam splitter the lower path contributes / to the amplitude just past the second beam splitter in route to O. Thus, we have / / for the amplitude going into the spin selector just before O. The spin selector effect on the amplitude is given by, so the amplitude at O when the magnetic field is placed in path II is given by / 1 2 2 / Thus, we have / /, their Eq (14). If we had omitted the magnetic field and phase shifter, we would have gotten /, which they refer to as their reference intensity I REF. To find the amplitude at H for this case, we need only change the effect of the last beam splitter and omit the last spin selector on our work above. Changing the upper path contribution to reflect off the last beam splitter (instead of transmit) and changing the contribution from the lower path to transmit through the last beam splitter (instead of reflect) we have / 2 2 2 / 2 / for the amplitude at H when the magnetic field is placed in path II. Thus, we have / /, their Eq (16).
Putting B z in the lower path (I) and following the same algorithm as above using we have / 2 2 / 1 2 / for the amplitude at O when the magnetic field is placed in path I. Thus, / / 3 4, their Eq (13). As with the previous case, to find the amplitude at H we need only change the effect of the last beam splitter and omit the last spin selector on our work above. Changing the upper path contribution to reflect off the last beam splitter (instead of transmit) and changing the contribution from the lower path to transmit through the last beam splitter (instead of reflect) we have / 2 2 2 / 2 / for the amplitude at H when the magnetic field is placed in path I. Thus, / /, their Eq (15). We now compare these theoretical intensities with the corresponding experimental results. IV. Discussion Again, the results of Figure 3 (partially blocking a path) are straightforward, so it is only those of Figure 4 (adding a magnetic field to a path) that we need to analyze and discuss. The authors give the χ = 0 reference value for intensity in counts per second at O as I REF = 11.25(5). Since / we have, for α = 20o (given in their paper), a theoretical prediction of 10.91(5) c.p.s., which agrees with their measured value of 10.936. [See page 5 of their paper.] In theory, we should see no oscillation in χ at O and Figure 4 shows it is very small (the sinusoidal fit for this case is omitted altogether in the arxiv version of the paper (5) ). Thus, when the magnetic field is introduced in the upper path (II) with small α (small magnetic field intensity), we see a small decrease in the χ = 0 intensity at O (where, again, only > is sampled) due to some of the > amplitude in the upper path being converted to + > amplitude by the
magnetic field there. The decrease is / 1, or about a 3% reduction in I REF, i.e., 11.25 0.34 = 10.91. What we see at H in this case is a sinusoidal oscillation in χ due to interference between / (created from > by the magnetic field in the upper path) and / from the lower path. This is why Corrêa et al. write, the results can be explained as simple quantum interference (although we see there is a non-interference (χ = 0) component to the experiment that our analysis also explains). When the magnetic field is introduced on the lower path (I), we have / 3 for χ = 0, which for I REF = 11.25(5) and α = 20 o gives a theoretical prediction of 11.59(5) c.p.s. in agreement with their measured value of 11.576. So, the reason we see an increase in the intensity at O when the magnetic field is placed in the lower path is because it generates an additional > term in the amplitude when operating on + > and > is what is measured at O. This slight increase in > created by the magnetic field on path I is added to that from the upper path going to O. The increase is / 1, so the increase is the same as the decrease created by converting some of the > amplitude in the upper path to + > amplitude, i.e., 11.25 + 0.34 = 11.59. Unlike the previous case, we do have oscillation in χ at O for this case, i.e., / 3 4. This oscillation is caused by interference between / in the upper path and / created from + > by the magnetic field in the lower path. A similar interference creates an oscillation in χ at H that is approximately out of phase with that at O, since /. This oscillation in χ at O and H is clearly evident in Figure 4. The so-called weak values presented by the authors in support of their spectacular claim can be easily understood in the context of the intensities computed here. As they show in their Eq (10), the weak values are nothing more than expansion coefficients in the weak field approximation to the χ = 0 intensities. That is, their Eq (10), 1 Π Π, with, Π 0, Π 1, Π 1, and Π 0 is nothing more than / 1 and / 1 obtained above. So, we see
that the weak values they find follow tautologically from the exact functional form of the intensities obtained above. And, therefore, these weak values can in no way contradict what is readily apparent from the derivation of these intensities. In conclusion, the authors have a beautiful marriage of experiment with simple quantum mechanics theory. However, we are all aware that, even for researchers with the best of intentions, when it comes to interpreting quantum mechanics or the results of quantum experiments, it is very easy to conflate phenomenology and interpretation. We are also aware that researchers in the foundations of quantum mechanics sometimes overstate the certainty or superiority of their interpretation, e.g., claiming an experimental set-up demands retrocausal processes, multiple histories or what have you. In those cases, the merits of relative interpretations can be debated. In this case however, there is no room for debate, the experiment by Denkmayr et al. does not show that a particle and one of its properties have been spatially separated. The calculational details provided here explain the experimental results unambiguously. The spin results obtain because the magnetic field converts some > amplitude to + > amplitude when placed in the upper path, and it converts some + > amplitude to > amplitude when placed in the lower path. It is clearly not the case that the magnetic field in path I is operating on the spin of the neutrons in path II, or vice-versa. Since the weak values are nothing more than expansion coefficients in the weak field approximation to the χ = 0 intensities, their values in no way alter what is understood per the calculation of these intensities. Finally, since the amplitude squared at any location gives the neutrons per second at that location, and the amplitude carries the spin designations > and/or + > at each location, and the spin designations are never independent of the neutrons-per-second aspect of the amplitude, it is not even the case that neutrons behave as if they are separated from their spins in this experiment. Therefore, according to this analysis, there is no reason to conclude that neutrons and their spins are spatially separated in this experiment. *stuckeym@etown.edu silbermd@etown.edu mcdevittt@etown.edu
Figure 3
Figure 4
References (1) Denkmayr, T., Geppert, H., Sponar, S., Lemmel, H., Matzkin, A., Tollaksen, J., & Hasegawa, Y.: Observation of a quantum Cheshire Cat in a matter-wave interferometer experiment. Nature Communications, DOI: 10.1038/ncomms5492 (29 Jul 2014). (2) Tollaksen, J.: Quantum Reality and Nonlocal Aspects of Time. PhD thesis, Boston University. (2001); Matzkin, A. & Pan, A. K.: Three-box paradox and Cheshire cat grin : the case of spin-1 atoms. Journal of Physics A: Mathematical and Theoretical 46, 315307 (2013); Aharonov, Y., Popescu, S., Rohrlich, D. & Skrzypczyk, P.: Quantum Cheshire Cats. New Journal of Physics 15, 113015 (2013). (3) Carroll, L.: Alice s Adventures in Wonderland. MacMillan Press, United Kingdom (1865). (4) Corrêa, R., Santos, M.F., Monken, C.H., & Saldanha, P.L.: Quantum Cheshire Cat as Simple Quantum Interference. http://arxiv.org/abs/1409.0808. (5) ibid Denkmayr et al., http://arxiv.org/pdf/1312.3775.pdf p 5.