An Improved Experiment to Determine the `Past of a Particle' in the Nested Mach-Zehnder Interferometer
Alon Ben-Israel, Lukas Knips, Jan Dziewior, Jasmin Meinecke, Ariel Danan, Harald Weinfurter, Lev Vaidman
AAn Improved Experiment to Determine the ‘Past of a Particle’ in the NestedMach-Zehnder Interferometer
Alon Ben-Israel , Lukas Knips , , Jan Dziewior , , JasminMeinecke , , Ariel Danan , Harald Weinfurter , , and Lev Vaidman Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 69978, Israel Max-Planck-Institut f¨ur Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany Department f¨ur Physik, Ludwig-Maximilians-Universit¨at, 80797 M¨unchen, Germany
We argue that the modification proposed by Li et al . [Chin. Phys. Lett. , 050303 (2015)] tothe experiment of Danan et al. [Phys. Rev. Lett. , 240402 (2013)] does not test the past ofthe photon as characterised by local weak traces. Instead of answering the questions: (i) Were thephotons in A ? (ii) Were the photons in B ? (iii) Were the photons in C ? the proposed experimentmeasures a degenerate operator answering the questions: (i) Were the photons in A ? (ii) Were thephotons in B and C together? A negative answer to the last question does not tell us if photonswere present in B or C . A simple variation of the modified experiment does provide good evidencefor the past of the photon in agreement with the results Danan et al. obtained. Li et al . [1] recently proposed an ‘ideal’ experiment de-signed to determine the past of a particle passing throughthe nested interferometer analyzed by Danan et al . [2].They proposed to use an alternative method for observ-ing the location of the photon based on Kerr media tochallenge and refute Danan’s claim that the past of a pho-ton in this interferometer is described by disconnectedpaths.In this Letter we analyze the method of Li et al . andfind that their proposed experiment is not a good testof the past of the photon. However, a modification oftheir experiment does provide a correct alternative mea-surement of the past of the photon, which, as we believe,will reveal the disconnected paths that Danan et al . havecharacterized.First, we ask in what way the proposed experimentis ‘ideal’. In standard quantum mechanics there is noconcept of the particle path or the past of a particle.The past of a particle is not defined, and so there cannotbe an ‘ideal’ way to find it. The approach which doesnot allow to talk about particles at intermediate timesbetween measurements saves us from having to considerseemingly paradoxical results, but at the same time lim-its the possible insight we may gain by considering thisconcept.Several approaches have been suggested that allow usto discuss the past of particles in quantum mechanics,associating trajectories to each particle. One of those isthe de Broglie-Bohm interpretation of QM, in which thetrajectories of particles are determined by the wavefunc-tion via a guiding equation [3]. If the wavefunction of theparticle is a well-localized wave packet, the Bohmian tra-jectory of the particle coincides with the trajectory of thewave packet. For an evolving wave packet that splits intoseveral wave packets, of which only one reaches the finaldestination via a continuous path, the trajectory of thispacket can be defined as the path of the particle. Thisis the ‘common sense’ approach advocated by Wheeler [4]: the particle went through this path because it couldnot come through any other path. Recently, Vaidman [5]proposed another definition: the past of the particle isdescribed by the locations where a particle leaves a weaktrace. Danan’s experiment was designed to measure thisweak trace.The measurement of the trace in the experiment ofDanan et al . invariably spoils the perfect interferenceof the inner interferometer and creates some leakage inits dark port. Apparently, this leakage is what madethe original experiment ‘not ideal’ in the eyes of Li etal . This view is supported by the fact that the leak-age is crucial for explaining the results of Danan et al .:the meter of their experiment was a transversal degree offreedom of the photon itself. The trace, ‘written’ on thewave function of the photon, could not be observed bythe quad-cell detector placed outside the interferometerwithout the leakage towards it. From this perspectivethe proposal of Li et al . to place the meter inside theinterferometer is a desired change. The trace is recordedwhere it is created. Therefore, we need not to confrontthe question: How does the external detector get the in-formation about the trace inside the inner interferometerif only a tiny leakage passes from the place with the tracetoward the detector?However, the conduction of a measurement which de-tects the weak trace of the photon inside the interfer-ometer without testing the traces in each of its armsseparately, is a step in the wrong direction. The setupwith the nested interferometers is analogous to a threebox paradox [6], where the paths of the interferometercorrespond to the three boxes . We know that if we lookin arm A we find the photon with certainty and also,if we look at arm C instead, we find it there with cer-tainty too. But if we test the presence of the photonanywhere in B or C without resolving these two paths,we are certain not to find it, since it is equivalent to test-ing its presence in A . It has been proven [6] that if a usual a r X i v : . [ qu a n t - ph ] J a n (strong) measurement of an observable performed on pre-and post-selected system yields a particular eigenvaluewith certainty, a weak measurement of this observablemust yield the same value. The experiment of Li et al . issuch a weak measurement of the projection onto B and C together, so it must yield null result.The outcomes of weak measurements are weak values,and the experiment can be understood also in this lan-guage. In the three-box setup, the weak values of theprojection operators on different boxes are:( P C ) w = 1 , ( P A ) w = 1 , ( P B ) w = − . (1)The weak values are additive, so( P B + P C ) w = ( P B ) w + ( P C ) w = − . (2)Vaidman’s principle is that the pre- and post-selectedphoton was in every place where it left a local trace.Any nonvanishing weak value of a local operator in aparticular place leads to a local weak trace. Li et al .’sexperiment does not observe all these local traces. Itweakly measures the projection onto B and C together.Even though according to the definition proposed byVaidman the photon was in B , and also was in C , theinfluences of the photon in the two places on the meter ofLi et al . cancel each other. The meter in their experimentis the phase acquired by the probe photon passing in theKerr media in the middle of the inner interferometer, seeFig. 1a. The photon influences the probe photon due toits presence in both arms, B and C , but the influencesare in opposite directions resulting in the null outcome.This is possible because contrary to the case of a photonthat is pre-selected only in a superposition of being indifferent arms of the interferometer causing a mixtureof evolutions of the probe photon, the pre- and post-selected photon yields a superposition of the evolutionsof the probe photon [7] which can cancel each other.A small modification of the proposed experiment is suitable for measuring the local trace inside the inter-ferometer. We just have to move the path of the probephoton near the place where we want to observe the trace,see Fig. 1b. Repeating the experiment with a probe pho-ton passing in different regions inside the nested interfer-ometer (or adding more photon-meter interferometers)will provide the full information about the past of thephoton. These local measurements will necessarily de-stroy the perfect interference of the inner interferometerleading to some unavoidable leakage. However, the weaktrace left by this leakage is vanishingly small. Indeed, anidentical coupling in all arms of the interferometer whichcauses the traces of order (cid:15) in arms A , B , and C willlead to the trace in the dark port proportional to (cid:15) . Inthe weak limit of (cid:15) → (cid:15) can be neglected. In this sense, the photons are present (leave a trace) in the arms B and C inside the inner in-terferometer, but not in the arms leading in and out ofit. For more discussion, see [8–11].The modified proposal of Li et al . is conceptually abetter experiment for observing the past of a photon de-fined as the regions where it leaves a weak trace. It is adirect measurement with an external device. Moreover,it is a genuinely quantum experiment since its resultscannot be explained by Maxwell’s equations of the elec-tromagnetic field of the laser, as they were explained byDanan et al . in their experiment. However, it is muchmore challenging. In view of a recent proposal [12], itis on the verge of technological feasibility. Still, the ex-periment of Danan et al ., even if it has an alternativeexplanation, is a good demonstration of the past of apre- and post-selected photon.In conclusion, the null result claimed by Li et al . isobtained not because there was no effect, but becausein their measurement the effects of the photon on themeter interferometer from the arms B and C of the in-ner interferometer cancel each other. Shifting the pathof the meter interferometer from the center of the innerinterferometer would reveal the weak trace of the photonthere. Such a modified experiment will be an improve- FIG. 1. (a) The experimental setup proposed by Li et al . inwhich the probe photon’s path runs through the middle of theKerr media. (b) Our proposed modification, in which the pathof the probe photon is moved to a region in the Kerr media inclose proximity of the arm of the interferometer wherein thepresence of the photon is tested. ment over the experiment by Danan et al ., which is worthperforming.This work has been supported in part by the German-Israeli Foundation for Scientific Research and Develop-ment Grant No. I-1275-303.14. [1] L. Fu, F. A. Hashmi, Z. Jun-Xiang, and Z. Shi-Yao, Chin.Phys. Lett. , 050303 (2015).[2] A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman,Phys. Rev. Lett. , 240402 (2013).[3] D. Bohm, Phys. Rev. , 166 (1952). [4] J. A. Wheeler, Mathematical Foundations of QuantumTheory. Academic Press, New York, pp. 9–48 (1978).[5] L. Vaidman, Phys. Rev. A , 052104 (2013).[6] Y. Aharonov and L. Vaidman, Journal of Physics A:Mathematical and General , 2315 (1991).[7] Y. Aharonov, J. Anandan, S. Popescu, and L. Vaidman,Phys. Rev. Lett. , 2965 (1990).[8] V. Potoˇcek and G. Ferenczi, Phys. Rev. A , 023829(2015).[9] L. Vaidman, Phys. Rev. A , 017801 (2016).[10] H. Salih, Frontiers in Physics , 47, (2015).[11] L. Vaidman, A. Danan, D. Farfurnik, and S. Bar-Ad,Frontiers in Physics , 48, (2015).[12] A. Feizpour, X. Xing, and A. M. Steinberg, Phys. Rev.Lett.107