The probability of an encounter of photons in nested and double-nested Mach-Zehnder interferometers
TThe probability of an encounter of photons in nested and double-nestedMach-Zehnder interferometers
E. Schmidt ∗ and A. Dubroka Faculty of Science, Masaryk University, Kotl´aˇrsk´a 2, 61137 Brno, Czech Republic (Dated: July 18, 2019)We present the results of a theoretical work discussing the propagation of an electromagneticwave through nested Mach-Zehnder interferometers using classical optics and standard quantumtheory. We show that some seemingly surprising effects at first sight, which are often explained inthe literature using the two-state vector formalism (TSVF), are a direct consequence of destructiveor constructive interference and thus there is no need for the unconventional TSVF formulation.We show that the probability of a photon detection derived from the weak value used in TSVF canbe interpreted as the probability of an encounter of two opposing photon fluxes.
I. INTRODUCTION
The literature about photon behavior is very rich andextensive. One of the often recurring problems concernsthe question “where was the photon?” passing an opti-cal system, e.g., an interferometer. Our paper is inspiredby two experimental articles of Danan et al. and therecently published article of Zhou et al. that explaintheir experiment using the two-state vector formalism(TSVF), which is based on the theoretical paper of L.Vaidman . The basic idea of the TSVF approach is thatfor a more complete description of a quantum system, oneshould be concerned not only with the forward-evolvingwave function | ψ (cid:105) but also the backward-evolving wavefunction (cid:104) ϕ | . A knowledge of both should give some in-formation about the system which can be tested by theso-called weak measurement, which should only weaklyaffect photon behavior. The publication of the TSVF ap-proach was followed by a prompt response and a longseries of concurring or critical opinions . In this paper,we discuss the propagation of photons in Mach-Zehnderinterferometers (MZI) using classical optics and standardquantum-mechanical approach. We compare the resultsof this approach with the TSVF.We proceed along the lines of Li et al. and Hashmi etal. and consider the often used arrangement of theMach-Zehnder interferometer (MZI) with one smallernested MZI, see Fig. 1. For the sake of simplicity, weconsider four identical ideal achromatic beam splittersBS with reflection coefficient r and transmission coeffi-cient t : r = i √ , t = 1 √ . (1)We consider three ideal mirrors M with reflectivity equalto 1, five modulators A , A , B , C , E that can changethe phase and amplitude of the wave function. We sup-pose that it is possible to measure the intensity of thebeam propagating in the forward direction (from topto bottom in Fig. 1) and also in the backward direc-tion (from bottom to top). For the forward direction,we consider one source of light S and three detectorsD , D , D . For the backward direction, we consider M L D M BS L L L BS D S D S E B S D BS A C Q Q Q A L L L M L BS D D S FIG. 1. The scheme of the single-nested Mach-Zehnder in-terferometer. S and D i with i = 1 , , ii and D ii with ii = 11 , ,
33) denote sources and detectors, respectively,for the forward (backward) propagating beam. BS denotebeam splitters, M denote mirrors and A , A , B , C , E de-note modulators. There are three optical paths (or quantumchannels) Q k ( k = 1 , ,
3) shown by solid, dashed and dash-dotted blue lines, respectively. There are eight stages L m ( m = 1 . . .
8) shown by the horizontal dashed lines where weevaluate the forward and backward evolving wave functionsand related intensities. three sources S , S , S and three detectors D , D ,D . We consider three quantum channels (or opticalpaths) Q k ( k = 1 , ,
3) depicted in Fig. 1 by solid, dashedand dash-dotted lines, respectively, and eight stages L m ( m = 1 . . .
8) in order to distinguish the state before andafter the modulation. We calculate the forward and back-ward evolving wave functions | ψ (cid:105) and (cid:104) ϕ | , respectively,at the intersections of Q k and L m .The experimental arrangement of Danan et al. is verysimilar to the one presented in Fig. 1; it involves the a r X i v : . [ qu a n t - ph ] J u l source S and the detector D . As modulators, vibratingmirrors with a specific frequency were used in positionsequivalent to A , A , B , C , E . We believe the differ-ence in type of modulation is not substantial and shouldnot affect the following discussion and conclusions. Themain result of Danan et al. is that the signal measured atthe detector D exhibited the frequency from the pertur-bation A , B , C but not from A and E . The authorsof Ref. interpreted the experiment using the TSVF anddraw the nontrivial conclusion that photons come to thedetector D via the channel Q and at the same timeare present inside the nested interferometer (in the sec-tions with the modulators B and C ) but not outsidethe nested interferometer (in the sections with the mod-ulators A and E ). The latter results were associatedwith the conclusion that the past of the photons is notrepresented by continuous trajectories .There are several other experimental works on MZI.For example, single photon variant of the experiment ofDanan et al. was realized by Zhou et al. . There existalso other variants, see Li et al. and Ben-Israel et al. who used a combination of a standard MZI with a nestedMZI. Another modification of the nested MZI with Doveprism inside the nested part was tried by Alonso et al. followed by a comment . An alternative to the workof Danan et al. was reported by Bula et al. where aslightly modified spectral distribution of light was usedfor modulation instead of vibrating mirrors. The exper-iment of Len et al. on the MZI using a single-photonsource was recently published. Another variant of thesingle photon experiment using the MZI and nested MZIwas suggested by Englert et al. .The main part of this paper presented in Sec. II A isdevoted to calculations of the beam intensity at differentstages and quantum channels of the single nested MZIby means of classical optics and standard quantum the-ory. We also calculate the weak value, related probabil-ity and propose its interpretation as the probability ofan encounter of the forward and backward-propagatingphoton fluxes. We show that the discontinuous patternsoccurring in the weak value are direct consequence of de-structive or constructive interference. In Sec. II B, weshortly repeat the calculations for the symmetric double-nested MZI, and for completeness we consider the stan-dard (simple) MZI in Sec. II C. In Sec. II D, we discussthe results for the nested MZI with modulators. II. RESULTSA. Single nested Mach-Zehnder interferometer
First we state our assumptions concerning light inten-sity and detection. We suppose that during the measure-ment, the photon is absorbed in an irreversible processand the detector is capable of measuring a large num-ber of photons as well as individual photons. Formallywe proceed in the same way as in the work of P. L. Sal- danha , i.e., we use a classical description of waves whichis valid also for the wave function of photons. We as-sume that the beam is a plane wave, however, the resultsshould be valid also for a Gaussian beam. We assumethat the intensity of the beam is measurable at the out-put of the interferometer and at any stage inside the in-terferometer. Following Duarte , we assume that thebeam of light is a high power laser beam consisting of alarge number of photons. We adopt the Dirac’s idea that each photon interferes with itself and assume thatthe results of such an experiment and the interpretationof results are independent of the number of photons, thatis, they should be the same for an intensive flux of pho-tons as well as for a single photon state. In the lat-ter, the quantitative evaluation of the intensity shouldbe treated statistically as a number of photons detectedper a sufficiently long integration time. This should notbe confused with the so-called “weak measurement” or“weak value” mentioned below. In this section, we treatthe nonmodulated beam; the effects of modulation arediscussed in Sec. II D.In the TSVF, the forward- and backward-evolvingquantum states of light play an important role. Forus, this was an inspiration to consider both directionsof propagation. In the forward direction (denoted by theindex f ), the intensity distribution of photons (or en-ergy flux density) I fm,k is proportional to the probabilitydistribution p fm,k I fm,k = Kp fm,k = K | ψ m,k | , (2)which is calculated according to standard quantum the-ory as the square modulus of the corresponding wavefunction amplitude ψ m,k . K is the proportionality con-stant; for simplicity we take K = 1 and so the termsintensity distribution and probability distribution are in-terchangeable in this paper. As a consequence of theenergy conservation law the following sum rule over alloptical paths k holds at any stage m (cid:88) k =1 p fm,k = 1 . (3)Results of our calculations for the forward directionare schematically shown in Fig. 2(a). The intensity ofthe beam at a particular stage of the interferometer isschematically represented by the thickness of the line.Note that in the forward direction in the section betweenBS and BS , the intensity is zero due to destructiveinterference and the whole intensity from the nested in-terferometer propagates towards the detector D . In or-der to fully describe the interference including the energyconservation law, we consider detectors in all three quan-tum channels (D , D , and D ).For the backward direction (denoted by index b ), theintensity distribution of photons I bm,k,s is proportional tothe probability distribution p bm,k,s I bm,k,s = Kp bm,k,s = K | ϕ m,k,s | , (4) S D D D b)M M M BS BS BS BS S D D D M M M BS BS BS BS a) S D D D c)M M M BS BS BS BS S D D D d)M M M BS BS BS BS D S e)M M M BS BS BS BS S f)M M M BS BS BS BS D S g)M M M BS BS BS BS D S S h)M M M BS BS BS BS S i)M M M BS BS BS BS S S j)M M M BS BS BS BS S p f p b ( A ) w P FIG. 2. Results for the nested Mach-Zehnder interferometer. M denotes a mirror and BS a beam splitter (see also Fig. 1).Panel (a) schematically shows intensities (or probabilities p f ) of the forward-propagating beam from the source S (red lines).Panels (b), (c) and (d) show intensities (or probabilities p b ) for the backward-propagating beam from the sources S , S andS , respectively (red lines). The directions of the incoming beam are denoted by the arrows. Panels (e), (f), (g) show theweak value ( A ) w for the case of the forward-propagating beam coming from the source S and the backward-propagating beampost-selected by detectors D , D and D , respectively. Panels (h), (i), (j) show the encounter probability P = p f p b for thecase of the forward-propagating beam coming from the source S and the backward-propagating beam coming from the sourcesS , S and S , respectively. Blue lines denote positive and pink lines (in panels (e) and (f)) denote negative value. Themagnitude of all quantities is schematically represented by the thickness of the line. The black dashed lines denote the pathwith zero value of the corresponding quantity. which is calculated as the square modulus of the corre-sponding wave function amplitude ϕ m,k,s . In the follow-ing, we again choose K = 1 for simplicity. The index s = 1 , , , S and S , respectively. The in- tensity is detected at the end of the MZI by the detec-tors D , D , D . The corresponding intensity distribu-tion is displayed in Figs. 2(b), 2(c) and 2(d), respectively.Thanks to the energy conservation law, the sum of theprobability over all optical paths k yields (cid:88) k =1 p bm,k,s = 1 (5)for any value of m and s . Because of symmetry reasons,the following sum rule over the experimental configura-tions s holds as well: (cid:88) s =1 p bm,k,s = 1 (6)for any value of m and k .Next, we confront these results with those of TSVF.Following Ref. , the weak value of the projection operator A m,k onto a section with the coordinates m and k isdefined as ( A m,k,s ) w = (cid:104) ϕ s | A m,k | ψ (cid:105)(cid:104) ϕ s | ψ (cid:105) , (7)where | ψ (cid:105) is the forward-evolving wave function (corre-sponding to the source S ) and (cid:104) ϕ s | is the backward-evolving wave function post-selected by the detector D s .Following Li et al. , the square modulus of the nomi-nator of Eq. (7) P m,k,s = | (cid:104) ϕ s | A m,k | ψ (cid:105) | (8)is the probability of the detector s clicking under the con-dition that the photon is found at the section with coor-dinates m and k . In our paper, we call this quantity an encounter probability for the reasons that are discussedbelow. Expressed with the amplitudes of the vectors,Eq. (8) yields P m,k,s = | ϕ m,k,s | · | ψ m,k | (9)and it is obviously equal to the product of probabili-ties (2) and (4) of the forward and backward-propagatingwaves, respectively, P m,k,s = p bm,k,s p fm,k . (10)Note that within the TSVF framework, (cid:104) ϕ s | is thewave function evolving backward in time which is post-selected by the detector D s . In our calculations, thisbackward-time evolution influences only the results of theweak value (7). Since the encounter probability (9) isgiven by the square modulus of the wavefunctions, itsvalue is independent of the time direction and it can bethus calculated (and measured) using the beam that ispropagating forward in time from sources S ss to detectorsD ss .Using Eq. (6) and (10), the following sum rules can bederived (cid:88) s =1 P m,k,s = p fm,k (cid:88) s =1 p bm,k,s = p fm,k (11) and using Eq. (3) we obtain (cid:88) k =1 3 (cid:88) s =1 P m,k,s = 1 (12)for any value of m . The latter is a summation rulewhich is formally similar to the law of conservation ofenergy, see, e.g., Eq. (3), however here it is obtained onlywhen the encounter probability P m,k,s is summed up overall experimental arrangements s . For the sake of com-pleteness, we can define normalized encounter probability¯ P m,k,s using the Aharonov-Bergmann-Lebowitz rule ¯ P m,k,s = | (cid:104) ϕ s | A m,k | ψ (cid:105) | (cid:80) k (cid:48) | (cid:104) ϕ s | A m,k (cid:48) | ψ (cid:105) | = P m,k,s (cid:80) k (cid:48) P m,k (cid:48) ,s (13)that can be interpreted as a relative distribution of en-counter probability in individual channels.Numerical results of the weak value (7) for the threevariants of s = 1 , , P m,k,s are schemati-cally shown in Figs. 2(h), 2(i) and 2(j), for the three pos-sibilities of the backward-evolving wavefunction, s = 1 , P (we omit below the indexes m, k, s for brevity) in terms of the multiplication of prob-abilities p f and p b [see Eq.(10)] is straightforward. Bothprobabilities p f and p b are given using the standardquantum mechanical interpretation by the square mod-ulus of corresponding amplitudes and meet summationrules (3) and (5). Both experiments in the forward andthe backward direction are independent, so the productof the probabilities has also meaning of a probability, butrather as a probability of an “encounter” of photons oralternatively a probability of an independent presence ofboth forward and backward photons at the same stageat any time. Based on this interpretation, we call P an encounter probability. The encounter probability, seeEq. (10), in contrast with probabilities used in Eqs. (2),(4) does not meet the summation rule similar to Eq. (3)or (5). This interpretation of P allows us to understandeasily the obtained intensity patterns. In the case of thebeam propagating in the forward direction [see Fig. 2(a)],the intensity I f is zero (and p f = 0) between BS andBS due to destructive interference. In the case of thebeam propagating in the backward direction from thesources S and S , see Fig. 2(b) and (c) respectively,the intensity I b is zero (and p b = 0) between BS andBS also due to destructive interference. In case of thesource S [see Fig. 2(d)], the intensity I b is zero (and p b = 0) in the sections BS -M -BS and BS -BS forgeometric reasons. Consequently, the probability of en-counter P given by the product of p f and p b is zero inall these sections because of the corresponding reasons.The interpretation in terms of TSVF is different. Itis claimed that a photon trajectory can be calculatedfrom the forward and the backward-propagating wavefunctions of the emitted and detected photon and thatthe weak value is proportional to the trace the photonleaves . Alternatively, according to the work of Li etal. , the modulus square of the weak value nominator,see Eq. (8), represents propability of the detector clickingunder the condition that the photon is found at the corre-sponding section. The problems of both of these interpre-tations clearly shows up in cases displayed in Fig. 2(e),2(f) and Fig. 2(h), 2(i), respectively, where the trajec-tories in the nested interferometer are completely dis-connected from the source and detector since it raisesquestions, e.g., “how did the photon get to the nestedinterferometer?” etc. This effect is within the TSVF in-terpreted as a discontinuity of photon trajectory . How-ever, if it is interpreted as a probability of encounter ofphotons propagating in the opposite directions, then thediscontinuous sections are naturally understood as dis-cussed above. We do not doubt numerical evaluation ofthe TSVF calculation reported, e.g., in , but we raisequestions about the interpretation, legitimacy or physi-cal meaning of the procedure, similarly to discussions andobjections of Refs. . B. Double-nested Mach-Zehnder interferometer
The single-nested MZI that was discussed above isasymmetric, consequently, the first channel (via mirrorM ) is always open. Therefore, in this section we con-sider a symmetric four-channel interferometer with twonested MZI. We proceed formally in the same way as inSec. II A.Figure 3(a) schematically shows the intensity of thebeam propagating in the forward direction from thesource S to four detectors D i , ( i = 1 , , , S ii to the detectors D ii , where ii = 11 , , , , respec-tively. The corresponding encounter probabilities calcu-lated using Eq. (10) are presented in Figs. 3(f), 3(g), 3(h)and 3(i), respectively. Similarly here, this results can beeasily visualized by a “graphical multiplication” of thecorresponding intensity patterns shown in Fig. 3(a) andFigs. 3(b), 3(c), 3(d) and 3(e), respectively. In Fig. 3, wedo not show the weak value because the denominator ofEq. (7) is zero for this symmetric double nested MZI andthe weak value diverges .The encounter probabilities shown in Figs. 3(f) and3(i) exhibit continuous trajectories similarly to standardbeam paths, e.g., shown in 3(a). In contrast, the en-counter probabilities shown in Figs. 3(g) and 3(h) areparticularly interesting since P is non zero only insidethe nested parts of the interferometer. The interpreta- tion of these results in terms of the encounter probabilitycan be easily explained similarly as above using either de-structive interference or geometric reasons. However, interms of the TSVF, the paths are discontinuous withoutany input nor output and typically the interpretation in-volves claims about a discontinuous photon trajectory . C. Simple Mach-Zehnder interferometer
In the context of the discussion of the TSVF vs. theencounter probability, it useful to return to the simplestcase of the MZI without any nested part that is very of-ten discussed in literature, e.g., in Refs. . Usingthe same procedure as in the case of the single or thedouble nested MZIs, we calculate the intensities of theforward and the backward-propagating beam and the en-counter probability. Figure 4(a) displays the intensity ofthe forward-propagating beam from the source S to thedetectors D and D . There is zero signal on the detec-tor D due to destructive interference and a unit signalon D due to constructive interference. Figures 4(b) and4(c) display the intensity of the backward-propagatingbeam from the sources S and S , respectively. Thedetectors D and D detect zero signal as shown inFigs. 4(b) and 4(c), respectively, due to destructive in-terference. Figures 4(d) and 4(e) display the encounterprobability, that can be as above visualized as the prod-uct of the probabilities shown in Fig. 4(a) and Figs. 4(b),4(c), respectively. Note that again the denominator ofthe weak value, see Eq. (7), is zero for this symmetricMZI and the weak value diverges .According to the TSVF, Figs. 4(d) and 4(e) show a tra-jectory of photons when the photon was detected by D or D , respectively. Again here we see a similar situationas above where some of the patterns exhibit continuoustrajectories, see Fig. 4(e), but the other, see Fig. 4(d),shows zero probability neither on input nor on the outputof the interferometer, however, there is still a non-zeroprobability inside of the interferometer. The interpreta-tion of P shown in Fig. 4(d) in terms of the encounterprobability involves simple arguments about destructiveor constructive interference, however, the interpretationin terms of the TSVF has to involve speculations abouta discontinuous photon trajectory . Already this mostsimple MZI exhibits the key properties that occur in themore complex nested MZIs discussed above. D. The nested Mach-Zehnder interferometer withperturbations
We believe that the issue is essentially explained above.However, since perturbations of MZI are often used in ex-periments, we devote this section to the discussion of thistopic. Consider modulators A , A , B , C , E shown inFig. 1. Inspired by the modulation spectroscopy , weprefer modulators based on the change of amplitude or M BS BS BS BS M BS BS BS M M M BS D BS BS b) D D S S D S S S D BS BS BS D D D S BS MM BS MM MM a) M BS BS BS BS BS BS M S f) M e) M BS BS BS BS M M BS S BS g) S c) D S M BS BS BS BS BS BS d) M M M BS BS BS BS BS BS M M D D D D D D D S S i) M BS BS BS BS BS BS M M S BS BS M h) p f p b P M BS BS BS BS BS BS M M D D D D FIG. 3. Results for the double nested Mach-Zehnder interferometer. M denotes a mirror and BS denotes a beam splitter. MirrorM is a mirror reflecting on both sides. Panel (a) schematically shows intensities (or probabilities p f ) of the forward-propagatingbeam (red lines). Panels (b), (c), (d) and (e) schematically show intensities (or probabilities p b ) of the backward-propagatingbeam (red lines). The directions of the incoming light are denoted by the arrows. Panels (f), (g), (h) and (i) schematicallyshow encounter probability P = p f p b (blue lines) for the case of the forward-propagating beam coming from the source S andthe backward-propagating beam coming from the source S , S , S and S , respectively. The magnitude of all quantitiesis schematically represented by the thickness of the line. The black dashed lines denotes the path with zero value of thecorresponding quantity. phase of the beam rather than a change of the beamdirection. For simplicity we assume transmission modu-lators that modulate the amplitude and in principle thephase. The transmission coefficient of the modulator fora wave function (or plane wave) is assumed in the form τ X = [ τ X − (cid:15) X cos(2 πf X t )]e i δ X (14)where τ X is the unperturbed transmission coefficient(for simplicity we choose τ X = 1), (cid:15) X (cid:28) f X isthe frequency of the modulation and δ X is a phase shift.For an easy comparison with the literature we use zerophase shifts δ X = 0, however the calculations can be eas- ily extended for non-zero phase shifts; see, e.g., Ref. .One experimentally often used way of distinguishingthe influence of different modulators in the detected sig-nal is by using different modulation frequencies f X ; see,e.g., Ref. . Consequently we examine the Fourier trans-form of the detected signal and we take the Fouriercoefficient with the corresponding frequency as a mea-sure of the detected perturbation amplitude, see alsoRefs. . In the following, we call it the perturbationamplitude. We describe in detail here only the single-nested MZI; for the double-nested and simple MZIs, theconclusions are analogous. We choose the frequencies ofthe modulators A , A , B , C , E in the following or- D BS BS BS M M S S D D D D D BS BS BS BS BS BS BS M M M M M M M M S S S S S b)a) e)d) c) p f p b P FIG. 4. Results for the simple Mach-Zehnder interferome-ter. M denotes mirrors and BM denotes beam splitters. Panel(a) schematically shows intensities (or probabilities p f ) of thebeam propagating in the forward direction (red lines). Pan-els (b) and (c) schematically show intensities (or probabilities p b ) of the beam propagating in the backward direction (redlines) from the sources S and S , respectively. The direc-tions of the incoming light are denoted by the arrows. Thethickness of the lines on each path is proportional to the in-tensities of light. Panels (d) and (e) schematically show theencounter probability P = p f p b (blue lines) on each pathfor the forward-propagating beam coming from the source S and the backward-propagating beam coming from the sourcesS and S , respectively. The magnitude of all quantities isschematically represented by the thickness of the line. Theblack dashed lines denote the path with zero value of the cor-responding quantity. der: 3, 5, 7, 11 and 17 Hz. The choice of frequencies isnot essential; however it is advisable to avoid overlaps intheir mutual combinations. All modulators work simul-taneously.The results for the perturbation amplitude in the for-ward direction are schematically displayed in Fig. 5(a).For the sake of visibility, the same line thickness is used for all signals with non-zero intensity and the order ofperturbation in (cid:15) X is differentiated by the line type(solid line for first order, dashed line for the second orderand dotted line for the third order). The total signal is afunction of frequency with the first order signals havingthe fundamental frequencies. The frequency of the sec-ond and third order signals is given by the combinationsof their corresponding fundamental frequencies. For sim-plicity, we display the signal only with the lowest orderfrom a given modulator.Figure 5(a) depicts that the detector D detects signalfrom the modulators A , B , C in the first order andA , E in the second order; the same situation is on thedetector D . The latter was already described in workof M. Wiesniak . The detector D detects only the firstorder signals from A , B , and C . The section of theoptical path with the modulator E (between BS andBS ) is particularly interesting. Recall that it exhibitsthe complete destructive interference in the case of theunperturbed MZI, see Fig. 2(a). However, in the caseof the perturbed MZI, there is a signal but only in thesecond order from the modulators B , C and in the thirdorder from the modulators A , E depicting the imperfectdestructive interference.If the transmission coefficients τ B1 and τ C1 are equal,the complete destructive interference in the section withmodulator E is restored. This situation is displayed inFig. 5(b) and depicts that there are neither signals A ,B , C , E on the detectors D and D nor on the op-tical path in the section with the modulator E . Theseresults are obvious in terms of classical optics and stan-dard quantum theory.In the work of Danan et al. they reported the casewhen the optical path at the modulator A is blocked andno modulated signal on the detector D was observed,see Fig. 2(c) in Ref. . In our case this situation wouldcorrespond to τ A = 0 and the corresponding intensity inthe forward direction is shown in Fig. 5(c). We see thatthe signals from A , B , C , E can be detected on thedetectors D and D but only in the second and the thirdorder of the detected signal. This signal was probably notresolved in Ref. because the higher order signals havemuch lower intensity. Similar conclusion (the absence ofdetection of higher order signals) can be made for thesimilar experimental work of Zhou et al. that used thesingle photon source.So far in the case of perturbations we calculated onlythe intensities of the forward propagated beam. Formallywe can calculate also the backward-propagating wave andcalculate the encounter probability P using Eq. (10). Theresults for the encounter probability calculated for per-turbations shown in Figs. 5(a) and 5(b) and for the back-ward beam with s = 1, (i.e., coming from the source S )are shown in Fig. 5(d) and 5(e), respectively. We can seethat the encounter probability corresponding to the caseof the imperfect destructive interference, see Fig. 5(d),looks continuous, at least in some higher order signalsin perturbation, however, in case of the perfect destruc- AA AA BB CC EE a) SS DD DD DD AA AA BB CC EE b) SS DD DD DD AA AA BB CC EE d) SS SS AA AA BB CC EE e) SS SS AA AA BB CC EE c) SS DD DD DD FIG. 5. Results for the nested Mach-Zehnder interferometer with active modulators A , A , B , C and E denoted by blue,green, red, cyan and magenta circles, respectively. Some symbols shown in Fig. 1 are omitted for clarity. The thick linesdenote Fourier perturbation amplitudes of intensity (or the encounter probability) with color corresponding to the modulators.The order of perturbation is differentiated by the line type (solid line for first order, dashed line for the second order anddotted line for the third order). Thin dotted lines represents the signals with zero Fourier amplitudes. Panel (a) displays theFourier perturbation amplitudes of the beam propagating in the forward direction from the source S when all modulatorswork at different frequencies, i.e., are independent. Panel (b) displays the same as (a) except that modulators B and C areidentical. Panel (c) displays the situation of panel (a) with modulator A blocking the beam. Panels (d) and (e) display theFourier perturbation amplitudes of the encounter probability for the forward beam shown in (a) and (b), respectively, with thebackward-propagating beam coming from the source S . tive interference, see Fig. 5(e), it exhibits discontinuoussections. Analogous conclusions can be obtained for thedouble nested or simple MZI. III. SUMMARY
We have presented calculations of the intensity of thebeam propagating in a nested, a double nested and a sim-ple Mach-Zehnder interferometer calculated using classi-cal optics and standard quantum theory. Qualitatively,the results can be understood as a result of constructiveor destructive interference. We show that the probabilityof detection of photon derived from the weak value usedin the TSVF formalism can be interpreted as the prob- ability of encounter of two opposing photon fluxes. Thisinterpretation does not need to involve any discontinuousphoton trajectories often used in TSVF interpretations.We discussed also the perturbations of the nested Mach-Zehnder interferometer and showed that the signals fromthe modulators can propagate in the case of the imper-fect destructive interference in the second or the thirdorder in perturbation.
ACKNOWLEDGMENTS
We acknowledge helpful discussions with D. Munzarand J. Rusnaˇcko. This work was supported by the MEYSof the Czech Republic under the project CEITEC 2020(LQ1601). ∗ [email protected] A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman, Phys.Rev. Lett. , 240402 (2013). Z.-Q. Zhou, X. Liu, Y. Kedem, J.-M. Cui, Z.-F. Li, Y.-L.Hua, C.-F. Li, and G.-C. Guo, Phys. Rev. A , 042121(2017). L. Vaidman, Phys. Rev. A , 052104 (2013). Z.-H. Li, M. Al-Amri, and M. S. Zubairy, Phys. Rev. A ,046102 (2013). L. Vaidman, Phys. Rev. A , 046103 (2013). J. Lundeen, Physics , 133 (2013). H. Salih, Front. Phys. , 47 (2015). A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman,Front. Phys. , 48 (2015). L. Vaidman, Phys. Rev. A , 024102 (2014). Z.-Q. Wu, H. Cao, J.-H. Huang, L.-Y. Hu, X.-X. Xu, H.-L.Zhang, and S.-Y. Zhu, Opt. Express , 10032 (2015). P. L. Saldanha, Phys. Rev. A , 033825 (2014). K. Bartkiewicz, A. Cernoch, D. Javurek, K. Lemr, J. Sou-busta, and J. Svozil´ık, Phys. Rev. A , 012103 (2015). V. Potocek and G. Ferenczi, Phys. Rev. A , 023829(2015). L. Vaidman, Phys. Rev. A , 017801 (2016). L. Vaidman, Phys. Rev. A , 036103 (2016). K. Bartkiewicz, A. Cernoch, D. Javurek, K. Lemr, J. Sou-busta, and J. Svozil´ık, Phys. Rev. A , 036104 (2016). F. A. Hashmi, F. Li, S.-Y. Zhu, and M. S. Zubairy, J. Phys.A , 345302 (2016). F. Li, F. A. Hashmi, J. X. Zhang, and S.-Y. Zhu, Chin.Phys. Lett. , 050303 (2015). A. Ben-Israel, L. Knips, J. Dziewior, J. Meinecke, A.Danan, H. Weinfurter, and L. Vaidman, Chin. Phys. Lett. , 020301 (2017). M. A. Alonso and A. N. Jordan, Quantum Stud.: Math.Found. , 255 (2015). L. Vaidman, I. Tsutsui, Entropy , 538 (2018). M. Bula, K. Bartkiewicz, A. Cernoch, D. Javurek, K.Lemr, V. Michalek, and J. Soubusta, Phys. Rev. A , 052106 (2016). Y. L. Len, J. Dai, B.-G. Englert, and L. A. Krivitsky, Phys.Rev. A , 022110 (2018). B.-G. Englert, K. Horia, J. Dai, Y. L. Len, H. K. Ng, Phys.Rev. A , 022126 (2017). D. Sokolovski, Phys. Lett. A , 1593 (2016). D. Sokolovski, Phys. Lett. A , 227 (2017). R. B. Griffiths, Phys. Rev. A , 032115 (2016). G. N. Nikolaev, Jetp Lett. , 152 (2017). Q. Duprey and A. Matzkin, Phys. Rev. A , 032110(2017). L. Vaidman, Phys. Rev. A , 066101 (2017). R. B. Griffiths, Phys. Rev. A , 066102 (2017). L. Vaidman, Jetp Lett. , 473 (2017). G. N. Nikolaev, Jetp Lett. , 475 (2017). D. Sokolovski, Phys. Rev. A , 046102 (2018). U. Peleg and L.Vaidman, Phys. Rev. A , 026103 (2019). B.-G. Englert, K. Horia, J. Dai, Y.L. Len and H. K. Ng,Phys. Rev. A , 026104 (2019). Y. Aharonov and L. Vaidman. The Two-State Vector For-malism of Quantum Mechanics. In:
Time in QuantumMechanics , edited by J. G. Muga, R. Sala Mayato, andI. L. Egusquiza, Springer 2002, p. 369. C. H. Holbrow, E. Galvez and M. E. Parks, Am. J. Phys. , 260 (2002). F. J. Duarte,
Quantum Optics For Engineers (CRC Press,Boca Raton, FL, 2014). P. A. M. Dirac,
The Principles of Quantum Mechanics , 4thed. (Oxford University Press, Oxford, 1999). A. Feizpour, X. Xing, and A. M. Steinberg, Phys. Rev.Lett. , 133603 (2011). M. Cardona,
Modulation Spectroscopy (Academic PressNew York, New York, 1969). Z.-Q. Wu, H. Cao, J.-H. Huang, L.-Y. Hu, X.-X. Xu, H.-L.Zhang, and S.-Y. Zhu, Opt. Express 23, 10032 (2015). M. Wiesniak, Phys. Lett. A382