---
abstract: 'Recently, a scheme based on the method of weak measurements to register the trajectories of photons passing through a nested Mach-Zehnder interferometer was proposed \[L. Vaidman, Phys. Rev. A **87**, 052104 (2013)\] and then realized \[A. Danan, D. Farfurnik, S. Bar-Ad, et al., Phys. Rev. Lett. **111**, 240402 (2013)\]. Interpreting the results of the experiment, the authors concluded that “the photons do not always follow continuous trajectories”. It is shown in this work that these results can be easily and clearly explained in terms of traditional classical electrodynamics or quantum mechanics implying the continuity of all possible paths of photons. Consequently, a new concept of disconnected trajectories proposed by the authors of work \[Phys. Rev. Lett. **111**, 240402 (2013)\] is unnecessary.'
author:
- 'G. N. Nikolaev'
title: |
    Paradox of Photons Disconnected Trajectories\
    Being Located by Means of “Weak Measurements”\
    in the Nested Mach-Zehnder Interferometer
---

\[sec:Introduction\]INTRODUCTION
================================

Quantum mechanics had deeply impact on the possibilities of measurements of physical quantities. On one hand, it revealed the limits of measurement of a number of physical quantities because of inherent “quantum noise” caused by the probabilistic nature of quantum laws. On the other hand, the quantization (discreteness) of certain physical quantities has made it possible to increase significantly the accuracy and to avoid the manifestations of thermal noise of measurements (Mössbauer effect, quantum Hall effect, superconductivity, etc.). A new type of quantum measurements, the so-called “weak measurements,” was proposed about three decades ago [@Aharonov1988; @Duck1989; @Aharonov1990; @Aharonov2005; @Dressel2014; @Struppa2014]. Such measurements have a number of paradoxical properties: the measured “weak quantities” of Hermitian operators can be significantly beyond the spectrum of their values, weak quantities are generally complex, weak measurements of noncommuting observables can be performed simultaneously, and “weak probabilities” of projection operators can have negative values. Because of such unusual properties, weak measurements are still under discussion. Despite this discussion, weak measurements were realized soon after their prediction [@Ritchie1991]. Many hundreds of works have been performed on this subject. The unique properties initiated a great interest in weak measurements as promising metrological tools. In particular, owing to a specific property of enhancement of weak signals, weak measurements can be used to estimate small changes in parameters such as the deviation of a beam of light and frequency, phase, velocity, temperature, and time shifts. The complexity of weak quantities whose real and imaginary parts can be measured simultaneously underlies methods of direct measurement of quantum states, wavefunctions, and geometric phases (see review [@Dressel2014] and references therein).

Using the concept of weak measurements, the authors of [@Danan2013] performed a fine experiment in order to reveal the past of photons passing through a nested Mach-Zehnder interferometer (Fig. \[fig:setup\]). The results were so unexpected that the authors arrived at the necessity of rejecting the commonsense treatment of the past of a quantum particle. They concluded that the past of photons is not described by a continuous trajectory or a set of possible trajectories, as is commonly accepted in traditional quantum mechanics [@Feynman2011ru]. The idea of such an experiment was proposed in [@Vaidman2013a], where it was predicted that weak measurements at the output of the nested Mach-Zehnder interferometer should indicate both the presence of photons in the inner Mach-Zehnder interferometer, which is tuned to destructive interference, and the absence of them at its input and output. This conclusion was based on the two-state vector formalism [@Aharonov1964; @Aharonov1990] and under the assumption that perturbation of the Mach-Zehnder interferometer in the process of weak measurements can be neglected (which was also used when interpreting the results of the experiment reported in [@Danan2013]). However, the authors of [@Li2013] commenting on work [@Vaidman2013a] qualitatively showed that weak measurements in the inner Mach-Zehnder interferometer partially destroy the interference between waves passing through it.

The surprising results of the experiment reported in [@Danan2013] were actively discussed in [@Wiesniak2014; @Salih2014; @Svensson2014a; @Huang2014; @Saldanha2014; @Li2015; @Bartkiewicz2015], where their interpretation was criticized and various modifications of the experiment and methods of data processing were proposed. The authors of [@Danan2013] also reported the calculation of detected signals according to the classical theory of electromagnetic waves but did not find any simple interpretation of the signals within this approach. The authors of [@Bartkiewicz2015] proposed to describe the evolution of the state of photons within a modified scheme according to traditional quantum mechanics. A numerical-analytical description of the discussed experiment was given in [@Saldanha2014] on the basis of the paraxial optical approximation and angular spectrum. The aim was to give a clear physical interpretation of paradoxical results of the experiment reported in [@Danan2013]. However, this aim was achieved incompletely. It was emphasized that interference of all beams of light in the nested Mach-Zehnder interferometer is important for the formation of a signal on the detector. This was explicitly declared in the introduction in [@Saldanha2014] and was used to calculate the amplitude of the angular spectrum at the output of the nested Mach-Zehnder interferometer in the case of destructive tuning of the inner Mach-Zehnder interferometer. However, this fundamental aspect was disregarded when deriving expressions for signals on the photodetector, thus reducing the credibility of the results. The considered feature is brightly indicated by an additional paradoxical result of the discussed experiment: in the case of destructive tuning of the inner Mach-Zehnder interferometer, all three signals (from mirrors $A$, $B$ and $C$) disappear when the beam of light in the outer arm of the outer Mach-Zehnder interferometer (from mirror $C$) is blocked. This result cannot be explained within the treatment of the two-state vector formalism by the authors of [@Danan2013]. As a whole, the results of the experiment reported in [@Danan2013] were quite clearly interpreted in [@Saldanha2014] (including numerical calculations), but with a number of uncertainties and ambiguities. In particular, in the case of destructive interference in the inner Mach-Zehnder interferometer, the absence of a signal from the mirror F was attributed to a small deviation of the beam induced by this mirror (as compared to its width) with the same order of deviation from other mirrors. It was stated that the nature of the absence of signals from mirrors F and E under these conditions is the same, which is incorrect, as will be shown below. The accuracy of absence of signals from mirrors F and E, as well as the dependence of this accuracy on the feature of the beam, was not analyzed. In the case of constructive interference in the inner Mach-Zehnder interferometer, a physical reason for the doubling of the amplitude of modulation of signals from mirrors F and E as compared to signals from other mirrors was not ascertained. Finally, the following important question remained unanswered: Are the discussed paradoxical results the consequences of the features of the used beams (with a Gaussian profile) and their symmetry or are they valid at any profile of beams?

The aim of this work is to clarify the listed uncertainties using a more direct general approach differing from that used in [@Saldanha2014]. It is shown that perturbations of the interferometer when performing weak measurements in the discussed experiment cannot be neglected; moreover, just these perturbations are entirely responsible for the results of these measurements. The extremely clear explanation of paradoxical results of the experiment reported in [@Danan2013] will be given within traditional wave theory of light or quantum mechanics, which implies continuous trajectories of light. Consequently, an additional concept of discontinuity of trajectories of photons is unnecessary.

\[sec:Analisys\]ANALYSIS OF THE EXPERIMENT [@Danan2013]
=======================================================

Figure \[fig:setup\] shows the simplified layout of the experiment. Identically tuned polarizers P1 and P2 and polarization beam splitters PBS1 and PBS2 are placed at the input and output of the nested Mach-Zehnder interferometer. The polarizer P1 and polarization beam splitter PBS1 are tuned so that 2/3 and 1/3 of the total intensity of incident light are incident on mirrors E and C, respectively.

![(Color online) Simplified layout of a setup for detection of paths of photons [@Danan2013]. Elements BS1, A, B, and BS2 form the inner Mach-Zehnder interferometer, whereas elements P1, PBS1, C, E, F, PBS2, and P2 form the outer Mach-Zehnder interferometer, where A, B, C, E, and F are mirrors; P1 and P2 are polarizers; BS1 and BS2 are ordinary beam splitters; PBS1 and PBS2 are polarization beam splitters; and QCD is the quad-cell photodetector.[]{data-label="fig:setup"}](Fig1_e){width="\columnwidth"}

Light beams transmitted through PBS2 and P2 interfere and are detected on the quad-cell photodetector (QCD). Mirrors A, B, C, E, and F undergo small vibrations around their horizontal axes with individual frequencies resulting in vertical shifts of the beam (along the $y$ axis) on the QCD. The frequency power spectrum of the difference between intensities of light in the upper and lower parts of the surface of the photodetector is recorded. The ratio of frequencies of vibration of the mirrors to the frequency of light is $\sim10^{-12}$. The ratio of the angular deflection of the beam ($\sim300$ nrad) caused by the modulation of mirrors to the diffraction divergence of the beam is $\sim10^{-4}$. Consequently, the modulation of mirrors can be neglected when calculating optical paths. Therefore, the total amplitude of the electric field $E(x,y)$ at the point ($x,y$) of the detector has the form

$$\begin{aligned}
\label{eq:amplitude}
E(x,y)&=\dfrac{\sqrt{I_{0}}}{3}\left[f(x,y-\delta_{E}-\delta_{A}-\delta_{F})e^{i\varphi_{A}}+ \right. \notag \\
&+f(x,y-\delta_{E}-\delta_{B}-\delta_{F})e^{i\varphi_{B}}+\\
&+\left. f(x,y-\delta_{C})e^{i\varphi_{C}}\right], \notag\end{aligned}$$

where $I_{0}$ is the intensity of light at the input of the Mach-Zehnder interferometer; $\varphi_{C}$, $\varphi_{A}$ and $\varphi_{B}$ are the phase increments of light fields transmitted through mirrors C, A, and B, respectively, at their motion from the input of the Mach-Zehnder interferometer to the detector; $\delta_{C}$, $\delta_{E}$, $\delta_{A}$, $\delta_{B}$, and $\delta_{F}$ are the vertical shifts of beams of light because of vibrations of mirrors C, E, A, B, and F, respectively; and $f(x,y)$ is the amplitude profile of the incident beam of light along the horizontal and vertical directions normalized as $\iint f^{2}(x,y)dxdy=1$.

The QCD records the difference $D$ between the integral intensities of light from regions $y>0$ and $y<0$: $$D\equiv\iint_{y>0}\left|E(x,y)\right|^{2}dxdy-\iint_{y<0}\left|E(x,y)\right|^{2}dxdy.\label{eq:detec_diff}$$

Since all deviations are small ($\delta\ll1$), $f(x,y-\delta)$ can be represented in the form of the sum of the first two terms of the Taylor series, which are the unmodulated initial profile and the term linearly modulated in $\delta$. In this case, the modulated difference between the integral intensities of light given by Eq. can be represented in the form

$$\begin{aligned}
\label{eq:detec_diff_expl}
D&=\dfrac{2}{9}I_{0}\left\{ \delta_{C}+\left[\delta_{A}+\delta_{B}+2\left(\delta_{E}+\delta_{F}\right)\right]\left[1+\cos\left(\phi_{AB}\right)\right]+\right. \notag \\ &+\left[\delta_{A}+\delta_{C}+\delta_{E}+\delta_{F}\right]\cos\left(\phi_{AC}\right)+\\ &+\left.\left[\delta_{B}+\delta_{C}+\delta_{E}+\delta_{F}\right]\cos\left(\phi_{BC}\right) \right\} \int{f^{2}(x,0)dx},  \notag\end{aligned}$$

$$\phi_{ij}\equiv \varphi_{i}-\varphi_{j}.$$ When the profile of the beam of light is symmetric, i.e., $f(x,-y)=f(x,y)$, which corresponds to the conditions of the experiment reported in [@Danan2013], Eq. is the total signal on the QCD photodetector. In the case of an arbitrary profile, there is also an additional constant component. The first line in the curly brackets in Eq. is the sum of self-interferences of the modulated and unmodulated parts of all beams of light moving through different arms of the composite Mach- Zehnder interferometer. Here, the coefficient 2 of the term $\left(\delta_{E}+\delta_{F}\right)$ appears because mirrors E and F modulate beams of light moving through both arms of the inner Mach-Zehnder interferometer. The second and third lines in the curly brackets in Eq. are due to the interference of beam C with beams A and B, respectively.

In the case of constructive coherence of beams of light passing through mirrors A, B, and C, i.e., $\varphi_{A}=\varphi_{B}=\varphi_{C}$, the expression in the curly brackets in Eq. becomes the form $$\left\{ \cdots\right\} =3\left\{ \delta_{A}+\delta_{B}+\delta_{C}+2\left(\delta_{E}+\delta_{F}\right)\right\} .\label{eq:constr_interf}$$

The power spectrum of the difference signal given by Eqs. and rather than the signal itself was detected in the experiment reported in [@Danan2013]. For this reason, according to Eq. , the intensity of spectral components of the signal given by Eq. that correspond to vibrations of mirrors E and F is four times larger than the others (see Fig. 2a in [@Danan2013]).

When the phase $\varphi_{B}$ is changed by $\pi$ in the inner Mach-Zehnder interferometer ($\varphi_{A}=\varphi_{C}=\varphi_{B}\pm\pi$), which corresponds to the complete destructive interference of light at its output, the expression in the curly brackets in Eq. becomes the form $\left\{ \cdots\right\} =\left\{ \delta_{A}-\delta_{B}+\delta_{C}\right\}$. In this case, the intensities of spectral components of the signal in the detector that are due to vibrations of mirrors A, B, and C will be identical (see Fig. 2b in [@Danan2013]). At the same time, the spectral components at the frequencies of vibrations of mirrors E and F are absent. The authors of [@Danan2013] interpreted this extraordinary result as the discontinuity of trajectories of photons, which are as if present in the inner Mach-Zehnder interferometer but are absent at its input and output. The authors believe that a simple and intuitively clear explanation of such an extraordinary result is possible only within the two-state vector formalism of quantum mechanics [@Aharonov1964; @Aharonov1990]. Within the framework of this formalism, it is asserted that photons are present only where there is both a direct quantum wave (from the source) and an inverse one (from the detector) exist (see Fig. 3 in [@Danan2013]).

If light is blocked immediately behind mirror F, the detector naturally records a signal only from light in the lower arm of the outer Mach-Zehnder interferometer. In this case, only the last term remains in the square brackets in Eq. and only the first term in the curly brackets in Eq. is nonzero: $\left\{ \cdots\right\} =\left\{
\delta_{C}\right\} $.

![(Color online) Transverse distribution of the amplitude of light at the output of the inner Mach-Zehnder interferometer (in front of mirror F, see Fig. \[fig:setup\]), tuned to complete destructive interference of rectangular beams with the width $\mathrm{d}$ passing through it. (a) Mutual transverse arrangement of the beam passing through mirrors B and A (the case of the deviation of only one beam by $\delta_{A}$ caused by deflection of mirror A is shown for simplicity). (b) Result of their destructive interference. The vertical arrows mark the instantaneous positions of electric-field vectors of the corresponding beams.[]{data-label="fig: profile_on_mirror_F"}](Fig2a_e "fig:"){width="\columnwidth"}\[fig: profile\_on\_mirror\_F\_a\] ![(Color online) Transverse distribution of the amplitude of light at the output of the inner Mach-Zehnder interferometer (in front of mirror F, see Fig. \[fig:setup\]), tuned to complete destructive interference of rectangular beams with the width $\mathrm{d}$ passing through it. (a) Mutual transverse arrangement of the beam passing through mirrors B and A (the case of the deviation of only one beam by $\delta_{A}$ caused by deflection of mirror A is shown for simplicity). (b) Result of their destructive interference. The vertical arrows mark the instantaneous positions of electric-field vectors of the corresponding beams.[]{data-label="fig: profile_on_mirror_F"}](Fig2b_e "fig:"){width="\columnwidth"}\[fig: profile\_on\_mirror\_F\_b\]

If light is blocked between mirror C and polarization splitter PBS2, the last term in the square brackets in Eq. is absent and only the second term in the curly brackets in Eq. is nonzero: $\left\{ \cdots\right\} =\left\{
\left[\delta_{A}+\delta_{B}+2\left(\delta_{E}+\delta_{F}\right)\right]\left[1+\cos\left(\phi_{AB}\right)\right]\right\}
$. In the case of the complete destructive interference of light at the output of the inner Mach-Zehnder interferometer ($\varphi_{A}=\varphi_{C}=\varphi_{B}\pm\pi$), the expression in the curly brackets vanishes; i.e., the intensities of spectral components of the signal of the detector that are caused by vibrations of all mirrors are zero (see Fig. 2ñ in [@Danan2013]).

Such paradoxical results in the complete destructive interference of light ($\varphi_{A}=\varphi_{C}=\varphi_{B}\pm\pi$) at the output of the inner Mach-Zehnder interferometer can be simply and clearly explained within both the wave theory of light and traditional quantum mechanics.

Indeed, spectral components with the frequency of vibration of mirror E are absent because the vibration of this mirror identically displaces beams of light passing through mirrors A and B. For this reason, interference of these beams at the output of the inner Mach-Zehnder interferometer does not change at the displacement of mirror E. Consequently, light at the output of this Mach-Zehnder interferometer is absent both in the case of immobile mirror E and in the case of its vibrations.

![(Color online) Transverse distribution of the amplitude of light in front of the photodetector. For simplicity, the direction of the polarization of fields in this figure and in Fig. \[fig: profile\_on\_detector\_final\] remains unchanged (in reality, polarizer P2 changes this direction). (a) Mutual transverse arrangement of the beams transmitted through mirrors C and F on the photodetector. The latter mirror additionally displaces a beam by $\delta_{F}$. (b) Result of their destructive interference. The shaded symmetric regions do not contribute to a signal of the quad-cell photodetector QCD.[]{data-label="fig: profile_on_detector"}](Fig3a_e "fig:"){width="\columnwidth"}\[fig: profile\_on\_detector\_a\] ![(Color online) Transverse distribution of the amplitude of light in front of the photodetector. For simplicity, the direction of the polarization of fields in this figure and in Fig. \[fig: profile\_on\_detector\_final\] remains unchanged (in reality, polarizer P2 changes this direction). (a) Mutual transverse arrangement of the beams transmitted through mirrors C and F on the photodetector. The latter mirror additionally displaces a beam by $\delta_{F}$. (b) Result of their destructive interference. The shaded symmetric regions do not contribute to a signal of the quad-cell photodetector QCD.[]{data-label="fig: profile_on_detector"}](Fig3b_e "fig:"){width="\columnwidth"}\[fig: profile\_on\_detector\_b\]\

On the contrary, at the displacement of any of mirrors A and B, destructive interference of beams of light from these mirrors at the output of the inner Mach-Zehnder interferometer is incomplete because the amplitudes of interfering beams are no longer equal to each other at any point of the cross section of the outgoing beam. More precisely, in the case of $\delta\ll 1$ and a symmetric profile of beams, this destructive difference is an antisymmetric function of $y$ in the first order in $\delta$. In turn, a change in this antisymmetric function caused by small vibrations of mirror F has the second order of smallness and is a symmetric function of $y$. A fraction of light incident on the photodetector, which is modulated with the frequency of deviations of mirror F, is the result of interference of the modulated fraction of light from mirror F and the unmodulated fraction of light from mirror C. Both of these components are symmetric functions of $y$. Consequently, their interferential sum is also symmetric with an accuracy up to the third order in $\delta$. This is the reason why the QCD does not record it. In the case of an arbitrary profile of the beam, the signal from mirror F is not detected with an accuracy of the second order of smallness.

![(Color online) Detected part of the distribution of amplitude of light on the QCD detector. The amplitude obviously depends only on the deflection of mirror A. The deflection of mirror F does not affect a detected signal.[]{data-label="fig: profile_on_detector_final"}](Fig4_e){width="\columnwidth"}

The signal detected in this case is due to both the modulated part of the light from mirror C and the result of interference of its unmodulated part fraction with the light coming from the inner Mach-Zehnder interferometer and being the differential of the profile of the initial light beam. The result of this interference is equivalent (with an accuracy up to the second order regard to $\delta$) to the initial unmodulated beam of light from mirror C displaced at the QCD on the shift ${\delta}_A$ $-{\delta}_B$ of the beams of light transmitted through the inner Mach-Zehnder interferometer.

A similar physically transparent interpretation can be given for the absence of signals from all mirrors at $\varphi_{A}=\varphi_{B}\pm\pi$ and at blocking of light between mirror C and polarization splitter PBS2. The absence of a signal from mirror C is obvious and the absence of signals from mirrors E and F was discussed above. Signals from mirrors A and B are absent because the amplitude profile of light in this case is an antisymmetric function of $y$ (with an accuracy of the second order in $\delta$) not only at the output of the inner Mach-Zehnder interferometer but also on the QCD. At the same time, the QCD measures the difference of integral intensities given by Eq. , which vanishes in this case (with an accuracy of the third order in $\delta$). In the case of an arbitrary profile of the beam, any signal from mirrors A and B is not detected with an accuracy of the second order of smallness.

An important feature of discussed signals should be emphasized. Such weak measurements of the presence of photons in various places of the embedded Mach-Zehnder interferometer are completely due to the interference between the modulated and unmodulated parts of beams of light, because modulated parts of the amplitude given by Eq. , as well as detected signals given by Eq. , are proportional to small deviations $\delta$. In the general case, each modulated part interferes with all unmodulated parts.

The marked “interference” feature is an inherent part of any weak measurements [@Duck1989]. To illustrate the indicated interference features, it is convenient and instructive to consider beams of light with a symmetric step profile where the amplitude of light is constant at any point of the cross section of the beam. A remarkable feature of such a profile is that all above results are exact for it if the total deviation of the beam is no more than its half-width. The reason is that the difference between the displaced and initial beams with such a profile is an exactly antisymmetric function on ($y-\delta/2$) at an arbitrary transverse deviation less than the half-width of the beam. Figures \[fig: profile\_on\_mirror\_F\] – \[fig: profile\_on\_detector\_final\] show the amplitude profiles of beams of light in various places of the embedded Mach-Zehnder interferometer and on the QCD in the case of complete destructive interference of light at the output of the inner Mach-Zehnder interferometer (for clarity, it is accepted that mirror C is at rest). It is clearly seen (in Figs. \[fig: profile\_on\_detector\] and \[fig: profile\_on\_detector\_final\]) that a nonzero integral difference signal specified by Eq. appears only at the interference of light from the inner Mach-Zehnder interferometer and unmodulated light from mirror C. A signal from mirror F does not appear for the same reason that was discussed above for an arbitrary symmetric profile of the beam and small deviations $\delta$. Indeed, the difference between the amplitudes at the output of the inner Mach-Zehnder interferometer and behind mirror F is a symmetric function of ($y-\delta_{A}/2-\delta_{F}/2$). The detector records the result of the interference of this difference and the symmetric unmodulated component of light from mirror C. The integral contribution of this interference vanishes if the total deviation $\delta_{A}+\delta_{F}$ is no more than the half-width of the beam.

\[sec:Conclusions\]CONCLUSIONS
==============================

To summarize, it has been shown that paradoxical results of the experiment reported in [@Danan2013], which were interpreted by the authors as the *discontinuity* of possible trajectories of photons, can be simply and clearly explained within the traditional concepts of the wave and quantum natures of light, *which are based on the continuity of all possible paths of photons*. It has been established that extraordinary signals detected in [@Danan2013] are completely due to the perturbation of destructive interference of light in the inner Mach-Zehnder interferometer.

It has been shown that the absence of signals from mirrors at the input and output of the inner Mach-Zehnder interferometer at its destructive tuning is due not to *discontinuity of trajectories of light (photons)*, as was stated in \[8\], but to *the used method of detection of paths of photons* (harmonic deviation of the mirrors of the interferometer for a small deviation of trajectories of light). The modification of the scheme can reveal “traces” of photons at the indicated places. In particular, the path of photons through mirror E can be detected if this mirror does not deflect light but modulates its polarization and a birefringent plate and a polarizer are placed in one of the arms of the inner Mach-Zehnder interferometer.

Finally, the performed analysis has indicated that it is unnecessary to introduce a new physical concept of *discontinuity* of possible trajectories of photons, which was proposed by the authors in the discussed work on the basis of the results of the performed experiment and their interpretation within their treatment of the two-state vector formalism.

I am grateful to V.A. Sorokin for stimulating discussions. This work was supported by the Government of the Russian Federation (project no. 01201372518, Program of Basic Research for the State Academies of Sciences).

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