PHYSICAL REVIEW A 89, 033825 (2014)

Interpreting a nested Mach-Zehnder interferometer with classical optics

Pablo L. Saldanha*

Departamento de Fısica,Universidade Federal de Minas Gerais, Caixa Postal 701, 30161-970, Belo Horizonte, MG, Brazil(Received 28 December 2013; published 13 March 2014)

In an recent work with the title “Asking Photons Where They Have Been,” Danan et al. experimentallydemonstrate an intriguing behavior of photons in an interferometer [Phys. Rev. Lett. 111, 240402 (2013)]. Intheir words: “The photons tell us that they have been in the parts of the interferometer through which they couldnot pass.” They interpret the results using the two-state vector formalism of quantum theory and say that, althoughan explanation of the experimental results in terms of classical electromagnetic waves in the interferometer ispossible (and they provide a partial description), it is not so intuitive. Here we present a more detailed classicaldescription of their experimental results, showing that it is actually intuitive. The same description is valid forthe quantum wave function of the photons propagating in the interferometer. In particular, we show that it isessential that the wave propagates through all parts of the interferometer to describe the experimental results. Wehope that our work helps to give a deeper understanding of these interesting experimental results.

DOI: 10.1103/PhysRevA.89.033825 PACS number(s): 42.25.Hz, 03.65.Ta, 42.50.−p

The wave-particle duality is one of the most intriguingfeatures of quantum mechanics. Quantum entities may behaveas particles, as waves, or as a strange combination of thesepossibilities. The affirmation that a quantum entity, such asan electron or a photon, is either a particle or a wave willalways imply a contradiction with experiments. So we can saythat these entities are not particles nor waves, but very strange“things” that we do not understand in an intuitive way. Thisduality is explicitly manifested, for instance, in delayed-choiceexperiments [1–3] and in quantum erasers [4–7]. Recentlydelayed-choice experiments were performed with quantumbeam splitters [8–12] following the proposal of Ref. [13],showing even more intriguing behaviors.

In an interesting recent work, Danan et al. demonstratedanother experiment in which the wave-particle duality playsan important role in the nonintuitive experimental results [14].This experiment was inspired by recent discussions about thepast of a quantum particle in an interferometer [15]. In the ex-perimental arrangement, there is an inner interferometer in oneof the arms of a large interferometer [14]. They demonstratedthat even when the inner interferometer is adjusted to producedestructive interference to the direction of the output port ofthe large interferometer, a tilting of some mirrors in the innerinterferometer affects the average detection position of photonsat the exit of the large interferometer. It is in this sense thatthe authors say that the photons “have been in the parts of theinterferometer through which they could not pass,” since thedependence of the average detection position of the photons onthe tilting of the mirrors shows that the photons have beenin that arm, while the destructive interference in the innerinterferometer should imply that the detected photons couldnot have passed through that arm. They also showed that thetilting of some other mirrors in this same arm of the largeinterferometer does not affect the average detection positionof the photons at the exit. Danan et al. argue that this is astrange behavior, since the dependence on the tilting of somemirrors shows that the photons have been in that arm, while theindependence on the tilting of the other mirrors would imply

*saldanha@fisica.ufmg.br

that they have not been in that arm. The authors provide anexplanation using the two-state vector formalism of quantumtheory [16,17] and claim that this would be the most intuitiveexplanation of the phenomenon. They also discuss that anexplanation in terms of classical electromagnetic waves iscertainly possible, since the experiment was done with a stronglaser beam, and provide a partial classical description of theirresults [14].

Here we provide a more detailed classical description of theexperiments of Ref. [14], showing that it is actually intuitive.The same description is valid for the propagation of the wavefunction of the photons in the interferometer. In particular,we show that the wave (be it classical or quantum) must passthrough both arms of the large interferometer to explain theexperimental results, and that the fact that some mirrors affectthe average photon detection position and some do not can beunderstood in terms of wave interference in a simple way. Sowe hope to give a contribution for a better understanding ofthe interesting results of Ref. [14] with this work.

The experimental setup of Ref. [14] is depicted in Fig. 1. Alarge interferometer, with entrance and exit beam splitters BS1

and BS4, has an inner (nested) interferometer with entranceand exit beam splitters BS2 and BS3 in one of its arms.The modulus of the reflection and transmission coefficientsof BS1 and BS4 are

√23 and

√13 respectively, while the

modulus of the reflection and transmission coefficients ofBS2 and BS3 are 1/

√2. The mirrors A, B, C, E, and F

vibrate around their horizontal axes, causing an oscillationof the vertical position of beam reaching the detector D. Theamplitude of this oscillation is much smaller than the beamdiameter, and each mirror vibrates at a different frequency.The detector D is a quad-cell photodetector and the registeredsignal is proportional to the difference between the numberof photons detected in the upper and lower cells. The powerspectrum of the measured signal contains peaks at the vibrationfrequencies of the mirrors whose angular positions affectthe beam position at D. Since the vibration frequencies aredifferent, the influence of each mirror on the beam position atD can be extracted from the data.

In the experimental situation of Fig. 1(a), the innerinterferometer is aligned to generate constructive interference

1050-2947/2014/89(3)/033825(5) 033825-1 ©2014 American Physical Society

PABLO L. SALDANHA PHYSICAL REVIEW A 89, 033825 (2014)

E

F

A

B

C

BS1

BS2

BS3

BS4

D

(a)

E

F

A

B

C

BS1

BS2

BS3

BS4

D

(b)

E

F

A

B

C

BS1

BS2

BS3

BS4

D

(c)

FIG. 1. (Color online) Interferometer setup. A laser beam prop-agates through a complex interferometer and is detected by aquad-cell photodetector D in one of its exits. The registered signalis proportional to the difference between the number of photonsdetected in the upper and lower cells. The large interferometer,with entrance and exit beam splitters BS1 and BS4, has an innerinterferometer with entrance and exit beam splitters BS2 and BS3

in one of its arms. The mirrors A, B, C, E, and F vibrate aroundtheir horizontal axes, causing an oscillation of the vertical positionof beam reaching the detector. Each mirror has a different oscillationfrequency and the power spectrum of the signal is obtained. (a) Theinner interferometer is aligned to generate constructive interferenceto the direction of mirror F and the large interferometer is alignedto generate constructive interference to the direction of the detectorD. (b) The inner interferometer is aligned to generate destructiveinterference to the direction of mirror F . (c) The inner interferometeris aligned to generate destructive interference to the direction ofmirror F and the arm of mirror C is blocked.

to the direction of mirror F and the large interferometer isaligned to generate constructive interference to the directionof the detector D. The experimental results show that theangular position of all mirrors A, B, C, E, and F affectthe beam position at D [14]. The interesting and surprising

result happens in the situation of Fig. 1(b), where the innerinterferometer is aligned to generate destructive interferenceto the direction of mirror F and the large interferometer isaligned to generate constructive interference to the direction ofthe detector D. In this case the experimental results show thatthe angular positions of mirrors A, B, and C affect the beamposition at D, but the angular positions of mirrors E and F donot [14]. This result seems to imply that the photons interactwith mirrors A and B without interacting with mirrors E andF , what is, of course, impossible. In the experimental situationof Fig. 1(c), we have the same picture of Fig. 1(b) but the armof mirror C is blocked. In this case, the angular positions ofnone of the mirrors affect the beam position at D [14].

Let us start our classical description of the experiments withthe situation of Fig. 1(b), which is the most interesting one. Thephotons come from a Gaussian laser beam, such that their am-plitude can be written as �(x,y,z,t) ∝ e−(x2+y2)/W 2

cos(kz −ωt) in a region close to its waist, where ω is the angularfrequency of light and k = ω/c the modulus of the wave vector[18]. This amplitude may refer to a component of the electricor magnetic field of the beam, or to a component of the real orimaginary part of the Bialynicki-Birula–Sipe wave functionof the photons in the beam [19–21]. If the experiment isperformed with electrons, neutrons, or other massive particles,the amplitude would be of the real or imaginary part of theSchrodinger wave function of these particles. So our treatmentis valid for a classical electromagnetic wave and for quantumparticles. Since the beam is in the paraxial regime, we canwrite its angular spectrum in terms of the components kx andky of the wave vectors in the xy plane as the Fourier transformof the amplitude [18], obtaining

�T (kx,ky) = �(kx)�(ky), with �(ki) = Ne−k2i /σ

2, (1)

with σ = 2/W and N being the normalization factor of thecorresponding wave. On this way, the beam amplitude can bewritten as

�(x,y,z,t) ∝ Re

[ ∫dkx

∫dky�T (kx,ky)

×ei(kxx+kyy+

√k2−k2

x−k2yz−ωt

)], (2)

in a decomposition in terms of the wave vectors, the paraxialregime meaning that kx � k and ky � k for all non-negligiblevalues of �T (kx,ky) [18]. From now on we will consideronly the y dependence of the beam state, �(ky), since thex dependence does not change with the tilting of the mirrors.

Considering the z direction as the propagation direction ofthe beam in each part of the interferometer for simplicity, ifmirror i is tilted, all wave vector components of the beamchange with the reflection, such that the state of the beam afterthe reflection changes like �(ky) → �i = �(ky − κi), where�i refers to the state of the beam just after mirror i and κi isproportional to the inclination amount. With this notation, thestate of the beam just after mirror F is

�F = 1√6

[�(ky − κE − κA − κF )

− �(ky − κE − κB − κF )]. (3)

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INTERPRETING A NESTED MACH-ZEHNDER . . . PHYSICAL REVIEW A 89, 033825 (2014)

FIG. 2. (Color online) Beam state after mirror F , given by �F

from Eq. (3), for different values of κA, κB , κE , and κF . The verticalscale in the plots is arbitrary, but is the same in all plots.

In Fig. 2 we plot �F for different values of κA, κB , κE , andκF . The vertical scale in the plots is arbitrary, but is the samein all plots. When κA = κB , as in Fig. 2(a), no light goes tomirror F . This is expected, since there is perfect destructive

FIG. 3. (Color online) Two Gaussian functions �A and �B withthe same amplitude and width, but different centers.

interference in the inner interferometer in this case. WhenκA − κB > 0, as in Fig. 2(b), there are almost no wave vectorswhose y components are close to 0, since there is an almostperfect destructive interference. But for wave vectors whose y

components are close to ±σ , we have some amplitude. Thiscan be understood with the help of Fig. 3, that represents thebeam states that come from mirrors A (�A) and B (�B) forexaggerated values of κA (positive) and κB (negative). Whenwe compute �F ∝ �A − �B for the graphs of Fig. 3, it is clearthat a curve like Fig. 2(b) is obtained. When κA − κB < 0, as inFig. 2(c), there is an inversion of the positive and negative partsof the amplitude. The amplitude maximum is proportional to|κA − κB |. The values of κE and κF almost do not change theamplitude of the wave that is reflected by mirror F , as canbe seen comparing Figs. 2(b) and 2(d), that have the samevalues for κA and κB and different values for κE and κF . Thereis no perceptible difference between Figs. 2(b) and 2(d). Thereason is that the angular tilting of mirrors E and F does notchange the interference behavior of the inner interferometer,they only displace the wave vectors distribution of the beamafter mirror F . This displacement is very small, since in theplots of Fig. 2 and in the experiments of Ref. [14] we haveκi at least three orders of magnitude smaller than σ , such thatthey do not significantly influence the state of the beam aftermirror F .

The beam that goes in the direction of the detector D isa superposition of the beam that comes from mirror C, withstate �C = �(ky − κC)/

√3, with the beam that comes from

mirror F , with state given by �F from Eq. (3). �F has a muchsmaller amplitude than �C due to the destructive interferencein the inner interferometer. We can see that, when κA > κB ,as in Figs. 2(b) and 2(d), the superposition of �C with �F

will increase the components of �C with positive ky anddecrease the components with negative ky . This results in aup displacement of the resultant beam in the far field. WhenκA < κB , as in Fig. 2(c), the opposite happens and we havea down displacement of the resultant beam in the far field.And when κA = κB , as in Fig. 2(a), there is no displacementof the beam. Since κE and κF do not appreciably change thestate �F from Eq. (3), they do not affect the resultant beamdisplacement in the far field.

The amplitude of the resultant beam in the positionof the detector D, considered to be in the far field, is

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PABLO L. SALDANHA PHYSICAL REVIEW A 89, 033825 (2014)

proportional to the Fourier transform of the field amplitudein the interferometer, thus being proportional to the angularspectrum of the field at the exit of the interferometer [18]. Soit is proportional to �D(ky) = 1/

√3 �C(ky) + √

2/3 �F (ky)with the substitution ky → αy, α being a constant. The detectorsignal S, being proportional to the difference between the lightintensity in the upper and lower cells, is then proportional tothe integral of the modulus squared of the angular spectrumof the resultant beam at the exit of the large interferometer forky > 0 minus the same integral for ky < 0. So we obtain

S ∝∫ ∞

0

∣∣�D(ky)∣∣2

dky −∫ 0

−∞|�D(ky)|2dky, (4)

with

�D(ky) = 13 [�(ky − κC) + �(ky − κE − κA − κF )

−�(ky − κE − κB − κF )]. (5)

Since we are considering the limit κi � σ , we can write

�(ky − κi) ≈ �(ky) − κi

∂�(k)

∂k

∣∣∣∣k=ky

. (6)

Substituting this approximation for the terms on the right sideof Eq. (5), we obtain

�D(ky) ≈ 1

3

[�(ky) − (κC + κA − κB)

∂�(k)

∂k

∣∣∣∣k=ky

]

≈ 1

3�(ky − (κC + κA − κB)). (7)

According to Eq. (1), the function �D(ky) from the above equa-tion is symmetric around the maximum at ky = κC + κA − κB .So, considering initially that we have κC + κA − κB > 0, theintegral of the first term on the right side of Eq. (4) from2(κC + κA − κB) to ∞ cancels the second integral in thisequation. In this case we have

S ∝∫ 2(κC+κA−κB )

0|�D(ky)|2dky. (8)

Since the function �D(ky) is approximately constant on theabove integral, we have

S ∝ κC + κA − κB. (9)

It is easy to see that the above equation is also valid forκC + κA − κB < 0. So when the inner interferometer causes adestructive interference for the light propagation in the direc-tion of mirror F , as in Fig. 1(b), the vibrations of the mirrorsA, B, and C make the signal S oscillate, while the vibrationsof mirrors E and F do not. For this reason, if all mirrors vibrate

at different frequencies, the power spectrum of the signal willpresent peaks at the vibration frequencies of mirrors A, B, andC, but not at the vibration frequencies of mirrors E and F .This is shown in the experiments of Ref. [14].

Considering now the situation depicted in Fig. 1(a), wherethere is constructive interference in the inner interferometerfor the photons to go out in the direction of mirror F , thecalculations are basically the same. But now the angularspectrum of the resultant beam at the exit of the interferometercan be written as

� ′D(ky) = 1

3 [�(ky − κC) + �(ky − κE − κA − κF )

+�(ky − κE − κB − κF )]. (10)

Substituting in Eq. (4) in the limit κi � σ and following thesame steps as before in the calculation, we obtain

S ′ ∝ κC + κA + κB + 2κE + 2κF . (11)

Now the vibrations of all mirrors make the signal S oscillate,such that the vibration frequencies of all mirrors should bein the power spectrum. Since the tilting of mirrors E andF causes twice the beam displacement than a tilting of thesame amount of the other mirrors, the power spectrum at thevibration frequencies of mirrors E and F should have a peakfour times the value of the peaks at the vibration frequencies ofmirrors A, B, and C, since the power spectrum is proportionalto the square of the oscillation amplitude at each frequency.This is shown in the experiments of Ref. [14].

Considering the situation depicted in Fig. 1(c), all lightthat goes to the detector comes from mirror F . We can seethat, according to the plots in Fig. 2, the light intensity willbe always (almost) symmetric around the origin, such that thevibration of the mirrors should not affect the signal S and nopeak should be observed in the power spectrum. This is alsoshown in the experiments of Ref. [14].

To summarize, we have presented a description of theexperiments of Ref. [14] using a classical description of lightpropagating in the interferometer. The same description is validfor the propagation of the wave function of quantum particlesin the interferometer. This interpretation, which clearly showsthat it is essential that the wave propagates through all partsof the interferometer to describe the experimental results, iscomplementary to the one using the two-state vector formalismof quantum theory [16,17] presented in Ref. [14]. We hope thatour results can give more physical insight to the behavior ofphotons (and electromagnetic waves) in this interesting kindof interferometer.

This work was supported by the Brazilian agencies CNPqand PRPq/UFMG.

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