112 Linear and angular momenta of photons in the context of which path experiments of quantum mechanics Masud Mansuripur

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Linear and angular momenta of photons in the context of “which path”

experiments of quantum mechanics

Masud Mansuripur

James C. Wyant College of Optical Sciences, The University of Arizona, Tucson

[Published in the Proceedings of SPIE 12198, Optical Trapping and Optical

Micromanipulation XIX, 1219807 (3 October 2022); doi: 10.1117/12.2632866]

Abstract. In optical experiments involving a single photon that takes alternative paths through an

optical system and ultimately interferes with itself (e.g., Young’s double-slit experiment, Mach-

Zehnder interferometer, Sagnac interferometer), there exist fundamental connections between the

linear and angular momenta of the photon on the one hand, and the ability of an observer to

determine the photon’s path through the system on the other hand. This paper examines the

arguments that relate the photon momenta (through the Heisenberg uncertainty principle) to the

“which path” (German: welcher Weg) question at the heart of quantum mechanics. We show that

the linear momenta imparted to apertures or mirrors, or the angular momenta picked up by

strategically placed wave-plates in a system, could lead to an identification of the photon’s path

only at the expense of destroying the corresponding interference effects. We also describe a

thought experiment involving the scattering of a circularly-polarized photon from a pair of small

particles kept at a fixed distance from one another. The exchange of angular momentum between

the photon and the scattering particle in this instance appears to provide the “which path”

information that must, of necessity, wipe out the corresponding interference fringes, although the

fringe-wipe-out mechanism does not seem to involve the uncertainty principle in any obvious way.

1. Introduction. In the historic Bohr-Einstein debates concerning the foundations of quantum

mechanics,1 the Young double-slit experiment of classical optics played a central role. In a nutshell,

Einstein argued that the passage of a single photon could be attributed to one slit or the other, since

the mechanical momentum picked up by the plate housing the slits contains the information needed

to identify the path taken by the photon through the system. Bohr’s counter argument was that

acquiring the “which path” information by way of monitoring the plate’s momentum would, in

accordance with Heisenberg’s uncertainty principle, perturb the position of the plate, thus wiping

out the expected interference fringes.2 In other words, acquisition of the “which path” information,

while revealing the particle-like nature of the photon, would blur the anticipated interference fringes

to the extent that all evidence for the photon’s wave-like behavior will be lost.

The goal of the present paper is to examine a few variations on the theme of the Bohr-Einstein

argument, and to explore the extent to which Heisenberg’s uncertainty principle needs to be relied

upon in the context of “which path” experiments. Feynman has emphasized that, in the absence of

“which path” information, it is the (complex) probability amplitudes of an event that must be added

together, whereas the existence of such information, even in principle, compels one to explain the

outcome of an experiment by directly adding the probabilities associated with individual paths that

could have led to a specific event.3 Thus, wave-like behavior is exhibited when the probability

amplitudes are required to be added together, whereas particle-like behavior emerges in situations

where the probability of occurrence of a multi-path event turns out to be the sum of the probabilities

of its individual paths. We will see in the next section how the availability of “which path”

information (in the form of a photon’s polarization state in the double-slit experiment) wipes out the

interference fringes, and also how the “erasure” of this information causes the fringes to reappear.

A photon of frequency  propagating in a given direction in free space, say, along the -axis, is

a wavepacket of energy  and linear momentum ( 

) in the number state |1; here  is the

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reduced Planck constant and  is the speed of light in vacuum.4 In addition, a circularly-polarized

photon carries angular momentum in the amount of ±, with the plus or minus sign depending on

the sense (or helicity) of its circular polarization. While deflection of a photon’s trajectory at a slit,

or its reflection from a conventional mirror, involves an exchange of linear momentum, its passage

through a birefringent plate could entail an exchange of angular momentum as well.5 We will

discuss examples of systems where the angular momentum transferred to a wave-plate or to a

scatterer can be relied upon to provide the desired “which path” information.

The organization of the paper is as follows. In Sec.2, we examine the canonical Young’s

double-slit experiment6,7 in the special case when a single photon passes through the pair of slits

and interferes with itself at the observation plane. Reconstructing Bohr’s argument in his debates

with Einstein, we show how a straightforward application of Heisenberg’s uncertainty principle

confirms the incompatibility of observing wave-like behavior by the photon (i.e., appearance of

interference fringes) with particle-like behavior (namely, acquisition of information about the slit

through which the photon has passed).1 We proceed to extend the analysis to cases where a  phase-

shifter is placed in one of the slits, or when a birefringent window is placed in a slit, to shed

additional light on the nature of Bohr’s complementarity principle.

Section 3 is devoted to an analysis of a single photon passing through a Mach-Zehnder

interferometer,6,7 where the probability amplitudes associated with its passage through one arm or

the other of the device combine to reveal the possibility of observing the effects of interference at

the output ports of the interferometer. It will be seen once again that the acquisition of “which path”

information (by monitoring the mechanical momenta picked up by the mirrors within the device)

disturbs the optical path lengths just enough to eliminate the possibility of observing the anticipated

interference. An alternative method of acquiring the “which path” information (based on placing a

quarter-wave plate in each arm of the interferometer, then monitoring their angular momenta before

and after the passage of the photon) will be shown to similarly ruin the observability of interference.

In the case of the Sagnac interferometer8-10 examined in Sec.4, the mechanical momenta

imparted to the mirrors by a passing photon (or any angular momentum picked up by the device as a

whole) do not contain the desired “which path” information. However, a pair of properly oriented

and strategically placed quarter-wave plates (QWPs) in the system can, in principle, pick up angular

momenta in the amount of ± from their interaction with the passing photon. Once again,

application of the uncertainty principle reveals the impossibility of acquiring “which path”

information without disturbing the necessary conditions for observing the interference effects

associated with the photon’s wave-like behavior.

In Sec.5, we describe a thought experiment involving a right-circularly-polarized (RCP) photon

being scattered from a pair of small particles that are kept at a fixed distance from each other. The

existence of a nonzero probability for the scattered photon to become left-circularly-polarized

(LCP) indicates that the difference  between the angular momenta of the incident and scattered

photons must be picked up by one of the two scatterers. The availability of the “which path”

information in this instance does not appear to disturb the state of the scattered photon in ways that

would prohibit the formation of interference fringes at the observation plane. While invoking the

Heisenberg uncertainty principle in this case does not bring about the disturbances needed to

prevent fringe formation, the entanglement of the state of the pair of particles with that of the

scattered photon guarantees that interference will not occur and that, therefore, the core principle of

wave-particle duality at the heart of quantum mechanics remains inviolate.

The paper closes in Sec.6 with a summary and a few concluding remarks. An Appendix

provides a concise derivation of the Heisenberg uncertainty relation between a pair of non-

commuting observables.

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2. Young’s double-slit experiment. Figure 1 shows a slight variation on the Young double-slit

arrangement,6,7 where the incident light is a single-photon wavepacket in the number state |1,

having frequency , -vector =( 

), and linear polarization 

=

. (In terms of the vacuum

wavelength , the magnitude of  may be written as = 2 

.) The slits of width  are carved

into two separate plates, which are symmetrically positioned in the -plane such that the center-to-

center distance between the slits along the -axis is . The splitting into two separate parts of the

plate that houses the slits is intended to simplify the forthcoming analysis of momentum transfer

from the photon to the plates for purposes of identifying the slit through which the photon has

passed. Let the -field amplitude of the light transmitted through the pair of slits be written as

(,= 0)=rect( 

)[(+ ½)+(  ½)], (1)

where () is Dirac’s delta-function, and the standard function rect() equals 1.0 when ||< ½,

zero when ||> ½, and ½ when ||= ½; the asterisk represents the convolution operation, and

the Fourier transform of rect() is rect()d



 = sin() ()=sinc(). Thus, the

Fourier transform of the emergent -field amplitude immediately after the slits is given by



()=()d



 = 2 sinc()cos(). (2)

The product  of the incident wavelength  and the Fourier variable  appearing in Eq.(2)

represents sin  for an emergent plane-wave (corresponding to a geometric-optical ray) at an angle

 relative to the -axis; see Fig.1. Thus, in the observation plane in the far field, the ±1st peaks of

the fringe pattern appear at sin = ±  

. To ensure that these fringes do not get blurry (or washed

out), the positions of the two plates that house the slits must each have an uncertainty  along the

-axis much less than , say,  = with   0.1. The visibility of the fringes that are further

away from the center of the observation plane requires progressively smaller values of .

A photon arriving at the center of a first bright fringe on one side or the other of the central

fringe will have the following -component of momentum:

( 

)sin = ±(2 

)( 

)= ±2 

. (3)

Fig.1. A slight variation on Young’s double-slit experiment, where the plate containing the slits (width

=, separation =) is split in the middle, so that, in the wake of a single photon’s passage through

either slit, the -component of momentum transferred to the corresponding half-plate can be monitored.



 



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1/12Linearandangularmomentaofphotonsinthecontextof“whichpath”experimentsofquantummechanicsMasudMansuripurJamesC.WyantCollegeofOpticalSciences,TheUniversityofArizona,Tucson[PublishedintheProceedingsofSPIE12198,OpticalTrappingandOpticalMicromanipulationXIX,1219807(3October2022);doi:10.1117/12.2632866]...

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112 Linear and angular momenta of photons in the context of which path experiments of quantum mechanics Masud Mansuripur

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