Interference of a single photon in an interferometer

[weezy]

I think we can all agree that when we are shooting many photons one by one, through an interferometer, we can eventually land up with the interference pattern. This can be explained by saying that two photons combining in some areas to give four photons and in some places annihilating each other. This violates the conservation of energy so a more plausible reasoning would be to assume that every photon interferes with itself only to preserve local conservation of energy.

However, I have a problem with this for in the interferometer set up where we are splitting one beam into two parts, we still get an interference pattern with arms of different lengths. This would mean that no single photon that gets "split" into two states of motion ( two beams traveling in different arms ), arrives at the same time, at the screen or sensor. So how is it even possible to get the interference pattern? ( which we do get experimentally )

In his book, principles of quantum mechanics by Dirac, he mentions that when the beam is split in two different components we should still regard them as being one entity with a superposition of two wavefunctions (for two beams); But doesn't the very act of splitting disturb the photon and cause it to already collapse or make it's mind to go in one of the arms? Is it just that we're unaware of the direction the photon choose and so we use superposition to address our uncertainty of the situation?

In case of the double slit experiment I think a similar situation arises since the most points on the screen are at different path lengths from the slits from which wavefunction gets split.

TLDR:

How do we explain single photon interference in an unequal arm length interferometer? Do we simply accept that it is a fact of nature that when we do the experiment with a large no. of photons we get the familiar interference pattern because single photon interference has no meaning?

 

[zonde]

The idea that photon interferes with itself is experimentally falsified by this experiment:
Interference of Independent Photon Beams
[Mentor's note: This post has been edited to prevent this thread from being hijacked and turned into an interpretational debate. We don't need to go there quite yet, as this thread is about accuratelly and completely describing the many strange and counterintuituve things that happen in multi-photon systems]

 

[weezy]

The idea that photon interferes with itself is experimentally falsified by this experiment:
Interference of Independent Photon Beams

Doesn't that violate local conservation of energy? The very thing that led to the idea of the photon interfering with itself in the first place?

 

[zonde]

Doesn't that violate local conservation of energy? The very thing that led to the idea of the photon interfering with itself in the first place?

I'm not sure what do you mean. How experiment can violate conservation of energy? It is model that can do that.
Maybe you mean that this experiment can't be explained by any model that conserves energy?

 

[.Scott]

Doesn't that violate local conservation of energy? The very thing that led to the idea of the photon interfering with itself in the first place?

The theory that the photon is interfering with itself does not violate conservation of energy. The theory is not that two photons exist at one time, but that there is a superposition of the photon. In the two slit experiment, the whole notion that the photon follows a trajectory is discounted. The photon is emitted and is then detected at the screen.

Once detected, there is no evidence remaining that would indicate which slit it had traveled through, and thus it should not be viewed as having transited either slit. It should certainly not be viewed as there being two photons, one each for each slit.

A better example is the interferometer experiment where "nulls" in the interference pattern can be eliminated by blocking one of the two interferometer paths. The photons that land in those nulls do so because they could have hit the block, but didn't. That should clearly demonstrate that there is only one photon and that it's final detected position is not the result of a simple trajectory.

 

[Nugatory]

Doesn't that violate local conservation of energy? The very thing that led to the idea of the photon interfering with itself in the first place?

It does not - Pfleegor and Mandel's work, cited by Zonde above, is sound. It also doesn't falsify the statement that a photon interferes with itself; it falsifies the statement "a photon only interferes with itself". Googling for "Pfleegor Mandel interference" will bring up many of these subtleties.

 

[noir1993]

In 'Quantum Mechanics, 3rd Ed by S. Gasiorowicz, the author discusses the relevant portion of Dirac's text in the very beginning of Chapter 2 to motivate the validity of Schrodinger's equation. I will quote the relevant text here as it can shed some light sn the matter at hand:

We also know from classical optics that a beam of light consisting of many photons
will exhibit wavelike properties—that is, diffraction and interference. An experiment
carried out by G. I. Taylor in 1909 was the first to show that a beam of light gave rise to a
diffraction pattern around a needle even when the intensity of the light was so low that
only one photon at a time passed by the needle. Since then, many more experiments
showed that the interference and diffraction properties cannot be due to the collective
effect of the many photons in a beam. This raises new problems. Consider a thought
experiment, which is a variant of the Taylor experiment, in which a very low intensity beam of
light is directed at a screen with two slits in it. The photons are then detected at a second
screen (Fig. 2-1). The intensity is such that at a given time no more than one photon
passes through the two-slit screen. After very many photons have passed by, we see the
classically expected diffraction pattern. Classically this is well understood: If the electric
fields at a particular point r on the detecting screen due to electromagnetic waves crossing
slits 1 and 2 are E;(r, t) and E2(r, t) respectively, then the total field at the point r at the
time t is the sum of the fields. This is a consequence of the superposition rules for electric
fields, which in turn is a consequence of the fact that Maxwell's equations for the
electromagnetic fields are linear. The intensity at the screen is proportional to the square of the
total electric field, and thus to (E^r, t) + E2(r, i))2. The interference pattern is due to the
presence of the E;(r, t) • E2(r, t) cross term in the square of the sum of the fields. If only
slit 1 were open, the intensity would be proportional to E,(r, t)2, and if only slit 2 were
open, the intensity would be proportional to E2(r, t)2. If we now translate intensity into
probability, as suggested by our discussion about polarization, we find that if only slit 1 is
open, the probability of finding a photon at r is P{(r, t), and if only slit 2 is open, the
probability of finding a photon at r is P2(r, t). However, if both slits are open, the probability
is not the sum of the probabilities associated with each slit.
The only way to resolve these difficulties is to assume that each photon interferes
with itself. This can be handled by assuming that each photon is described by its own
electric field, e(r, t), and that in the presence of two slits, the photon field at the detector is the
sum of two terms. These are associated with the presence of two slits, so that
e(r, t) = e^r, t) + e2(r, t)
just as for a classical light wave. Note that we are still talking about a single photon. The
only real requirements are (1) that the field e(r, t) obeys a linear equation and that (2) in
the classical limit a large collection of photons acts in accordance with Maxwell's
equations. The actual formulation of a quantum theory of photons is somewhat complicated,
and we leave the discussion of this to Supplement 18-A.

Supplement 18 A can be accessed in the link below and provides a quantum field theoretic treatment of the photon.
http://bcs.wiley.com/he-bcs/Books?action=resource&bcsId=1533&itemId=0471057002&resourceId=1342

 

[.Scott]

Just noting that we have yet to address the the OP's 1st question:

How do we explain single photon interference in an unequal arm length interferometer?

 

[zonde]

In 'Quantum Mechanics, 3rd Ed by S. Gasiorowicz, the author discusses the relevant portion of Dirac's text in the very beginning of Chapter 2 to motivate the validity of Schrodinger's equation. I will quote the relevant text here as it can shed some light sn the matter at hand:

...
The only way to resolve these difficulties is to assume that each photon interferes
with itself.
...

If the photon self interference is the only way how to explain double slit interference then we are left with no explanation for Pfleegor Mandel interference experiment.

 

[zonde]

Just noting that we have yet to address the the OP's 1st question:
How do we explain single photon interference in an unequal arm length interferometer?

Well, if we could explain Pfleegor Mandel interference experiment then the same explanation should work for unequal arm length interferometer.
So I think that discussion is closely related to OP question.

 

[Lord Jestocost]

...
How do we explain single photon interference in an unequal arm length interferometer? ...

I recommend to read: "Interactive tutorial to improve student understanding of single photon experiments involving a Mach–Zehnder interferometer" by Emily Marshman and Chandralekha Singh (Eur. J. Phys. 37 (2016) 024001 (22pp)) (http://iopscience.iop.org/article/10.1088/0143-0807/37/2/024001)

 

[vanhees71]

The explanation of the experiment is given in terms of standard QED in the above cited paper. As to be expected QED works.

Strictly speaking the experiment tests the interference effect not between two independent single photons but between two low-intensity coherent states. Nowadays the experiment should also be doable with true single-photon sources. I do not know, whether this has been done yet.

 

[Lord Jestocost]

In his book, principles of quantum mechanics by Dirac, he mentions that when the beam is split in two different components we should still regard them as being one entity with a superposition of two wavefunctions (for two beams); But doesn't the very act of splitting disturb the photon and cause it to already collapse or make it's mind to go in one of the arms? Is it just that we're unaware of the direction the photon choose and so we use superposition to address our uncertainty of the situation?

Unless there is a measurement device on either arm of the experiment, we cannot say that the photon has either been transmitted or reflected, we thus have an "entity" which can formally be "described" as a superposition of two wavefunctions. A quantum mechanical superposition has nothing to do with the classical understanding that something is - with some probability - either there or there. The quantum mechanical “ ǀphoton in arm#1> AND ǀphoton in arm#2> “is fundamentally different from the classical “ ǀphoton in arm#1> OR ǀphoton in arm#2> “. There is no way to bridge this abyss. Take it as it is.

 

[weezy]

The theory that the photon is interfering with itself does not violate conservation of energy. The theory is not that two photons exist at one time, but that there is a superposition of the photon. In the two slit experiment, the whole notion that the photon follows a trajectory is discounted. The photon is emitted and is then detected at the screen.

Once detected, there is no evidence remaining that would indicate which slit it had traveled through, and thus it should not be viewed as having transited either slit. It should certainly not be viewed as there being two photons, one each for each slit.

A better example is the interferometer experiment where "nulls" in the interference pattern can be eliminated by blocking one of the two interferometer paths. The photons that land in those nulls do so because they could have hit the block, but didn't. That should clearly demonstrate that there is only one photon and that it's final detected position is not the result of a simple trajectory.

That is not what I meant by violation of conservation of energy. It was in the Dirac's book where I read that the motivation for considering photons interfering with themselves comes from the fact that we cannot have two photons meet up at a screen to produce 0 photons or 4 photons at the minima and maxima respectively, as would be explained by a classical physicist. The trouble I have is addressing the question that If I have a system where I can split a beam of light into two components with different transit times to the screen and obtain the same interference pattern, how would this make sense when I am shooting photons one by one. One of the split component always reaches the screen after its partner has arrived.

 

[weezy]

If the photon self interference is the only way how to explain double slit interference then we are left with no explanation for Pfleegor Mandel interference experiment.

Consider the double slit experiment for a single photon source, if we are to just imagine the probability wave goes through both slits then the distance to a point at the screen will be at unequal intervals too from the two slits. Then the probability wave arising from slit 1 and from slit 2 must interfere because that is what we do in quantum mechanics. We take state 1, take state 2, add them, square them and say that's the probability of finding the photon there. But notice how if we are to assume state 1 and state 2 follows the same time evolution laws i.e. the same speed of propagation then they never make it to the same place on the screen on time.

 

[weezy]

Consider the double slit experiment for a single photon source, if we are to just imagine the probability wave goes through both slits then the distance to a point at the screen will be at unequal intervals too from the two slits. Then the probability wave arising from slit 1 and from slit 2 must interfere because that is what we do in quantum mechanics. We take state 1, take state 2, add them, square them and say that's the probability of finding the photon there. But notice how if we are to assume state 1 and state 2 follows the same time evolution laws i.e. the same speed of propagation then they never make it to the same place on the screen on time.

I know it is not wise to consider the probability waves as something analogous to a real wave propagating through space but most quantum mechanics books address the double slit experiment through this way. As for the QED explanation I'm not there yet to fully understand it.

 

[vanhees71]

I never understood what's meant by the phrase "a photon (or a massive particle) interferes with itself". Usually what people mean is a single-photon/particle state which they write as the superposition of other single-photon states. There's nothing mysterious about this.

 

[weezy]

I never understood what's meant by the phrase "a photon (or a massive particle) interferes with itself". Usually what people mean is a single-photon/particle state which they write as the superposition of other single-photon states. There's nothing mysterious about this.

The confusion I think comes from introducing the idea of photon self-interference in the first place.

 

[Nugatory]

take state 1, take state 2, add them, square them and say that's the probability of finding the photon there. But notice how if we are to assume state 1 and state 2 follows the same time evolution laws i.e. the same speed of propagation then they never make it to the same place on the screen on time.

You are making the assumption that there's a single moment of emission so that both paths are starting at the same precisely defined point in space and time; this is basically classical thinking sneaking back in. If you don't make this assumption there's room for overlap at the detector even when the path lengths are different.

 

[weezy]

You are making the assumption that there's a single moment of emission so that both paths are starting at the same precisely defined point in space and time; this is basically classical thinking sneaking back in. If you don't make this assumption there's room for overlap at the detector even when the path lengths are different.

But the emission started out as a one photon given by one state vector at the laser source.

 

[weezy]

You are making the assumption that there's a single moment of emission so that both paths are starting at the same precisely defined point in space and time; this is basically classical thinking sneaking back in. If you don't make this assumption there's room for overlap at the detector even when the path lengths are different.

I think you meant that the act of "splitting" the probability wave by the two slits cannot be regarded as simultaneous.

 

[rrogers]

I haven't seen it mentioned but https://arxiv.org/pdf/quant-ph/0603048.pdf mentions
"We report an observation of non-classical interference of two single photons originating from two independent, separate sources, which were actively synchronized with an r.m.s. timing jitter of 260 fs. "
Notice the word synchronized. To me, this means the probability wave functions could interfere by some quantum linkage.
IMO (probably worthless): A tell-tale sign that we really had the interference of two "independent" solution/probability waves would be the appearance of "beat frequencies" when one of the sources was tuned off frequency. Better yet oscillated back and forth on the nanoscale order of magnitude, just to be practical, to induce a Doppler shift that could be extracted, or not, from the detector signal. The beat frequencies would from the squaring of the added "probability waves" to get the real probability.

 

[Gary A Gordon]

My thoughts are similar in essence to those expressed just above by Nugatory. A photon can interfere with itself at the detector when it has the choice of two unequal (free space) path lengths to travel to the detector, provided that the time that the photon was emitted is not measured. For example, the instant of photon emission might be measured by some effect (a very, very, very, sensitive measurement of the reaction momentum of the emitter), in which case the photon could not interfere with itself after traveling two different free space path lengths, and no interference would result. In the absence of such measurement, the photon's time of emission is just another parameter of its motion to the detector, so the interference is completely consistent with other such single-photon interference results.

 

[weezy]

is it correct to say that just after the photon is ejected from our source, the wavefunction, which may have a speed greater than the speed of light ** or have no speed at all and is present at all points in space, from the moment the photon was released, has already interfered at the detector before the photon would make it to the detector at velocity and it's only after time {t=path length / c} that we get the measurement/click at the detector?

 

[Gary A Gordon]

Hi Weezy - My point of view is that the wavefunction for a photon of unknown and unmeasured ejection time is not produced when a photon is ejected. It is not a property of any individual photon, but a property of the total configuration in which photons are emitted at random points in time. The wavefunction changes if the configuration changes, for example if some means of producing an observable trace is added to record the approximate time when each photon is subsequently released. Suppose that the configuration change can be effected by the flick of a switch by the experimenter. If he flicks the switch at time t so that photons subsequently emitted leave an observable trace indicating the approximate time of their emission, then the wavefunction corresponding to subsequent photon emissions must change to reflect the change in the configuration. The new wavefunction change spreads out, starting at time t, from the point of emission to the rest of space at the speed of light. At the detection surface, the probability distribution of photon impact changes as a function of time, starting from the point at which the wavefunction change first reaches it. Subsequently, there would be a smooth transition from the probability distribution corresponding to unmeasured photon emission time to that corresponding to measured emission time. The initial probability distribution would reflect the possibility that photons that have the choice of two unequal path lengths have a greater opportunity to interfere to a final distribution that reflects a greatly reduced interference probability. Note that the change does not depend on whether or not the experimenter actually examines the observable traces of photon emission times produced. And consciousness certainly plays no role. The only relevant factor is that the changed configuration responds with an observable trace indicating the approximate time of each subsequent photon release. In the point of view of Carlo Rovelli relational quantum mechanics, the response of the changed system to photon emission changes the wave function whether or not the response record is examined by the experimenter.

 

[Derek P]

If the photon self interference is the only way how to explain double slit interference then we are left with no explanation for Pfleegor Mandel interference experiment.

[Deep breath] I think you are mistaken.

Now it is quite possible that I am missing something about Pfleegor Mandel interference. Perhaps it's something more than interference between separate sources even at low photon rates. But if that's all it is I think there is a very simple explanation that needs nothing more than superposition. Which QM has already.

Photon emission is usually stated as occurring at random. But the quantum description of a system before measurement is not a mixed state, it's a pure superposition of possible outcomes - emitted and not emitted in this case. And the emitted states are superpositions of all possible emission-time states. Since this is true for every atom, a given photon (i.e. at a specific time) is always a superposition of all possible source-atom states. It's only when you decide to measure the source that you "collapse the superposition", and identify a particular source atom. Otherwise the photon state is a superposition of all of them. And therefore delocated across across all atoms in the lasing medium or, more simply, across both lasers. And therefore we have interference even if the photons are an hour apart.

 

[DrChinese]

Here is a reference from 2004 showing single-photon interference (determined by coincidence) and discussing the theory behind it. See section III:

http://webedit.colgate.edu/portalda...lery/interference-with-correlated-photons.pdf

"The probability amplitude, a complex number, is a key idea in quantum mechanics. Interference arises from squaring the sum of the probability amplitudes for alternative ways to the same observational outcome. Interference can occur if two or more different ways to produce the same result cannot be distinguished with the apparatus. If the apparatus yields information that can distinguish between alternatives, interference will not occur."

It then goes on to discuss Feynman's approach to explaining this (always worth considering) as well as a more modern treatment (theory and experiment).

Hopefully this will go a bit futher to the OP's question.