Young's slits with incandescent light source

[flippiefanus]

You are putting up a strawman. Nobody here ever stated that " there are some forms of light that cannot be modelled in terms of photons". You just made that up.
The very question raised in the initial question of this thread is whether one MUST consider double slit interference as a quantum phenomenon. The answer is a clear no.

No, go back and look at #7, #9 and #16 again. It is not a strawman.

 

[flippiefanus]

See, e.g., this peer-reviewed paper (Phys. Rev. A 109, 022216 (2024)) that clearly states the common consensus:
"Consequently, any photonic test whose measurements constitute only first-order coherence can be simulated with the classical theory of coherence". This is so important that I typically started my own quantum optics lectures with this point in the last few years.

That quote is also significant for what it does not say: it does not say that one cannot use quantum theory for such scenarios.

 

[TheHutch]

in the quantum-optics sense the term "photon" is usually understood to be a single excitation of the light field, while it is simultaneously also common to refer to a decomposition of the light field modes into monochromatic frequencies as"photon modes". If one talks about photon number distribution, one considers the first meaning. If one talks about "photons with different wavelengths" one means the latter.

Many thanks all for your patient replies. I am learning a lot. It does seem that the use of the word 'photon' in different senses has contributed much to my confusion.

Relying on wikipedia might not be the ideal approach, but I had in mind this on Fock spaces and states (sorry the formatting has not come out very well):

"Let

be an orthonormal basis of states in the underlying one-particle Hilbert space. This induces a corresponding basis of the Fock space called the "occupancy number basis". A quantum state in the Fock space is called a Fock state if it is an element of the occupancy number basis."

...so I think for the light field we are discussing there must be an 'underlying one-particle Hilbert space' - I was assuming this one-particle is the traditional QM photon and the Hilbert space is its wave functions for a particular experimental set up - an abstract photon as opposed to a particular photon in an experiment.

 

[bhobba]

...so I think for the light field we are discussing there must be an 'underlying one-particle Hilbert space' - I was assuming this one-particle is the traditional QM photon and the Hilbert space is its wave functions for a particular experimental set up - an abstract photon as opposed to a particular photon in an experiment.

The QM EM fIeld is relativistic from the start because Maxwell's equations are relativistic; i.e., it is a quantum field theory (in fact, the firs† one discovered: https://en.wikipedia.org/wiki/Quantum_electrodynamics ).

It requires a multiparticle Fock space:
https://en.wikipedia.org/wiki/Fock_space

The idea of a single photon is somewhat problematic:


For the gory details of the Quantum EM field, see:
https://digitalcommons.usu.edu/cgi/viewcontent.cgi?article=3211&context=physics_facpub

Thanks
Bill

 

[PeterDonis]

what is the reason why we want such a probability interpretation? ... The underlying reason is because we work with quanta.

The underlying reason is that QM cannot make exact predictions for measurement results, it can only make probabilistic predictions. But that does not mean every such prediction is a prediction about "quanta" as you are using the term. Not all observables have a discrete spectrum, and not all observables with a discrete spectrum have equal spacing between each eigenvalue. There are lots of observables besides the number operator.

 

[Cthugha]

No that is not the case. See #7, #9, and #16 for example. Those are the replies that I started making comments to.

Okay, I see your point, but I did not (and I think I still do not) interpret these posts in the way you do. Taking into account that for these states a description in the photon picture is harder than the classical one, I considered "useful description" to mean a description that adds to the description of the phenomenon at hand beyond what the classical picture can do. I did not understand it as meaning "wrong" or "impossible". But okay, I think we can agree that one can always use photons to describe the scenarios.

True, I've expressed myself poorly. Let me clarify.

If $|\psi\rangle$ and $|\phi\rangle$ are single photon states, then we can have a superposition $|\psi\rangle a+|\phi\rangle b$ that can lead to interference. However, the tensor product $|\psi\rangle|\phi\rangle$ does not give us interference.

As you correctly point out, the situation with a coherent state is more complicated. We can have a tensor products of coherent states and still get interference from it. How does that work? One needs to expand it in terms of Fock states to see what is going on. After the first beam splitter we get
$$ c_0 |\alpha\rangle_A|\alpha\rangle_B = |0\rangle_A|0\rangle_B
+ c_1(|1\rangle_A|0\rangle_B+|0\rangle_A|1\rangle_B) +c_2( ... ) + ... , $$
where the $c$'s are the coefficients. For the interference, we send it through another beam splitter. What then happens is the the same kind of tensor product state is produced, except that each Fock state is replaced by a tensor product of superpositions of two single-photon Fock states. Another way to see this is to do the analysis directly in terms of creation operators. What it shows it that the interference on a quantum level happens on a photon per photon basis.

I would disagree with the wording that interference happens on a photon by photon basis. I get what you intend to get across, but ironically, the off-diagonal matrix elements between different Fock states are the ones which carry the phase information and dominate all phase-dependent phenomena - also for the matrix elements beyond n=1.
Also, I would not say that one needs (emphasis on purpose) to expand this in terms of Fock states to see what is going on. I consider it much easier to treat this in phase space (via the Glauber-Sudarshan P distribution or Wigner/Husimi functions), but I agree that it is worthwhile to do this at least once explicitly in the Fock basis. However, as soon as you treat a real-life problem including finite detector efficiencies and losses, treatments in Fock space become extremely cumbersome for Gaussian light fields.

So what about those tensor products that represent different mutually incoherent sources? A way to model such mutual incoherent sources is to introduce an unknown phase into the superpositions of photons from the different sources. The unknown phase then destroys the interference effect.

Yes, indeed, but is is worthwhile to mention that one always needs to do this in the ensemble limit over long times or many repetitions. This is equivalent to removing the off-diagonal matrix elements, which turns the pure state into a mixed state. In principle, there are no fully mutually incoherent light fields without doing the ensemble averaging. Due to the superposition principle all light fields always interfere. However, in almost all situations, the interference pattern differs in individual shots (when considering ensembles) or is stable only for insanely short amounts of time shorter than the integration time of any suitable detector.
Being mutually coherent (in first-order coherence) is not a "yes/no?" criterion, but rather a "how long?" criterion. This means that,e.g., if you take two completely different light sources emitting rather short pulses and put each of them to one slit of a double slit, each of the individual shots will show interference but all of the patterns will be completely different, so all interference gets lost in the averaging process (which is exactly due to the random phase you mention). However, one cannot remove the interference in the individual shots which is always there and you will find correlations between the count rates at different positions

 

[flippiefanus]

the off-diagonal matrix elements between different Fock states are the ones which carry the phase information and dominate all phase-dependent phenomena

Yes, I agree that off-diagonal terms carry the phase information, but not in the Fock basis. One needs a basis involving the other degrees of freedom, like the spatiotemporal degrees of freedom.

 

[flippiefanus]

Also, I would not say that one needs (emphasis on purpose) to expand this in terms of Fock states to see what is going on. I consider it much easier to treat this in phase space (via the Glauber-Sudarshan P distribution or Wigner/Husimi functions), but I agree that it is worthwhile to do this at least once explicitly in the Fock basis. However, as soon as you treat a real-life problem including finite detector efficiencies and losses, treatments in Fock space become extremely cumbersome for Gaussian light fields.

Then we are in agreement. Personally I also prefer to work on phase space, but one needs to work through the basics to have a good understanding.

 

[flippiefanus]

...so I think for the light field we are discussing there must be an 'underlying one-particle Hilbert space' - I was assuming this one-particle is the traditional QM photon and the Hilbert space is its wave functions for a particular experimental set up - an abstract photon as opposed to a particular photon in an experiment.

The Wikipedia statement probably intended to explain how the one-particle Hilbert space is used as a basic element to build up multiple particle states as tensor products. The Fock basis includes states with multiple particles, for which the occupation number is larger than 1.

 

[Cthugha]

Yes, I agree that off-diagonal terms carry the phase information, but not in the Fock basis. One needs a basis involving the other degrees of freedom, like the spatiotemporal degrees of freedom.

The (mean) phase $\langle \phi \rangle$ is the argument of the complex amplitude $a$ of the light field, where $a$ is the complex value you get by applying the photon annihilation operator:
$$\langle \phi \rangle=arg \langle a \rangle=arg (Tr(\rho a))=arg \left( \sum_n \sqrt{n+1} \rho_{n+1,n} \right).$$

Therefore, the first-order off-diagonal matrix elements directly translate into the phase. The second-order off-diagonal elements then go into squeezing and so on. You can become a bit more rigorous by calculating the circular mean phase using the Pegg-Barnett approach to phase, but then you will get a phase distribution. As a density matrix is a statistical approach, this makes sense.

Of course, one can also define relevant phases for a different choice of the basis, but the Fock basis also gives you the (or at least a) relevant phase. Consider, e.g., the attempt to excite a two-level system (a qubit) into an intermediate superposition state between the ground state and the excited state using a resonant $\pi$/2 pulse (which is a light field that can be described by a density matrix). Irrespective of the phase distribution of the light field, all $\pi$/2 pulses will create such a superposition state. However, only if you use a pulse with well-defined phase, you will create a pure intermediate state - otherwise it will be a mixed state.

The noticeable difference is that if you want to use a second $\pi$/2 pulse to excite the qubit from the intermediate superposition state to the excited state and then perform a measurement to find it in the excited state, you can only do so if you had a pure intermediate state. If the state is mixed, the measurement will give you a 50%/50% mixture of finding the qubit in the ground and excited states, respectively because you cannot know where on the equator of the Bloch sphere your state actually is.

This behavior is fully encoded in the phase within the density matrix in Fock space (and essentially a measure of how well you can define a relative phase with respect to the first light field).

 

[flippiefanus]

The (mean) phase $\langle \phi \rangle$ is the argument of the complex amplitude $a$ of the light field, where $a$ is the complex value you get by applying the photon annihilation operator:
$$\langle \phi \rangle=arg \langle a \rangle=arg (Tr(\rho a))=arg \left( \sum_n \sqrt{n+1} \rho_{n+1,n} \right).$$

Therefore, the first-order off-diagonal matrix elements directly translate into the phase. The second-order off-diagonal elements then go into squeezing and so on. You can become a bit more rigorous by calculating the circular mean phase using the Pegg-Barnett approach to phase, but then you will get a phase distribution. As a density matrix is a statistical approach, this makes sense.

Of course, one can also define relevant phases for a different choice of the basis, but the Fock basis also gives you the (or at least a) relevant phase. Consider, e.g., the attempt to excite a two-level system (a qubit) into an intermediate superposition state between the ground state and the excited state using a resonant $\pi$/2 pulse (which is a light field that can be described by a density matrix). Irrespective of the phase distribution of the light field, all $\pi$/2 pulses will create such a superposition state. However, only if you use a pulse with well-defined phase, you will create a pure intermediate state - otherwise it will be a mixed state.

The noticeable difference is that if you want to use a second $\pi$/2 pulse to excite the qubit from the intermediate superposition state to the excited state and then perform a measurement to find it in the excited state, you can only do so if you had a pure intermediate state. If the state is mixed, the measurement will give you a 50%/50% mixture of finding the qubit in the ground and excited states, respectively because you cannot know where on the equator of the Bloch sphere your state actually is.

This behavior is fully encoded in the phase within the density matrix in Fock space (and essentially a measure of how well you can define a relative phase with respect to the first light field).

OK so it depends on what state you have. The expectation value of $a$ for a single Fock state is zero. But you get a nonzero value when you have for example a coherent state. So now you expand the state in a Fock basis and then consider the phase of off-diagonal terms in its density operator. OK I get it.

However, in a physical system you don't only have the particle-number degree of freedom. So, the amplitude and phase that your coherent state has becomes a complex function when you include the spatiotemporal degrees of freedom. That was my (poorly expressed) point.

 

[Cthugha]

However, in a physical system you don't only have the particle-number degree of freedom. So, the amplitude and phase that your coherent state has becomes a complex function when you include the spatiotemporal degrees of freedom. That was my (poorly expressed) point.

You mean like when you do not only consider Gaussian spatial modes,but also more complicated Laguerre-Gaussian modes carrying orbital angular momentum and possibly also their superpositions? Yes, I fully agree. The relative phases appearing there are very relevant.

I just intended to point out that also for a single mode light field, there is some phase that can be determined from the density matrix.