I Is there a difference between EM waves and photon wavefunctions?

[sciencejournalist00]

Matter has a wavefunction associated to it. But what about light? Does it have both a electromagnetic wave described by Maxwell's equation and a wavefunction described by Schroedinger's equation?

Or is the electromagnetic wave considered to be the wavefunction of the photon?

I read somewhere that in interference patterns, the highest intensity peaks of a EM wave correspond to greatest probability of observing a photon. Elsewhere that photons have no wavefunction associated to them because massless particles have no position eigenstate.

Which one is true?

Intensity of EM waves = probability distribution for photon location

No wavefunction for masless particles

or

Both EM wave and wavefunction for photons

 

[sciencejournalist00]

From Wikipedia

Both (photons and material) particles such as electrons create analogous interference patterns when passing through a double-slit experiment. For photons, this corresponds to the interference of a Maxwell light wave whereas, for material particles, this corresponds to the interference of the Schrödinger wave equation. Although this similarity might suggest that Maxwell's equations are simply Schrödinger's equation for photons, most physicists do not agree. For one thing, they are mathematically different; most obviously, Schrödinger's one equation solves for a complex field, whereas Maxwell's four equations solve for real fields. More generally, the normal concept of a Schrödinger probability wave function cannot be applied to photons. Being massless, they cannot be localized without being destroyed; technically, photons cannot have a position eigenstate

, and, thus, the normal Heisenberg uncertainty principle

does not pertain to photons.

 

[blue_leaf77]

Or is the electromagnetic wave considered to be the wavefunction of the photon?

No, it is not. At least, from the fact that a wavefunction is not an observable quantity on the other hand EM field is, you should see that they are two different things. In the case of photons having certain wavefunction, the observed electric and magnetic fields are the average values of the field operators.

Elsewhere that photons have no wavefunction associated to them because massless particles have no position eigenstate.

By "wavefunction", I believe that source was referring to the wavefunction in position representation, photons do not have this kind of wavefunction. But they do have states, as any quantum objects do.

 

[A. Neumaier]

most obviously, Schrödinger's one equation solves for a complex field, whereas Maxwell's four equations solve for real fields.

This is a very superficial difference.

One of the nice representations of the electromagnetic field is in terms of Silberstein vectors, which are 3-dimensional complex vectors, for which field equations can be derived. With an appropriate inner product, these are precisely the functions making up the 1-photon Hilbert space, and hence define the pure states of a single photon.

Intensity of EM waves = probability distribution for photon location

This is not correct. it is well-known that photon location cannot be described by a probability distribution.

See the slides of my lectures Classical and quantum aspects of light and Optical models for quantum mechanics for a more appropriate correspondence.

 

[Charles Link]

The probability of finding a photon, i.e. the photon density, is proportional to the second power of the electric field amplitude E. The total energy is the number of photons multiplied by the energy of each photon, Ep=h*f. (The letter Ep is used for photon energy, but E for electric field is not energy.) The intensity "I" and energy density U are both proportional to the square of the electric field amplitude E. The "I", U, and N*Ep where N is number of photons are all proportional to the second power of the electric field amplitude E. The electric field is proportional to the the quantum mechanical wave function "psi". (Greek letter). The psi*psi also gives a probability density for photons.

 

[A. Neumaier]

The probability of finding a photon, i.e. the photon density, is proportional to the second power of the electric field amplitude E. The total energy is the number of photons multiplied by the energy of each photon, Ep=h*f. (The letter Ep is used for photon energy, but E for electric field is not energy.) The intensity "I" and energy density U are both proportional to the square of the electric field amplitude E. The "I", U, and N*Ep where N is number of photons are all proportional to the second power of the electric field amplitude E. The electric field is proportional to the the quantum mechanical wave function "psi". (Greek letter). The psi*psi also gives a probability density for photons.

Can you substantiate your claims with a reference? The first few sentences flatly contradict well-known results about the nonexistence of a position operator and therefore of a Schroedinger-type probability interpretation. The remaining part is also incorrect - A wave function is necessarily complex while the electric field is real.

 

[blue_leaf77]

The probability of finding a photon, i.e. the photon density, is proportional to the second power of the electric field amplitude E.

This is what I would call technical understanding of photons. Theoretical and quantum physicists have their own, which is the correct, description of photons. For this last one, I would call theoretical understanding of photons. Many scientists like to use the term "photon density" to actually refer to EM wave density - in any textbooks which uses these words in particular instances within the book, I am almost sure that they are actually referring to the equation $\rho = 1/2\epsilon_0|E|^2+1/2|B|^2/\mu_0$. The reason why many scientists mix up the use of the word "photon" with "EM radiation" is most likely because a matter of convenience. Since the introduction of photons as light particles by Max Planck (if I am not mistaken), the easy way of thinking is that "light are made up of photons". This has made the hand-waving, but blunt association between EM radiation and photons came into existence among scientists. The only thing in your comment which is unheard of to me is

The psi*psi also gives a probability density for photons.

I have never heard of this claim though.

Can you substantiate your claims with a reference?

As I have a little bit elaborated above, there are actually physicists who indeed make use of the word "photons" in a broader sense than it should have been. While such description is not correct, that's just the way some of us understand EM radiation as being made up of a stream of photons. If you are lucky, you can find technical physics textbooks (some of them must be those in the interdisciplinary category) which abuse the literal use of "photon density", which should not exist in the correct theoretical view. Some online examples are and http://www.medilexicon.com/medicaldictionary.php?t=23602. The last one even establishes it as one of their vocabulary collection.

 

[Charles Link]

Can you substantiate your claims with a reference? The first few sentences flatly contradict well-known results about the nonexistence of a position operator and therefore of a Schroedinger-type probability interpretation. The remaining part is also incorrect - A wave function is necessarily complex while the electric field is real.

My post is for the most part simply Classical Electrodynamics results, (see for example J.D. Jackson's Classical Electrodynamics textbook), but the electric field can be written as a complex quantity. The Q.M. wavefunction for the photons is of a many-particle type boson form. One obstacle that is encountered is if there are multiple photons in the same photon mode (e.g. at the same wavelength) as in a laser, how can the individual electric fields add in phase?, and the answer to this is that apparently the phases are random for multiple photons in the same mode. (The phasor diagram is a 2-D random walk.) The complete quantum mechanical description of the electromagnetic field with second quantized field operators (see e.g. textbooks on relativistic quantum field theory by Bjorken and Drell) is rather difficult to follow-(I struggled with the concepts as a graduate student), but the items contained in my previous reply are considerably simpler, and I believe them to be correct.

 

[Charles Link]

Additional item: A photodiode measures photons giving typically one electron of photocurrent for each photon with typically about an 85%-90% quantum efficiency. In working in the visible and IR parts of the spectrum, it is photons (=photocurrent from a photodiode) that are measured rather than a time-varying electric field (e.g. as with radio waves). The wave nature of the light shows up when using instrumentation such as diffraction gratings (diffraction grating spectrometers), and interferometers (e.g. the Michelson interferometer.) Incident visible light of irradiance S(watts/cm^2)=1 milliwatt/cm^2 (using the nomenclature of J.D. Jackson textbook where S=Poynting vector for electromagnetic wave ), contains billions of photons per second per square centimeter. S=c*E^2/(4*pi) (c.g.s. units) where E is electric field amplitude, c=speed of light.) Localizing the energy and/or the photons seems to be straightforward. The derivation of the Planck blackbody radiation formula is one very good example of the use of both the wave nature of the light along with the particle nature of the photon. In this derivation, the photon modes are first counted inside a cavity, the numbers of photons is then computed using the mode mean occupancy number for each mode for bosons (as a function of the temperature), and the rate of photons that effuse out of an aperture is computed along with their energy to arrive at the spectral formula for blackbody radiation. (see e.g. Statistical Physics by F. Reif)

 

[A. Neumaier]

Since the introduction of photons as light particles by Max Planck (if I am not mistaken),

Einstein reintroduced in 1905 the particle picture of light already used by Newton, in a modified quantum form. The name photon appeared much later (1936), and even then it took a very long time to be generally accepted as a valid description of light.

 

[A. Neumaier]

My post is for the most part simply Classical Electrodynamics

No, there is no notion of photons in classical ED.

he complete quantum mechanical description of the electromagnetic field with second quantized field operators (see e.g. textbooks on relativistic quantum field theory by Bjorken and Drell) is rather difficult to follow-(I struggled with the concepts as a graduate student), but the items contained in my previous reply are considerably simpler, and I believe them to be correct.

How you relate the notion of photons to CED is completely nonstandard and mistaken.

The correct quantum mechanical description can be found in the quantum optics book by Mandel and Wolf (Optical Coherence and Quantum optics), which is easy to follow, though it requires some serious study, including background material. It draws a picture substantially different from yours.

 

[blue_leaf77]

The Q.M. wavefunction for the photons is of a many-particle type boson form. One obstacle that is encountered is if there are multiple photons in the same photon mode (e.g. at the same wavelength) as in a laser, how can the individual electric fields add in phase?, and the answer to this is that apparently the phases are random for multiple photons in the same mode. (The phasor diagram is a 2-D random walk.) The complete quantum mechanical description of the electromagnetic field with second quantized field operators (see e.g. textbooks on relativistic quantum field theory by Bjorken and Drell)

That's right, that's what some of us have been talking about. However, the problem with post #5 is that you associated EM wave energy density with photon's spatial probability density which is supposed to be non-existent. From this and

but the electric field can be written as a complex quantity.

I have got the impression that you associate fields with photon wavefunction. They are fundamentally different, photon wavefunction (or more properly photon state) is mathematical object introduced as a part of quantum description of light and it can be complex vector in the Fock state basis, but a state is not an observable physical quantity. Fields, on the other hand, can be physically measured. Despite sometimes being written complex, it is usually done so to simplify the calculation; every time a field is written as a complex number, the complex conjugate of this number with the right sign must also be written (or at least implied) to make the entire field expression real. Most importantly, in QM description of light, fields are assigned with operators, as any other physical quantities in quantum mechanics are, so they are by no means the state of the photon itself. Although in some instances, the time dependency of the fields do carry signature of which kind of photon state these observed fields are subject to. For example, photons in Fock state, coherent state, and squeezed state have their own forms of the time-dependent fields.

 

[vanhees71]

Can you substantiate your claims with a reference? The first few sentences flatly contradict well-known results about the nonexistence of a position operator and therefore of a Schroedinger-type probability interpretation. The remaining part is also incorrect - A wave function is necessarily complex while the electric field is real.

Take any textbook on quantum optics. Obviously is the detection probability for photons taken as proportional to the energy density of the em. (quantum) field a very succuessful model. It's to a certain extent taken from classical electromagnetics, where the intensity of light is also taken as the (time-averaged) em. field-energy density.

Also more complicated observables (like HBT correlations) are defined via electromagnetic field-strength-tensor correlators. See, e.g.,

M. O. Scully, M. S. Zubairy, Quantum Optics, Cambridge University Press 1997

It's of course true that photons have no position operator but still you have intensity measures and detection probabilities for single photons at the place where your detector is located, and that's usually defined via the expectation value of the field-energy density or correlation functions of the electromagnetic-field operator.

 

[A. Neumaier]

Obviously is the detection probability for photons taken as proportional to the energy density of the em. (quantum) field a very successful model.

It is not the detection probability but the photodetection rate that is proportional to the energy density of the electromagnetic field. Since it doesn't matter whether the latter is a classical or a quantum field, the photodetection rate it is not a property of photons. Certainly not a photon density as specified in post #5.

 

[vanhees71]

Ok, it's the photodetection rate (although I'd rather say that's the modulus of the (expectation value of) the Poynting vector, i.e., the energy-flow density, but that's semantics). A photon-number density is indeed problematic to define, because there's no conserved photon number, i.e., no photon-number-density four-current that obeys the continuity equation, and thus photon numbers, defined in a naive way with the "photon number operators" are not a Lorentz-invariant (scalar) quantity, while for total field energy and momentum that's the case.

 

[Charles Link]

Ok, it's the photodetection rate (although I'd rather say that's the modulus of the (expectation value of) the Poynting vector, i.e., the energy-flow density, but that's semantics). A photon-number density is indeed problematic to define, because there's no conserved photon number, i.e., no photon-number-density four-current that obeys the continuity equation, and thus photon numbers, defined in a naive way with the "photon number operators" are not a Lorentz-invariant (scalar) quantity, while for total field energy and momentum that's the case.

Most of the time, when measuring light sources, you can simply count photons using a photodiode-i.e. there are billions of them each second. Normally there is no concern about photon creation or photon annihilation processes, where the photon number changes. For a semi-classical approach the photons are proportional to the energy and the photon number can be considered to be conserved. In the case of a Doppler shift, I think the photon number can be considered to be conserved. It appears you are taking a more advanced and detailed approach.

 

[A. Neumaier]

a semi-classical approach the photons are proportional to the energy and the photon number can be considered to be conserved.

In a semiclassical treatment the electromagnetic field is classical and there are no photons. What is proportional to the energy density is the number of detector clicks, which can be counted, not the number of photons, which cannot exist in a model only containing quantum electrons and a classical external electromagnetic field.

 

[Charles Link]

In a semiclassical treatment the electromagnetic field is classical and there are no photons. What is proportional to the energy density is the number of detector clicks, which can be counted, not the number of photons, which cannot exist in a model only containing quantum electrons and a classical external electromagnetic field.

Undoubtedly, to get a completely accurate treatment of the electromagnetic field, a detailed quantum mechanical approach is necessary. I believe Bjorken and Drell's textbooks, Gordon Baym's Quantum Mechanics, and J.J. Sakurai's Advanced Q.M. all treat this subject, but I found this topic very difficult to get a good handle on.

 

[A. Neumaier]

Undoubtedly, to get a completely accurate treatment of the electromagnetic field, a detailed quantum mechanical approach is necessary. I believe Bjorken and Drell's textbooks, Gordon Baym's Quantum Mechanics, and J.J. Sakurai's Advanced Q.M. all treat this subject, but I found this topic very difficult to get a good handle on.

The book by Mandel and Wolf cited in #11 is much better, and very comprehensive. it explains both the semiclassical and the full quantum situation, including squeezed states and parametic downconversion, which are true quantum phenomena. But even then, the energy density cannot be equated with a photon density.

 

[Charles Link]

The book by Mandel and Wolf cited in #11 is much better, and very comprehensive. it explains both the semiclassical and the full quantum situation, including squeezed states and parametic downconversion, which are true quantum phenomena. But even then, the energy density cannot be equated with a photon density.

If you consider each photon to have energy e(w)=h/(2*pi)*w and the photon density n(w) per unit volume, the energy density at frequency w is U(w)= n(w)*e(w). I do believe it will agree precisely with the energy density U(w)=E(w)^2/(4*pi) (see J.D. Jackson) where E is the electromagnetic field amplitude at frequency w (c.g.s. units). (and the total energy density is of course a summation/integral over all w). One item that is somewhat tricky here is that if one considers individual photons to each have a sinusoidal electric field, and if they are in the same Bose mode (such as in a laser), the individual electric fields can not superimpose in phase, because the energy is proportional to the second power of the E-field. A model that considers the individual photon phases to be random (but constant) would satisfy the energy requirement.

 

[vanhees71]

The book by Mandel and Wolf cited in #11 is much better, and very comprehensive. it explains both the semiclassical and the full quantum situation, including squeezed states and parametic downconversion, which are true quantum phenomena. But even then, the energy density cannot be equated with a photon density.

The energy density is of course the energy density and not a photon-number density. Nevertheless the assumption that it reflects the "intensity" of electromagnetic radiation is very successful, or do you know any example, where the measured photone-detection rates deviate from the predictions based on this assumption? I've to check, how Mandel&Wolf treat the photon-detection issue, but I'm pretty sure it doesn't deviate too much from the treatment in Scully&Zubairy in Sect. 6.5, which gives the full quantum theory of single-photon and double-photon excitation of an atom, which is the most simple model for photon detection. Of course, it uses the usual dipole approximation to describe the interaction between the quantized em. field with the the atom and first-order time-dependent perturbation theory. Then you immediately get that the single-photon-detection rate is proportional to the expectation value of the energy density of the em. field and the double-photon excitation function to the 2nd-order correlation function of the electric field.

Mandel&Wolf is a great book, of course, but often (as a non-expert in quantum optics) I'm thankful to have a more compact source like Scully&Zubairy ;-)).

 

[A. Neumaier]

the assumption that it reflects the "intensity" of electromagnetic radiation is very successful, or do you know any example, where the measured photon-detection rates deviate from the predictions based on this assumption?

Under natural idealization assumptions that are usually satisfied to good accuracy, the predicted and measured single electron-ionization rate is proportional to the intensity of the electromagnetic radiation field (not the electric field alone). This is the only unambiguous statement. Since this statement is valid both in the semiclassical treatment where there are no photons and in a full quantum field treatment (where - see below - the question of what a single photon is is ambiguous), the interpretation of this in terms of photons, though widely used, is questionable. Photodetection is measuring the intensity of the electromagnetic radiation field by measuring the electron-ionization rate, not by counting the number of emitted photons. If the intensity goes down, one simply needs to wait longer to get a reliable measurement of the intensity to a given accuracy. In my opinion, the detector events have no other meaning than this.

It is in some way analogous to measuring the output flow rate of a water faucet by (observing from far apart - not seeing the details) how many cups are filled in a given time. This works accurately in a short time if a lot of water flows, but if the faucet is only dripping it takes a long time before the number of cups accurately represents the water flow rate. I find this analogy helpful although it is only a classical analogy with a limited explanatory value as it lacks the randomness observed in the quantum case.

The problem with the notion of a single photon in a quantum field setting is that the quantum field typically is in some Heisenberg multiphoton state, and this single Heisenberg state gives rise to an electron-ionization rate rather than a probability. For example, in an ordinary laser the electromagnetic field is in a fixed coherent Heisenberg state, and by waiting long enough this fixed state gives rise to as many electron ionizations as one likes. This even holds when the coherent state has very low intensity. This is quite unlike what is assumed in the typical quantum optical experiments where everything is interpreted as if photons were particles just like nonrelativistic electrons, except that they move with the speed of light. There is a nontrivial interpretation step in going from the former to the latter description, one never analyzed in the literature (as far as I can tell).

The Heisenberg description (QFT) has a straightforward, almost classical interpretation of the state. To go from it to the interaction picture (QM) with the interpretation of an ensemble of particles (prepared or actually being - depending on the interpretation of QM used) in a corresponding state one has to ''invent'' individual photons with ghostlike properties that are completely unobservable until detection events destroy the ghosts and thereby prove their alleged existence. Extremely weird, this assumed picture of individual photons. One is left wondering why these ghosts are completely absent in the semiclassical description (which is quantitatively correct in the case of coherent light) although the detection events still herald their existence (if they herald anything). But if they herald nothing in the semiclassical setting, they also shouldn't herald anything in the quantum case. This then implies that the ghosts are completely unobservable. They can be eliminated without making any observable differences, and using Ockham's razor, they should be eliminated. My language is chosen accordingly.

 

[Charles Link]

Under natural idealization assumptions that are usually satisfied to good accuracy, the predicted and measured single electron-ionization rate is proportional to the intensity of the electromagnetic radiation field (not the electric field alone). This is the only unambiguous statement. Since this statement is valid both in the semiclassical treatment where there are no photons and in a full quantum field treatment (where - see below - the question of what a single photon is is ambiguous), the interpretation of this in terms of photons, though widely used, is questionable. Photodetection is measuring the intensity of the electromagnetic radiation field by measuring the electron-ionization rate, not by counting the number of emitted photons. If the intensity goes down, one simply needs to wait longer to get a reliable measurement of the intensity to a given accuracy. In my opinion, the detector events have no other meaning than this.

It is in some way analogous to measuring the output flow rate of a water faucet by (observing from far apart - not seeing the details) how many cups are filled in a given time. This works accurately in a short time if a lot of water flows, but if the faucet is only dripping it takes a long time before the number of cups accurately represents the water flow rate. I find this analogy helpful although it is only a classical analogy with a limited explanatory value as it lacks the randomness observed in the quantum case.

The problem with the notion of a single photon in a quantum field setting is that the quantum field typically is in some Heisenberg multiphoton state, and this single Heisenberg state gives rise to an electron-ionization rate rather than a probability. For example, in an ordinary laser the electromagnetic field is in a fixed coherent Heisenberg state, and by waiting long enough this fixed state gives rise to as many electron ionizations as one likes. This even holds when the coherent state has very low intensity. This is quite unlike what is assumed in the typical quantum optical elements where everything is interpreted as if photons were particles lust like nonrelativistic electrons, except that they move with the speed of light. There is a nontrivial interpretation step in going from the former to the latter description, one never analyzed in the literature (as far as I can tell).

The Heisenberg description (QFT) has a straightforward, almost classical interpretation of the state. To go from it to the interaction picture (QM) with the interpretation of an ensemble of particles (prepared or actually being - depending on the interpretation of QM used) in a corresponding state one has to ''invent'' individual photons with ghostlike properties that are completely unobservable until detection events destroy the ghosts and thereby prove their alleged existence. Extremely weird, this assumed picture of individual photons. One is left wondering why these ghosts are completely absent in the semiclassical description (which is quantitatively correct in the case of coherent light) although the detection events still herald their existence (if they herald anything). But if they herald nothing, they also shouldn't herald anything in the quantum case. This then implies that the ghosts are completely unobservable. They can be eliminated without making any observable differences, and using Ockham's razor, they should be eliminated.

Thank you for your very thorough response. Until very recently, with the advent of things such as correlated and entangled photons, there was not much reason to question the results from a photodiode. On occasion, e.g. with a fast/short laser pulse slight non-linearities might be observed, but otherwise the photodiode response to incident light (both visible and IR) has been found found to be very linear over several orders of magnitude. Experiments of blackbody sources (e.g. T=1273 K) and various aperture sizes using an optical chopper and lock-in amplifier have shown the measured photocurrent from the photodiode deviated from linearity by less than 2% over a considerable range. Your analysis is quite interesting. I am going to need to study it further, but perhaps some of the experimental anomalies (e.g. in correlated photon experiments) are going to be found to be attributed to the nature of the photoionization process.

 

[f95toli]

Just to a add to what has been written above.
In many fields (including mine) we frequently measure energy in "units" of average number of photons. This means that <n> can take on any value (and is is frequently <<1) The reason why this is useful is that we often deal with ensembles of two-level systems(TLS), and each TLS can be exited by absorbing a single photon (and in principle follows J-C dynamics) . Hence, in order to understand the properties of the system it is useful to have some idea of the number of photons that are "available" compared to the number of TLS.
Hence, the EM field is (usually) in a coherent state but the photon concept is still useful for understanding the physics.

This also means that not much changes In the cases when we do deal with true single photon state (unless we get into situations where we need the Tavis-Cummings Hamiltonian) ; unless of course we are also dealing with a single TLS (in which case the whole systems is described by the usual Jaynes-Cummings Hamiltonian).

Also, I my view it is quite pointless to start analyzing the situation where photons are absorbed by a single atom etc since you get into details like ionization etc.
There are many situations -including many practical single photon detectors (e.g superconducting SPDs or KIDs) - where the photon is detected using other mechanisms; the easiest case to understand is actually that of a single solid-state qubit where the photon is absorbed the whole system that makes up the qubit (e.g. a superconducting circuit or a quantum dot). The physics of photodiodes is simply irrelevant.

 

[A. Neumaier]

Your analysis is quite interesting. I am going to need to study it further, but perhaps some of the experimental anomalies (e.g. in correlated photon experiments) are going to be found to be attributed to the nature of the photoionization process.

There is no difference between classical and quantum in the case of a single, localized detector.

However, for multi-point detection of correlated electromagnetic fields there is indeed a difference between the classical and the quantum description - not for coherent states where the two descriptions still match but for nonclassical states, for example for squeezed states (used, e.g., to describe parametric down-conversion). This difference is an expression of the fact that electromagnetism in Nature is described by QED and not by classical electrodynamics.

The main difference in the resulting detection statistics of the ionization events is usually described in terms of the notions of bunching and antibunching.

(see this page)
The above discussion makes it clear that the Hanbury Brown and Twiss (or photon bunching) effect can be entirely described by classical optics. [...] Antibunching, whether of bosons or of fermions, has no classical wave analog.

 

[A. Neumaier]

the easiest case to understand is actually that of a single solid-state qubit where the photon is absorbed the whole system that makes up the qubit (e.g. a superconducting circuit or a quantum dot).

Please give an (as you claim) easy to understand reference where this absorption process is quantitatively described, so that it can be discussed here.

I don't believe that the situation will be very different. What you call the mean photon number will again just be the mean number of detection events in some time interval, if these are countable. And this number will depend on the time interval chosen, hence is not a property of the coherent state alone...

 

[f95toli]

Not sure why you are asking about coherent states, I explicitly stated that I was referring to Fock states above when talking about detection of single events (which would make no sense for an coherent state) .

Anyway. there is a (now fairly old) review paper which describes the system I am most used to

Blais, Alexandre, et al. "Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation." Physical Review A 69.6 (2004): 062320.

These systems can be used to create and manipulate just about any photon state you can imagine, including obviously simple Fock states.

If you want experimental data and more recent results just look at some of the papers that refer to the article above.

 

[A. Neumaier]

Not sure why you are asking about coherent states, I explicitly stated that I was referring to Fock states above when talking about detection of single events (which would make no sense for an coherent state) .

Well, in a low intensity coherent state one can distinguish single detection event, and you had said

the EM field is (usually) in a coherent state

in a context where $<n>$ is frequently $<<1$.

But thanks for the reference. For those readers who don't have access to Phys.Rev., a public preprint is at http://arxiv.org/abs/cond-mat/0402216.

 

[andresB]

Article arguing about the existence of photon wave function

http://www.cft.edu.pl/~birula/publ/CQO7.pdf

 

[A. Neumaier]

Article arguing about the existence of photon wave function

http://www.cft.edu.pl/~birula/publ/CQO7.pdf

This is the representation in terms of Silberstein vectors mentioned in post #4.

 

[andresB]

This is the representation in terms of Silberstein vectors mentioned in post #4.

True.

I posted it more for the interpretation the author gives to the equations

 

[Charles Link]

There is no difference between classical and quantum in the case of a single, localized detector.

However, for multi-point detection of correlated electromagnetic fields there is indeed a difference between the classical and the quantum description - not for coherent states where the two descriptions still match but for nonclassical states, for example for squeezed states (used, e.g., to describe parametric down-conversion). This difference is an expression of the fact that electromagnetism in Nature is described by QED and not by classical electrodynamics.

The main difference in the resulting detection statistics of the ionization events is usually described in terms of the notions of bunching and antibunching.

Thank you. This is quite interesting. It is somewhat difficult to stay completely current with all of the new progress that has occurred, but this gives me a good starting point.

 

[vanhees71]

Under natural idealization assumptions that are usually satisfied to good accuracy, the predicted and measured single electron-ionization rate is proportional to the intensity of the electromagnetic radiation field (not the electric field alone). This is the only unambiguous statement. Since this statement is valid both in the semiclassical treatment where there are no photons and in a full quantum field treatment (where - see below - the question of what a single photon is is ambiguous), the interpretation of this in terms of photons, though widely used, is questionable. Photodetection is measuring the intensity of the electromagnetic radiation field by measuring the electron-ionization rate, not by counting the number of emitted photons. If the intensity goes down, one simply needs to wait longer to get a reliable measurement of the intensity to a given accuracy. In my opinion, the detector events have no other meaning than this.

It is in some way analogous to measuring the output flow rate of a water faucet by (observing from far apart - not seeing the details) how many cups are filled in a given time. This works accurately in a short time if a lot of water flows, but if the faucet is only dripping it takes a long time before the number of cups accurately represents the water flow rate. I find this analogy helpful although it is only a classical analogy with a limited explanatory value as it lacks the randomness observed in the quantum case.

The problem with the notion of a single photon in a quantum field setting is that the quantum field typically is in some Heisenberg multiphoton state, and this single Heisenberg state gives rise to an electron-ionization rate rather than a probability. For example, in an ordinary laser the electromagnetic field is in a fixed coherent Heisenberg state, and by waiting long enough this fixed state gives rise to as many electron ionizations as one likes. This even holds when the coherent state has very low intensity. This is quite unlike what is assumed in the typical quantum optical experiments where everything is interpreted as if photons were particles just like nonrelativistic electrons, except that they move with the speed of light. There is a nontrivial interpretation step in going from the former to the latter description, one never analyzed in the literature (as far as I can tell).

The Heisenberg description (QFT) has a straightforward, almost classical interpretation of the state. To go from it to the interaction picture (QM) with the interpretation of an ensemble of particles (prepared or actually being - depending on the interpretation of QM used) in a corresponding state one has to ''invent'' individual photons with ghostlike properties that are completely unobservable until detection events destroy the ghosts and thereby prove their alleged existence. Extremely weird, this assumed picture of individual photons. One is left wondering why these ghosts are completely absent in the semiclassical description (which is quantitatively correct in the case of coherent light) although the detection events still herald their existence (if they herald anything). But if they herald nothing in the semiclassical setting, they also shouldn't herald anything in the quantum case. This then implies that the ghosts are completely unobservable. They can be eliminated without making any observable differences, and using Ockham's razor, they should be eliminated. My language is chosen accordingly.

In short, to make sure to measure single photons you have to prepare single-photon states at the very beginning. A detector click from a laser (coherent state), may it as "dim" as you like, does not necessarily mean one has detected a single photon and only a single photon.

 

[A. Neumaier]

In short, to make sure to measure single photons you have to prepare single-photon states at the very beginning. A detector click from a laser (coherent state), may it be as "dim" as you like, does not necessarily mean one has detected a single photon and only a single photon.

And in the case of prepared 1-photon states one knows the presence of the single photon only because of the preparation, not because of the detection! The detection only informs one about the presence of a local burst of the energy density (sufficient to fill the cup, in the classical analogy). Together with the knowledge about the preparation one infers that the photon arrived.

 

[f95toli]

And in the case of prepared 1-photon states one knows the presence of the single photon only because of the preparation, not because of the detection! The detection only informs one about the presence of a local burst of the energy density (sufficient to fill the cup, in the classical analogy). Together with the knowledge about the preparation one infers that the photon arrived.

That depends on the detection method. There are CQED methods that allow you to "sense" the number of photons in a Fock state dispersivly. This can e.g. be done using a system where you have an effective single photon Kerr term.

See e.g. http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.110.107001
for one of the more recent results in this area. This is probably not the best reference,. but all I could think of off the top of my head. .

Thinking of "conventional" detectors such as PN diodes when discussing single photon states is -in my view- just confusing. It is much better both theoretically and experimentally to use some sort of qubit (i..e. non-linear two level) based detection system since you can then distinguish between true single photon states and just "bursts of energy" as mentioned above.
Non-classical states of light is quite a bit research field so there are a lot of references available for those who are interested.

Edit: Note also that is is possible to perform tomography of photon states using "classical" detectors by using methods borrowed from radio engineering (IQ mixing) . It is is of course an indirect method since you can't "see" individual events, but as long as you are able to average over many events you can re-construct the state of the photons generated by your source.

 

[A. Neumaier]

There are CQED methods that allow you to "sense" the number of photons in a Fock state dispersivly.

How do they respond to dim coherent laser light?

 

[vanhees71]

And in the case of prepared 1-photon states one knows the presence of the single photon only because of the preparation, not because of the detection! The detection only informs one about the presence of a local burst of the energy density (sufficient to fill the cup, in the classical analogy). Together with the knowledge about the preparation one infers that the photon arrived.

Exactly. Now we completely agree :-)).

 

[A. Neumaier]

Blais, Alexandre, et al. "Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation." Physical Review A 69.6 (2004): 062320.

These systems can be used to create and manipulate just about any photon state you can imagine, including obviously simple Fock states.

I was talking about photons in a beam extending over a laboratory desk, and their detection. Your setting is quite different.
How do you get the photons from a photon beam into the cavity to be detected by your solid state devices?

 

[A. Neumaier]

Note also that is is possible to perform tomography of photon states using "classical" detectors by using methods borrowed from radio engineering (IQ mixing) . It is is of course an indirect method since you can't "see" individual events, but as long as you are able to average over many events you can re-construct the state of the photons generated by your source.

This can be done for the state of any source - one doesn't need any assumptions on whether or not it contains ghostlike particles.

The density matrix of an $n$-level system has $n^2-1$ independent degrees of freedom, and it is not difficult to find this many independent observables such that, from the measurement of their mean (by repeated observation of individual systems emitted by the source), the density matrix can be reconstructed by solving a linear system of equations.

 

[Elemental]

Sciencejournalist00,

On your original question,

Or is the electromagnetic wave considered to be the wavefunction of the photon?

consider the following. One performs a "box quantization" of the electromagnetic field by treating each cavity mode of the electromagnetic field as a harmonic oscillator. Here though you are actually starting with cavity modes of the electromagnetic field rather than modes of the Schrödinger wave function but it goes the same way. Singling out a particular mode, from Maxwell's equations it is found one can treat the electric field as the generalized coordinate and the magnetic field as the conjugate momentum. (Of course other approaches are possible including treating the magnetic field as the coordinate or more commonly quantizing the vector potential, but this approach illustrates the relation of wave function to EM field.)

You write the Hamiltonian for the EM field energy, then re-express it in terms of annihilation/creation operators exactly as is done for the harmonic oscillator. Presto, you obtain EXACTLY the same Schrödinger equation and equations of motion as you get for the harmonic oscillator. This has a wave function whose allowed modes have the same frequency as those of the electromagnetic field you started with! One can then define coherent states like you get from a laser, Fock (number) states, squeezed states, etc.

So the modes of the wave function sure look a lot like electromagnetic field modes. But what about its interpretation? For the harmonic oscillator, the magnitude squared of the wave function gives the probability of the position (coordinate) of the particle. For the quantized electromagnetic field as derived here, it does not give position probability, it gives the probability that the electric field will have a particular value. Not the same thing. However I've noticed in many quantum optics papers that the difference in interpretation between the Schrödinger wave function and the electromagnetic filed is glossed over.

 

[vanhees71]

One should stress that there is no consistent interpretation of relativistic wave equations in the sense of the "first quantization formalism", except in certain low-energy cases, where the physics is close to non-relativistic dynamics, i.e., only for massive quanta (e.g., bound states, where the binding energy is small compared to the masses of the bound quanta).

For massless quanta, and photons are with amazing accuracy observed as massless quanta, no such "wave-function interpretation" makes any sense. Since photons are described as massless spin-1 fields, they do not admit the definition of a position operator, and thus the position of a photon is not describable as an observable. So the localization of a photon is not even definable.

Another reason is that photons are very easily produced and destroyed. This leads to the description of relativistic quanta in terms of a quantum many-body, where quanta can produced and destroyed, and thus the number of quanta is not conserved. This is precisely what quantum field theory does in a very natural way and as explained in the previous posting #40.

The "box quantization" described in this posting is not only an interesting real physical situation in "nano physics", where one investigates single photons or states with a small number of photons in cavities ("cavity QED"), but also a convenient mathematical technique to quantize the (completely gauge fixed, using the radiation gauge) electormagnetic field canonically. In this case, you rather use periodic than rigid boundary conditions, because then you have a well defined momentum operator for the photons in the box-regularized version. At the end of the calculation you can take the infinite-volume limit, leading to the usual continuous momenta of photons in free space.

For a very detailed explanation, including a very good and intuitive introduction to the necessity for the field-quantization approach for relativistic particles, of this approach to photons, see Landau&Lifshitz, Course of Theoretical Physics, vol. IV.

 

[tade]

This is not correct. it is well-known that photon location cannot be described by a probability distribution

What about the double-slit experiment, and the interference fringe pattern?

Aren't the bright fringes bright because photons have a higher probability of being there, and the dark fringes are dark because the photons have a low probability of being there?

What about double-slit experiments where they reduce the intensity to one photon at a time, yet there is still an interference pattern?

 

[A. Neumaier]

What about the double-slit experiment, and the interference fringe pattern?

The point where a beam hits a screen determines only the transversal position, not the longitudinal one. Transversal position is reasonably well-determined in the paraxial approximation.

 

[tade]

The point where a beam hits a scree determines only the transversal position, not the longitudinal one. Transversal position is reasonably well-determined in the paraxial approximation.

I'm sorry, I don't understand this terminology.

How does this relate to probability waves, or that photons cannot be described by a probability distribution?

 

[A. Neumaier]

I'm sorry, I don't understand this terminology.

If a beam goes into the z-direction, the transversal position are the x,y coordinates only, while z is the longitudinal position. The paraxial approximation defines what a beam is in term of a field.

How does this relate to probability waves, or that photons cannot be described by a probability distribution?

There are commuting operators for transversal position but not for longitudinal position.

 

[tade]

If a beam goes into the z-direction, the transversal position are the x,y coordinates only, while z is the longitudinal position. The paraxial approximation defines what a beam is in term of a field.There are commuting operators for transversal position but not for longitudinal position.

Ok, is that considered a transversal probability distribution?

My understanding so far is described by this Wikipedia passage:

Sending particles through a double-slit apparatus one at a time results in single particles appearing on the screen, as expected. Remarkably, however, an interference pattern emerges when these particles are allowed to build up one by one (see the image to the right).

This demonstrates the wave-particle duality, which states that all matter exhibits both wave and particle properties: the particle is measured as a single pulse at a single position, while the wave describes the probability of absorbing the particle at a specific place of the detector. This phenomenon has been shown to occur with photons, electrons, atoms and even some molecules, including buckyballs.

This is the electron build up over time. What's the fundamental difference between electrons and photons in this scenario?

I posed a question about wavefunction and EM waves in my thread: https://www.physicsforums.com/threa...ld-and-probability-amplitude-of-waves.869530/
which is what led me to this thread.

 

[Nugatory]

Aren't the bright fringes bright because photons have a higher probability of being there, and the dark fringes are dark because the photons have a low probability of being there?

No, and the problem is in the words "being there" - wording it that way is already assuming that the photon has a position and hence a probability distribution for that position. There's no substitute for actually learning and using quantum electrodynamics, but if you want a non-rigorous and intuitive sort of understanding, you could say that the bright fringes are bright because there is a higher probability that the electromagnetic field will deliver energy to those areas, or you could say "appearing there" instead of "being there".

 

[tade]

No, and the problem is in the words "being there" - wording it that way is already assuming that the photon has a position and hence a probability distribution for that position. There's no substitute for actually learning and using quantum electrodynamics, but if you want a non-rigorous and intuitive sort of understanding, you could say that the bright fringes are bright because there is a higher probability that the electromagnetic field will deliver energy to those areas, or you could say "appearing there" instead of "being there".

I see.
Am I right to say that a photon cannot be described by a scalar wavefunction? (a wavefunction similar to the wavefunctions of the Schrödinger equation) That only a vector field can describe it?

 

[A. Neumaier]

I see.
Am I right to say that a photon cannot be described a scalar wavefunction?

The wave function is given by the Silberstein vector field.
You may wish to browse my theoretical physics FAQ.

 

[tade]

The wave function is given by the Silberstein vector field.

Ahhh, thank you.