Built on Facts

In every cop drama there’s a scene where a suspect is being questioned in an interrogation room. The room contains a large mirror, and behind that mirror the detectives and district attorneys are observing and arguing about the progress of the case. The mirror is a two-way mirror.

These kinds of mirrors aren’t complicated. Light shines on them, and some fraction is reflected back while some fraction passes through. The suspect in the brightly lit room can’t see the dark room beyond the mirror because the bright room light washes out the much smaller amount coming from the adjacent dark room.

We use the same concept in the lab. A device called a beam splitter is essentially a one-way mirror. They come in various types and styles for various purposes, but let’s pretend we have one that reflects exactly half the incoming field and transmits the other half. I’ll draw a very simplified schematic of this sort of mirror:

i-662c10d08ced2ff5d945915972a5d32d-bs1.png

Here the incoming light is a, the half that’s reflected is b, and the half that’s transmitted is c. Dig into the classical electrodynamic details of this and you’d find that the transmitted wave will generally undergo a pi/2 phase shift, but that’s details. For classical light as described by Maxwell’s equations this works just fine. Since pretty much all light is produced either incoherently by light bulbs and the sun or whatnot, or coherently by lasers and related devices, our description of the beam splitter can pretty much stop there.

But if we’re dealing with very small amounts of light – down to the level of individual photons – it turns out this description fails to work properly. A photon is a discrete quantum of light, so it can’t be split in two. Any single photon you shoot at the beam splitter will instead have a 50:50 chance of going one way or another. For large numbers of photons this reproduces the classical case, but for small numbers of photons this statistical description is required.

That’s not so strange. What’s strange is what happens if you shoot two photons of the same frequency at the beam splitter at the same time, one from the left on the picture and one from the top. Then there’s three possibilities: both photons might exit via path b, both photons might exit via path c, or one photon could take b while the other takes c. So the situation we have is this:

i-0886fbb51e19c574c8913af0559ea08a-bs2.png

In our particular situation we have a 1-photon state at input “a”, a 1-photon state at input d, and unknown states that we have to calculate at outputs b and c. Now we have one of those cases where doing the math is easy, but for the math to make any sense I’d have to teach you how to quantize an electric field. That’s is a very math-intensive project, so we’ll skip it for now. What we find is that the two possibilities “both photons are reflected” and “both photons are transmitted” have opposite signs in their contributions to the process “one photon in each output”. As such the amplitude of that process ends up being zero. The only possibilities left are “both exit via b” and “both exit via c”.

But until you actually measure where the photons are, you have a superposition of those two states. If you find that one of the photons is in path b, you know for a fact that you will also find the other photon in path b. And if you find one in path c, you know the other one is also in path c. Until you measure, you can’t know which one of those cases it will be.

Commonly these processes are described by breathless popular press articles like the article we talked about earlier this week as though simultaneously both photons are in path b and both photons are in path c. After all, the article described the oscillator in the experiment as both vibrating and not vibrating. While to some extent it’s just a matter of semantics, I feel that kind of description is more confusing than helpful. A quantum superposition is not two contradictory things magically happening at the same time, it’s a combination of different and sometimes mutually exclusive states – only one of which will actually end up being observed. Now these states are real; each state in the superposition can interact with the other states and affect their probability of being observed once the system actually interacts with something. But the actual observation will only involve one of these states on a probabilistic basis. Quantum mechanics is an odd and frequently counterintuitive theory, but like every other accepted theory in physics it is rigidly bound by the rules of mathematics and logic. It’ll give you strange results, but it’ll never give you impossible ones.

UPDATE: Reworded this paragraph slightly as per suggestion in the comments.

Comments

  1. #1 Alan B
    April 9, 2010

    If I did want to learn the mathematics of “how to quantize an electric field”, would you be able to recommend a book or two that would cover that? I have an EE background, and a solid understanding of classical EM, so I think I could handle an upper-level undergraduate physics textbook.

    I’ve always been curious about this, and not-quite-satisfied with pop-science books like Brian Greene’s or similar – actually, until reading this post I was under the impression that the wave-particle duality was simply a mystery. I thought we had perfect math for waves OR particles, but not both.

  2. #2 Anne
    April 9, 2010

    I don’t think I agree with the claim that “A quantum superposition is not two contradictory things magically happening at the same time, it’s two contradictory things having each a certain probability of being observed once the system actually interacts with something, in this case a photodetector. That’s all.”

    If it were just a question of two different things happening, in which the question of which happened wasn’t resolved until you observed the system, then there’d be nothing weird about it, and philosophers could argue endlessly about whether both things “really” happened. But quantum superpositions are weird because we can measure the effects of both things happening. In the double-slit experiment, it’s not just an academic argument about whether the electron is in a superposition of “gone through left slit” and “gone through right slit”; we can measure interference between the two possibilities (so that, for example, there are places the electron will never hit). We can’t dismiss superpositions as a philosopher’s abstract distinction because they have concrete effects.

    Whether you describe it as “two contradictory things really happening at the same time” or not, it’s clear that something peculiar is happening: the two contradictory things affect each other, and the system does not behave as if either one of them had just happened by itself.

  3. #3 Matt Springer
    April 9, 2010

    Alan, I’d recommend Introductory Quantum Optics, by Gerry & Knight. I’d say a basic knowledge of quantum mechanics would be very helpful, though you would probably get the gist of it even with just a classical E&M background.

    Anne, your points are entirely correct. But what I’m trying to drive home is the fact that though these mutually exclusive states exist and effect each other, the actual observable quantities can never be two contradictory things at once. For instance in your double slit example, though the two slit probabilities interfere you can never actually observe a single electron in both slits simultaneously.

    But it’s true that the vibrating/not vibrating oscillator really is in a superposition rather than just in a single eigenstate of which we happen to have imprecise knowledge. The point is that once we observe it, we’ll find it to be in one state or the other. I was probably not quite clear in the post, so I may reword it.

    EDIT: I’m still not super happy with my rewording, because the double slit experiment is not easily described by the representation I used. There the observation picks from a continuum of states of the form “electron hits the detecting surface at a point (x,y)” rather than just from the possibilities “electron went through slit 1″ and “electron went through slit 2″. The relationship between states and observables there is more complicated than in the Fock state basis I’ve used to describe the beam splitter.

  4. #4 Peter Morgan
    April 9, 2010

    “A photon is a discrete quantum of light, so it can’t be split in two.” I beg to disagree, insofar as we can superpose a vacuum state with a state that is created by the action of a single creation operator a_f^\dagger on the vacuum in continuously varying fractions,

    (\lambda+\mu*a_f^\dagger)|0>
    —————————————-
    sqrt[|\lambda|^2+|\mu|^2*(f,f)],

    where f(x) is a test function that describes the wave number distribution of the “single photon state”, and (f,f) is the inner product <0|a_f a_f^\dagger|0>. [I note that this equation also reveals that a state in the Fock space of free electromagnetic field states that is created by the action of a single creation operator on the vacuum can have a spread of wave numbers, so we can’t talk of a photon having a single wave number.]

    Of course producing such a state in the lab is not simple to do, which is why my disagreement is qualified. Your statement that “It’ll give you strange results, but it’ll never give you impossible ones” might need a qualification that we have to use the “right” Hilbert space and the “right” Hamiltonian to get a QM model that is empirically useful as a model of a real experiment. Whether we can construct an experiment that produces experimental results that correspond as closely as we like to any Hilbert space and any unitary evolution is surely a deep question in the foundations of QM.

    In terms of an over-simple analogy, we might argue that in principle we can produce any initial condition for an infinite (classical) string that has finite energy (QFT is associated with infinite space-time, so an infinite length string is more comparable than a finite string). The fact that we can decompose that initial condition into Fourier components does not make the initial condition discrete. The introduction of probability stretches this analogy to breaking point, in naive mathematical terms by introducing tensor products, but the possibility of constructing continuously varying superpositions and mixtures of arbitrary states leaves the quantized electromagnetic field with no essential discreteness.

    For spinor fields, the situation is different, so that there are superselection sectors that cannot be continuously superposed and there is a more robust discreteness, corresponding in the simplest case to electric charge.

    Concerning Anne#2, I also have reservations about the aspect she picks up on.

  5. #5 Matt Springer
    April 9, 2010

    Also good points. My description of the single-wavenumber Fock state ideal beamsplitter is not a complete one. And even so, of course there exist processes like parametric down-conversion that can take a single photon to a two-photon state of half the energy each. Still, insofar as we’re willing to accept the simplification it’s a good description of experiment.

    I have updated the post as per Anne’s concerns. As I mentioned above I’m still not thrilled with my own wording, but I think it’s better to err against the misconception that quantum mechanics allows contradictory things to be observed simultaneously.

  6. #6 Peter Morgan
    April 9, 2010

    “there exist processes like parametric down-conversion that can take a single photon to a two-photon state of half the energy each”. Right (apart from the quibble –which could be substantial in another context– that usually the down-conversion is a transformation of a coherent superposition of many-particle states that is emitted by a laser into an approximately two-photon state), but it’s as well to ask what this does.

    A “two-photon state” is orthogonal to all “one-photon state”s, so a two-photon state does not cause a detection event in a “one-photon detector”. In practice, therefore, a detector is more a “one-photon extractor”, insofar as it responds to energy that is part of a two-photon state as well as responding to energy that is present as a one-photon state. A two-photon state + initial internal degrees of freedom of the detector changes, perhaps, into a one-photon state + modified internal degrees of freedom of the detector. The “perhaps” gives notice that, inter alia, we generally don’t know in detail what we measure using a given apparatus, we have to use other preparation and measurement apparatuses and construct models for many experimental apparatuses that are mutually consistent.

    An Avalanche Photo Diode (APD) exhibits different statistics of thermodynamic transitions if it is placed near a down-conversion crystal than if it is isolated from light sources, but considerable power from the mains is used by the APD to enable it to make discrete transitions at all (the avalanche events are, in some sense, amplification events, which uses external power to amplify modulations of the fluctuations of the quantized electromagnetic field into more-or-less discrete thermodynamic transitions). Supposing that we can construct a perfect two-photon state, how do we know whether the state of the electromagnetic field after a thermodynamic transition of an APD is a pure one-photon state or a non-trivial superposition of many two-photon states, one-photon states, and the vacuum? Remember that we can’t call an APD anything as simple as “a single-photon detector”, because it is responding here to a “two-photon state”, so there is no straightforward a priori expectation of what the electromagnetic field state after the APD event might be. It is true that we have observed a single event, but that single event would not have happened if we hadn’t carefully placed the APD, so the event is at least as much a property of the APD and its power supplies as of the quantized electromagnetic field that it measures.

    From a different point of view, the output from a down-conversion crystal also has the property that one may observe correlations between events in two different APDs that one does not observe from a generic light source. In particular, we may observe events that are closely correlated in time. That is, a down-conversion crystal transforms the electromagnetic field output from a coherent electromagnetic field, with the various correlations that implies we would observe using a single or multiple APDs, into a state in which the spectra of the correlation functions between APD event timings are different.

    What the down-conversion does: it transforms the spectra of correlation functions between whatever compatible measurements we make. The correlation functions in question are a function of multiple dimensions, describing the locations and orientations of multiple measurement devices, so that the state space is substantially different from the state space of (non-stochastic) classical electromagnetism.

    Obviously this is work in progress. I called it a day at this point because I can feel the account getting (even more) out of control, blog comments being what they are. You make good tries at explaining complex things well, which, as you see, I do not. Congratulations if you read through this and got something from it. I hope you discovered some doubts.

  7. #7 Nathan Myers
    April 9, 2010

    The danger of popular interpretations isn’t that people will think impossible events may be observed. Regular people can understand that anything observed is manifestly not impossible, just (possibly) weird. Furthermore, as noted, systems really do behave as if an impossible outcome were in play. What people need to be guided away from is the very natural inclination to inchoate hidden-variable theories — that what we observe is merely a shadow of unobservable reality. I have frequently caught professional physicists espousing cryptic versions of such models.

  8. #8 Ryan
    April 9, 2010

    When I saw the headline, I thought it was going to be about Tony Shaloub using Heisenberg’s Uncertainty Principle to concoct his defense of Frances McDormand in the excellent Coen Brothers film The Man Who Wasn’t There. It’s a pretty entertaining scene in which he butchers the physics and says that the harder you look at something, the less you know so how can we really know what happened. The Coens revisit the same theme in a slightly different manner in their last movie A Serious Man.

    This is all very interesting however.

  9. #9 deadatheists
    April 10, 2010

    DEATH TRAP

    ******************************************

    abcnews.go.com/Nightline/FaceOff/
    *****************************************

    THE REAL QUESTION:

    DOES ATHEISM HAVE A FUTURE?

    AND THE ANSWER – NO!

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    but you LOST…

    THE DEATH OF ATH*ISM – SCIENTIFIC PROOF OF GOD

    engforum.pravda.ru/showthread.php?t=280780

    Einstein puts the final nail in the coffin of atheism…

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    atheists deny their own life element…

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  10. #10 TheDude
    April 10, 2010

    How do you send two photons at the same time? What happens in the experiment if they are timed slightly different?

  11. #11 Peter Morgan
    April 10, 2010

    TheDude, there are so many interpretations of QM that it is presumptuous of me, but I recommend that you don’t think of “sending” photons, whether one or two, but instead think of there being an almost amorphous “quantum field state” in the space between the parts of the apparatus that prepare the quantum field state and the parts of the apparatus that from time to time manifest events. The events that people commonly say are caused by particles are only discrete because those parts of the apparatus are designed to manifest discrete events and the events happen where they happen because those parts of the apparatus were put just there and not in other places. [Those interested in taking this kind of approach can try Art Hobson’s “Teaching quantum physics without paradoxes”, The Physics Teacher, Feb 2007, pp. 96-99, available at http://physics.uark.edu/hobson/papers.html

    You ask, “What happens in the experiment if [the photons] are timed slightly different?” If you know your relativity, you’ll remember that there are only three possibilities, space-like separation, light-like separation, and time-like separation of events. In classical special relativity, if something slower than light could travel from one event to another then the two events are time-like separated, if only light could travel from one event to another then the two events are light-like separated; otherwise, two events are space-like separated. Whether two events are “timed slightly different” is not the question to ask, in a relativistic theory, because if two events are space-like there is an inertial observer for whom the events are simultaneous.

    If we insist on speaking in terms of particles, as Physicists commonly do, with the unspoken proviso that we’re really talking about quantum field states and observables, then if we see discrete events happening consistently at space-like separation from each other (that is, at the same time in some relativistic coordinate system), then they must have been caused by two different particles, because particles by definition cannot travel faster than light. It is precisely that we observe two (or N) events at space-like separation from each other that causes us to say that a two (or N) particle state has been observed. If at different times we observe different numbers of events at space-like separation, it may be that the state is a superposition or mixture of two-, three-, or N-particle states, or that the devices that manifest events are not 100% efficient (although the idea of efficiency suggests that there is something really there, which is contrary to older interpretations of QM), or there may be some other reason for the varying number of more-or-less simultaneous events seen at different times.

  12. #12 Anonymous Coward
    April 11, 2010

    Re: TheDude at #10:

    Yes, the photons have to be overlapping in time.

    Re: The original post:

    > Dig into the classical electrodynamic details of this and you’d find that the transmitted wave will generally undergo a pi/2 phase shift.

    Huh? At what locations are you measuring the phases that you have the transmitted waving having a pi/2 phase shift? It’s true that by choosing your two locations you can get any phase shift you want, but pi/2 is a pretty unconventional choice for a transmitted wave, no? Most intro textbooks I’ve seen choose a phase shift of 0 for transmitted waves.

    Also, re: the original post:

    Just to try to clarify a subtle point: the distinction between Hong-Ou-Mandel-type interference at a beamsplitter and “regular” interference is NOT due to the intensity of the interfering beams – NOT “if we’re dealing with very small amounts of light” – but instead due to the TYPES of beams. If I interfere two coherent or thermal states (even if they each have a mean photon number of one) the effect you mention will not be observed. If two Fock states are interfered on a beamsplitter, Hong-Ou-Mandel-type effects will be observed even if the photon number is large (although in practice it’s difficult to generate Fock states of large N).

  13. #13 Dov Henis
    May 23, 2010

    Science-Religion Entanglement?

    A. From “Physicists entangle five particles, each existing in two states simultaneously.”
    http://www.sciencenews.org/view/generic/id/59217/title/Record_number_of_photons_lassoed_into_a_quantum_limbo%C2%A0

    – Record number of photons lassoed into a quantum limbo, entangling five light particles possessing the weird feature of “superposition.”

    – Superposition is a condition in which a light particle, or photon, exists in both of two possible states at once.

    B. From “N a t u r e o f R e a l i t y”
    http://www.norlabs.org/qm.html

    – Richard Feynman observed: A philosopher once said that it is necessary for the very existence of science that the same conditions always produce the same results. Well, they don’t!.

    – Quantum and macro realities are one and same. There is only one physical structure in the universe and this physical structure cannot follow two different sets of law.

    – It appears to be very difficult task to construct a (unified, TOE) theory, but then it is not easy to understand the truth.

    – We believe that a physical entity can r e m a i n in two different physical states simultaneously even though we know that our field theories rule out any such possibilities.

    – Presently we must limit ourselves to computing probabilities. We suspect very strongly that it is something that will be with us forever, that this is the way nature really is.

    C. Heraclitus: “You cannot step twice into the same river.”

    D. EOTOE: E=Total[m(1 + D)]

    – Since the cosmic E/m superposition resolution (the BB) cosmic m and D vary/flow continuously. We do not know if the E/m superposition “remained” more than 10^-35 seconds. We do know that since inflation there are never “same conditions”.

    – Cosmic “physical structures” do not “follow sets of law”. Their observed repeated behaviours are summarized by us as “nature’s laws”.

    – When we “believe that a physical entity can “r e m a i n in two different physical states simultaneously” we ENTANGLE OURSELVES in science-religion states.

    – It appears to be difficult to convince people that science and religion are not entangled and that EOTOE is indeed an Embarrasingly Obvious Theory Of Everything that advances understanding of nature’s reality.

    Dov Henis
    (Comments From The 22nd Century)
    03.2010 Updated Life Manifest
    http://www.the-scientist.com/community/posts/list/54.page#5065
    Cosmic Evolution Simplified
    http://www.the-scientist.com/community/posts/list/240/122.page#4427
    “Gravity Is The Monotheism Of The Cosmos”
    http://www.the-scientist.com/community/posts/list/260/122.page#4887

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