Re: Two-slit experiment
- From: "FrediFizzx" <fredifizzx@xxxxxxxxxxx>
- Date: Fri, 7 Jul 2006 22:27:26 -0700
"nightlight" <nightlight@xxxxxxxxxxxxxx> wrote in message
news:1152305215.661809.5310@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Timo A. Nieminen wrote:http://groups.google.com/group/sci.physics.research/browse_frm/thread/29a7934b4fb64637
http://groups.google.com/group/sci.physics.research/msg/386f48731520d145
If photons are waves, and massive particles are waves,
then why is the exchange of energy between matter and EM
fields quantised?
The exchange is quantized only for bound electrons, not for
free electrons. In the case of bound or spatially constrained
electrons, the available frequencies for the electron matter
field are discrete for the same reason that the frequencies
of constrained guitar wires are discrete. The discrete
spectrum is a property of the solutions of those types
of PDEs with those boundary conditions.
Of course, you may object that since Schrodinger equation
is linear, if Psi(x) is a solution, then a function C*Psi(x),
for any constant C is also a solution (with the same bound
state boundary conditions, zero Psi at infinity). Hence the
dynamical equations with their boundary conditions are
not sufficient mechanism capable of producing discrete
energy spectrum.
This objection to Schrodinger' s original interpretation
of his 'wave mechanics' was raised by Tomonaga (cf. [7]
pp. 25-31, where Dorling discusses and answers Tomonaga's
objection). In other words, there is an _additional_
condition, the normalization to 1 of the integral of
|Psi(x)|^2, which is used in order for Schrodinger equation
to yield discrete spectrum for bound states. Hence, this
is a question as to why is the electron matter field in
the atom normalized to have charge n*e, where n is an
integer (since without this discrete normalization the
solutions would have continuum for the energy spectrum,
with different energies corresponding to different total
charge integrals). There is presently no dynamical theory
(other than isolated speculations, including those
by Dirac and Barut) of charge quantization which could
deduce this quantization and the particular observed
charge values, so we have to leave this question alone,
even though it is certainly relevant for the origin of
the discrete energy spectra of bound states.
The answer by Schrodinger to this objection is that the
correct equations are not linear, but rather a nonlinear
PDE system of coupled Maxwell-Schrodinger/Dirac equations.
It is then easy to see that, for example, if this PDE
system has a solution Psi(x), then no other value C*Psi(x)
is a solution, unless |C|=1, hence the correct equations
for the full system (EM+matter fields) do have a preferred
value for the charge within the family of functions C*Psi.
While invalidating the objection at the fundamental level,
this answer still falls short of providing the possible
values (the n*e values) of the total charge. Some progress
toward a more complete answer has been made in 1990s, when
the existence of solition solutions of coupled Maxwell-Dirac
system with some electron properties was demonstrated
(cf. [8] and references there).
The photons emitted by the excited atom are simply the beats
between the (quasi-)stationary electron matter field frequencies.
There is nothing in the derivation of the spontaneous emissions,
in QED or semi-classically, that describes a localized or
pointlike object, or any kind of 'individual' hv kind of object,
departing the atom. All that is departing is an EM field
disturbance with the spectrum concentrated around the beat
frequencies of the electron field. Since the combined EM
and matter fields satisfy the energy-momentum conservation
(which follows from the invariance properties of the combined
EM + matter fields Lagrangian), if the atom state changed
between the two stationary levels, the total energy carried
by the EM field beats must be the difference of the energies
of these two stationary levels.
In the inverse process, which is simply a time reverse of the
emission, the electron matter field resonantly absorbs the beat
frequencies between its quasi-stationary levels from the EM field,
and the same energy-momentum conservation holds here as well.
Further, the EM field of spontaneous emission described in QED is
superposed of infinitely many free field modes (which are the QED
photons, described by plane waves with sharp energy-momentum
eigenvalues). The individual modes, or individual QED photons,
appear in those derivation only as mathematical artifacts
of particular basis arbitrarily chosen for perturbative expansion,
in the same manner that the plane waves appear in Fourier
expansion used in the semi-classical treatment.
The above physical picture was the original Schrodinger's view
of his wave mechanics (as described in his 1926 series of papers
introducing the wave mechanics). The (first) quantization was
in his view merely a transition from the classical particles
to classical matter fields, a natural completion of the process
initiated by Faraday and Maxwell. The matter fields are in
that picture real objects on par with EM field and the two
coupled fields are nonlinear classical system.
Although Schrodinger tried modeling the spontaneous emission
and radiative corrections based on the coupled Maxwell-KG
fields (he discovered KG eqations independently from Klein
& Gordon, along with several others), the numbers he obtained
were wrong even for basic spectrum of Hydrogen, so he gave
up on that approach and turned to the multiparticle QM
formalism (previously discovered by Heisenberg). It was only
in 1985-1992 that Schrodinger original program was carried out
to near completion by A. Barut and collaborators, under the
name "Self-Field Electrodynamics" (SFED). A brief intro with
key references (most available online) to SFED was posted
in an earlier s.p.r thread (and a followup):
http://groups.google.com/group/sci.physics.research/msg/bb1225c256c34c02
non-linear
The review of Schrodinger's original attempt (and reasons
it failed) is given in ref [3a] there, discussion of charge
quantization and how one might solve it within SFED is in [3h],
and meaning of the energy quantum hv from the wave mechanics
/ SFED perspective in [3j].
The proposal is just to address the claim that the photon
has a size equal to the spread of its wavefunction,...
The size or location of photon can only be defined as a
matter of arbitrary convention. QED photon is quantized mode
of the _free_ EM field with sharp energy-momentum, appearing
in perturbative treatment, just like the equivalent Fourier
transforms appear in the semiclassical or SFED treatments.
The QED photons have infinite spatio-temporal extent, thus they
are not elements of the formalism which map operationally to
the coincidence measurements in Quantum Optics. The Quantum
Opticians have their own QO photon, which is a different
entity than QED photon. The QO photon is a superposition
(and sometimes even a mixture) of infinitely many QED photons
with different energy-momenta, but with sharp photon number
eigenvalue (as discussed some more later). They are more
meaningful theoretical objects (EM field states) for QO since
they can be approximately localized and thus brought into the
operational correspondence with QO coincidence setups. But
the interactions between EM field and the matter fields,
such as photodections, are still described via QED photons.
The "single" QO photon can trigger multiple remote detectors
(each trigger caused by absorption of one QED photon), while
a single QED photon can be absorbed only once, thus trigger
only one detector.
The confusion on this key distinction (and its operational
interpretation) is the root of the miscommunications and
confusions between Quantum Opticians and the rest of physicists
as to what the experimental facts are. For example, while one
can say that a PDC source emits a single pair of QO photons
in a given time window, it is also true that the very same
EM field state contains infinitely many QED photons, hence
the number of possible photo-ionisations (which are the result
of the QED photon absorptions by electron's matter field)
per sampling window is not limited to one per member of the
PDC QO photon pair, as the QO photon terminology would suggest
(and which sometimes gets stated explicitly).
... and that detection probability is simply proportional
to E.
Informally speaking, the detection probability _is_ proportional
to the EM energy (the integral of intensity) incident on
the cathode in a given sampling time window. More formally,
it is proportional to the time integral of the expectation
value of the normally ordered product [E-][E+] of electric
field operators (negative and positive frequency components)
on the cathode. The normal ordering requirement is due to
the removal of the vacuum photon terms (1/2 photon per mode
in the free EM field Hamiltonian). This removal of vacuum
photons is operationally mapped by the QO counting rules
to the subtraction of the vacuum fluctuations effects
(the 'dark currents', which are partially suppressed by
the design which minimizes the dark rates, and partially
by the explicit subtractions of the background rates
from all actual counts & correlations).
Note though, that the electric field operator in Heisenberg
picture [E(x,t)]=[E+] + [E-] evolves via Maxwell equations
and it does not "collapse" or vanish from detector A cathode
when a remote detector B triggers, even though the combined
EM field state corresponds to a single QO photon. Its evolution
on the A cathode, thus the trigger probability of detector A,
is entirely independent of the events on a remote detector B,
or of anything outside of the backward light cone of the
A cathode in that time window.
The point of the experiments such as [1] is precisely to
try demonstrating that the probability is not proportional
simply to E, but that it drops to zero on detector A,
as soon as detector B triggers. As explained in [2], neither
experiment [1] nor any other has demonstrated such phenomenon.
Nor has it been shown to exist as a theoretical prediction
of QED model for PDC source. As demonstrated in the series
of papers [5], PDC source is completely classical source
regarding the photon statistics in any coincidence setup,
hence no such anticorrelation phenomenon can be observed
with PDC pairs even in principle (e.g. by trying to deduce
it via the full QED treatment).
You are welcome to bring up an experiment or a QED
prediction (the formalism plus the proper operational
mapping of the elements of the formalism to events in
a QO coincidence setup, counts, etc) that shows how
is the actual (non-adjusted) photo-count statistics
different in a beam-splitter experiment, such as [1],
from what a classical field model (such as Stochastic
Electrodynamics, cf. [5]) would predict based on simple
thresholding of the field energy measurements.
For experiment, you need to bring in the non-adjusted
data and how were the coincidence units configured,
so one can check whether any classical inequality was
violated by the actual counts properly collected.
For the theory part, show how the full QED dynamical
treatment of detection reduces the probability of
trigger on A, when a remote detector B triggers.
Glauber did that kind of calculation for a general
EM field state and any number of detectors in his
1964 Les Houches lectures, and no such reduction
in probability of A trigger comes out at the end
(see [2] for references and the discussion of the
Glauber's theory). Or try at least showing how such
QED treatment could do it in principle, even though
we may not be able to actually carry out such detailed
calculations. What is the device or mechanism in
the formalism which could do it?
In short, the original poster was essentially correct,
however informal his statement may have been.
Have a weak enough source so that in some time interval longer
than the time required for the detector to click and recover,
you only expect one photon to be emitted, or else there's
no point to the experiment. It's not intended as a "collapse
of the wavefunction" experiment.
Without "collapse" all you have is equivalent to measuring
EM field energy in some sampling windows and thresholding
the result to a binary value 0=no trigger 1=trigger.
To show that there is anything going on that would surprise
a 19th century physicist you need to show:
a) The EM field sampling windows are selecting coherent
wave packet fragments (called here fragments A and B).
b) The remote fragments A and B are equal (other than spatial
translation & rotation), hence they can interfere with nearly
perfect visibility (the "dark" regions have negligible counts,
while the maxima have nearly full sum of the A and B counts).
c) The probability P(AB) of two coincident triggers in a sampling
window is smaller than the product of individual detectors
trigger probabilities in the same sampling window
P(AB) < P(A) P(B). The probabilities are estimated from
the counts:
N=# of sampling windows,
N(AB)=# of coincident triggers of DA and DB
N(A)=# of triggers of DA,
N(B)=# of triggers of DB,
via P(x)=N(x)/N, for x=A,B,AB. Hence the classical counts
must satisfy inequality (cf. eqs. (7),(14) in [1]):
g2 = N*N(AB) / N(A)*N(B) >= 1 .... (1)
The violation of (1) is the "collapse" criterium i.e. a mere
trigger of a detector DA makes the trigger of DB in the same
sampling window less likely (smaller value N(AB)), as if the
trigger DA has collapsed the remote wave packet B. This is
the conjectured phenomenon that [1] was trying to demonstrate.
Experiment [1] added a third coincidence unit to count N(AB)
(even though the results collected from DA and DB already
provide the results for AB coincidence), then, as shown
in [2], the signal delays feeding this unit were configured
so that the signals fell well outside of the 'ready' state of
the unit, resulting in virtually zero counts reported for N(AB).
{ Further, for whatever reason, this particular incorrect delay
value of 6 ns for the third unit, which is the critical parameter
responsible for the N(AB) count coming out as essentially zero,
was explictly brought up 11 times in the article [1], making it
by far the most repeated and emphasized single figure describing
the entire experiment and all of its settings. It seems the
authors wanted very badly to make sure it "worked" the way they
imagined those finicky "single photons" ought to work, and not
the way they actually work.}
The properties (a) and (b) are verified by bringing the two fragments
into a common region to interfere. The property (c) is verified by
counting triggers & non-triggers of detectors DA and DB
within the sampling windows. Only the combination of all three (a)-(c)
would surprise a 19th century physicist. Important details to watch:
1. The sampling windows used to verify (a) & (b) via interference
must be the "same" sampling windows as those used in (c)
i.e. they are the "same" with at most the spatio-temporal
translation (for given optical paths) which is required to bring
the non-overlapping fragments A and B from the experiment (c)
into the same region needed for interference experiment (a)-(b).
Further, in case of three coincidence units (such as the setup
of experiment [1]) the sampling windows used to count N(A)
and N(B) must be exactly the same sampling windows as those
used to count N(AB).
Otherwise it is trivial to create a perfect interference (a)-(b)
and a perfect anticorrelation (c), if one misaligns the sampling
windows used for counting N(AB) in (c) so that the signal peaks
of A and B are partially shifted (for the N(AB) coincidence unit)
from the center of the sampling window, in opposite directions.
That will reduce N(AB). If one simultaneously does use well
centered (on the signal peaks of A and B) sampling windows for
counting N(A) and N(B), one can obtain an apparent violation
of (1) (which is a perfectly classical 'violation').
In practice one would create the sampling windows using photon
pairs, such PDC or atomic cascade pairs, where one photon of
the pair defines the EM field sampling window (usually 1-3 ns
centered around its trigger), while the other photon is used
for measurements (a)-(c). See [1] for how this can be done.
2. In order to measure the relevant counts N, N(A), N(B) and N(AB)
one cannot discard cases when neither DA nor DB trigger in the
sampling window (see [1]). Otherwise, a perfectly classical
sequence of pair results (A,B), such as the results obtained
by tossing a pair of coins A and B (and assuming large enough
N, so that the statistical fluctuation ~sqrt(N) is negligible
compared to N, when the counts are used in (1)):
#(0,0)=N/4
#(1,1)=N/4
#(1,0)=#(0,1)=N/4
where N is the number of sampling windows (or pair tosses) and
#(A,B) is the number of occurrences of the result pair (A,B),
will violate the classicality condition (1). Namely, with all
sampling windows properly counted (and defining heads=1 and
tails=0 as trigger and non-trigger):
N(A) = #(1,1) + #(1,0) = N/2
N(B) = #(1,1) + #(0,1) = N/2
N(AB) = #(1,1) = N/4
which satisfies (1) as equality: N(AB)*N / N(A)*N(B) = 1.
On the other hand, with rejection of (0,0) events, we have a
new count of sampling windows: N' = N - #(0,0) = 3*N/4, while
the other 3 counts #(1,1), #(0,1) and #(1,0) remain unchanged.
Using this adjusted N' as the total count of sample N in (1),
we get: N'*N(AB)/N(A)*N(B)= 3/4 < 1. Hence by discarding (0,0)
events, we can get a "violation" of classicality by tossing a
pair of coins. Therefore, QO experiments aiming to show the
violation of classical inequality (1) cannot discard the
(0,0) results either, otherwise even the plain coin tossing
yields a non-classical statistics.
Your suggestion to observe single gamma photon detections shows
nothing that would surprise a 19th century physicist. If P(A) and
P(B) are small enough (weak source or low yield detectors or short
sampling window), and you discard (0,0) data, you can have g2
arbitrarily close to 0 i.e. you get an apparent perfect particle-like
anticorrelation (whenever there is a trigger, it is almost always
a single trigger), while still getting a perfect interference in
the (a)-(b) experiment.
Note that PDC source cannot, even in principle, violate (1), since
the laser pump used is at best a Poissonian source, hence the number
of pairs in each sampling window is at best Poissonian (cf. [3],[4]
for actual PDC pair statistics, experiment and theory). For the same
reason, PDC source cannot produce violation of Bell inequality (as
demonstrated in a series of papers by T. Marshall, E. Santos, et al,
[5]), or do anything non-classical in Quantum Computing/Encryption.
More general problem is that a classical laser pump which yields
photon pairs via some interaction Hamiltonian (modeling some
optical medium), cannot even in principle (due to the linearity of
the interaction Hamiltonian operator acting on a coherent state)
produce sub-Poissonian pair statistics. Blotting out events randomly
from a sequence of Poissonian events, always yields another Poissonian
sequence, just sparser.
One needs to read carefully the claims by Quantum Opticians, though,
since they use different concept of "photon" than QED photon (the
one used to mediate the fermion interaction in QED perturbation
expansion). While QED Photon is a quantized mode of free EM field
with sharp energy-momentum values, the QO photon is any superposition
of infinitely many QED photons with the same value of photon number
operator (and otherwise arbitrary energy-momentum). Thus when Quantum
Opticians claim they have demonstrated "single photon" source which
uses classical pump (laser), that single QO photon is still at best
a coherent superposition of infinitely many QED photons. Since the
photon relevant for describing the results of the interaction (such
as photo-ionisation, or generally any detection) is the QED photon,
the actual detector counts remain at best Poissonian, hence anything
deduced from these counts or their correlations is classical. The
term "non-classical" in QO thus refers to violation of classical
inequalities, such as (1) or Bell inequality, by the adjusted counts
(via post-selection i.e. after seeing the results), such as N' in the
coin tossing example. Even though the coincidence counts are adjusted
after seeing the outcomes (from remote detectors), the Quantum
Opticians specializing in the "non-classicality" demonstrations have
declared that this procedure represents a "fair sampling" (cf. [6]
on absurdity of such wishful labeling), hence they are free to
extrapolate the actual Poissonian results to the hypothetical "ideal"
setup in which the EM field states have sharp number of QED photons
(hence, yield the sub-Poissonian detector counts).
Hence, at best, one should take QO "non-classicality" as a
convenient term of art describing certain states of EM field,
but which has no relation to the genuine non-classicality,
as understood by physicists.
Generally, though, these QO non-classicality claims are ambiguous,
relying on omissions in describing the counting & subtraction
procedure, and publishing only the adjusted figures without a
caveat, or even a mention of subtractions/adjustments or the
"fair sampling" assumption, rationalized (when questioned) that
it is all a well known QO counting convention. Unfortunately,
if you're a student or a physicist not specializing in QO,
trying to learn what the actual empirical facts on quantum
"non-classicality" are, you don't know the "well known QO
counting conventions". You take that "counts" are what you
get when you count the detector triggers and then end up
wondering how can that be. The simple answer is, once the
peculiar QO terminology is translated into the language of
conventional physics -- it indeed can't be and it is not
what is being observed in Quantum Optics.
Wow! Very nice. I only have four words to say. Quantum Vacuum Charge
rules!
FrediFizzx
Quantum Vacuum Charge papers;
http://www.vacuum-physics.com/QVC/quantum_vacuum_charge.pdf
or postscript
http://www.vacuum-physics.com/QVC/quantum_vacuum_charge.ps
http://www.arxiv.org/abs/physics/0601110
http://www.vacuum-physics.com
-- Referenceshttp://arxiv.org/find/quant-ph/1/AND+au:+Santos_E+au:+Marshall_T/0/1/0/all/0/1
1. J.J. Thorn, M.S. Neel, V.W. Donato, G.S. Bergreen,
R.E. Davies, M. Beck
"Observing the quantum behavior of light in an
undergraduate laboratory"
Am. J. Phys., Vol. 72, No. 9, 1210-1219 (2004).
Preprint:
http://marcus.whitman.edu/~beckmk/QM/grangier/Thorn_ajp.pdf
Experiment Home Page: http://marcus.whitman.edu/~beckmk/QM/
2. Physics Forum thread with discussion of [1] and more references:
http://www.physicsforums.com/showthread.php?t=71297
3. T. S. Larchuk, M. C. Teich, and B. E. A. Saleh
"Statistics of Entangled-Photon Coincidences in
Parametric Downconversion"
Ann. N. Y. Acad. Sci. 755, 680-686 (1995)
http://people.bu.edu/teich/pdfs/ANYAS-755-680-1995.pdf
4. A. Joobeur, B. E. A. Saleh, T. S. Larchuk, and M. C. Teich
"Coherence Properties of Entangled Light Beams Generated
by Parametric Down-Conversion: Theory and Experiment"
Phys. Rev. A 53, 4360-4371 (1996).
http://people.bu.edu/teich/pdfs/PRA-53-4360-1996.pdf
Other M.C. Teich papers of interest:
http://people.bu.edu/teich/cv.html#TECHNICAL
5. T. Marshall & E. Santos arXiv preprints (with peer
reviewed versions cited there):
6. Emilio Santos "Optical tests of Bell's inequalities not
resting upon the absurd fair sampling assumption"
arXiv quant-ph/0401003 http://arxiv.org/abs/quant-ph/0401003
Additional preprints by Adenier & Khrennikov:
http://arxiv.org/find/quant-ph/1/ti:+EXACT+fair_sampling/0/1/0/all/0/1
7. J. Dorling "Schrodinger's original interpretation of the
Schrodinger equation: a rescue attempt"
in "Schrodinger, Centenary celebration of a polymath"
ed. C. W. Kilmister, Cambridge Univ. Press 1989
8. C. S. Bohun, F. I. Cooperstock
Dirac-Maxwell Solitons
arXiv preprint physics/0001038
http://arxiv.org/abs/physics/0001038
.
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