Is the interference of Fullerene due to the center-of-mass approximation?

[craigi]

Can anyone offer a description of what's actually going on in the observed interference of Fullerene?

We're talking about molecules with a lot of complexity, with angular momentum, many internal degrees of freedom and that emit thermal radiation.

How do we explain that they remain coherent in order to produce intereference patterns?

What do we even mean by a coherent molecule?

Can we breakdown the wave equation of the molecule, in principle at least, and expect to see matching interference from its constituent particles or do we need a more complex picture?

How do we reconcile the fact that these consituent parts can have different relative positions with respect to each other on the different paths as well as potentially different frequencies and phases?

Do we expect that each constituent particle self-interferes or do we have sufficient flexilbility that different particles interfere with each other at the detector, rather than themselves?

Does the rotational symmetry of Fullerene play a significant role in this?

To what exetent do we consider the consitutent nuclei and electrons distinguishable?

Why do we not see infrared photon emission causing decoherence of the molecule? These experiments are conducted with the molecule beam encased in metal. Those photons can't be getting very far before being obsorbed by a macroscopic system, right?

I understand that some of this isn't necessairily well understood yet and is likely to be an area of ongoing research, but any insight would be appreciated.

 

[Maui]

Can anyone offer a description of what's actually going on in the observed interference of Fullerene?

Isolated molecules behave quantum mechanically, though in this particular case the isolation is of a different type and only potential information of the whichpath is isolated.

In quantum interference experiments, coherent superposition
only arises if no information whatsoever can be obtained, even in
principle, about which path the interfering particle took. Interaction
with the environment could therefore lead to decoherence.We
now analyse why decoherence has not occurred in our experiment
and how modifications of our experiment could allow studies of
decoherence using the rich internal structure of fullerenes.
In an experiment of the kind reported here, ‘which-path’ information
could be given by the molecules in scattering or emission
processes, resulting in entanglement with the environment and a
loss of interference. Among all possible processes, the following are
the most relevant: decay of vibrational excitations via emission of
infrared radiation, emission or absorption of thermal blackbody
radiation over a continuous spectrum, Rayleigh scattering, and
collisions.
When considering these effects, one should keep in mind that
only those scattering processes which allow us to determine the path
of a C60 molecule will completely destroy in a single event the
interference between paths through neighbouring slits. This
requires lpd; that is, the wavelength l of the incident or emitted
radiation has to be smaller than the distance d between neighbouring
slits, which amounts to 100nm in our experiment. When this
condition is not fulfilled decoherence is however also possible via
multi-photon scattering7,8,17.
At T < 900 K, as in our experiment, each C60 molecule has on
average a total vibrational energy of Ev < 7 eV (ref. 18) stored in 174
vibrational modes, four of which may emit infrared radiation at
lvib < 7–19mm (ref. 10) each with an Einstein coefficient of
Ak < 100 s21 (ref. 18). During its time of flight from the grating
towards the detector (t < 6 ms) a C60 molecule may thus emit on
average 2–3 such photons.
In addition, hot C60 has been observed19 to emit continuous
blackbody radiation, in agreement with Planck’s law, with a measured
integrated emissivity of e < 4:5 ð 6 2:0Þ 3 1025 (ref. 18). For
a typical value of T < 900 K, the average energy emitted during the
time of flight can then be estimated as only Ebb < 0:1 eV. This
corresponds to the emission of (for example) a single photon at
l < 10mm. Absorption of blackbody radiation has an even smaller
influence as the environment is at a lower temperature than the
molecule. Finally, since the mean free path for neutral C60 exceeds
100min our experiment, collisions with background molecules can
be neglected.
As shown above, the wavelengths involved are too large for single
photon decoherence. Also, the scattering rates are far too small to
induce sufficient phase diffusion. This explains the decoupling of
internal and external degrees of freedom, and the persistence of
interference in our present experiment.

http://qudev.ethz.ch/content/courses/phys4/studentspresentations/waveparticle/arndt_c60molecules.pdf

We're talking about molecules with a lot of complexity, with angular momentum, many internal degrees of freedom and that emit thermal radiation.

How do we explain that they remain coherent inorder to produce intereference patterns?

I would be much more worried about how that probability wave manages to approximate classical behavior.

What do we even mean by a coherent molecule?

Molecule wanna-be? No, seriously this is new stuff, it's totally puzzling.

 

[craigi]

http://qudev.ethz.ch/content/courses/phys4/studentspresentations/waveparticle/arndt_c60molecules.pdf

Excellent article. It offers a good description of why themal coupling to the environment doesn't introduce significant decoherence and goes on to suggest the role of rotational symmetry and indistinguishability.

Furthermore, we note the fundamental difference between iso-
topically pure C60, which should exhibit bosonic statistics, and the
isotopomers containing one13 C nucleus, which should exhibit
fermionic statistics. We intend to explore the possibility of obser-
ving this feature, for example by showing the different rotational
symmetry between the two species in an interferometer
23,24

I also, think that I can answer one of my own questions. Since the wavelengths of contituent particles differ from the de-Broglie wavelength of the molecule then the concept of the interference emerging from interference of the constituent particles seems to be invalidated.

This is leaves me with a bit of a knowledge gap as to how the wave equation, no matter how complex, can give rise to the composite particle with a wavelength and phase such that it can undergo interference and how the internal degrees of freedom don't distrupt the interference.

Does anyone know if such a theoretical understanding has been put forward or if it's still an open problem?

I would be much more worried about how that probability wave manages to approximate classical behavior.

Indeed. Whereas propogating the centre of mass and orientation of the composite particle isn't a problem, how the constituent particles retain the necessary structure in order to hold the molecule together is puzzling.

 

[DrChinese]

This is leaves me with a bit of a knowledge gap as to how the wave equation, no matter how complex, can give rise to the composite particle with a wavelength and phase such that it can undergo interference and how the internal degrees of freedom don't distrupt the interference.

Not sure there is a theoretical problem that is particularly "open" here. The article describes what processes cause the kind of decoherence that disrupts interference on 681, right column.

Or are you asking how atoms and molecules have quantum observables such as phase and wavelength?

 

[kith]

This is leaves me with a bit of a knowledge gap as to how the wave equation, no matter how complex, can give rise to the composite particle with a wavelength and phase such that it can undergo interference and how the internal degrees of freedom don't distrupt the interference.

De Broglie wavelengths are derived from momentum eigenstates or superpositions of such states. For a composite particle, you can construct the total momentum operator by simply combining the single particle momentum operators. Sure, the interactions between the constituents will affect non-stationary states of the composite particle. Whether this is a problem for interference experiments is a question of time scales.

Maybe this paper contains the treatment you are looking for (haven't read it).

 

[craigi]

http://arxiv.org/abs/1305.2417

Thanks.

From the abstract, that sounds like exactly what I'm talking about. I'll try to wade through the maths.

From a quick skim through it, they seem to ignore the internal structure of the molecule and I'm still not aware of the justification for that.

 

[craigi]

Not sure there is a theoretical problem that is particularly "open" here. The article describes what processes cause the kind of decoherence that disrupts interference on 681, right column.

Or are you asking how atoms and molecules have quantum observables such as phase and wavelength?

Yes. That's one part of what I'm asking. Without going into the maths, I can crudely justify this with beat frequency and flipping the sign on one of the components of momenta (presumably from squaring the wavefunction), or perhaps some other phoenonmena from superposition of waves. Though conceptually, I can't understand what is actually interfering with what, when we consider composite particles. Obviously, the composite particle is interfering with itself, but is it right to presume that this is a manifestation of a more microscopic interference?

I'm happy with Arndt et al's treatment of decoherence for their experimental arrangement. That answers that part of my question at least as well as I could've expected.

Most of the remainder of the question is about how we can treat molecules in the same way that we treat elementary particles, rather than as the distributed aggregate of their component elementary particles, evolving under much more complex dynamics. It's in this regard, that I'm wondering how a molecule can remain coherent to internal degrees of freedom, rather than through environmental decoherence.

Part of the problem may be that I've still got some kind of stuff kicking around in my head when I visualise these things, that is too much like the Bohr model of the atom, but I'm not sure what that is, or perhaps I'm visualising counterfactual definiteness, where there may be none.

 

[Maui]

Indeed. Whereas propogating the centre of mass and orientation of the composite particle isn't a problem, how the constituent particles retain the necessary structure in order to hold the molecule together is puzzling.

I like how the electrons around the nucleus of the constituent atoms are not moving but exist in a cloud and the cloud has the potential to show movement in any direction if something comes along to 'measure' that movement. As the experiment shows, whatever measurement is, it must be truly significant, if not fundamental and it's as if the act of measurement produces the movement(quite in line with your question on how the constituent particles retain the necessary structure in order to hold the molecule together). Heisenberg once said that "What we observe is not nature itself, but nature exposed to our method of questioning" which always made me ponder the nature of the questioning.

 

[craigi]

I like how the electrons around the nucleus of the constituent atoms are not moving but exist in a cloud and the cloud has the potential to show movement in any direction if something comes along to 'measure' that movement. As the experiment shows, whatever measurement is, it must be truly significant, if not fundamental and it's as if the act of measurement produces the movement(quite in line with your question on how the constituent particles retain the necessary structure in order to hold the molecule together). Heisenberg once said that "What we observe is not nature itself, but nature exposed to our method of questioning" which always made me ponder the nature of the questioning.

The electron cloud around an atom demonstrates the indistinguishability that I'm talking about. In a molecule, this cloud would also include the valence electrons. The nuclei, if all the same isotope, are indistinguishable from each other too.

In quantum physics, we regularly see angular momentum and angular velocity, but we don't see discussion of orientation. Presumably because it's not relevant to an elementary particle. Perhaps even with Fullerene, orientation isn't particularly relevant, due to its rotational symmetry, but there's only so far up the classical scale that you can go before orientation becomes a consideration.

Perhaps, what we might find, is that if we sum for our paths, over the probability density distribution for orientation and all configurations of internal excited states, then the wavefunction of the electron cloud exhibits exactly the interference that we get by treating the molecule as we would an elementary particle and similarly so for the nuclei. I'm yet to be convinced of that for either Fullerene or for molecules with no such rotational symmetry.

 

[mitchell porter]

I think it must have to do with the validity of a center-of-mass approximation.

Suppose you have a quantum object made of N interacting particles. Classically, instead of describing the location of the particles using N position vectors, you can instead use a position vector for the center of mass, and then relative position vectors for the locations of the N particles relative to the center of mass. You can perform the same change of variables for the N-particle quantum wavefunction, it just means that you rewrite psi so it depends on center-of-mass position and particle relative position.

Now suppose you're sending this N-particle object through a diffraction grating. The N-particle wavefunction will evolve in some way. I would think that you get recognizable interference effects if and only if the behavior of that N-particle wavefunction, expressed in center-of-mass and relative-position variables, can be closely approximated by a simpler wavefunction just employing the center-of-mass variable. This would depend partly on the internal dynamics of the N-particle object, and partly on the interaction between the N-particle object and the diffraction grating.

I'm not sure what the precise mathematical condition would be. Perhaps something like, lack of dispersion of the wavefunction, in the state-space directions corresponding to the relative-position degrees of freedom. This would mean that wavepackets which converged on the same value of the center-of-mass coordinate, would also be peaked around roughly the same values of the relative-position coordinates, so the two wavepackets would be converging on the same region of the combined coordinate space, and would therefore recombine and interfere.

 

[Jilang]

I think it must have to do with the validity of a center-of-mass approximation.

Suppose you have a quantum object made of N interacting particles. Classically, instead of describing the location of the particles using N position vectors, you can instead use a position vector for the center of mass, and then relative position vectors for the locations of the N particles relative to the center of mass. You can perform the same change of variables for the N-particle quantum wavefunction, it just means that you rewrite psi so it depends on center-of-mass position and particle relative
... This would mean that wavepackets which converged on the same value of the center-of-mass coordinate, would also be peaked around roughly the same values of the relative-position coordinates, so the two wavepackets would be converging on the same region of the combined coordinate space, and would therefore recombine and interfere.

Sorry if this is a bit off-thread, but is this the origin of Born's postulate? With the probability of detection of an electron ( by another electron always?) given by their joint amplitude in the centre of mass frame?