# Violating conservation of momentum and its resolution

• A
Gold Member

## Main Question or Discussion Point

The process is known as counter-propagating Spontaneous Parametric Down-Conversion (CP-SPDC).

In regular SPDC, a photon from a (pump) laser enters a transparent nonlinear crystal at rest, and gets converted into a pair of photons whose total energy and momentum add up to that of the original (pump) photon.

$E_{p}=\hbar \omega_{p} = \hbar \omega_{1} + \hbar \omega_{2}$
$\vec{p}_{p}=\hbar \vec{k}_{p} = \hbar \vec{k}_{1} + \hbar \vec{k}_{2}$

In CP-SPDC, the energy of the photon pair adds up to the energy of the original pump photon, but the momenta of the photon pair adds to zero since they are propagating in opposite directions. Since the total energy of the field is the same before and after the interaction, but the momentum is different, it would look like there is a violation of conservation of momentum, since the crystal gets a momentum kick without changing its energy.

This is a process that really exists, and has been demonstrated in the laboratory.

I can work through the math and show that the technique known as quasi-phase matching (a way of periodically switching the crystal structure of the medium) allows one to add a term that offsets the momentum imbalance, but that much is theoretical.

If anyone out there is intimately familiar with this process or has a photon-level understanding of quasi-phase matching, what is physically going on here? I can imagine the kinetic energy of the crystal remaining the same, while internal kinetic energy is converted to center-of-mass kinetic energy, allowing for a momentum kick, but that would be a first for me. Thermodynamically, it always goes the other way.

Thoughts?

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PeterDonis
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In CP-SPDC, the energy of the photon pair adds up to the energy of the original pump photon, but the momenta of the photon pair adds to zero since they are propagating in opposite directions.
Why do you think this? I thought that the two down converted photons would come out in directions that were different, but not opposite.

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PeterDonis
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using quasi phase matching, it is possible to structure the medium so that the down-converted photons actually go in completely opposite directions
From what I can see from these papers, the pump beam comes in at an angle to the direction of the counter-propagating photons, and there is a difference in the frequency (and therefore in the momentum) of the two down-converted photons that depends on the pump beam angle. In the limit where the pump beam angle goes to 90 degrees (i.e., perpendicular to the down converted propagation direction), the two photon frequencies become equal. So I don't think there is an issue with conservation of momentum; the net momentum of the two photons is nonzero for a non-perpendicular pump beam because of their different frequencies, even though their directions of propagation are opposite.

• bhobba
Gold Member
From what I can see from these papers, the pump beam comes in at an angle to the direction of the counter-propagating photons, and there is a difference in the frequency (and therefore in the momentum) of the two down-converted photons that depends on the pump beam angle. In the limit where the pump beam angle goes to 90 degrees (i.e., perpendicular to the down converted propagation direction), the two photon frequencies become equal. So I don't think there is an issue with conservation of momentum; the net momentum of the two photons is nonzero for a non-perpendicular pump beam because of their different frequencies, even though their directions of propagation are opposite.
With the CP-SPDC process happening in a single nonlinear waveguide, the momentum of the two created photons adds up to a vector pointing parallel to the waveguide. If the pump photon has a momentum that's not parallel to the waveguide, there would be a mismatch between the total momentum before and after the SPDC process, even though energy is conserved. Having the final momentum add up to zero is not as important to the conceptual problem as having them add up to anything not exactly the same as the initial pump momentum.

PeterDonis
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If the pump photon has a momentum that's not parallel to the waveguide, there would be a mismatch between the total momentum before and after the SPDC process, even though energy is conserved.
Yes, you're right; I would expect the waveguide to experience a recoil perpendicular to the direction of the down converted photons in this case, but I don't see that discussed in the paper.

Cthugha
With the CP-SPDC process happening in a single nonlinear waveguide, the momentum of the two created photons adds up to a vector pointing parallel to the waveguide. If the pump photon has a momentum that's not parallel to the waveguide, there would be a mismatch between the total momentum before and after the SPDC process, even though energy is conserved. Having the final momentum add up to zero is not as important to the conceptual problem as having them add up to anything not exactly the same as the initial pump momentum.
In principle you already have the same problem already in standard SPDC as the created photons leave at an angle with respect to the incoming beam and so the vector addition of the momenta equals the initial momentum vector of the incoming photon, but the sum of the magnitudes of the momenta does not add up.

In general, linear momentum conservation follows from Noether's theorem and is a consequence of the invariance of a system with respect to spatial translations. However, a crystal (and of course also a waveguide) is spatially finite system and usually at least one (and for waveguides even two) of the dimensions has a finite size. As with any finite size system, momentum along the finite-size direction is not a conserved quantity anyway and if you do the math, you will find that the momentum mismatch is small compared to the momentum uncertainty due to the finite size. Effectively, instead of the requirement for perfect momentum conservation, you will get a term that scales as $sinc(\frac{\Delta k_z L_z}{2})$.

If you tried to get this SPDC process to work in a huge bulk crystal, it would not work.

• kith and bhobba
In CP-SPDC, the energy of the photon pair adds up to the energy of the original pump photon, but the momenta of the photon pair adds to zero since they are propagating in opposite directions. Since the total energy of the field is the same before and after the interaction, but the momentum is different, it would look like there is a violation of conservation of momentum, since the crystal gets a momentum kick without changing its energy.
The energy of the photon pair is reduced by the energy of the kick.

If a stream of identical photons is getting down converted, the upstream photons will be Doppler- shifted according to the accelerating crystal.

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...Effectively, instead of the requirement for perfect momentum conservation, you will get a term that scales as $sinc(\frac{\Delta k_z L_z}{2})$.

If you tried to get this SPDC process to work in a huge bulk crystal, it would not work.
If you used a huge periodically-poled bulk crystal, it would work.
This is the crux of the problem. Periodic poling allows down-conversion to occur where the momentum of the signal and idler (the daughter photons) no longer add to that of the pump, even though their energies do add correctly.

Cthugha
If you used a huge periodically-poled bulk crystal, it would work.
Well, I know what you mean, but I would not call a spatially extended non-linear optical element such as a piece of periodically poled lithium niobate "bulk". In order to cope with the group velocity mismatch, you include oppositely poled domains, such that the mismatch runs off and then runs back towards zero mismatch in the next domain. However, the concept is the same as for the small-size crystal: You deliberately break invariance with respect to continuous spatial translation so that momentum conservation becomes replaced by some quasi-momentum conservation condition.

With respect to symmetry, the periodically poled crystal is rather similar to a heterostructure or a metamaterial, which has spatially modulated material properties, than to a bulk system, which I would consider to have homogeneous properties throughout the whole crystal.

(By the way, the concept also works for energy mismatch if you use quasi-phase-matching in the time domain, see Bahabad, Murnane and Kapteyn, Nature Photonics 4, 570 (2010)).

• kith
Gold Member
Well, I know what you mean, but I would not call a spatially extended non-linear optical element such as a piece of periodically poled lithium niobate "bulk". In order to cope with the group velocity mismatch, you include oppositely poled domains, such that the mismatch runs off and then runs back towards zero mismatch in the next domain. However, the concept is the same as for the small-size crystal: You deliberately break invariance with respect to continuous spatial translation so that momentum conservation becomes replaced by some quasi-momentum conservation condition.

With respect to symmetry, the periodically poled crystal is rather similar to a heterostructure or a metamaterial, which has spatially modulated material properties, than to a bulk system, which I would consider to have homogeneous properties throughout the whole crystal.

(By the way, the concept also works for energy mismatch if you use quasi-phase-matching in the time domain, see Bahabad, Murnane and Kapteyn, Nature Photonics 4, 570 (2010)).
This is an interesting point, and I'm very grateful for the journal reference. I still don't think that arguments about translation invariance (a la Noether's theorem) allow one to dispense with traditional momentum or energy conservation in SPDC when using quasi phase matching. A lot of the more interesting physics I expect is swept under the rug with the hugely simplifying assumption that we only need the optical susceptibility tensors (to various orders) to describe the materials. At an elementary level, I think that invalidating traditional momentum conservation would require space and time itself to be distorted, just as energy and momentum conservation are modified in general relativity. I'm not aware of any process in quantum field theory or quantum optics that serves as a counter-example to conservation of total momentum.

With this, we'd be led back to the crystal experiencing a momentum kick without changing total energy, meaning perhaps that internal kinetic energy is getting transformed into center-of-mass kinetic energy, with the total of the two being the same. Would this cool the crystal down?