# Why is a nonlinear crystal necessary to stimulate quantum fluctuations that entangle photons?

I've been reading about spontaneous parametric down conversion (SPDC). The Wikipedia article on it says:

A nonlinear crystal is used to split photon beams into pairs of photons that, in accordance with the law of conservation of energy and law of conservation of momentum, have combined energies and momenta equal to the energy and momentum of the original photon and crystal lattice, are phase-matched in the frequency domain, and have correlated polarizations... SPDC is stimulated by random vacuum fluctuations, and hence the photon pairs are created at random times [...]

Why is a crystal necessary for these fluctuations to occur, and how do the fluctuations entangle the incoming photons?

Maxwellian electrodynamics, in vacuum, is a linear theory: that is, it obeys the principle of superposition, and the sum of any two given solutions will still be a solution, so that e.g. two beams crossing each other will pass by without affecting each other in any way.

Moreover, most materials that you meet in everyday life (at the intensities of EM radiation that you get in everyday life), are also linear: more specifically, unless they're opaque, they are dielectrics that are characterized by an electric polarization density $\mathbf P$ that depends linearly on the local electric field, $$\mathbf P = \epsilon_0\chi\mathbf E,$$ for a constant $\chi$ called the electric susceptibility of the material, and this polarization density then feeds back linearly into the response of the material to the light (going into e.g. the refractive index). Because of this linear constitutive relation, linear dielectrics also obey the superposition principle, exactly as in vacuum.

Now, here's the thing: the superposition principle is all well and good for finding solutions and so on, but ultimately it is a boring property for a system to have. Why? because under linear conditions, the modes of the radiation are fixed and there is no way for the state of any given mode to 'talk to' the state of any other mode, at all, and that precludes any interesting dynamics from happening with the photons.

As a more relevant example, in a linear material, a light beam of frequency $\omega$ propagating down the material is a solution of the Maxwell equations (plus the constitutive relation), or, to put it another way: parametric down-conversion, where the beam's energy is transferred into modes of frequencies $\omega_\mathrm{s}$ and $\omega_\mathrm{i}$ (such that $\omega_\mathrm{s}+\omega_\mathrm{i}=\omega$) is completely impossible. Similarly, any kind of 'gating' behaviour, where the phase or propagation of one beam is affected by a second beam, like you might want in a photonic computer, is also completely ruled out.

This is why we turn to nonlinear optical components. These have nonlinear constitutive relations, where the dielectric polarization depends on higher powers of the electric field which break the linearity, in the form $$\mathbf P = \epsilon_0\chi\mathbf E + \epsilon_0\chi^{(2)} \cdot \mathbf E^{\otimes 2} + \epsilon_0\chi^{(3)} \cdot \mathbf E^{\otimes 3} + \cdots$$ (where the $\chi^{(n)}$ and $\mathbf E^{\otimes n}$ are tensors and the dots are contractions, none of which is essential here) where now if the medium is subjected to the superposition of two beams, its response will differ from the vector addition of the individual responses. That then allows the modes to affect each other and it breathes dynamics back into optics.

Like I said in a previous question of yours, nonlinearity is a key requirement to be able to do anything interesting, and particularly for computational purposes. As far as quantum computing goes, nonlinearity in the interactions between components is a key resource to be sought and treasured, because it enables the whole game to play. (This is also true of classical computing, which only became possible on electronic substrates when nonlinearity, in the form of vacuum tubes, and later transistors, became available. Classical computing using only linear circuit elements is impossible.)

So, what about parametric down-conversion? This is a second-order nonlinear process, which means that it rides on the $\chi^{(2)} \cdot \mathbf E^{\otimes 2}$ term. To see how it works, suppose that we have a medium that has a nonzero $\chi^{(2)}$ (so, typically a BBO or LiNbO$_3$ crystal) along the $\chi^{(2)}_{zzz}$ directions, and that we apply to it two fields: a driver field $$\mathbf E_\mathrm{d}(t) = \hat{\mathbf e}_z E_{\mathrm{d},0} \cos(\omega_\mathrm{d} t),$$ and a signal field $$\mathbf E_\mathrm{s}(t) = \hat{\mathbf e}_z E_{\mathrm{s},0} \cos(\omega_\mathrm{s} t),$$ and we look at the nonlinear polarization: \begin{align} P^{(2)}_z(t) &= \epsilon_0 \chi^{(2)}_{zzz} (E_{\mathrm{d},z}(t) + E_{\mathrm{s},z}(t))^2 \\ & \cong 2\epsilon_0 \chi^{(2)}_{zzz} E_{\mathrm{d},z}(t)E_{\mathrm{s},z}(t) \\ & = 2\epsilon_0 \chi^{(2)}_{zzz} E_{\mathrm{d},0} E_{\mathrm{s},0} \cos(\omega_\mathrm{d} t)\cos(\omega_\mathrm{s} t) \\ & = 2\epsilon_0 \chi^{(2)}_{zzz} E_{\mathrm{d},0} E_{\mathrm{s},0} \left[ \cos((\omega_\mathrm{d} -\omega_\mathrm{s})t) + \cos((\omega_\mathrm{d} +\omega_\mathrm{s}) t) \right] \\ & \cong 2\epsilon_0 \chi^{(2)}_{zzz} E_{\mathrm{d},0} E_{\mathrm{s},0} \cos((\omega_\mathrm{d} -\omega_\mathrm{s})t) , \end{align} where $\cong$ means that I'm neglecting terms that don't contribute to the process I want to describe. The important thing to notice here is that the polarization (contains a term that) depends on the product $E_{\mathrm{d},z}(t)E_{\mathrm{s},z}(t)$, and that this is a product of cosines that decomposes into trigonometrics at different frequencies: to wit, the idler frequency $$\omega_\mathrm{i} = \omega_\mathrm{d}-\omega_\mathrm{s}.$$ This is the down-conversion process, where we've taken light at frequency $\omega$ and diverted some of its energy into frequency $\omega_\mathrm{i}$, with an amplitude that can be quite big even if $E_{\mathrm{s},0}$ is small. If you do the math in full, it also turns out that a similar amount of energy ends up reinforcing the signal field $E_{\mathrm{s},z}(t)$.

To be a bit more precise, this process is stimulated parametric down-conversion, because we required an initial seed on $E_{\mathrm{s},z}(t)$, however small, to fix the phase (a.k.a. time of emission) of the signal and idler beams, onto which the energy from the driver could 'congeal'.

In addition to this, there is also a spontaneous parametric down-conversion process, where (if the phase-matching conditions are right) light at the driver frequency will split into beams at the signal and idler frequencies without any external prompting. As described in Wikipedia, this cannot happen within classical nonlinear optics, and it requires the QED vacuum fluctuations to initiate the process, and therefore it is not surprising that (i) it happens on a per-photon basis, and (ii) it can produce highly entangled states of the signal and idler beams.

But, either way, it should be clear that without a way to have a physical response that's proportional to both the driver and signal fields that we can then see as having a different frequency content, i.e. without a nonlinear component of the dynamics, none of this would be possible at all.

• So, to summarize: a nonlinear crystal allows the vacuum fluctuations to interact with the photons, and it's the interaction of the photons with each other (initiated by the vacuum fluctuations) that causes the entanglement? – heather Jul 25 '17 at 15:28
• @heather I'm not a fan at all of the view that "vacuum fluctuations induce stuff to happen". The situation here is exactly the same as in spontaneous emission from an excited atomic state, in that it's classically forbidden, occurs at a random time, produces photons with ill-defined phase (though SPDC leaves one relative phase well-defined), and is described quantum-mechanically by the details of the action of creation & annihilation operators on the quantum vacuum. That last bit is often interpreted as "stimulated by vacuum fluctuations" but that's a very crass oversimplification. – Emilio Pisanty Jul 25 '17 at 15:46
• A more nuanced view is that the nonlinear crystal provides a setting (with the hamiltonian $\hat H = H_0 \hat{a}_\mathrm{s}^\dagger \hat{a}_\mathrm{i}^\dagger \hat{a}_\mathrm{d}$ instead of the usual quadratics) in which the driver, signal and idler modes can interact. This interaction can be with a nonzero input signal (as in an OPO), or on the QED vacuum $|0\rangle$. The entanglement comes from the simultaneous creation, but, as always with entanglement, it needs to be studied with a bit more care. – Emilio Pisanty Jul 25 '17 at 15:54

In order for photons to become entangled they must have interacted at some point. Photons don't interact in free space, but they do interact in nonlinear crystals. That is indeed why these special crystals are called nonlinear, they support some form of multi-photon interaction.