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Given a BBO (Beta barium borate) crystal, with a pump beam of 405nm passing through it, what is the mathematical representation for the transformation spontaneous parametric down conversion (type I) performs on the photons? Would the representation be dependent on the dimensions of the BBO crystal? I'm looking for a representation in the framework of linear algebra.

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  • $\begingroup$ Could you perhaps clarify what you mean by "matrix" here? The SPDC process is usually represented as a nonlinear interaction term. It contains the second order susceptibility tensor, which is a tensor of rank 3 that represents the nonlinear properties of the crystal. $\endgroup$ – flippiefanus Aug 15 '17 at 4:48
  • $\begingroup$ @flippiefanus, I'm thinking about this in the context of quantum computing, where all gates can be represented as matrices. $\endgroup$ – heather Aug 15 '17 at 15:07
  • $\begingroup$ SPDC is usually not regarded (or used) as a gate in quantum computing. Do you have any references for that? $\endgroup$ – Norbert Schuch Aug 16 '17 at 10:47
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    $\begingroup$ You might want to consult Sec.7.2 of the textbook by Gerry and Knight as a starting point. Briefly: the answer you want depends on how one treats the pump field. $\endgroup$ – ZeroTheHero Aug 17 '17 at 12:32
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Downconversion is a process whereby one photon is "split into two". In terms of creation and destruction operator, it is represented by a term like $$ ab^\dagger c^\dagger \tag{1} $$ which removes the original photon $a$ and creates photons $b^\dagger$ and $c^\dagger$. These act on the photon vaccuum state. Obviously (1) is not hermitian so the actual interaction Hamiltonian is $$ H_{int}=\chi \left(ab^\dagger c^\dagger + a^\dagger b c\right) $$ where $\chi$ is the Kerr coefficient of the medium. The frequencies of photons $b^\dagger$ and $c^\dagger$ are related to $a$ by conservation of energy and conservation of linear momentum basically $\vec k_a=\vec k_b+\vec k_c$.

In the parametric approximation, the incident photon is from a strong pump field, usually modelled as a coherent state. Since coherent states satisfy $a\vert \alpha\rangle = \alpha \vert \alpha\rangle$, one replaces $a$ and $a^\dagger$ by $\alpha$ and $\alpha^*$ to obtain $$ H_{int}\to H_{parametric}=\chi\left(\alpha b^\dagger c^\dagger + \alpha^* b c\right) \tag{2} $$ If the outgoing photons are degenerate (have same frequency), then $c^\dagger=b^\dagger$ etc.

Sticking to this case for simplicity, the difference between (1) and (2) is that (2) closes on the (non-compact, real) Lie algebra sp(2,$\mathbb{R}$), with generators $$ \hat K_+=\frac{1}{2}b^\dagger b^\dagger\, ,\qquad \hat K_-=\frac{1}{2} b\,b\, ,\qquad \hat K_0=\frac{1}{2}(b^\dagger b+ b b^\dagger) $$ whereas (1) does NOT close on any finite dimensional Lie algebra.

The unitary irreducible representations of sp(2,$\mathbb{R}$) are actually infinite dimensional, so it is not possible to write the "full" matrix representation for $H_{parametric}$: in the unitary evolution $e^{-i t H_{parametric}}$ one typically the series, usually after the first power in $\chi$ since the coefficients are typically small, and works in a truncated subspace of the infinite-dimensional representation.
Since the $b^\dagger$ and $b$ are just field creation operators, we have $$ b^\dagger b^\dagger \vert 0\rangle = \sqrt{2}\vert 2\rangle $$ etc. and $$ e^{-i t H_{parametric}}\vert\alpha\rangle\vert 0\rangle = \vert\alpha\rangle\vert 0\rangle -i t \chi\alpha \sqrt{2}\vert\alpha\rangle\vert 2\rangle+\ldots \tag{3} $$ (I'm being loose with the notation...) Here, $t$ is really the interaction time so for a thicker media the probability of downconversion increases: $t \approx \Delta x/c$, with $\Delta x$ the thickness. In this case truncating (3) to a single term may be overly drastic.

In general, $su(1,1)$ matrix elements for arbitrary representations (typically used in many-body systems and not limited to those having $\vert 0\rangle$ as "ground state") can be found in this open access paper by Ui. There is also quite a bit of more recent work in the context of quantum optics by C.C. Gerry (including a textbook with Sir Peter Knight).

Finally, I don't remember downconversion being used directly for gates (as pointed out by @NorbertSchuch) and a quick search was not successful. If you have a specific paper in mind where this would be exposed I would like to know about it.

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