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As mentioned in an earlier question of mine, I am interested in using type one spontaneous down conversion (SPDC) in optical quantum computing. However, SPDC is a somewhat low probability occurrence - most of the photons pass straight through the crystals unentangled. What methods, if any, are there to improve the probability of down conversion occurring, and therefore entanglement between photons?

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    $\begingroup$ This paper may be of interest to you: arxiv.org/abs/1712.07140. It deals with domain-engineering of PDC crystals to improve heralding, brightness and spectral purity of the emitted pairs (in the telecom band) $\endgroup$ – glS Mar 25 '18 at 18:03
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    $\begingroup$ In case this helps: An alternative to spontaneous parametric down conversion (SPDC) is two-photon emission from electrically driven semiconductors. Relevant paper: Observation of two-photon emission from semiconductors - Alex Hayat, Pavel Ginzburg & Meir Orenstein. According to the abstract it is "three orders of magnitude more efficient than the existing down-conversion schemes". I deleted my previous answer to this question because it didn't really answer how to improve photon entanglement in SPDC. It should rather be a comment. $\endgroup$ – Sanchayan Dutta Mar 25 '18 at 18:04
  • $\begingroup$ @Blue, interesting paper, I've never heard of that. $\endgroup$ – heather Mar 25 '18 at 18:10
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Your question asks two questions that are less-related than you might hope.

First, how do we increase the probability of down-conversion occuring?

This is fundamentally a question about material properties: the chance per unit length of down-conversion occurring is proportional to $\chi^{(2)}$; if our material of choice doesn't have good phase-matching conditions then we can "cheat" and use quasiphasematching at a hit of ~$2/\pi$. As it happens, PPLN is just about as good as you're going to do, at least to within an order of magnitude. A very long device, say with a waveguide, lets you get more unit lengths; things still scale linearly, so to get much higher conversion you'd need a much longer device, which brings its own problems (such as absorption).

(This is setting aside other options, such as spontaneous four-wave mixing and true single photon sources.)

Second, you ask "...and therefore entanglement between photons?".

This does not immediately follow from the first problem (increasing conversion from pump to daughter photons). Depending on what kind of entanglement one is interested in, it's not at all the case that making more photons per (pulse/unit time) increases the entanglement of those photons. Generally speaking, SPDC sources for (discrete-variable) optical quantum computing are limited by the multi-fold emission rate, not by the pump power; this literally cannot be avoided by an SPDC scheme. If you're looking at achievable squeezing for CV experiments that's another kettle of fish that I'm not an expert in; my understanding is that the limits in total squeezing at present are not due to limited $\chi^{(2)}$, but other noise. (I may be mistaken on this count.)

As it turns out, this leads to people trying to make unentangled photons pairs and heralding one of them to try and simulate a "true" single photon source; a multiplexing scheme using space, delay lines, or a memory can then be used to put your known-good photon in the mode of interest. Once you have a device that can spit out a photon in a particular mode in a pure state, you're good to go.

If, on the other hand, you're interested in large amounts of time-frequency entanglement, that too is not set by the conversion efficiency, but by the phase-matching conditions in the crystal: the opposite problem from the papers posted by glS.

To conclude, while there are reasons to want to increase conversion efficiency (for one thing it seems awfully wasteful to have conversion efficiencies as low as we do), it's not immediately obvious that doing so is a rate-limiting step in most experiments at present. It's unfortunately also not at all clear how to do so in a more than marginal way; leading to people working on related problems such as increasing heralding efficiency and purity of the single-photon states instead.

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Here's some relevant work in Optimizing type-I polarization-entangled photons-Radhika Rangarajan, Michael Goggin, Paul Kwiat.

Abstract:

Optical quantum information processing needs ultra-bright sources of entangled photons, especially from synchronizable femtosecond lasers and low-cost cw-diode lasers. Decoherence due to timing information and spatial mode-dependent phase has traditionally limited the brightness of such sources. We report on a variety of methods to optimize type-I polarization-entangled sources - the combined use of different compensation techniques to engineer high-fidelity pulsed and cw-diode laser-pumped sources, as well as the first production of polarization-entanglement directly from the highly nonlinear biaxial crystal BiB3O6 (BiBO). Using spatial compensation, we show more than a 400-fold improvement in the phase flatness, which otherwise limits efficient collection of entangled photons from BiBO, and report the highest fidelity to date (99%) of any ultrafast polarization-entanglement source. Our numerical code, available on our website, can design optimal compensation crystals and simulate entanglement from a variety of type-I phasematched nonlinear crystals.

Apart from that I'd like to mention another interesting development in methods to entangle photons efficiently: An alternative to spontaneous parametric down conversion (SPDC) is two-photon emission from electrically driven semiconductors. According to Wikipedia:

The newly observed effect of two-photon emission from electrically driven semiconductors has been proposed as a basis for more efficient sources of entangled photon pairs. Other than SPDC-generated photon pairs, the photons of a semiconductor-emitted pair usually are not identical but have different energies. Until recently, within the constraints of quantum uncertainty, the pair of emitted photons were assumed to be co-located: they are born from the same location. However, a new nonlocalized mechanism for the production of correlated photon pairs in SPDC has highlighted that occasionally the individual photons that constitute the pair can be emitted from spatially separated points.

Relevant paper: Observation of two-photon emission from semiconductors - Alex Hayat, Pavel Ginzburg & Meir Orenstein.

According to the abstract it is "three orders of magnitude more efficient than the existing down-conversion schemes".

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