Anisotropic purity of entangled photons from Cooper pairs in heterostructures

It was theoretically proposed that a forward-biased $p-n$ junction with a superconducting layer (P-N-S) would produce pure, polarization-entangled photons due to inherent spin-singlet pairing of electrons. However, any heterostructure interface generically induces Rashba spin-orbit coupling, which in turn generates a mixed singlet-triplet superconducting order parameter. Here we study the effect of triplet pairing on the purity of photons produced through Cooper pair recombination. A unique directional dependence of the state purity is found for a triplet superconductor with fixed $\mathit{d}$: pure, entangled photons are produced when the photon polarization axis is parallel to $\mathit{d}$. Induced triplet pairing in a singlet superconductor is shown to degrade the state purity, while induced singlet pairing in a triplet superconductor is shown to enhance the production of entangled pairs. These considerations may aid the design of functional devices to produce entangled photons.

Figure 1

(a) Schematic P-N-S heterostructure, consisting of a $p$-type semiconductor and an $n$-type semiconductor in which pairing is induced by the superconductor through the proximity effect. The heterostructure normal direction is defined by $\widehat{\mathit{n}}$; the photon polarization axis, by $\widehat{\mathit{z}}$; and a particular $\mathit{d}\left(\mathit{k}\right)$ vector direction, by $\widehat{{\mathit{d}}_{\mathit{k}}}$. The angle between $\widehat{\mathit{z}}$ and $\widehat{{\mathit{d}}_{\mathit{k}}}$ is taken to be ${\theta }_{\mathit{k}}$. (b) Schematic band structure of the semiconductors in the quantum well structure, showing the splitting of heavy-hole and light-hole bands.

Figure 2

(a) Calculated purity of photon pairs produced by spin-triplet Cooper pairs as a function of the polar angle $\theta$ between the photon polarization axis $\widehat{\mathit{z}}$ and the $\mathit{d}$ vector and (b) full angular dependence. (c) Purity of photon pairs for triplet fraction $t=0.77$ as a function of the polar angle $\theta$ between the photon axis $\widehat{\mathit{z}}$ and the $\mathit{d}$ vector and (d) full angular dependence.

Figure 3

Illustration of the induced $\mathit{d}$ vector from $s$-wave pairing due to Rashba SOC, shown as blue arrows on the Fermi surface in red. When the photon axis lies in the plane of the heterostructure, there are two parallel $\mathit{d}$ vectors at $\pm {\mathit{k}}_{\parallel }$, and two perpendicular at $\pm {\mathit{k}}_{\perp }$. All Cooper pairs, except those with $\pm {\mathit{k}}_{\parallel }$, make a finite angle ${\theta }_{\mathit{k}}$ with $\widehat{\mathit{z}}$ and thus degrade the state purity.