Photon Sorting, Efficient Bell Measurements, and a Deterministic Controlled-$Z$ Gate Using a Passive Two-Level Nonlinearity

Although the strengths of optical nonlinearities available experimentally have been rapidly increasing in recent years, significant challenges remain to using such nonlinearities to produce useful quantum devices such as efficient optical Bell state analyzers or universal quantum optical gates. Here we describe a new approach that avoids the current limitations by combining strong nonlinearities with active Gaussian operations in efficient protocols for Bell state analyzers and controlled-sign gates.

Figure 1

(a) Possible physical implementations of an efficient TLS exploiting unidirectional coupling obtained by implementing a two-level emitter in a chiral photonic-crystal waveguide (left) [8], a whispering-gallery resonator (middle) [7], or a photonic-crystal cavity (right) [6]. (b) $\eta$ as a function of the spectral width of the incoming pulse in the case of a Lorentzian spectral response and for three different values of the ${\beta }_{\text{dir}}$ factor (${\beta }_{\text{dir}}=1$ is the top solid line). The dashed lines show ${\epsilon }_1^2/2$ for the different values of ${\beta }_{\text{dir}}$. Where the solid and corresponding dashed lines meet are the optimal operation points. In order to minimize loss, the crossing points at higher spectral widths should be chosen as the operation points.

Figure 2

Components of the photon sorter and Bell measurement device. (a) The photon sorter is constructed from a TLS followed by SFG, where the classical pump is in the mode $f^{\prime }$, followed by a dichroic beam splitter that separates the single-photon component at the sum frequency from the two-photon component at the original frequency. (b) A Bell measurement can be implemented with linear optics and four-photon sorters as shown. The four Bell states are unambiguously determined by the measurement of photons at particular detector combinations. In particular, $|{\psi }^{+}⟩$: 1, 4, or 3, 2; $|{\psi }^{-}⟩$: 1, 2 or 3, 4; $|{\varphi }^{+}⟩$: 5, 8 or 6, 7; $|{\varphi }^{-}⟩$: 5, 7 or 6, 8.

Figure 3

Components of the ns gate and deterministic cz gate. (a) The ns gate is constructed from a TLS followed by SFG, where the classical pump is in the mode $f^{\prime }$. This is followed by a $\pi$ phase shift which is imposed only on the two-photon term. If ${\beta }_{\text{dir}}<1$ then the loss is also applied only to the two-photon term. A second SFG then reverses the frequency shift, followed by short term storage in a GEM that inverts the pulse shape. Finally, the inverted pulse is sent through a second TLS that recombines the one- and two-photon terms back into the same mode. If the initial mode shape was time symmetric then the overall effect will be that of an ns gate, i.e., to impose a $\pi$ phase shift only on the two-photon terms while leaving the mode shapes unchanged. (b) The resultant ns gates can be incorporated in a simple linear optical circuit to produce a cz gate. Beam splitters on the outer arms are required if ${\beta }_{\text{dir}}<1$ in order to resymmetrize the state.