Direct interferometric test of the nonlinear phase-shift gate

Probabilistic quantum gates permeate nearly all aspects of modern linear optical quantum computing. This is largely due to the measurement induced nonlinearities of which they are capable. The tradeoff between the cost of resources required to boost success probabilities to scalable levels and the realization of effective photon-photon interactions required grows ever more favorable to this paradigm with rapidly evolving advancements in integrated nanophotonics. As quantum circuits generating cluster states become increasingly complex, component analysis is critical. In this paper, we propose and demonstrate the experimental viability of testing a probabilistic gate. Specifically, as an illustrative example of the technique, we propose a direct test of the nonlinear phase-shift gate (NLPSG), an essential component of a Knill-Laflamme-Milburn (KLM) controlled-not (cnot) gate. We develop our analysis for both the case of the original bulk optical KLM NLPSG and the case of the scalable integrated nanophotonic NLPSG based on microring resonators (MRRs) that we have proposed very recently. We consider the interference between the target photon mode of the NLPSG along one arm of a Mach-Zehnder interferometer (MZI) and a mode subject to an adjustable linear phase along the other arm. Analysis of triple-photon coincidences between the two modes at the output of the MZI and the success ancillary mode of the NLPSG provides a signature of the conditionally successful operation of the NLPSG. We examine the triple coincidence results for experimentally realistic cases of click or no-click detection with subunity detection efficiencies. Further, we compare the case for which the MZI input modes are seeded with weak coherent states with that for which the input states are those resulting from colinear spontaneous parametric down conversion (cl-SPDC). In particular, we show that, though more difficult to prepare, cl-SPDC states offer clear advantages for performing the test, especially in the case of relatively low photon detector efficiency. We develop the interferometric analysis in terms of a general four-mode unitary so that comparison and contrast of the signatures of the specific NLPSG implementation (i.e., KLM and MRR) can be carried out easily by substituting in the appropriate unitary matrix elements at the end of the calculation. The analysis is readily extendable to other several-mode low photon number quantum circuits the successful operation of which is heralded by the measurement of ancilla photons, under nonideal detection conditions germane to laboratory experiments.

Figure 1

The MRR NLPSG, with the ordinary three beam splitters used in the KLM implementation (see Fig. 4 in Appendix pp1) replaced by microring resonators (MRRs) (black circles). The input state $|{\mathrm{\Psi }}^{\left(0\right)}\rangle$ entering the upper and lower leg of the MZI are weak coherent states (w-CS) of the form ${\left({\alpha }_0\right|0\rangle }_1+{\alpha }_1{|1\rangle }_1+{\alpha }_2{|2\rangle }_1)$ and ${\left({\alpha }_0^{\prime }\right|0\rangle }_4+{\alpha }_1^{\prime }{|1\rangle }_4+{\alpha }_2^{\prime }{|2\rangle }_4)$, respectively, with the modes of the NLPSG in the upper leg of the MZI labeled (top down) as 1,2,3. Modes 2 and 3 are the ancilla modes to the NLPSG that are initially in the state ${|1,0\rangle }_{2,3}$. The lower leg, mode 4, of the MZI contains a phase-shift element $e^{i\phi a_4^{†}{a}_4}$ which effectively sends ${\alpha }_k^{\prime }\to {\alpha }_k^{\prime }{e}^{ik\phi }$. The action of ${\text{NLPSG}}_{123}\otimes {\text{phase}}_4$ produces the intermediate state $|{\mathrm{\Psi }}^{\left(1\right)}\rangle$. The final element of the MZI is a BS of angle $\theta$ (such that $\theta =\pi /2$ is a 50:50 BS), the output of which is $|{\mathrm{\Psi }}^{\left(2\right)}\rangle$. Coincidence detection, producing the un-normalized state $|{\tilde{\mathrm{\Psi }}}^{\left(2\right)}\rangle ={\mathrm{\Pi }}^{1234}|{\mathrm{\Psi }}^{\left(2\right)}\rangle$, is performed on the exiting modes 1 and 4, conditioned on the NLPSG ancilla modes 2 and 3, and occurs with probability $\langle {\tilde{\mathrm{\Psi }}}^{\left(2\right)}|{\tilde{\mathrm{\Psi }}}^{\left(2\right)}\rangle$. The KLM NLPSG is obtained by replacing each black circle in the NLPSG subdiagram by a single beam splitter with reflectivities $0\le {\eta }_i\equiv r_i^2\le 1$, where $-1\le r_i\le 1$ are reflection coefficients for $i\in \{1,2,3\}$.

Figure 2

Plot of $P_{1234}^{\prime }$ from Eq. (38a) for the (left) KLM and (right) MRR NLPSG for ${\beta }^2=1/4$ for (dotted) w-CS input states and (dot-dashed) cl-SPDC input states with detection efficiencies (gray) $40\%$ and (black) $85\%$. The left and right graphs are identical.

Figure 3

Plots of the physical MRR transmissivities (solid) ${\eta }_{1=3}^2$ vs ${\tau }_{1=3}^2$ and (dashed) ${\eta }_2^2$ vs ${\tau }_2^2$ for fixed values of the fictitious KLM reflectivities $r_1^{*2},r_2^{*2}$ yielding ${\beta }^2=1/4$.

Figure 4

The KLM NLPSG using three ordinary beam splitters of reflectivities $R_i=r_i^2$, and optical path delays of ${\delta }_i$. Mode 1 is the primary input state in a weak coherent state (w-CS) containing up to two photons. Modes 2 and 3 are ancilla modes, initially in the state ${|1,0\rangle }_{23}$. Success of NLPSG is heralded by the detection of the output ancilla modes in their initial state.

Figure 5

Plots of the physical MRR transmissivities (solid) ${\eta }_{1=3}^2$ vs ${\tau }_{1=3}^2$ and (dashed) ${\eta }_2^2$ vs ${\tau }_2^2$ for fixed values of the fictitious KLM reflectivities $r_2^{*2}$ yielding ${\beta }^2=1/4$ (Fig. 3 repeated here for clarity).

Figure 6

Two optical modes $a$ and $b$ mixing on a BS of reflectivity $R$. Note that in this (not always standard) representation the $a_{\text{in}}$ mode is the top-left input, while the $a_{\text{out}}$ mode is defined via where $a_{\mathrm{in}}$ transmits to, i.e., as the lower right output, and similarly for $b_{\text{in}}$ and $b_{\text{out}}$. Here, the bottom of the BS imparts a $\pi$ phase shift of $-1$ upon reflection.