Optical Synthesis of Large-Amplitude Squeezed Coherent-State Superpositions with Minimal Resources

We propose and experimentally realize a novel versatile protocol that allows the quantum state engineering of heralded optical coherent-state superpositions. This scheme relies on a two-mode squeezed state, linear mixing, and a $n$-photon detection. It is optimally using expensive non-Gaussian resources to build up only the key non-Gaussian part of the targeted state. In the experimental case of a two-photon detection based on high-efficiency superconducting nanowire single-photon detectors, the freely propagating state exhibits a 67% fidelity with a squeezed even coherent-state superposition with a size $|\alpha {|}^2=3$. The demonstrated procedure and the achieved rate will facilitate the use of such superpositions in subsequent protocols, including fundamental tests and optical hybrid quantum information implementations.

Figure 1

Schemes for generating $\alpha |n-2⟩+\beta |n⟩$ and theoretical simulations. (a) Two squeezed vacua are mixed on a tunable beam splitter. The detection of $n$ photons in one of the outputs heralds the generation. Equivalently, the scheme can be implemented directly from a two-mode squeezed vacuum. The two orthogonally polarized modes are separated with a tunable mixing $\epsilon$ induced by a rotated half-wave plate (HWP) and a polarizing beam splitter (PBS), with $\epsilon =\sin (2\theta )$. The angle $\theta =0°$ corresponds to the perfect separation, leading to the generation of a $n$-photon state. (b) Two-photon detection: fidelity with a squeezed even CSS as a function of the ratio $\epsilon /\lambda$. The fidelity is optimized by the size $|\alpha {|}^2$ and the squeezing of the target state. The inset gives the weights of the vacuum and two-photon components. (c) Three-photon detection and fidelity with a squeezed odd CSS.

Figure 2

Experimental setup. (a) A two-mode squeezed vacuum generated by a type-II OPO is split on a tunable polarizing beam splitter. One of the outputs, which is frequency filtered by an interferential filter (IF) and a Fabry-Perot cavity (FP), impinges on a fiber beam splitter. Coincidence events from high-efficiency superconducting single-photon detectors herald the preparation. The resulting state is characterized by homodyne detection, with an overall efficiency $\eta =85\%$. (b) Two-photon state generation when signal and idler are separated without mixing ($\theta =0°$). The first plot gives the measured quadrature distribution while the second one provides the diagonal elements of the density matrix, with and without correction for detection losses. The right figure displays the corresponding Wigner function without correction, with a negative ring value $W=-0.13\pm 0.01$. The heralding rate reaches 200 Hz.

Figure 3

Experimental results. The plots give the density matrices (absolute values) and the associated Wigner functions for incremental values of the half-wave plate angle $\theta$ ($\epsilon$ between 0.017 and 0.16). The results are corrected for the $\eta =85\%$ detection efficiency. For $\theta =1.5°$, the negative peaks of the Wigner function reach $W=-0.27\pm 0.01$.

Figure 4

Fidelity between the generated state and a squeezed even CSS, for (a) $\theta =1.5°$ and (b) $\theta =2.5°$. The plots give the calculated fidelity as a function of $|\alpha {|}^2$ and the squeezing in dB. The green crosses indicate the maximal fidelities. For $\theta =1.5°$, the fidelity reaches 0.67 with an even CSS with a size $|\alpha {|}^2=3$ and a 4-dB squeezing. For $\theta =2.5°$, the fidelity reaches 0.68 with an even CSS with a size $|\alpha {|}^2=1.9$ and a 3-dB squeezing.