Production of superpositions of coherent states in traveling optical fields with inefficient photon detection

We develop an all-optical scheme to generate superpositions of macroscopically distinguishable coherent states in traveling optical fields. It nondeterministically distills coherent-state superpositions (CSS’s) with large amplitudes out of CSS’s with small amplitudes using inefficient photon detection. The small CSS’s required to produce CSS’s with larger amplitudes are extremely well approximated by squeezed single photons. We discuss some remarkable features of this scheme: it effectively purifies mixed initial states emitted from inefficient single-photon sources and boosts negativity of Wigner functions of quantum states.

Figure 1

A schematic of the simplification of the CSS generation. PC represents photon counting and HD represents homodyne detection. (a) Conditional production using QND with photon counting, (b) conditional production using QND with homodyne detection, and (C) deterministic production only by squeezing a single photon.

Figure 2

The fidelity between an odd CSS and squeezed single photon. The odd CSS is extremely well approximated by the squeezed single photon for a small coherent amplitude, $\alpha ⩽1.2$.

Figure 3

(Color online) The Wigner functions of odd CSS’s (left) and squeezed single photons (right). The amplitudes of CSS’s are (a) $1∕\sqrt{2}$, (b) 1, and (c) 2. The degrees of squeezing of squeezed single photons are (a) 0.164, (b) 0.313, and (c) 0.853, which are chosen for maximum fidelity with CSS’s. It is evident from the figure that only small amount of squeezing makes a single photon a good approximate CSS.

Figure 4

A schematic of the nondeterministic CSS-amplification process. Two small CSS’s at modes $a$ and $b$ are added to produce a larger CSS at mode $f$ by a conditional measurement using detectors $A$ and $B$. See text for details.

Figure 5

A schematic of successive applications of the CSS amplification processes. CA represents the CSS amplification process depicted in Fig. 4. As the first step, eight CSS’s of initianl amplitude ${\alpha }_i$ are fed into the four CA processes. If all the eight detectors in the four CA processes click, the resulting states are selected for the second step and so on. In this example, 14 detectors and 14 beam splitters are required with 7 auxiliary coherent states to distill a CSS of amplitude $2\sqrt{2}{\alpha }_i$ from smaller CSS’s of amplitude ${\alpha }_i$.

Figure 6

The success probabilities of the CSS-amplifying process in Fig. 4 for the input fields of two identical odd CSS’s (solid line), two identical even CSS’s (dashed line), and even and odd CSS’s (dotted line).

Figure 7

section of the Wigner function for an initial even CSS of amplitude $1∕2$ (solid line), the resulting CSS after the first iteration (dashed line), and the resulting state after the second iteration (dotted line). The maximum negative values are shown in the figure. One can see a radical increase of negativity of the Wigner function.

Figure 8

(a) The maximum fidelity obtained in our scheme vs the coherent amplitude. (b) The number of iterations which gives the maximum fidelity vs the coherent amplitude. The improvements of the fidelity are remarkable when compared with Fig. 2.

Figure 9

(a) Purity and (b) fidelity of the input state (solid lines), the input output state after the first iteration (dot-dashed lines), and of the output after the second iteration (dashed line) as functions of the photon production inefficiency $p$. The input states are mixed squeezed single photons intended to be approximate odd CSS’s of amplitude $\alpha =1∕2$. The final output states should be therefore approximate CSS’s of $\alpha =1$.