Implementation of a Nondeterministic Optical Noiseless Amplifier

Quantum mechanics imposes that any amplifier that works independently on the phase of the input signal has to introduce some excess noise. The impossibility of such a noiseless amplifier is rooted in the unitarity and linearity of quantum evolution. A possible way to circumvent this limitation is to interrupt such evolution via a measurement, providing a random outcome able to herald a successful—and noiseless—amplification event. Here we show a successful realization of such an approach; we perform a full characterization of an amplified coherent state using quantum homodyne tomography, and observe a strong heralded amplification, with about a 6 dB gain and a noise level significantly smaller than the minimal allowed for any ordinary phase-independent device.

Figure 1

Conceptual layout of the noiseless amplifier. A single photon is split on an asymmetric beam splitter (A-BS). The input state $|\alpha ⟩$ is superposed with reflected output of the A-BS on asymmetric beam splitter (S-BS). A successful run of the amplifier is flagged by a single-photon event on detector $D_1$ and no photons on detector $D_2$. The transmitted mode constitutes the output mode of the amplifier, and is approximately in an amplified state $|g\alpha ⟩$, conditioned on the right detection events, as described by Eq. (1).

Figure 2

Experimental results for the Wigner functions illustrating the evolution of the output state. For each value of $\Vert \alpha \Vert$ we show a 3D and a contour plot of the Wigner function. These results are obtained directly from raw homodyne data, without corrections for the detection efficiency. The value of $\alpha$ is arbitrarily chosen to be real and positive, but the results would be the same for any other choice, since the amplifier gain is phase independent.

Figure 3

Experimental test of the noiseless optical amplifier. (a) effective phase-independent gain as a function of the input state amplitude. The solid line is the prediction from a model accounting for the main imperfection of our setup. The relevant values of $\Vert \alpha \Vert$ are smaller than 0.1. (b) since the noise of the amplified state is not fully circular in phase space (see Fig. 2), homodyne detections with different phases will see different noises. Therefore, we plot the average EIN (■), maximal EIN (▲), and minimal EIN (●) as a function of the input state amplitude $\Vert \alpha \Vert$. Our model predicts the solid line for the average EIN, and dashed lines for the minimal and maximal EIN in the output state. We also report as a reference (dotted line) the minimal noise attained with a nondegenerate parametric amplifier [1] for $g_{\mathrm{eff}}>1$ and with a beam splitter model [25] for $g_{\mathrm{eff}}<1$. For small $\Vert \alpha \Vert$, $N_{\mathrm{eq}}$ is clearly negative.