Heralded noiseless amplification and attenuation of non-Gaussian states of light

We examine the behavior of non-Gaussian states of light under the action of probabilistic noiseless amplification and attenuation. Surprisingly, we find that the mean-field amplitude may decrease in the process of noiseless amplification—or may increase in the process of noiseless attenuation, a counterintuitive effect that Gaussian states cannot exhibit. This striking phenomenon could be tested with experimentally accessible non-Gaussian states, such as single-photon added coherent states. We propose an experimental scheme, which is robust with respect to the major experimental imperfections, such as inefficient single-photon detection and imperfect photon addition. In particular, we argue that the observation of mean-field amplification by noiseless attenuation should be feasible with current technology.

Figure 1

Noiseless attenuator using a beam splitter with amplitude reflectance $\nu$. The lower input mode is set to vacuum, and we postselect on vacuum in the lower output mode.

Figure 2

Proposed experimental setup. Coherent states are injected into signal and idler ports of a nonlinear crystal where parametric downconversion with a low gain $\lambda$ occurs. Conditional photon addition is heralded by a click of the trigger avalanche photodiode (${\mathrm{APD}}_T$). ${\mathrm{BS}}_1$ is a beam splitter with amplitude reflectance $\nu$, and noiseless attenuation is heralded by a no click of the single-photon detector APD. Imperfect detection with efficiency $\eta <1$ is modeled by coupling to an auxiliary mode $C$ prepared in a vacuum state where $\eta$ is the transmittance of ${\mathrm{BS}}_2$.

Figure 3

Noiseless attenuation of a non-Gaussian state Eq. (52). The amplitude gain $G_{\mathrm{eff}}$ is plotted as a function of the attenuation factor $\nu$ for four different values of detection efficiency $\eta =1$ (solid line), $\eta =0.75$ (long dashed line), $\eta =0.5$ (dot-dashed line), and $\eta =0.25$ (short dashed line). The other parameters read $\alpha =0.25,\delta =-0.55$, and $p=1$.

Figure 4

Dependence of the (a) input amplitude $A$ and (b) amplitude gain $G_{\mathrm{eff}}$ on the displacement parameter $\delta$. The other parameters read $\alpha =0.25,\nu =1/\sqrt{2}$ and $\eta =1,p=1$ (solid line) or $\eta =0.25,p=0.75$ (dashed line).

Figure 5

The same as Fig. 3 but $p=0.75$ and $\delta =-0.65$.