Noiseless amplification of weak coherent fields exploiting energy fluctuations of the field

Quantum optics dictates that amplification of a pure state by any linear deterministic amplifier always introduces noise in the signal and results in a mixed output state. However, it has recently been shown that noiseless amplification becomes possible if the requirement of a deterministic operation is relaxed. Here we propose and analyze a noiseless amplification scheme where the energy required to amplify the signal originates from the stochastic fluctuations in the field itself. In contrast to previous amplification setups, our setup shows that a signal can be amplified even if no energy is added to the signal from external sources. We investigate the relation between the amplification and its success rate as well as the statistics of the output states after successful and failed amplification processes. Furthermore, we also optimize the setup to find the maximum success rates in terms of the reflectivities of the beam splitters used in the setup and discuss the relation of our setup with the previous setups.

Figure 1

Schematic illustration of the noiseless amplification of a weak coherent field. First, a single photon is subtracted from the field, then added back to the field, and finally again subtracted from the field. This sequence is also described by the operator $\widehat{a}{\widehat{a}}^{†}\widehat{a}$.

Figure 2

(a) Effective gain as a function of the input field amplitude for three different beam splitter reflectivities and for the analytic low reflectivity approximation used by Zavatta et al. [2]. (b) The effective fidelity calculated with respect to a coherent field $|g_{\mathrm{eff}}\alpha \rangle$. (c) The fidelity calculated with respect to an ideally amplified field $|2\alpha \rangle$ for comparison with the results obtained by Zavatta et al. [2]. (d) The contour plots of the Wigner functions for three amplified coherent fields with different input amplitudes when the beam splitter reflectivity is $r=0.4$.

Figure 3

Probability of successful amplification was maximized in an effective gain constrained optimization problem. The optimal (a) probability of successful amplification $P_{\mathrm{opt}}$, (b) the input field amplitude ${\left|\alpha \right|}_{\mathrm{opt}}$, (c) the beam splitter reflectivity $r_{\mathrm{opt}}$, and (d) the output field fidelity $F_{\mathrm{opt}}$ are plotted as a function of the required minimum effective gain parameter $g_{\mathrm{eff}},0$.