Non-Gaussianity of quantum states: An experimental test on single-photon-added coherent states

Non-Gaussian states and processes are useful resources in quantum information with continuous variables. An experimentally accessible criterion has been proposed to measure the degree of non-Gaussianity of quantum states based on the conditional entropy of the state with a Gaussian reference. Here we adopt such a criterion to characterize an important class of nonclassical states: single-photon-added coherent states. Our studies demonstrate the reliability and sensitivity of this measure and use it to quantify how detrimental is the role of experimental imperfections in our implementation.

Figure 1

Layout of the experiment. An optical parametric amplifier (OPA) is injected with a coherent state of variable amplitude $\Vert \alpha \Vert$ in the range $[0,1.5]$. This is realized by a 100-$\mu$m-thick slab of potassium niobate, pumped by a frequency-doubled mode-locked Ti:sapphire laser (${\lambda }_p=425$ nm, pulse duration 230 fs, repetition rate 800 kHz). Our OPA is driven in the frequency-degenerate and noncollinear regime to generate an idler at the same wavelength $\lambda =2{\lambda }_p$ as the coherent seed; this is then spatially filtered with a single-mode fiber, spectrally filtered by a diffraction grating and a slit, indicated as F in the figure. Finally, the idler is detected by an avalanche photodiode APD. The observation of the output conditioned by an APD count results in single-photon addition. The quantum state of the output is reconstructed by homodyne detection (HD). Mode matching with the local oscillator (LO) employs polarization: the signal and the LO are first matched on a polarization beam splitter, and then combined using a half waveplate and a second polariser so to realize an accurate 50 : 50 intensity splitting.

Figure 2

Non-Gaussianity $\delta [\varrho ]$ (upper panel) and nonclassicality $\nu [\varrho ]$ (lower panel) as a function of the amplitude $\left|\alpha \right|$ of the input coherent state for different values of the squeezing parameter $r$ (dashed lines); from top to bottom $r=\{0.15,0.30,0.45\}$. The black solid line corresponds to the non-Gaussianity of the ideal photon-added coherent state; that is, to the limit $r\to 0$.

Figure 3

Experimental Wigner functions for increasing values of $\alpha$. The output states are reconstructed by a maximal likelihood algorithm [50] which interpolates 800 000 data points sorted according to their phase into 12 histograms. The effectiveness of the sorting algorithm sets a lower bound $\left|\alpha \right|~0.5$, so that the oscillations due to interference are much larger than low-frequency noise fluctuations. Notice that, for $\Vert \alpha \Vert =0$, this noise can be compensated by using a moving-average technique.

Figure 4

Non-Gaussianity $\delta [\varrho ]$ as a function of the noise parameters of the experimental setup for fixed amplitude $\left|\alpha \right|=0.5$ and squeezing parameter $r=0.15$. In details: the blue dot-dashed line corresponds to $\delta [\varrho ]$ as a function of $\gamma$, the green dotted line as a function of $\xi$, and the red dashed line as a function of $\eta$. The solid lines refer to the non-Gaussianity of the ideal photon-added coherent state with $\left|\alpha \right|=0.5$ (upper black line) and to the non-Gaussianity of the state obtained by considering $\left|\alpha \right|=0.5$, squeezing parameter $r=0.15$, and no imperfections (lower grey line).

Figure 5

(Top) Non-Gaussianity $\delta [\varrho ]$ as a function of the amplitude $\left|\alpha \right|$ of the input coherent state. (Bottom) Nonclassicality $\nu [\varrho ]$—related to the minimum value of the Wigner function of $\varrho$—as a function of $\alpha$. The red points are the experimental values from the reconstructed matrices. The black dashed line is obtained from our model including the main experimental imperfections of our implementation. The parameters are chosen in such a way as to fit the data of the non-Gaussianity: $r=0.105$, $\gamma =0.425$, $\xi =0.96$, and $\eta =0.71$.