Temporal limitations and digital data processing in continuous variable measurements of non-Gaussian states

Abstract

Non-Gaussian quantum states and operations are essential tools for multiple quantum information protocols exploiting light as information career. In this context, a key role is played by schemes operating with continuous wave light, in which non-Gaussian states are obtained by photon subtraction/addition and eventually reconstructed by quantum state tomography. In these configurations, the temporal resolution of the homodyne detection and the digital data processing critically affect our ability to faithfully reconstruct the produced non-Gaussian states. In this work, we apply digital data processing to experimental data to study how the temporal performances of the detection chain affect the acquisition and treatment of tomographic data. This allows understanding how these features impact the quality of quantum states observed by non-ideal detection chains. By doing so, we discuss the actual constraints on the acquisition and reconstruction of non-Gaussian states by taking into account the limitations of realistic experimental resources.

Figures (4)

• Figure 1: a) Schematic diagram of an heralding scheme. ESS stands for entangled state source. The heralding path contains a filtering stage and single photon detection. On the heralded state detection, the homodyne detection and a digital sampling part have finite bandwidths and sampling rate, respectively. b) Temporal profile of the ideal detection mode in the case of an optical cavity used as filter on the heralding path and of a single pass ESS (i.e. a source without a feedback or resonator mechanism). c) Digitalized trace at the output of the CV detection chain that measures the heralded state. Each heralding event leads to the recording of one of such traces of duration $\Delta T$. The heralding path detector is taken to be ideal; the effect of non ideal heralding on the quality of produced NG states has been already treated in a previous paper Gouzien_2018.
• Figure 2: Temporal modes $u(t)$ in a.u. extracted from experimental data post-treated to simulate the effect of different HD cutoff frequency (a) and sampling rates (b). All profiles are normalised to have $\int u(t)^2dt=1$. The peak height is indicated for each $u(t)$. The profile corresponding to $f_c=301$ MHz and $f_s=5$ Gsps (in red) is the one obtained with native experimental data and it fully matches the analytically computed $u_\text{id}(t)$melalkia_plug-and-play_2022. The thinner lines in figure (a) show the temporal profiles that are simulated by applying the Butterworth filter to the ideal detection mode, i.e.$u_\text{fc}(t)= FT^{-1}[H(f,f_c)\cdot \tilde{u}_\text{id}(f)]$.
• Figure 3: Negativity of the Wigner function at the origin as a function of the CV detector bandwidth ($f_c$) and of the sampling rate ($f_s$). The values of $f_s$ have been limited to values below 2.5 Gsps as no relevant change is observed in $W_{0,0}$ between $f_s$= 2.5 Gsps and 5 Gsps.
• Figure 4: Reconstructed Wigner functions $W(x,y)$ of the heralded NG state corrected by the homodyne detection efficiency $\eta_{HD}=0.72$. (a) Optimal state at $f_c=301$ MHz and $f_s$=5 Gsps (untreated experimental data). (b) Reconstructed state at $f_c=301$ MHz and $f_s=238$ Msps. (c) Reconstructed state at $f_c=31$ MHz and $f_s=5$ Gsps. (d) Reconstructed state at $f_c=31$ MHz and $f_s=238$ Msps.