Tomographic markers and photon addition to coherent states of light: Comparison with experiment

TL;DR

This work addresses how to identify and quantify photon addition to a single-mode coherent state using optical tomograms rather than full density-matrix reconstruction. The authors define and compute three tomographic distance markers, the Wasserstein distance $W_{1}$, the Kullback–Leibler divergence $D_{\rm KL}$, and the Bhattacharyya distance $D_{\rm B}$, directly from tomograms to track changes induced by photon addition. They apply the method to $m=1,2$ photon-added coherent states $|\alpha,m\rangle$ and validate against a recent experimental report, showing that tomogram-based markers reproduce fidelity trends with $|\alpha|$ and that $D_{\rm KL}$ and $D_{\rm B}$ often outperform $W_{1}$ as discriminants at larger $|\alpha|$, while quadrature variances from tomograms agree with reconstructed results. The results demonstrate a practical, reconstruction-free approach to characterizing nonclassical light and suggest extensions to more complex or multimode systems for broader applicability.

Abstract

Photon addition to quantized light is of immense interest, both experimentally and theoretically. We identify a set of markers that play an important role in the context of photon addition to coherent states of light. These markers are directly computable from optical tomograms. We calculate the amplification gain due to photon addition, and the dependence of quadrature variances on relevant parameters, from the tomograms and compare them with results obtained after state reconstruction in a recent experiment. Our results match well with the fidelity plots reported by the experimenters. Our approach which circumvents state reconstruction could provide a viable procedure to identify specific aspects of photon addition to nonclassical light as well, from the tomograms themselves.

Figures (3)

• Figure 1: Fidelities of $n$-photon-added coherent states with exact coherent states. Red solid lines represent the theoretical dependence of fidelity with the coherent state with amplitude $\beta_{\rm opt}$ given by Eq. (11) of Fadrny:2024. Blue dashed lines depict theoretical fidelities with the coherent states with amplitude ${\sf g}_{n}(\alpha)\alpha$, where ${\sf g}_{n}(\alpha)$ is the amplification gain (see Eq. (7) of Fadrny:2024). Red circles and blue squares are the corresponding experimental fidelities determined from the reconstructed density matrices of the generated states. Data are plotted for $n = 1$ ( a) and $n = 2$ ( b). This figure (with caption) has been taken from the publication "Experimental preparation of multiphoton-added coherent states of light" by J. Fadrný, M. Neset, M. Bielak, M. Ježek, J. Bílek, and J. Fiurášek, npj Quantum Information 10, 89 (2024) Fadrny:2024, DOI:10.1038/s41534-024-00885-y.
• Figure 2: (a) $\langle W_{1} \rangle$, (b) $\langle D_{\rm KL} \rangle$ and (c) $\langle D_{\rm B} \rangle$ are the average values of the Wasserstein distance, Kullback-Leilber divergence, and Bhattacharyya distance respectively as functions of $|\alpha|$. The averages are obtained by computing these distances between $|\alpha,m\rangle$ and $|\beta_{\rm opt}\rangle$ ($m=1$: black circles and $m=2$: blue asterisks) for $5$ different tomographic slices (i.e., $5$ different values of $\theta$ which are equally spaced between $0$ and $\pi$). Increasing the number of tomographic slices does not change the values of these three averages significantly. The three averages increase with increase in $m$ (from $1$ to $2$) for any given $|\alpha|$. With increase in $|\alpha|$, the three averages decrease, corresponding to an increase in the fidelity between $|\alpha,m\rangle$ and $|\beta_{\rm opt}\rangle$, which is borne out by the experimental plots (Figs. 5(a) and (b) in Fadrny:2024, reproduced in Fig. \ref{['fig:Fadrny2024_figure_5']} here). (d) The relative difference $(\langle \cdot \rangle_{m=2} - \langle \cdot \rangle_{m=1}) / (\langle \cdot \rangle_{m=2} + \langle \cdot \rangle_{m=1})$ for $W_{1}$ (black circles), $D_{\rm KL}$ (blue asterisks), and $D_{\rm B}$ (red triangles) plotted as functions of $|\alpha|$. The plots corresponding to $\langle D_{\rm KL} \rangle$ and $\langle D_{\rm B} \rangle$ are essentially similar. Overall, $\langle W_{1} \rangle$, $\langle D_{\rm KL} \rangle$ and $\langle D_{\rm B} \rangle$ computed directly from the tomograms, mirror the trends in the fidelity computed after state reconstruction. Since the relative differences are more pronounced with increasing $|\alpha|$ for $D_{\rm KL}$ and $D_{\rm B}$ compared to $W_{1}$, they are better discriminators of photon addition to the CS for $|\alpha| > 1$.
• Figure 3: Top panel: $\langle W_{1} \rangle$ between $|\alpha,m\rangle$ and $|{\sf g}_{m}(\alpha)\alpha\rangle$ ($m=1$: green triangles and $m=2$: red squares), and between $|\alpha,m\rangle$ and $|\beta_{\rm opt}\rangle$ ($m=1$: black circles and $m=2$: blue asterisks), as functions of $|\alpha|$. Bottom panel: $\langle D_{\rm KL} \rangle$ between $|\alpha,m\rangle$ and $|{\sf g}_{m}(\alpha)\alpha\rangle$ ($m=1$: green triangles and $m=2$: red squares), and between $|\alpha,m\rangle$ and $|\beta_{\rm opt}\rangle$ ($m=1$: black circles and $m=2$: blue asterisks), as functions of $|\alpha|$.