Completely characterizing multimode nonlinear-optical quantum processes

Abstract

Complete characterization of a multimode optical process has paved the way for understanding complex optical phenomena, leading to the development of novel optical technologies. Until now, however, characterizations have mainly focused on a linear-optical process, despite the plethora of multimode nonlinear-optical processes crucial for photonic technologies. Here we report the experimental characterization of multimode nonlinear-optical quantum processes by obtaining the full information needed to describe them while satisfying the necessary physical condition. Specifically, to characterize a second-order nonlinear-optical process of parametric downconversion, we determine the amplification and noise matrices of multimode field quadratures. The full information allows us to factorize the multimode process, leading to the identification of eigenquadratures and their associated amplification and noise properties. Moreover, we demonstrate the broad applicability of our method by characterizing various nonlinear-optical quantum processes, including cluster state generation, mode-dependent loss with nonlinear interaction, and a quantum noise channel. Our method, by providing a versatile technique for characterizing a nonlinear-optical process, will be beneficial for developing scalable photonic technologies.

Figures (5)

• Figure 1: A multimode nonlinear-optical quantum process.a, Unlike a linear-optical process, a second-order nonlinear-optical process ($\chi^{(2)}$) induces optical amplification and noise in multimode fields. This nonlinear process is fully characterized by the amplification matrix A and the noise matrix N in Eq. (\ref{['eq1']}). b, Our method for completely characterizing a multimode nonlinear-optical quantum process. We first inject coherent states in sequence as a probe into an unknown multimode nonlinear process, and measure the output mean quadrature vector (left). Next, we use the multimode vacuum state as the probe, and measure the output state by using homodyne detection (right). The former provides information about A, and the latter provides additional information about N, enabling the complete characterization of the nonlinear-optical process and thereby facilitating the prediction of the evolution of general multimode quantum states.
• Figure 1: Input eigenquadratures.a, Eigenquadratures obtained by decomposing the amplification matrix (i.e., the column vectors of $\textbf{V}$). b, Experimentally generated eigenquadratures for probing the multimode nonlinear process. Eigenquadratures are represented by superpositions of the wavelength modes ($m=1,\dots,16$), where amplitudes and phases are obtained by $\sqrt{(q_m)^2+(q_{m+16})^2}$ and $\tan^{-1}{(q_{m+16}/{q_m}})$, respectively.
• Figure 2: Experimental characterization of a 16-mode nonlinear-optical quantum process.a,b, The nonlinear process is completely characterized by obtaining (a) amplification matrix A and (b) noise matrix N. $\hat{\textbf{x}}=(\hat{x}_1, \hat{x}_2, ..., \hat{x}_{16})^T$, $\hat{\textbf{p}}=(\hat{p}_1, \hat{p}_2, ..., \hat{p}_{16})^T$, and the inset shows the wavelength distribution of the 16 modes. c, A singular value decomposition of A into $\textbf{U}\textbf{D}\textbf{V}^T$. The decomposition allows us to understand the multimode process $\textbf{A}$ as independent amplification processes (see the inset): the input (output) eigenquadratures $\{\vec{v}_m\}$ ($\{\vec{u}_m\}$) are the column vectors of $\textbf{V}$ ($\textbf{U}$), where the associated amplifications $\{d_m\}$ are the diagonals of $\textbf{D}$. The eigenquadratures are plotted in more detail in Extended Data Fig. \ref{['ext_fig1']} and \ref{['ext_fig2']}. d, Amplifications of the eigenquadratures: prediction from the decomposition result $\{d_m\}$ and direct experimental measurement by injecting probes in eigenquadratures. e, The overlap between experimentally generated eigenquadratures (input: $\{\vec{v}_m^{\textrm{(exp)}} \}$, output: $\{\vec{u}_m^{\textrm{(exp)}} \}$) and the eigenquadratures obtained by the decomposition, quantified by $( \vec{v}_m^{\textrm{(exp)}} \cdot \vec{v}_m )^2$ and $( \vec{u}_m^{\textrm{(exp)}} \cdot \vec{u}_m )^2$.
• Figure 2: Output eigenquadratures.a, Eigenquadratures obtained by decomposing the amplification matrix (i.e., the column vectors of $\textbf{U}$). b, Experimentally observed eigenquadratures as a result of the multimode nonlinear process. Eigenquadratures are represented by superpositions of the wavelength modes ($m=1,\dots,16$), where amplitudes and phases are obtained by $\sqrt{(q_m)^2+(q_{m+16})^2}$ and $\tan^{-1}{(q_{m+16}/{q_m}})$, respectively.
• Figure 3: Experimental characterization of various multimode nonlinear-optical quantum processes.a,b, Cluster state generation. a and b represent the experimentally obtained A and N, respectively. The inset in a shows the theoretical prediction for comparison. c-f, Mode-dependent loss with nonlinear interaction (c: A, d: N). N exhibits correlated noise having non-vanishing off-diagonals. e,f, Eigendecomposition of N (e: eigenvalues, f: eigenquadratures). In f, eigenquadratures are represented by complex superpositions of the four optical modes. g,h, Quantum noise channel (g: A, h: N). The off-diagonals in N are sufficiently large to introduce quantum-correlated noise to the output. For the vacuum input, the output exhibits a negative partial-transposition eigenvalue of $\hbox{[}1.0]{$-$} 0.37$Duan:2000fw, making the process an entanglement-generating channel. See Supplementary Information for detailed methods for realizing the multimode nonlinear-optical processes.