Phase-space open-systems dynamics of second-order nonlinear interactions with pulsed quantum light

TL;DR

This work develops a phase-space framework for open quantum systems undergoing χ^{(2)} nonlinear interactions with pulsed, multimode light. It introduces the generalized Bloch-Messiah decomposition (GBMD) to map differing numbers of input and output modes to a common, reduced set, enabling a tractable input-output relation for Wigner functions. The central result expresses the output Wigner function as a convolution of a transformed reduced-input Wigner with a Gaussian, followed by a rescaling, naturally capturing decoherence and thermalization from mode-trace. Two key examples—THz-to-optical up-conversion of a single-mode Fock state and of a two-mode squeezed vacuum—illustrate how regime (beam-splitting vs squeezing) and entanglement affect the output state, including von Neumann entropy as a measure of thermalization. The framework supports optimization of ultrafast frequency conversion and measurement of quantum features (e.g., Wigner negativity) in broadband pulsed states, with potential applications in electro-optic sampling and ultrafast quantum tomography.

Abstract

The theoretical description of broadband, multimode quantum pulses undergoing a second-order $χ^{(2)}$-nonlinear interaction can be quite intricate, due to the large dimensionality of the underlying phase space. However, in many cases only a few broadband (temporal) modes are relevant before and after the nonlinear interaction. Here we present an efficient framework to calculate the relation between the quantum states at the input and output of a nonlinear element in their respective relevant modes. Since the number of relevant input and output modes may differ, resulting in an open quantum system, we introduce the generalized Bloch-Messiah decomposition (GBMD), reducing the description to an equal number of input and output modes. The GBMD enables us to calculate the multimode Wigner function of the output state by convolving the rescaled Wigner function of the reduced input quantum pulse with a multivariate Gaussian phase-space function. We expand on this result by considering two examples input states: A Fock state in a single broadband mode and a two-mode squeezed vacuum, both in the THz-frequency regime, up-converted to a single output broadband mode of optical frequencies. We investigate the effect, the convolution and thermalization due to entanglement breakage have on the output Wigner function by calculating the von Neumann entropy of the output Wigner function. The methods presented here can be used to optimize the amplification or frequency conversion of broadband quantum states, opening an avenue to the generation and characterization of optical quantum states on ultrafast time scales.

Figures (8)

• Figure 1: A schematic representation of the scenario considered in this work. In the most general setting, a quantum pulse of light, composed of multiple temporal modes with profiles $u_i(t)$, traverses a nonlinear medium with a second-order nonlinear susceptibility $\chi^{(2)}$. After the nonlinear interaction some, possibly different, modes $v_i(t)$ are relevant, e.g., for detection. An example of such a setting is electro-optic sampling or nonlinear homodyne detection. Since the set of input and output modes is not necessarily closed under the transformation by the nonlinear interaction, some additional modes $h_i(t)$ in the ground state (vacuum) are required for the description. The main achievement of this work is a relation between the input quantum pulse and the quantum state occupying the output modes based on the (multimode) Wigner function in Eq. \ref{['eq:outputWignerfunction1']}. This relation only includes the modes $u_i(t)$ and $v_i(t)$, but not $h_i(t)$, which we achieve by introducing the generalized Bloch-Messiah decomposition (GBMD) in Sec. \ref{['sec:gbmd']}.
• Figure 2: (a) The input modes of $\hat{\vec{\Gamma}}_\text{in}$ in the frequency domain. (b) The effective input modes, which are obtained from the generalized Bloch-Messiah decomposition in Eq. \ref{['eq:gbmd']}, by first transforming the input modes using the symplectic matrix $S_R$ and afterwards reducing the number of effective modes using $P_R$ to match the number, $M$, of output modes $v_i$ (in this case we consider one output mode, $M=1$, centered at $\omega_\text{out}$). We call the modes which are kept system modes (labeled by $s$) and label the residual modes by $r$. The effective modes are up-converted to the output modes in $\hat{\vec{\Gamma}}_\text{out}$, shown in (c), by the $\chi^{(2)}$-nonlinear interaction resulting in Eq. \ref{['eq:in_out_quadratures']}. Here we used the nonlinear interaction introduced in Sec. \ref{['sec:first_order_unitary']} as an example.
• Figure 3: An example of the kernels $J(\Omega, \omega)$ and $K(\Omega, \omega)$ describing the nonlinear interaction mediating between the input and output modes, which is defined in Eq. \ref{['eq:nl_unitary_first_order']}. The kernel $J(\Omega, \omega)$ describes the strength of a squeezing interaction between the continuous (angular) frequency mode at $\Omega$ and the mode at $\omega$, simultaneously creating or annihilating a photon in each mode. The kernel $K(\Omega, \omega)$ corresponds to a beam splitter interaction between the continuous frequency modes and creates a photon in the mode $\omega$, while annihilating one at $\Omega$. The nonlinear crystal is assumed to be zinc tellurid with a refractive index $n(\omega)$Marple1964, length $L = 20µm$ and an electro-optic coefficient $r_{41} = 4\cdot 10^{-12}sA\per kgm$Boyd2019. The classical coherent pulse driving the nonlinear interaction is assumed to be centered at $\omega_\text{p} / (2\pi) = 200THz$, to have a bandwidth of $\Delta \omega_{\text{p}} / (2\pi) = 118THz$ and a coherent amplitude of $\alpha$. We assume the nonlinear crystal is in free-space, the coherent pulse is in the fundamental paraxial mode with beam waist area $A=28µm\squared$ and polarized orthogonal to the mode of the quantum interaction. Note that for one frequency, $\Omega$ or $\omega$, fixed to $\omega_\text{p}$, most of the contribution from the squeezing interaction is below the central frequency of the coherent pulse, but above for the beam splitting contribution.
• Figure 4: (a) Wigner function of a pulsed THz three-photon Fock state used as the input to the nonlinear interaction. (b) and (c) show the output Wigner function for the parameter sets (i) and (ii) defined in the main text, and visualized in Fig \ref{['fig:oneInOneOut']}, corresponding to the beam-splitter regime introduced in Sec. \ref{['sec:first_order_unitary']}. In (b) the amplitude $\alpha$ of the coherent pulse driving the nonlinear interaction is chosen such that no up-conversion of the THz quantum pulse to the optical output mode occurs, showing the non-monotonic dependence of the up-conversion on the amplitude $\alpha$ in the beam-splitter regime. Thus the Wigner function corresponds to the vacuum of the optical modes. While in (c) the point of optimal up-conversion for a given $\omega_\text{out}$ is used, resulting in an output Wigner function resembling the quadrature distribution of a Fock state, i.e., a Hermite-Gauss distribution. The parameters of (d) are in the squeezing regime, where no optimal point of up-conversion exists. The stronger the amplitude of the coherent pulse, $\alpha$, the more the Fock state gets amplified in phase space.
• Figure 5: The four parameters, independent of the input quantum state, determining the Wigner function of the output quantum state, as a function of the central frequency of the output mode $\omega_\text{out}$ and the amplitude $\alpha$ of the coherent pulse driving the nonlinear interaction. (a), (b) The nonzero elements $A_{11}$, $A_{22}$ of the matrix $A$, used in the input-output relation of Eq. \ref{['eq:in_out_quadratures']}. The matrix $A$ scales the phase space of the output Wigner function. (c), (d) The singular values $\sigma_x$ and $\sigma_p$ of the covariance matrix defining the Gaussian phase-space function in Eq. \ref{['eq:gaussian_equal_mode_num']} as a function of the same parameters. Since $\sigma_x$, $\sigma_p$ determine the width of the Gaussian, the input Wigner function is convolved with, they determine the smoothness of the output Wigner function. The inverse of $\sigma_x$ and $\sigma_p$ are plotted to avoid divergences. The horizontally dashed lines mark the transition from the squeezing to the beam-splitting regime, introduced in Sec. \ref{['sec:first_order_unitary']}, as $\omega_\text{out}$ is increased. The colormap is saturated in the squeezing regime.