Generating spatially separated correlated multiphoton states in nonlinear waveguide quantum electrodynamics

TL;DR

This work tackles the deterministic generation of strongly correlated multi-photon states by introducing cascaded inelastic scattering in a nonlinear waveguide, mediated by far-detuned two-level emitters. Central to the approach is the pseudo-giant atom (PGA) concept, which treats the nonlocal, bound-state–driven coupling as a multi-point scattering potential, enabling unidirectional S to D conversion and automatic spatiotemporal sorting of output light into distinct photon-number components. By realizing the PGA with real giant atoms and employing scattering theory, the authors demonstrate programmable generation of S, D, and T states, culminating in a cascaded architecture that yields spatially separated, entangled multi-photon resources. The resulting spatiotemporal multiplexing and photon-number entanglement have immediate implications for quantum metrology, distributed quantum computing, and scalable quantum networks, with practical feasibility in superconducting circuits using transmon-based nonlinear waveguides.

Abstract

Strongly correlated multi-photon states are indispensable resources for advanced quantum technologies, yet their deterministic generation remains challenging due to the inherent weak nonlinearity in most optical systems. Here, we propose a scalable architecture for producing correlated few-photon entangled states via cascaded inelastic scattering in a nonlinear waveguide. When a single photon scatters off a far detuned excited two-level emitter, it coherently converts into a propagating doublon, a bound photon pair with anomalous dispersion. This doublon can subsequently scatter off a downstream excited emitter to further convert into a triplon, and so on, thereby establishing a photon-number amplification cascade $|\cdot \rangle \!\! \rightarrow \!\! |\!\!: \rangle \!\! \rightarrow \! \! |\!\!\therefore \rangle \!\! \to \!\! ...$ Central to this process is the concept of a pseudo-giant atom, which we introduce here to capture the non-local scattering potential emergent from the wave functions of bound states. By implementing this scheme using a real giant atom with multiple engineered coupling points, we achieve unidirectional and full controllable photon conversion without backscattering. The resulting output state forms a programmable superposition of spatially and temporally isolated photon-number components, automatically sorted by their distinct group velocities. This work opens a new paradigm in quantum state engineering, enabling on-demand generation of complex multi-photon resources for quantum simulation, metrology, and scalable quantum networks.

Figures (12)

• Figure 1: Schematic of cascaded scattering in a nonlinear waveguide. (a) A single photon incident from the left in the waveguide first scatters off a detuned excited emitter, generating a pair of correlated photons---a doublon. The doublon subsequently scatters off a second excited emitter, producing three correlated photons---a triplon. This cascaded scattering process can, in principle, be extended to generate $\mathcal{N}$ correlated photons. (b) Energy-level structure for single- ($E_{\mathrm{S}}$), two- ($E_{\mathrm{D}}$), and three-photon ($E_{\mathrm{T}}$) wave packets and the emitters with transition frequencies detuned by $\Delta_{e1}$ and $\Delta_{e2}$ from the waveguide cavity frequency $\omega_c$. The bandwidth of the single-photon band is 4J; the doublon and triplon bands are more narrow than that, and are not shown exactly to scale in this illustration. (c) Temporal evolution of the scattering process. The shading depth at times $\{t_{0^-}, t_0, t_{0^+}\}$ represents the instantaneous effective scattering strength.
• Figure 2: Schematic illustration of the scattering process at different stages. (a) Initial state at $t \to -\infty$. The left panel shows the physical setup in real space. The right panel depicts the effective "pseudo-giant atom" (PGA) description of the setup. (b) Final state at $t \to +\infty$, featuring the transmitted ($t$) and reflected ($r$) single-photon components, as well as right- and left-going doublon wavepackets $\ket{\pm K}$. (c) Scattering eigenstate structure, with right- and left-going single-photon wave functions denoted by $\varphi_{R,L}$. The labeled amplitudes indicate the spatial distribution of waves in different regions.
• Figure 3: Populations of (a) the emitter excitation, (b) single-photon transmission $|t|^2$ and reflection $|r|^2$, and (c) right- and left-going doublons $|u_{\pm}|^2$, as functions of the coupling strength $g$. Numerical results are indicated by diamond markers. Solid colored curves are obtained by solving Eq. \ref{['revised_f']} with the size of the PGA $\mathcal{S}_\text{PGA} = 7$, while thin gray dashed curves represent the pointlike scenario $\mathcal{S}_\text{PGA} = 1$. Parameters are $L_G = 0.002$, $U = 6J$, $\Delta_e = -6.633J$, and $k_0 = \pi / 2$.
• Figure 4: Schematic illustration of interference in doublon generation. (a) The pointlike scenario with equal distribution doublons in the right- and left-going modes. (b) PGA scenario: interference among seven pseudo-coupling points leads to an asymmetric doublon distribution.
• Figure 5: Scattering into spatially separated correlated photons. (a) Time evolution of the populations, including the single-photon state $P_{\rm I}$, the full two-photon state $P_{\rm II}$, the doublon state $P_D$, and the right-/left-going doublon components $|u_{\pm}|^2$, for different nonlinear strengths $U = 0$ and $U = 6 J$. The inset shows the initial bound state population. (b) Photon number $\langle \mathcal{N} \rangle$ as a function of position $n$ and time $J t$. Following scattering, the incident photon splits into four distinct components: transmitted ($t$) and reflected $(r)$ single photons, and right- and left-going doublon states $u_\pm$. The inset displays the natural spatial separation between single-photon and doublon states, arising from their distinct group velocities $v_{k_0}$ and $v_{K_r}$, respectively. The coupling strength is here set to $g = 0.5J$; all other parameters are the same as in Fig. \ref{['fig3']}.