Multiphoton Scattering Tomography with Coherent States

In this work we develop an experimental procedure to interrogate the single- and multiphoton scattering matrices of an unknown quantum system interacting with propagating photons. Our proposal requires coherent state laser or microwave inputs and homodyne detection at the scatterer’s output, and provides simultaneous information about multiple—elastic and inelastic—segments of the scattering matrix. The method is resilient to detector noise and its errors can be made arbitrarily small by combining experiments at various laser powers. Finally, we show that the tomography of scattering has to be performed using pulsed lasers to efficiently gather information about the nonlinear processes in the scatterer.

Figure 1

(a) A quantum scatterer with scattering matrix $U$ transforms an input state $|{\mathrm{\Psi }}_{\mathrm{in}}⟩$ of $m$ photons with generic quantum numbers $k_1,\dots ,k_m$ into an outgoing state $|{\mathrm{\Psi }}_{\mathrm{out}}⟩=U|{\mathrm{\Psi }}_{\mathrm{in}}⟩$ of $n$ photons with labels, $p_1,\dots ,p_n$. (b) Our experimental protocol for determining $U$ requires coherent state wave packets inputs $|{\alpha }_{\overrightarrow{k}}⟩$, prepared with a signal generator ($\sim$), and homodyne measurements at the output. Prior to measurement, the output signal is split evenly by an $N$-port beam splitter (BS), so as to measure all possible filtered correlations $⟨B_{p_1}\dots B_{p_n}⟩$.

Figure 2

A possible implementation of the protocol that works for one- and two-photon scattering matrices.

Figure 3

Reconstruction of the nonlinear two-photon scattering matrix for a two-level system weakly coupled to a 1D photonic channel. (a) Upper bound of the reconstruction error ${ϵ}_b^{(Z)}\ge |{ϵ}_2^{(Z)}|$, as derived in the Supplemental Material [30] when combining $q=1,\dots ,Z$ scattering matrix estimates $E(b^{q-1}|\alpha {|}^2)$ with $b=1.05$. The various curves show ${ϵ}_b^{(Z)}$ as a function of the smallest laser power $|\alpha {|}^2$ for $Z=1,2,\dots ,12$ (from top to bottom). (b) Predicted transmission measurement of $|T_{p_1{p}_2{k}_1{k}_2}{|}^2$ for a qubit with decay rate $\gamma$ and Gaussian wave packets of width $\sigma =0.8\gamma /c$. We vary the momentum differences ${\mathrm{\Delta }}_k=(k_2-k_1)/2$ and ${\mathrm{\Delta }}_p=(p_2-p_1)/2$, between incoming and outgoing photons, and fix the conserved average momentum to $\widehat{k}=({\omega }_0+1.5\gamma )/c$, with ${\omega }_0$ the qubit’s transition frequency. (c) Cross sections of $|T{|}^2$ measured at different widths, $\sigma =0.8\gamma /c$ (dashed blue) and $\sigma =0.4\gamma /c$ (solid brown), give the same exact result for $|\gamma \overline{\mathcal{T}}/c{|}^2$ after deconvolution (blue and brown dotted line). As marked in (a) and (d), the cross sections correspond to $|{\mathrm{\Delta }}_p|=1.5\gamma /c$. (d) Deconvolution of the measurements according to Eq. (19), to recover the two-photon interaction strength $|{\overline{\mathcal{T}}}_{p_1{p}_2{k}_1{k}_2}{|}^2$ of the two-level scatterer derived in Refs. [11, 12, 30].