Scattering in the Ultrastrong Regime: Nonlinear Optics with One Photon

The scattering of a flying photon by a two-level system ultrastrongly coupled to a one-dimensional photonic waveguide is studied numerically. The photonic medium is modeled as an array of coupled cavities and the whole system is analyzed beyond the rotating wave approximation using matrix product states. It is found that the scattering is strongly influenced by the single- and multiphoton dressed bound states present in the system. In the ultrastrong coupling regime a new channel for inelastic scattering appears, where an incident photon deposits energy into the qubit, exciting a photon-bound state, and escaping with a lower frequency. This single-photon nonlinear frequency conversion process can reach up to 50% efficiency. Other remarkable features in the scattering induced by counterrotating terms are a blueshift of the reflection resonance and a Fano resonance due to long-lived excited states.

Figure 1

Left: schematics for the considered system. Right: dispersion relation of the free photon band considered in this problem. The discontinuous line represents a linear dispersion with the same velocity at the qubit frequency $\mathrm{\Delta }=1$.

Figure 2

Energies of bound states. Dependence with coupling $g$ of ground state and bound excited state energies. $GS$ and $E_2$ have even parity, while $E_1$ and $E_3$ have odd parity. The inset shows the spacial profile of the number of excitations in each bound state, for $g=0.8$.

Figure 3

Time evolution. Evolution of $⟨n_x⟩$ (upper panels) and $⟨n_k⟩$ (lower panels) for ${\omega }_{\text{in}}=0.70$ (left panels) and ${\omega }_{\text{in}}=0.85$ (right panels). In both cases, at $t=0$ an initial wave packet is set centered at $x_0=-80$ and the coupling is $g=0.7$. For ${\omega }_{\text{in}}=0.70$ the scattering is elastic, while for ${\omega }_{\text{in}}=0.85$ there is an inelastic scattering channeling too.

Figure 4

Transmission as a function of both incident photon frequency ${\omega }_{\text{in}}$ and $g$. (a) Transmittance within the RWA. (b) Elastic transmittance in the full model. The black line marks the estimated frequency for the Fano resonance while the white line gives the estimated spectral position for the transmittance minimum (see text).

Figure 5

Inelastic transmittance. Transmittance in the full model in the inelastic channel as a function of both incident photon frequency ${\omega }_{\text{in}}$ and $g$. The white line is the estimated boundary for the region where the photon frequency conversion occurs. The inset presents, for $g=0.8$, the inelastic reflection spectra when the waveguide is terminated at position $\mathrm{\Delta }x=20$, showing that a 100% efficient Raman process is possible using one incoming photon.