Ultrastrong Coupling Few-Photon Scattering Theory

We study the scattering of individual photons by a two-level system ultrastrongly coupled to a waveguide. The scattering is elastic for a broad range of couplings and can be described with an effective $U(1)$-symmetric Hamiltonian. This simple model allows the prediction of scattering resonance line shapes, validated up to $\alpha =0.3$, and close to the Toulouse point $\alpha =1/2$, where inelastic scattering becomes relevant. Our predictions model experiments with superconducting circuits [P. Forn-Díaz et al., Nat. Phys. 13, 39 (2017)] and can be extended to study multiphoton scattering.

Figure 1

Ground state properties of the polaron Hamiltonian $H_p$ computed with MPS for different cutoffs. We plot (a) the excitation probability of the qubit $P_e=⟨{\sigma }^z+1⟩/2$ and (b) the total energy of photons $E_{\text{photon}}=\sum _k{\omega }_k⟨a_k^{†}{a}_k⟩/\mathrm{\Delta }$.

Figure 2

Spontaneous emission of the two-level system in the polaron-transformed model $H_p$. (a) Total number of excitations $N={\sigma }^{+}{\sigma }^{-}+\sum _k{a}_k^{†}{a}_k$ starting from state $|\uparrow ⟩|0⟩$. (b) Excitation probability $⟨\psi |{\sigma }^{+}{\sigma }^{-}|\psi ⟩$ as a function of time and (c) spectrum of emitted photons at $t\mathrm{\Delta }=30$. Lines correspond to $\alpha =0.01$ (solid), 0.07 (dashed), and 0.35 (dash-dot), simulated with Hamiltonian (3). Dots were obtained using Hamiltonian (4); the thin dotted line shows the lab frame values $⟨\psi |U_p^{†}{\sigma }^{+}{\sigma }^{-}{U}_p|\psi ⟩$ for $\alpha =0.35$.

Figure 3

(a) Reflection coefficient $R_k$ as a function of the spin-boson coupling strength $\alpha$ and the photon frequency. In solid, we plot the resonance frequency, ${\omega }_{\text{reson}}$, and in dashed-dot, we show the half-height lines. (b) Three cuts of the above plot show asymmetric line shapes for increasing $\alpha$. (c) The ratio between the experimental linewidth and the resonance frequency is a lower bound for $\alpha$.

Figure 4

Single-photon reflection and transmission coefficients as a function of the incident photon frequency at the Toulouse point $\alpha =1/2$, for an exponential cutoff ${\omega }_c={10}^8\mathrm{\Delta }$. We plot (a) the phases of the transmission and reflection amplitudes and (b) the reflection and transmission probabilities, both exact (solid) and with the polaron-RWA (6) (dashed). (c) Probability to detect one and three outgoing photons $P_1+P_3$ (dashed) and single-photon elastic scattering probability $P_1=R+T$ (solid).