Undamped Rabi oscillations due to polaron-emitter hybrid states in a nonlinear photonic waveguide coupled to emitters

The collective dynamics of two noninteracting two-level emitters, which are coupled to a structured wave guide that supports two-photon bound states, is investigated. Tuning the energy of the two emitters such that they are in resonance with the two-photon bound-state energy band, we identify parameter regimes where the system displays fractional populations and essentially undamped Rabi oscillations. The Rabi oscillations, which have no analog in the single-emitter dynamics, are attributed to the existence of a collective polaron-like photonic state that is induced by the emitter-photon coupling. The full dynamics is reproduced by a two-state model in which the photonic polaron interacts with the state $|e,e,\text{vac}\rangle$ (two emitters in their excited state and empty wave guide) through a Rabi coupling frequency that depends on the emitter separation. Our work demonstrates that emitter-photon coupling can lead to an all-to-all momentum space interaction between two-photon bound states and tunable non-Markovian dynamics, opening up exciting directions for emitter arrays coupled to a waveguide.

Figure 1

(a) Schematic of the Hamiltonian $\widehat{H}$. The cavity array and two-level emitters (2LE) are shown; the role of the different energy terms is illustrated. (b) Illustration of the Hilbert space structure of $\widehat{H}$ (left), ${\widehat{H}}^{\text{adia}}$ (middle), and ${\widehat{H}}^{\text{2-st}.}$ (right). The matrix element $M_b(k,n,K)$ is defined in Ref. [60]. Note that the energy difference $2\hslash {\omega }_e-E_{K,b}$, Stark shift $2{\mathrm{\Delta }}_e$, and polaron energy $E_{\text{pol}}$ are not shown to scale.

Figure 2

$|c_{ee}{\left(t\right)|}^2$ as a function of $Jt/\hslash$ for the initial state $|e,e,\text{vac}\rangle$, $U/J=-1$, $g/J=1/50$, and $\delta /J=0.0431$ (left) and $\delta /J=0.0011$ (right). Top, middle, and bottom rows are for $x/a=0$, 5, and 10, respectively. Black solid, red dotted, blue dashed, and green dash-dash-dotted lines are obtained using $\widehat{H}$, ${\widehat{H}}^{\text{adia}}$, ${\widehat{H}}^{\text{adia}}$ with $G_{K,K^{\prime }}=0$, and ${\widehat{H}}^{\text{adia}}$ with $G_{K,K^{\prime }}={\mathrm{\Delta }}_e=0$, respectively.

Figure 3

Static results ($U/J=-1$, $g/J=1/50$, and $\delta /J=0.0011$). (a) Black solid, red dotted, and magenta dash-dotted lines show the eigenenergies corresponding to hybridized states of $\widehat{H}$, ${\widehat{H}}^{\text{adia}}$, and ${\widehat{H}}^{\text{2-st.}}$, respectively, as a function of $x/a$. The gray dashed line shows $(E-E_{0,b})/J=0$. (b) The black solid line shows $E_{K,b}$ as functions of $Ka/\pi$ (main panel) and the state index (inset). The red circles show the eigenenergies supported by ${\widehat{H}}_b^{\text{adia}}$ (index 1 corresponds to the polaron-like state). (c) The squares, circles, and triangles show the dimensionless quantities $\text{Re}\left[G_{K^{\left(0\right)},K^{\left(0\right)}}\left(x\right)\right]/10$, $\text{Re}\left[F_{K^{\left(0\right)},b}\left(x\right)\right]$, and $\text{Im}\left[F_{K^{\left(0\right)},b}\left(x\right)\right]$ as a function of $x/a$ for $K^{\left(0\right)}a/\pi =0.0152$.

Figure 4

State composition of hybridized polaron-emitter states ($U/J=-1$, $g/J=1/50$, and $\delta /J=0.0011$). (a) Projection of $|e,e,\text{vac}\rangle$ onto ${\mathrm{\Psi }}_{+}$ (upper three lines) and ${\mathrm{\Psi }}_{-}$ (lower three lines) as a function of $x/a$. Black solid, red dotted, and magenta dash-dotted lines are obtained using $\widehat{H}$, ${\widehat{H}}^{\text{adia}}$, and ${\widehat{H}}^{\text{2-st.}}$, respectively. (b), (c) Projection of ${\mathrm{\Psi }}_{+}$ and ${\mathrm{\Psi }}_{-}$ onto $|g,g,K\rangle$ as a function of $Ka/\pi$ for $x/a=10$. The line styles are the same as in (a); black solid and red dotted lines are nearly indistinguishable on the scale shown.