Quantum optical impurity models in interacting waveguide QED

Abstract

We consider a generic model for interacting waveguide QED systems, where photons in a coupled-cavity array localize around atomic impurities while simultaneously interacting through local Kerr nonlinearities. This scenario appears naturally in nanophotonic crystals, circuit QED lattices, and ultracold atomic systems and is governed by the competition between attractive Jaynes-Cummings-mediated binding and intrinsic photon-photon repulsion. We analyze how this interplay affects the formation of localized few-photon bound states and determine the resulting many-body ground states for large periodic arrays of impurities and different filling factors. With the help of large-scale numerical simulations and approximate analytical models, we identify a rich phase diagram featuring Mott-like insulating states as well as superfluid phases with long-range correlations, which are mediated by an unbound, but strongly interacting photonic fluid.

Figures (12)

• Figure 1: Sketch of an interacting waveguide QED system. A set of TLAs with states $|g\rangle$ and $|e\rangle$ are coupled to photons propagating along an array of coupled cavities with nearest-neighbor hopping $J$. The TLAs couple to the local photonic mode with strength $g$ and photon-photon interaction within each cavity are modeled by Kerr-type process of strength $U$. In the considered periodic configurations, each unit cells highlighted by the dashed box contains one TLA and $d=L/N_{\rm a}$ cavities.
• Figure 2: Sketch of the full spectrum of the waveguide Hamiltonian given in Eq. \ref{['eq:Hamiltonian']} for $N_{\rm a}=1$ TLA and $N=2$ excitations. The energies are plotted as a function of $g/J$ and for the cases (a) $U=0$ and (b) $U/J=10$. The colors and labels indicate the different types of energy eigenstates, which are also illustrated in (c). These are: (1) localized two-photon bound states (green line), (2) states with one bound and one free photon (orange), (3) states with two free photons (dark blue) and (4) propagating repulsively bound photon pairs (light blue). See text for more details.
• Figure 3: Plot of the maximal number of photons that can be bound to a single TLA in the limit $J\rightarrow 0$. The boundaries between the different domains (blue lines) are determined by the minimal binding strength $g_{\rm b}(n)$, evaluated from Eq. \ref{['eq:U_critical_condition']}. The color indicates the value of the mixing angle $\theta_n$ of the corresponding ground state $\ket{n,-} = \sin(\theta_n/2)\ket{g,n} - \cos(\theta_n/2)\ket{e,n-1}$, which is more 'atom-like' when $\theta_n\approx 0$ and more 'photon-like' when $\theta_n\approx \pi$.
• Figure 4: Detachment of bound photons. (a) Plot of the number of bound photons at the impurity site, $\langle a_0^\dag a_0\rangle$, in the ground state of $H$ with $N_{\rm a}=1$ TLA and a total number of $N=5$ excitations. The exact numerical results (crosses) are compared with the predictions of Eq. \ref{['eq:cav_occ_pertbJ']} for the effective model $H_{\rm eff}$ for different $N$. The same comparison is shown in (b) for the width of the localized photon wavepacket, $\Delta X_{\rm ph}$, as defined in Eq. \ref{['eq:Xph']}. For this comparison, also the exact numerics is performed for different excitation numbers to avoid contributions from the remaining unbound photons. In both plots, the vertical gray dashed lines indicate the value of $U/g_{\rm b}(N)$ predicted from the single-site JC model in Sec. \ref{['subsec:JCmodel']}. For all simulations, we have assumed the same value of $J/g=0.01$ and used $L=13$ ($L=100$) for the exact (effective) model.
• Figure 5: Plot of the binding energy $E_{\rm b}(N=2)$ introduced in Eq. \ref{['eq:energy_difference']} for different JC couplings and interaction strengths and for $\Delta=0$. The dashed lines indicate the analytic expressions for $g_{\rm b}(N=2)$, as obtained from Eq. \ref{['eq:g_critical_0detuning']} and Eq. \ref{['eq:gb_weak']} in the strong-coupling and in the weak-coupling limit, respectively. For simulations we have used a system of size $L=40$ and replaced $E_{1,-}\rightarrow E_{1,-}+ U \cos{\theta_1}/L$ in the definition of $E_{\rm b}(N=2)$. This adaption is necessary to correctly account for the residual interaction between the unbound photon and the impurity site for any finite $L$.