Bound states and decay dynamics in $N$-level Friedrichs model with factorizable interactions

TL;DR

The paper extends the Friedrichs model to an $N$-level system coupled to a continuum with factorizable interactions, deriving bound and scattering states via projection formalism and an energy-dependent effective Hamiltonian. It provides explicit criteria for counting bound states outside the continuum and conditions for bound states inside the continuum (BICs). The decay dynamics are analyzed in both general and Markovian regimes, showing regimes of irreversible decay, saturation due to BICs, and oscillatory behavior from multiple bound states, with exceptional-point behavior in the resonant spectrum. The framework is applied to an atomic chain coupled to a photonic crystal waveguide, where BICs, multiple decay regimes, and an anti-PT-symmetric Markovian limit are demonstrated, highlighting a versatile approach to engineered dissipation in structured reservoirs.

Abstract

Considering an $N$-level system interacting factorizably with a continuous spectrum, we derive analytical expressions for the bound states and the dynamical evolution within this single-excitation Friedrichs model by using the projection operator formalism. First, we establish explicit criteria to determine the number of bound states, whose existence suppresses the complete spontaneous decay of the system. Second, we derive the open system's dissipative dynamics, which is naturally described by an energy-independent non-Hermitian Hamiltonian in the Markovian limit. As an example, we apply our framework to an atomic chain embedded in a photonic crystal waveguide, uncovering a rich variety of decay dynamics and realizing an anti-$\mathcal{PT}$-symmetric Hamiltonian in the system's evolution.

Figures (5)

• Figure 1: Graphical solution of Eq. (\ref{['sol']}). Roots $E_m$ (black crosses) of Eq. (\ref{['sol']}) are obtained from the intersection points between $K(E)$ (blue solid line) and $\Sigma^{-1}(E)$ (red dash-dotted line). Black hollow circles and blue solid circle represent the energy level $\epsilon_n$ of the discrete system and zeros of the function $K(E)$, respectively. The light yellow shaded region indicates the continuum band.
• Figure 2: Schematic of an atomic chain coupled to a photonic crystal waveguide. (a) Tight-binding model in the Wannier basis. The system consists an $N$ site chain of two-level atoms side-coupled to a one-dimensional semi-infinity photonic lattice at site $l$. (b) Corresponding energy spectrum in the Bloch basis. The spectral densities $J(\omega)$ are plotted for three coupling positions: $l=1$ (red solid line), $l=2$ (pink dashed-dot line), and the asymptotic limit $l\to\infty$ (purple dashed line).
• Figure 3: Number $M_{\rm out}$ of bound states of the tight-binding model for $N=1$ to $N=6$. $N_{\rm out}$ denotes the number of discrete energy levels $\epsilon_n$ lying outside the continuum, as determined by $N$ and $\kappa/\lambda$. The black star marks the parameter set $\kappa/\lambda=0.75$ and $\xi/\lambda=0.25$ employed in Fig. \ref{['sp']} for the case $N=3$.
• Figure 4: Decay dynamics of the survival probability $p(t)$ for parameters $N=3$, $\kappa/\lambda=0.75$, and $\xi/\lambda=0.25$. Analytical results (solid lines) and numerical data (open symbols) are shown for three waveguide geometries: coupling site $l=1$ (red line/circles), $l=2$ (orange line/squares), and the infinite-waveguide limit $l\to\infty$ (blue line/triangles).
• Figure 5: Complex eigenvalues and survival probability dynamics. (a) and (b) Real and imaginary parts, $\operatorname{Re}(z_i)$ and $\operatorname{Im}(z_i)$, of the two eigenvalues of the effective Hamiltonian as functions of the coupling strength $\xi$ for $N=2$ and $\kappa/\lambda=4$. Blue solid and red dashed lines correspond to $i=1$ and $i=2$, respectively. The shaded region for $\xi/\lambda >4$ indicates the $\mathcal{PT}$-symmetry-broken phase. (c)-(e) Survival probability $p(t)$ versus time $t$ for three representative couplings: $\xi/\lambda=2$ ($\mathcal{PT}$-symmetric phase), $\xi/\lambda=4$ (EP), and $\xi/\lambda=6$ ($\mathcal{PT}$-symmetry-broken phase). The solid purple line, dashed orange line and solid black circle represent the analytical, approximate, and numerical results, respectively.