Few-Photon Transport in Low-Dimensional Systems: Interaction-Induced Radiation Trapping

We present a detailed analysis of the dynamics of photon transport in waveguiding systems in the presence of a two-level system. In these systems, quantum interference effects generate a strong effective optical nonlinearity on the few-photon level. We clarify the relevant physical mechanisms through an appropriate quantum many-body approach. Based on this, we demonstrate that a single-particle photon-atom bound state with an energy outside the band can be excited via multiparticle scattering processes. We further show that these trapping effects are robust and, therefore, will be useful for the control of photon entanglement in solid-state based quantum-optical systems.

Figure 1

Time evolution (in units of $\hslash /J$) of transmission $⟨T⟩$, reflection $⟨R⟩$, and impurity occupation $⟨n_b⟩=⟨b^{†}b⟩=⟨{\sigma }_z+1/2⟩$ for a two-photon wave packet that scatters at a TLS. The TLS has a transition energy $\Omega =\sqrt{2}J$ and couples with coupling strength $V=J$ to the central lattice site $x_0=100a$ of a tight-binding lattice with total extent $L=199a$ and hopping element $J$. The photons are described via boson-symmetric wave packets that are constructed from single-particle Gaussian wave functions of width $s=6a$ with wave number $k=3\pi /4a$ and initial center $x_c=70a$ (see text and Ref. 9 for further details). All calculations are stopped at times not exceeding the transit time, i.e., the time the wave packet needs to pass through the waveguide, thus avoiding artificial reflections from the system’s boundaries.

Figure 2

Time evolution (in units of $\hslash /J$) of the impurity occupation $⟨n_b⟩$ for initial multiphoton states with different photon numbers $C$ that are constructed analogous to the two-photon states in Fig. 1. The corresponding system parameters are $L=99a$, $x_0=50a$, $\Omega =\sqrt{2}J$, and $V=J$. The photon parameters are $x_c=25a$, $s=5a$, and $k=3\pi /4a$. The results for photon numbers $C=3$ and $C=4$ have been obtained with a time-dependent density matrix renormalization group (DMRG) technique as described in Ref. 13.

Figure 3

Impurity occupation $⟨n_b⟩$ in the long-time limit (see Fig. 1) after scattering of two-photon states for different system parameters $V$ (in units of $J$) and $\Omega$ (in units of $J$). The fixed parameters are: $L=199a$, $x_0=100a$, $x_c=70a$, $s=12a$, and $k=3\pi /4a$.

Figure 4

Time evolution (in units of $\hslash /J$) of the impurity occupation $⟨n_b⟩$ for initial two-photon states (see Fig. 1) where a broadband loss waveguide has been introduced. The corresponding system parameters are (loss waveguide’s parameters are primed): $L=L^{\prime }=399a$, $x_0=x_0^{\prime }=200a$, $\Omega =\sqrt{2}J$, $V=J$, and $J^{\prime }=2J$. The strength of the coupling $V^{\prime }$ (in units of $J$) to the loss channel is varied. The photon parameters are $x_c=175a$, $s=6a$, and $k=3\pi /4a$.

Figure 5

Time evolution (in units of $\hslash /J$) of the impurity occupation $⟨n_b⟩$ for a system where two single-photon Gaussian wave packets of width $s=7a$ with different initial positions ($x_c^{(1)}$ and $x_c^{(2)}$) are launched from different sides towards the TLS. The corresponding system parameters are $L=199a$, $x_0=100a$, $\Omega =\sqrt{2}J$, and $V=J$. The photon parameters are $x_c^{(1)}=50a$, $k^{(1)}=3\pi /4a=-k^{(2)}$, and the initial position $x_c^{(2)}=150a+{\Delta }_x$ is varied.