Few-photon transport in low-dimensional systems

We analyze the role of quantum interference effects induced by an embedded two-level system on the photon transport properties in waveguiding structures that exhibit cutoffs (band edges) in their dispersion relation. In particular, we demonstrate that these systems invariably exhibit single-particle photon-atom bound states and strong effective nonlinear responses on the few-photon level. Based on this, we find that the properties of these photon-atom bound states may be tuned via the underlying dispersion relation and that their occupation can be controlled via multiparticle scattering processes. This opens an interesting route for controlling photon transport properties in a number of solid-state-based quantum optical systems and the realization of corresponding functional elements and devices.

Figure 1

Dispersion relations ${\epsilon }_k=-2J\cos (\mathit{ka})-2J_2\cos (2\mathit{ka})$ for tight-binding models with next-nearest-neighbor hopping term $J_2$.

Figure 2

Time evolution of the occupation of the TLS’s excited state $\langle n_b\rangle$ for an incoming two-photon wave packet and the dispersion relations depicted in Fig. 1. The parameters are (see text for details): $L=99a$, $x_0=50$, $V=J$, $x_c^{(1)}=x_c^{(2)}=20a$, $s^{(1)}=s^{(2)}=8a$, $k_0^{(1)}=k_0^{(2)}=\frac{\pi }{2a}$. The single-particle resonance condition, ${\epsilon }_{k_0^{(i)}}=\Omega$, is fulfilled. Black solid curve: $J_2=0$, $\Omega =0$. Blue dashed curve: $J_2=0.2J$, $\Omega =0.4J$. Green dash-dotted curve: $J_2=0.4J$, $\Omega =0.8J$. Note the finite amount of trapped occupation after scattering.

Figure 3

Cosine dispersion ${\epsilon }_k=-2J\cos (\mathit{ka})$ (black dashed line) in the first Brillouin zone linearized around $k=\pm \frac{\pi }{2a}$ (blue solid line). Both dispersion relations exhibit cut-offs due to upper and lower band edges that are situated at wave numbers $k=\pm \frac{\pi }{a}$ and $k=0$, respectively.

Figure 4

Time evolution of the occupation of the TLS’s excited state $\langle n_b\rangle$ for different incoming two-photon wave packets and the truncated linear dispersion relation depicted in Fig. 3. The parameters are (see text for details): $L=199a$, $x_0=100$, $V=J$, $x_c^{(1)}=x_c^{(2)}=70a$, $s^{(1)}=s^{(2)}=6a$. The single-particle resonance condition, ${\epsilon }_{k_0^{(i)}}=\Omega$, is fulfilled in all cases. Note the finite amount of trapped occupation after scattering.

Figure 5

Snapshots of the real-space occupation numbers $\langle n_x\rangle$ of the waveguide sites when two single-photon pulses are launched symmetrically from different sides toward the TLS (cf., the red solid curve in Fig. 6 for the occupation of the TLS). The red line denotes the position of the TLS at $x_0=100a$.

Figure 6

Time evolution of the TLS’s excited state occupation $\langle n_b\rangle$ for a setup where two single-photon wave packets are launched with different initial positions relative to the TLS. The parameters are: $L=199a$, $x_0=100a$, $V=J$, $\Omega =\sqrt{2}J$, $s^{(1)}=s^{(2)}=7a$, $k_0^{(1)}=-k_0^{(2)}=3\pi /4a$, $x_c^{(1)}=50a$. The initial position $x_c^{(2)}=150a+{\Delta }_x$ of the second single-photon wave packet is varied.

Figure 7

Time evolution of the TLS’s excited state occupation $\langle n_b\rangle$ for the interaction with two distinct photons, where only one fulfills the resonance condition prescribed to the cosine-shape dispersion and the other is energetically detuned from the TLS’s transition energy $\Omega$. The parameters are: $L=199a$, $x_0=100a$, $V=J$, $\Omega =\sqrt{2}J$, $s^{(1)}=s^{(2)}=7a$, $x_c^{(1)}=x_c^{(2)}=50a$, $k_0^{(1)}=3\pi /4a$. The wave number of the detuned photon, $k_0^{(2)}=k_0^{(1)}-{\Delta }_k$, is varied by an amount $\hslash {\Delta }_k$ relative to the resonant photon. For ${\Delta }_k=0.159\frac{\pi }{a}$, the detuned photon exhibits a detuning of $0.85J$ with respect to the atomic resonance.

Figure 8

Time evolution of the TLS’s excited state occupation $\langle n_b\rangle$ for the interaction with two distinct photons analogous to Fig. 7 but for the linearized dispersion relation (cf. Fig. 3). Parameters: $L=199a$, $x_0=100a$, $V=J$, $\Omega =2J(k_0-\pi /2a)$, $s^{(1)}=s^{(2)}=6a$, $x_c^{(1)}=x_c^{(2)}=50a$, $k_0^{(1)}=k_0=0.89\pi /a$, $k_0^{(2)}=k_0^{(1)}-{\Delta }_k$. For ${\Delta }_k=0.15\pi /a$, the detuned photon exhibits a detuning of $0.94J$ with respect to the atomic resonance.

Figure 9

Time evolution of the impurity occupation $\langle n_b\rangle$ for different values of the on-site repulsion $U$. Parameters: $L=99a$, $x_0=50a$, $V=J$, $\Omega =\sqrt{2}J$, $s^{(1)}=s^{(2)}=6a$, $x_c^{(1)}=x_c^{(2)}=16a$, $k_0^{(1)}=k_0^{(1)}=3\pi /4a$.

Figure 10

Time evolution of the atomic occupation $\langle n_b\rangle$ for few-photon wave packets of identical photons with different photon numbers $C$ and vanishing on-site interaction $U=0$. The parameters are: $L=99a$, $x_0^{(i)}=50a$, $V=J$, $\Omega =\sqrt{2}J$, $s^{(i)}=5a$, $x_c^{(i)}=25a$, $k_0^{(i)}=3\pi /4a$, $i=1,\dots ,C$. The single-particle resonance condition is fulfilled.

Figure 11

Occupation of the atom’s excited level as a function of time in a situation where photon wave packets with different photon numbers $C$ interact with an atom that exhibits a doubly degenerate upper level (cf., Fig. 10 for the relevant parameters).

Figure 12

Occupation of the atom’s excited level as a function of time in a situation where photon wave packets with different photon numbers $C$ interact with an atom that exhibits a threefold degenerate upper level (cf., Fig. 10 for the relevant parameters).

Figure 13

Snapshots of the momentum distribution $\left|{\Phi }_{k_1,k_2}\right|{}^2$ (in arbitrary units) corresponding to the atom being in its ground state and two propagating photons [cf. Eq. (14)]. Parameters: $L=199a$, $x_0=50a$, $V=0.8J$, $\Omega =0.1J$, $s^{(1)}=s^{(2)}=9a$, $x_c^{(1)}=x_c^{(2)}=35a$, $k_0^{(1)}=\pi /2a$, $k_0^{(2)}=3\pi /4a$. The distributions are boson-symmetric with respect to interchanging the wavenumbers $k_1$ and $k_2$ at all times (times are given in units of $\hslash /J$).

Figure 14

Snapshots of the momentum distribution $\left|e_k\right|{}^2$ (in arbitrary units) corresponding to the atom being in its excited state and one remaining propagating photon [cf. Eq. (15)]. (Note the differing scale of the $y$ axis at time $t=37.5\hslash /J$.)