Geometry-based circulation of local photonic transport in a triangular metastructure

A geometry-based mechanism for inducing circulation of photons is illustrated by a metastructure consisting of three quantum dots embedded in photonic structures and forming a triangle. The coupling between the photons and the excitons in the quantum dots leads to a photon blockade and limits the number of photons participating in the transport. In the steady state described by the quantum master equation of photons, the local photonic currents exhibit distinct circulation patterns, which originate from the wave nature in a multipath geometry. The geometry-based mechanism does not require an artificial gauge field from light-matter interactions. The phase diagrams showing the regions of different patterns of circulation with and without the effective photon interactions are presented. As the number of photons allowed per site increases, the regions saturate. By using the third-quantization formalism, we show the circulation survives without any photon blockade in the noninteracting case. Moreover, we demonstrate the decoupling of the direction of the local current from the density difference and propose possible applications of the local photonic transport.

Figure 1

The quantum dot–photonics metastructure for studying the geometry-based circulation of photons. The three quantum dots are embedded in the three photonic cavities labeled by 1, 2, and 3. The cavities are connected by photonic waveguides. The quantum dots provide excitons for coupling to the photons, and only the photons coupled to the excitons are transported via the waveguides. The photonic transport is described by the effective hopping of the photons and effective repulsion from the underlying electrons. Site 1 (3) of the system is connected to a photon pump (sink) via additional waveguides for maintaining a steady state.

Figure 2

Top panel: The local currents $J_{12}$ and $J_{13}$ as functions of time with $t_3/t_1=0.6\text{and}1.4, \gamma T_0=0.5$, and $M=1$ without the onsite interaction ($U=0$). The plateaus of the currents are the signatures of a steady state. Bottom panel: The steady-state values of the local currents $J_{12}\text{and}{J}_{13}$ and the total current $J_T$ as functions of $t_3/t_1$ with the same $\gamma T_0$ and $M$, showing opposite signs of $J_{12}$ and $J_{13}$ in certain regimes.

Figure 3

Phase diagrams showing the steady-state patterns of the photonic current with $U=0$ and $M=1,2,3,4,5$ (from left to right). Here the pink upside-down triangles denote the CW circulation, the black circles denote the UD flow, and the blue triangle denote the CCW circulation.

Figure 4

Phase diagrams showing the steady-state patterns of the photonic current for the cases with $U=t_1$ (the top row) and $U=5t_1$ (the bottom row). Here $M=2,3,4,5$ from left to right. The pink upside-down triangles denote the CW circulation, the black circles denote the UD flow, and the blue triangles denote the CCW circulation.

Figure 5

The dependence of the local steady-state currents $J_{12}$ (blue hollow hexagons), $J_{23}$ (green triangles), and $J_{13}$ (red upside-down triangles) on $1/M$ for $t_3/t_1=0.6,1.0,1.4$ from top to bottom. Here $\gamma T_0=0.5$ and $U=0$. $J_{12}$ and $J_{23}$ overlap in the steady state. The insets show the photon occupation numbers on the three sites, $n_1,n_2$, and $n_3$ (cyan circles, black squares, and brown diamonds), as functions of $1/M$ with the same parameters as those in the main panels.

Figure 6

Tuning the local currents by the system-reservoir coupling $\gamma$: The upper (lower) panel shows $J_{13}$ for $t_3/t_1=0.6$ and $M=5$ ($J_{12}$ for $t_3/t_1=1.4$ and $M=5$). Both cases show a change of the sign of the local current as $\gamma$ increases.

Figure 7

Phase diagram showing different steady-state patterns of the photonic current in a noninteracting ($U=0$) system in absence of the photon blockade ($M=\infty$). The pink upside-down triangles denote the CW circulation, the black circles denote the UD flow, and the blue triangles denote the CCW circulation.