Multiqubit subradiant states in $N$-port waveguide devices: ε-and-μ-near-zero hubs and nonreciprocal circulators

Quantum emitters interacting through a waveguide setup have been proposed as a promising platform for basic research on light-matter interactions and quantum information processing. We propose to augment waveguide setups with the use of multiport devices. Specifically, we demonstrate theoretically the possibility of exciting $N$-qubit subradiant, maximally entangled, states with the use of suitably designed $N$-port devices. Our general methodology is then applied based on two different devices: an epsilon-and-mu-near-zero waveguide hub and a nonreciprocal circulator. A sensitivity analysis is carried out to assess the robustness of the system against a number of nonidealities. These findings link and merge the designs of devices for quantum state engineering with classical communication network methodologies.

Figure 1

Conceptual sketch of an $N$-port waveguide device. Sketch of a device: two-level systems are coupled to $N$ single-mode waveguides connected via an $N$-port hub. The response of the device is characterized by its scattering ($S$-parameter) matrix, whose elements represent the reflection and transmission coefficients between the different ports.

Figure 2

Wave propagation through an $N$-port EMNZ hub. (a) Sketch of the simulation setup: a 2D EMNZ body ($\mu =\varepsilon =0.001$) of arbitrary shape is closed by a perfect electric conductor (PEC) wall and connected to the outside via four identical 2D waveguides (filled with air) of height ${\mathrm{h}}_{\mathrm{wg}}=0.25{\lambda }_0$ and length ${\mathrm{L}}_{\mathrm{wg}}=1.5{\lambda }_0$, terminated into waveguide ports supporting a TEM mode. (b) Snapshot, (c) magnitude, and (d) phase of the magnetic field excited when a wave is introduced into the system via the top port. The field distributions demonstrate that the waves outgoing the three remaining ports are identical in magnitude and phase.

Figure 3

$N$-port EMNZ 2D waveguide hubs. (a) Sketch and scattering matrix of a generic EMNZ waveguide hub. (b), (c) Snapshot of the magnetic field (${\mathrm{H}}_{\mathrm{z}}$ component), showing no emitted power exiting the system and thus evidencing the excitation of a dark state, when the QEs are replaced by a classical 2D point electric dipoles emitting in phase, for (b) $N=3$ and (c) $N=4$ port devices. (d), (e) Numerical (solid line) and analytical (markers) solutions to the time evolution of the probability amplitude of the individual QEs’ excited states, $c_n\left(t\right)$, located at a distance $k_{\mathrm{wg}}d=\pi$ of the ports of a EMNZ waveguide hub with (d) $N=3$ and (e) $N=4$ ports.

Figure 4

Idea for a possible all-dielectric implementation of a four-port EMNZ hub. (a) Sketch of the geometry: an EMNZ region may be considered with a photonic crystal (PC) exhibiting a Dirac cone at $k=0$ following Refs. [55, 57] (rectangular lattice with period $a_1=0.56{\lambda }_0,$ of dielectric rods of radius $r_1=0.2{\mathrm{a}}_1$ and relative permittivity ${\varepsilon }_r=12.5$). This region is closed by another PC ($a_2=0.392{\lambda }_0$, $r_2=0.18a_2$) exhibiting a complete band gap at the EMNZ frequency. The system is excited by four different sources (s1, s2, s3, and s4, shown as blue dots), modeled as out-of-plane current lines ($\mathbf{J}=\widehat{\mathbf{z}}\mathrm{I}$). (b) Snapshot of the electric field ${\mathrm{E}}_{\mathrm{z}}$ (top), and magnetic field magnitude $\left|\mathbf{H}\right|$, excited when only one source (s1, positioned in the top left waveguide) is activated. The field distributions confirm that the signals outgoing the other three ports are identical in magnitude and phase. (c) The same as panel (b), but when all sources are activated at the same time. In this case, the electric field distribution evidences the excitation of a “dark state,” with no outgoing waves (or power) exiting the system.

Figure 5

Entanglement generation in $N$-port (nonreciprocal) circulators. (a) Sketch and scattering matrix of the ideal device. (b), (c) Numerical (solid line) and analytical (markers) solutions to the time evolution of the probability amplitude of the individual QEs excited states located at distance $k_{\mathrm{wg}}d=\pi /2$ of the ports of a circulator with (b) $N=3$ and (c) $N=4$ ports.

Figure 6

Sensitivity analysis for a four-port EMNZ hub. (a) Comparison between the time evolution of the probability of the excitation remaining in the QE system, ${\mathrm{P}}_{\mathrm{e}}\left(\mathrm{t}\right)={\sum }_{\mathrm{n}=1}^{\mathrm{N}}|{\mathrm{c}}_{\mathrm{n}}\left(\mathrm{t}\right){|}^2,$ predicted in the ideal case and in the sensitivity analysis. Light blue background indicates the 95% confidence interval. For comparison we also include reference lines corresponding to the transient, subradiant, and combined exponential decays, ${\mathrm{e}}^{-{\mathrm{\Gamma }}_1\mathrm{t}}$, $\frac{1}{\mathrm{N}}{\mathrm{e}}^{-{\mathrm{\Gamma }}_2\mathrm{t}}$, and $\frac{\mathrm{N}-1}{\mathrm{N}}{\mathrm{e}}^{-{\mathrm{\Gamma }}_1\mathrm{t}}+\frac{1}{\mathrm{N}}{\mathrm{e}}^{-{\mathrm{\Gamma }}_2\mathrm{t}}$, respectively, with ${\mathrm{\Gamma }}_1=1.85{\mathrm{\Gamma }}_{\mathrm{wg}}$ and ${\mathrm{\Gamma }}_2=0.125{\mathrm{\Gamma }}_{\mathrm{wg}}$. (b) Comparison between the time evolution of the fidelity, $\mathcal{F}=|\langle {\mathrm{W}}_{\mathrm{N}}|\phantom{{\mathrm{W}}_{\mathrm{N}}\psi \left(\mathrm{t}\right)}\psi \left(\mathrm{t}\right)\rangle {|}^2$, predicted in the ideal case and average value in the sensitivity analysis. (c) Histograms of the fidelities predicted in the sensitivity analysis at the time points ${\mathrm{\Gamma }}_{\mathrm{wg}}t=1.9$, corresponding to the peak of the average fidelity, and ${\mathrm{\Gamma }}_{\mathrm{wg}}t=3$, corresponding to a point after the transient time. Average values are indicated with a vertical red line.