Chiral Quantum Optics in the Bulk of Photonic Quantum Hall Systems

We study light-matter interactions in the bulk of a two-dimensional photonic lattice system, where photons are subject to the combined effect of a synthetic magnetic field and an orthogonal synthetic electric field. In this configuration, chiral waveguide modes appear in the bulk region of the lattice, in direct analogy to transverse Hall currents in electronic systems. By evaluating the non-Markovian dynamics of emitters that are coupled to those modes, we identify critical coupling conditions, under which the shape of the spontaneously emitted photons becomes almost fully symmetric. Combined with a directional, dispersionless propagation, this property enables a complete reabsorption of the photon by another distant emitter, without relying on any time-dependent control. We show that this mechanism can be generalized to arbitrary in-plane synthetic potentials, thereby enabling flexible realizations of reconfigurable networks of quantum emitters with arbitrary chiral connectivity.

Popular Summary

Developing efficient ways to transmit quantum information between distant qubits is crucial for advancing quantum computing and other quantum technologies. This has created an intense activity on so-called chiral quantum networks, where information can travel unidirectionally from one qubit to another, without suffering from backscattering. Chiral quantum optical setups are already a reality, and a major focus is now on their implementation using one-dimensional (1D) topological edge modes of specific two-dimensional (2D) photonic lattices. However, these implementations still lack efficiency, and they are ultimately limited in scalability by building an entire 2D structure to only use its 1D edge.

Here we take inspiration from the quantum Hall effect of electronic systems to propose a new photonic platform for implementing a fully reconfigurable chiral quantum network in the 2D bulk of a topological system. We show that under a specific implementation, a single photon behaves effectively as an electron subjected to the combined effect of external magnetic and electric fields. In this synthetic analog quantum Hall setting for photons, the equipotential lines of the synthetic electric field become effective 1D chiral waveguides, with arbitrary connectivity for qubits located in the bulk of the lattice. Moreover, novel non-Markovian behaviors due to the interplay between quantum Hall and light-matter physics ensure high efficiency for transferring information among the qubits.

These operations can then be supplemented with other important elements such as beam splitters and demultiplexers, all based on the physical principles of the quantum Hall effect, to realize highly flexible chiral quantum networks for a diverse range of experimental platforms in the GHz and optical domain.

Figure 1

Photonic quantum Hall system. (a) Sketch of a 2D photonic lattice with a perpendicular synthetic magnetic field and an in-plane synthetic electric field. In analogy to Hall currents, this configuration results in the formation of chiral waveguide modes. Coupling of two-level emitters to the chiral waveguide modes leads to the directional emission and reabsorption of photons. (b) Schematic view of the photonic hopping amplitudes for the specific Landau-gauge, Harper-Hofstadter lattice configuration under consideration. (c) Spectrum of the photonic lattice Hamiltonian $H_{\mathrm{ph}}$ in Eq. (1), where the energy of each mode is plotted as a function of the mean displacement of its center of mass along the $x$ direction. In this representation, the tilting of the Landau levels by the electric field and the presence of edge modes at the boundaries are clearly visible. The parameters used for this plot are $\alpha =1/10$, $N_x=N_y=40$, and $U_0/J=0.05$ and we assume periodic boundary conditions (PBCs) along the $y$ direction.

Figure 2

Spontaneous emission dynamics of a single emitter in the weak-coupling regime. (a) Plot of the excited-state population $P_e=|c_e(t){|}^2$ as a function of time. The blue crosses represent the results from an exact numerical simulation, while the solid black line shows the prediction from Eq. (22). The green dashed line indicates an exponential decay with a rate ${\mathrm{\Gamma }}_e$ given in Eq. (27). (b) Snapshot of the photon density $|\phi (\overrightarrow{r},t){|}^2$ of the emitted wavepacket at a time $t>0$ well after the spontaneous emission process. The white arrow marks the direction of propagation. The parameters for both plots are $\alpha =1/20$, $N_x=31$, $N_y=100$, $U_0/J=0.001$, ${\mathrm{\Delta }}_e/J\approx 0$, and $\hslash g=0.4U_B/\sqrt{\alpha }\approx 0.003\hslash J$. The emitter is located at the position ${\overrightarrow{r}}_e/l_0=(15,50)$.

Figure 3

Evolution of the excited-state population $P_e=|c_e(t){|}^2$ of an initially excited emitter in the strong-coupling regime, where (a) $\hslash g=2U_B/\sqrt{\alpha }$ and (b) $\hslash g=8U_B/\sqrt{\alpha }$. The blue crosses represent the results from an exact numerical simulation, while the solid black line shows the prediction from Eq. (22) in the continuum limit. The other parameters assumed for both plots are $\alpha =1/20$, $N_x=N_y=40$, $U_0/J=0.01$, and ${\mathrm{\Delta }}_e/J\approx 0$.

Figure 4

(a) Excited-state population of a single emitter $P_e=|c_e(t){|}^2$ as a function of time. The blue crosses represent the results from an exact numerical simulation, while the solid black line shows the prediction from Eq. (22) in the continuum limit. The green dashed line indicates an exponential decay with a rate ${\mathrm{\Gamma }}_e$ given in Eq. (27). (b) A snapshot of the photon density $|\phi (\overrightarrow{r},t){|}^2$ of the emitted wavepacket at a time $t\approx 202J^{-1}\approx 23{\mathrm{\Gamma }}_e^{-1}$. The white arrow marks the direction of propagation. In both plots we assume $\alpha =1/10$, $N_x=N_y=31$, $U_0/(\hslash J)=0.1$, ${\mathrm{\Delta }}_e/J=0$, and $\hslash g=U_B/\sqrt{\alpha }\approx 0.4\hslash J$. The emitter is located at ${\overrightarrow{r}}_e/l_0=(15,15)$, as indicated by the green dot. (c) Integrated photon density profile $h_t(y)=\int dx|\phi (x,y,t){|}^2$ for a photon emitted in the Markovian regime with $\hslash g=0.3U_B/\sqrt{\alpha }$ (green line) and from a critically coupled emitter with $\hslash g=U_B/\sqrt{\alpha }$ (red line). The black arrow indicates the propagation direction while the green dashed line marks the emitter position at ${\overrightarrow{r}}_e/l_0=(10,70)$. The two snapshots are taken at the same time $Jt\approx 64$. In this plot we assume $\alpha =1/10$, $N_x=20$, $N_y=80$, $U_0/(\hslash J)=0.1$, and ${\mathrm{\Delta }}_e/J=0$. Both results in (c) are obtained from a numerical simulation of the full lattice dynamics.

Figure 5

Plot of the emitter excitation spectrum ${\mathcal{S}}_e$ as a function of the excitation frequency $\omega$ and the light-matter coupling strength $g$. The three panels show this spectrum for different values of the electric field, expressed in terms of the Landau voltage $U_B=eE{l}_B$. In each panel, the green dashed line marks the critical coupling value $\hslash g=U_B/\sqrt{\alpha }$. The gray-shaded area in the last panel is the weak-coupling (Markovian) regime, whose boundary is given by $\hslash g\approx U_B/(2\sqrt{2\alpha })$, as it is defined in traditional cavity QED setups with a Gaussian inhomogeneous broadening of the emitters [66, 67, 68]. The other parameters are $\alpha =1/10$, $N_x=N_y=30$, and ${\mathrm{\Delta }}_e/J\approx 0$. In all plots an artificial broadening ${\gamma }_{\delta }/J=0.015$ of all states is introduced to obtain a smooth spectrum.

Figure 6

Evolution of the excited-state population $P_e=|c_e(t){|}^2$ (left panels) and snapshot of the emitted photon density $|\phi (\overrightarrow{r},t){|}^2$ (right panels). In (a) the emitter is weakly coupled with $g\sqrt{\alpha }=0.3U_B/\hslash \approx 0.038J$, while (b) shows the case of a critically coupled emitter with $g\sqrt{\alpha }=U_B/\hslash \approx 0.126J$. In both cases $J{\tau }_{\mathrm{rev}}\approx 390$. The other parameters for these plots are $\alpha =1/10$, $N_x=31$, $N_y=61$, $U_0/(\hslash J)=0.1$, ${\mathrm{\Delta }}_e=0$ and the emitter is located at ${\overrightarrow{r}}_e/l_0=(15,53)$. All results have been obtained from a numerical simulation of the full lattice dynamics.

Figure 7

Maximum value of the revival probability $P_{\mathrm{rev}}$ of the excited-state population after a time $t\sim {\tau }_{\mathrm{rev}}$. This probability is plotted as a function of the light-matter coupling $g$ and the local detuning ${\mathrm{\Delta }}_e$. These results are derived from Eq. (21), using the continuum Green’s function with PBC and for $\alpha =1/10$ and $L_y/l_0=200$.

Figure 8

(a) Sketch of a lattice with PBC along the $y$ axis and OBC on the $x$ axis, and with an homogeneous out-of-plane magnetic field and an homogeneous in-plane electric field along the $x$ axis (left panel). In this case the photonic eigenmodes $f_{\lambda }({\overrightarrow{r}}_i)$ are approximately described by the Landau wave functions in Eq. (5), which are homogeneous stripes along the $y$ direction with lateral size approximately $l_B$ (right panel). (b) Sketch of the same lattice, but with OBC along both $x$ and $y$ (left panel). In such a lattice, photonic eigenmodes form closed loops along the edge (right panel). The lattice parameters assumed for both cases are $\alpha =1/20$ and $N_x=N_y=41$.

Figure 9

(a) Snapshots of the photon density $|\phi (\overrightarrow{r},t){|}^2$ at three different times after the excitation is released by an emitter located at the center of the lattice. (b) Excited-state population $P_e=|c_e(t){|}^2$ as a function of time in units of ${\tau }_{\mathrm{rev}}$. The three dashed lines mark the times of the snapshots shown in (a). The other parameters for theses plots are $\alpha =1/10$, $N_x=N_y=21$, $U_0/(\hslash J)=0.1$, ${\mathrm{\Delta }}_e/J=0$, and $\hslash g=U_B/\sqrt{\alpha }\approx 0.4\hslash J$.

Figure 10

(a) Example of a photonic lattice with a smooth edge given by a confining potential $V_{\mathrm{conf}}$ and a linear potential gradient along the $x$ axis. The equipotential lines of the total potential (black lines) are straight lines in the bulk, but bend into a closed loop along the edge. The white arrow marks the direction of the electric field in the bulk while the red arrows show an example of a photon trajectory along the red equipotential line. (b) Sketch of the proposed photonic demultiplexing device based on the guiding-center photonic motion along equipotential lines.

Figure 11

Chiral excitation transfer in different configurations. The left panels depict the locations of the two emitters in the lattice, while the right panels show the evolution of the excited-state populations, $P_e^{(n)}(t)=|c_n(t){|}^2$. (a) Excitation transfer via edge channels, where the two emitters are located on the edge (light green sites) at positions ${\overrightarrow{r}}_e^1/l_0=(20,40)$ and ${\overrightarrow{r}}_e^2/l_0=(20,0)$. The photon propagates along the edge from emitter $1$ to emitter $2$. The parameters in this example are $U_0/(\hslash J)=0$, $g_1=g_2=0.1{\omega }_B/\sqrt{\alpha }\approx 0.4J$, and ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=0.1\times {\omega }_B\approx 0.7J$. (b) Excitation transfer via the bulk, where the two emitters are located at positions ${\overrightarrow{r}}_e^1/l_0=(20,36)$ and ${\overrightarrow{r}}_e^2/l_0=(20,5)$. The photon propagates through the bulk from emitter $1$ to emitter $2$. The parameters in this example are $U_0/(\hslash J)=0.05$, $\hslash g_1=\hslash g_2=U_B/\sqrt{\alpha }\approx 0.2\hslash J$ and ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=0$. (c) Excitation transfer between two emitters in the bulk, but with the photon propagating along the edge from emitter $2$ to emitter $1$. The other parameters are the same as in (b). In all three simulations we assume $\alpha =1/10$ and $N_x=N_y=41$.

Figure 12

Disorder-averaged value of the transfer infidelity $1-\overline{\mathcal{F}}$ (color scale) as a function of the electric field, $U_0/(\hslash J)$, and the lattice disorder strength, ${\sigma }_p/J$. The coupling strength of the two emitters are fixed to the critical-coupling condition $\hslash g_1=\hslash g_2=U_B/\sqrt{\alpha }$. Emitter 1 is located at ${\overrightarrow{r}}_e^1/l_0=(10,17)$ while emitter 2 is located at ${\overrightarrow{r}}_e^2/l_0=(10,4)$. The other parameters for this simulation are $\alpha =1/10$, $N_x=N_y=21$, ${\mathrm{\Delta }}_e^1={\mathrm{\Delta }}_e^2=0$, and ${\gamma }_p/J={10}^{-5}$ (this reduced value of the losses is chosen to highlight the effect of disorder). The dashed white line indicates the condition $\hslash {\sigma }_p=U_0$. For each choice of parameters, the disorder average is performed over $N_{\mathrm{dis}}=100$ realizations.

Figure 13

Chiral state transfer in the randomly generated potential $V({\overrightarrow{r}}_i)$ shown in the two left panels. The two emitters are located along the red equipotential line, with the initially excited emitter 1 placed at ${\overrightarrow{r}}_e^1/l_0=(10,19)$. In (a) the second emitter is located at a position ${\overrightarrow{r}}_e^2/l_0=(10,1)$ with the same local gradient. In this case, a perfect transfer of the excitation is possible (right panel). In (b) the second emitter is located at ${\overrightarrow{r}}_e^2/l_0=(9,9)$ in a region with a different local electric field. In this situation, the transfer probability is strongly reduced. Other parameters for these plots are $\alpha =1/10$, $N_x=N_y=21$, $U_0/(\hslash J)=0.1$, $\hslash g_1=g_2=U_B/\sqrt{\alpha }\approx \hslash J$, and ${\tilde{\mathrm{\Delta }}}_1={\tilde{\mathrm{\Delta }}}_2=0$.

Figure 14

(a) Sketch of the lattice potential $V(x,y)=eE(|x-X_0|-|y-Y_0|)$ for the realization of a chiral beam splitter. The white arrows indicate the propagation direction along the specific equipotential line connected to an emitter located at ${\overrightarrow{r}}_e/l_0=(6,6)$. (b) Snapshots of the photon density $|\phi (\overrightarrow{r},t){|}^2$ for a photon that is emitted in the critical-coupling regime. The parameters for this simulation are $\alpha =1/10$, $N_x=N_y=31$, $U_0/(\hslash J)=0.1$, ${\mathrm{\Delta }}_e=0$, and $\hslash g=U({\overrightarrow{r}}_e)/\sqrt{\alpha }$, $X_0/l_0=Y_0/l_0=15$.