Light-matter interactions near photonic Weyl points

Weyl photons appear when two three-dimensional photonic bands with linear dispersion are degenerate at a single momentum point, labeled as Weyl point. These points have remarkable properties such as being robust topological monopoles of Berry curvature as well as an associated vanishing density of states. In this work, we report on a systematic theoretical study of the quantum optical consequences of such Weyl photons. First, we analyze the dynamics of a single quantum emitter coupled to a Weyl photonic bath as a function of its detuning with respect to the Weyl point and study the corrresponding emission patterns, using both perturbative and exact treatments. Our calculations show an asymmetric dynamical behavior when the emitter is detuned away from the Weyl frequency, as well as different regimes of highly collimated emission, which ultimately translate in a variety of directional collective decays. Besides, we find that the incorporation of staggered mass and hopping terms in the bath Hamiltonian both enriches the observed phenomenology and increases the tunability of the interaction. Finally, we analyze the competition between the coherent and dissipative components of the dynamics for the case of two emitters and derive the conditions under which an effective interacting spin model description is valid.

Figure 1

(a) Schematics of the analyzed system. (b) Top and side views of the three-dimensional lattice model mimicking the Weyl environment. Solid and dashed black lines correspond to a hopping term of magnitude $J$ and phase 0 and $\pi$, respectively, whereas cyan solid lines stand for the staggered hopping term. Orange arrows indicate the primitive vectors and the shadow gray area encloses the unit cell. (c) First Brillouin zone of the investigated lattice model. The associated dispersion relation is projected over the $k_y=\pm \pi /2$ planes. The inset shows a detailed view of these planes in which the position of the Weyl points (magenta dots) and the Berry curvature (vector map) is depicted for the case $m=0$ and $h/J=1$.

Figure 2

Dynamics of the single-emitter case for configuration in which $m=0$, $h/J=1$ and the light-matter coupling is chosen to be $g/J=0.5$. (a) Contributions to $|C_e{\left(0\right)|}^2$ as a function of the detuning of the emitter with respect to the Weyl frequency. (b) Markovian (dashed lines) and non-Markovian (solid lines) prediction for the temporal evolution of the emitter's excited-state population for various detuning values. The inset shows the long-time behavior of the emitter's excited population in a log-log scale for the case in which the emitter matches the van Hove singularity corresponding to $\mathrm{\Delta }/J=2$.

Figure 3

(a), (c), (e) Excited-state population of the emitter as a function of its detuning with respect to the Weyl frequency and time. (b), (d), (f) Comparison between the Markovian prediction for the decay rate and the density of states for several system configurations. Red arrows spot some selected van Hove singularities that are further explored in Fig. 4. We restrict ourselves to configurations where no staggered hopping term is assumed ($h/J=1$), but the staggered mass term is chosen to be $m/J=0$ (a), (b), $m/J=1$ (c), (d), and $m/J=2$ (e), (f). The light-matter coupling is $g/J=0.5$.

Figure 4

Emission pattern at fixed time ($t/J=25$) of an emitter coupled to a van Hove singularity for three different configurations of the system. For visualization, insets show special cuts along some selected directions: (a), (c) $z=0$ and (b) $x=0$.

Figure 5

(a) Cut along the $z=0$ plane of the photonic component associated to the Weyl bound state for the case $m=0$ and $h/J=1$. Note that in this case the Weyl bound-state confinement is mostly isotropic. (b) Power-law dependence of the bound-state profile along any of the principal directions, i.e., $(\pm 1,0,0)$, $(0,\pm 1,0)$, or $(0,0,\pm 1)$, for the case where $m=0$ and $h/J=1$. (c)–(f) Cuts along the $z=0$ and $x=0$ planes of the photonic component associated to the Weyl bound state for the case $m=0$ and $h/J=-1$ (c), (d) or $h/J=3$ (e), (f). (g), (h) Power-law dependence of the Weyl bound-state profile along some selected directions for the case where $m=0$ and $h/J=-1$ (g), (h) or $h/J=3$ (i), (j).

Figure 6

Dynamical behavior of the two-emitter problem for the case where no staggered mass nor staggered hopping terms are considered, i.e., $m=0$ and $h/J=1$. The light-matter coupling is chosen to be $g/J=0.5$. (a) Temporal evolution of the initially excited emitter (solid line) and of the initially deexcited one (dashed line) for $\mathrm{\Delta }/J=0$. (b) Spectral density function of the initially excited emitter associated to the dynamics shown in (a). (c) Temporal evolution of the initially excited emitter (solid line) and of the initially deexcited one (dashed line) for $\mathrm{\Delta }/J=3$. (d) Spectral density function of the initially excited emitter associated to the dynamics shown in (c).

Figure 7

Red crosses denote the ratio between the dissipative component of the dynamics ${\gamma }_{12}$ and the coherent component $J_{12}$ for the considered range of light-matter coupling values. Black dotted line stands for the corresponding linear fitting which yields a $\sim {(g/J)}^4$ dependence. As shown, the coherent component (yellow shaded area) is always much larger than the dissipative component (blue shaded area).

Figure 8

(a) Dynamics of the initially excited (solid line) and the initially deexcited emitter (dashed line) for a system configuration in which the emitters are coupled to two sites belonging to a different sublattice separated a distance equal to the lattice parameter along the $x$ direction. Both emitters are detuned towards the Weyl frequency and the parameters of the bath are chosen to be $m=0$ and $h/J=1$. (b) The same as in (a) but, in this case, the staggered mass term is chosen to be $m/J=1$. (c) The same as in (a) but, in this case, the staggered mass term is chosen to be $m/J=2$.

Figure 9

(a) Retarded ($C_{+}$) and advanced ($C_{-}$) contributions to the dynamics. (b) Outline of the integration path employed to compute the dynamics of the system for the configuration where $m=0$ and $h/J=1$. Solid brown lines stand for the detours that must be performed to avoid the nonanalytical regions of the problem's self-energy. Red crosses indicate the potential positions of the poles featured by the studied propagator.