Strong and long-range radiative interaction between resonant transitions

The radiative interaction between electronic quantum transitions changes both the energy and lifetime of transitions. It leads to phenomena such as F$\ddot{\mathrm{o}}$rster energy transfer [1, 2], super-radiance [3, 4, 5], and collective Lamb shift [6]. Because radiative coupling is short range, these phenomena can only be observed when the distance between transitions is not much longer than the wavelength of the light that mediates the interaction. Researchers have been interested in enhancing the strength and extending the range of radiative interactions, with applications in energy sciences and quantum technologies. While lots of progress has been made in the past decades, we observed that there has a been a trade-off in many methods studied in the literature [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. The trade-off occurs between the strength and the range: interactions with a long range are often weak in strength while strong interactions are very short range.

We first discuss how we define and quantify the range $\ell$ and strength $\Gamma_{m}$ of radiative interaction between quantum transitions. The radiative interaction includes two parts: $\Gamma(\mathbf{R})$ via propagating modes and $\Delta(\mathbf{R})$ via evanescent modes. Here we focus on the propagating modes because they are generally responsible for long-range interaction. We consider two quantum transitions in vacuum. Figure 1(b) plots the real photon part of radiative coupling coefficient as a function of the distance between two quantum transitions in vacuum (see Appendix C for detailed derivation on the radiative interaction), which describes the rate of energy exchange. The strength is defined by the maximum value of the coupling, which is commonly realized when the distance is close to zero. The range $\ell$ is defined by the distance at which the coupling decays to its half maximum value. For example, the interaction strength and range in vacuum are the spontaneous emission rate $\Gamma_{0}$ and $\ell=\ell_{0}\approx 0.6\lambda_{0}$, respectively.

Radiative interaction via real photon exchange is mediated through optical modes in the continuum. A general strategy to enhance radiative interaction strength and range is to place transitions in engineered photonic environments, where the optical modes can be drastically different from those in vacuum. This strategy was used in many interesting works that enhance the range [7, 8, 9, 10, 11, 4, 13] or the strength [20, 21, 28, 29]. For example, zero-index materials can extend the range of coupling [7, 8, 9, 10, 11] to $F\ell_{0}$, where the enhancement factor $F$ can be a few hundreds. However, the strength $\Gamma$ decrease by a factor of $F^{2}$. The greater the range is, the weaker the strength becomes. Another interesting example exploits local plasmonic resonance [26, 30, 31]. The strength can be enhanced. However, the range becomes shorter [26] due to the fast decay of plasmonic modes. There are other interesting works that use near fields [32, 15, 33, 34, 25] or far fields around the Weyl point [22] in photonic crystals. When extending the cases to three-dimensional space, they all exhibit similar trade-off between range and strength. The only exception is one-dimensional space, such as in a waveguide or along the one particular direction of photonic structures where the interaction range can be extended without compromising the strength [35, 13, 4, 5].

Figure 1(c) surveys recent works in which a trade-off between the interaction range and the strength can be observed. Although the specific mechanism for each case could vary, a common physics emerges in many of these cases. By identifying what contributes to the apparent trade-off, we show how to simultaneously enhance the strength and the range. The red hexagon in Fig. 1(c) for example has a $500$- and $8$-time enhancement of the strength and the range, respectively. Such enhancement could be very useful for quantum systems that use radiative interaction as the main mechanism for energy or information transduction.

We choose Dirac point because its isosurface can change rapidly while maintaining a relatively constant group velocity, which helps to isolate the effect of the contribution from isosurface. The Hamiltonian near a Dirac point is given by $H_{\mathrm{D}}=v(k_{x}\sigma_{x}+k_{y}\sigma_{y})$, where $\sigma_{x(y)}$ is the Pauli matrices and $v$ is the isotropic group velocity. This Hamiltonian results in a linear dispersion relationship as $\omega=v|\mathbf{k}-\mathbf{k}_{\mathrm{c}}|$, where $\mathbf{k}_{\mathrm{c}}$ is the location of the Dirac point in momentum space.

We study the interaction between two identical transitions placed inside the photonic crystal with the Dirac dispersion [36, 37] as shown in Fig. 1(a). We show three cases with their transition frequencies labeled in Fig. 2(c). From the point iii to point i in Fig. 2(c), the frequency gradually approaches the Dirac point. The radiative interaction as a function of the distance is shown in Fig. 2(d). The calculation is available in Appendix D. When the transition frequency is far away from the Dirac point, i.e. frequency-iii in Fig. 2(c), and the blue curve in Fig. 2(d), the strength and the range are not significantly different from those in vacuum. As we move the transition frequency close to the Dirac point, we see enhanced range at frequency-ii and frequency-i. In the case very close to the Dirac point (green curve in Fig. 2(d), the interaction decays slowly as the distance between the transitions increases. Conversely, the strength is weakened as the transition frequency approaches the Dirac point. Note that the green curve (point i) is displayed with a $9\times$ multiplication factor in Fig. 2(d). We summarize the strength and range of three cases in Fig. 2(b), and the trade-off is clearly observed: while the strength increases, the range decreases.

The trade-off observed here is quite general. We now discuss its origin. We consider a general continuum with a dispersion relationship $\omega=\omega(\mathbf{k})$, where $\mathbf{k}$ is the wavevector. We focus on optical modes around the transition frequency $\omega_{0}$ because they are the most important for mediating the radiative interaction. In the momentum space, these modes form an isosurface defined by $\mathbf{k}=\mathbf{k}(\omega_{0})$. For example, the isosurface in vacuum is a sphere with a radius of $k=\omega_{0}/c$. In engineered photonic environments, the shape and size of isosurface can be quite different. We recently showed that the isosurface plays a significant role in radiative interactions [22, 38].

The radiative interaction can be written in the following form [23, 22]

where the polarization factor $p_{\mathbf{k}}=\omega[\mathbf{d}_{1}\cdot\mathbf{E}_{\mathbf{k}}(\mathbf{r}_{1})]^{*}[\mathbf{d}_{2}\cdot\mathbf{E}_{\mathbf{k}}(\mathbf{r}_{2})]/16\pi^{2}\epsilon$ and $v_{k}=\sqrt{v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}$ is the group velocity. The integration is performed over the isosurface $\Sigma(\omega)$. $\mathbf{E}_{\mathbf{k}}$ is the electric field of mode, $\mathbf{k}$. $\mathbf{d}_{1,2}$ are the dipole moments of the two transitions, $\mathbf{R}=\mathbf{r}_{2}-\mathbf{r}_{1}$ is the distance between the two transitions, and $\epsilon=n^{2}\epsilon_{0}$ is the dielectric constant and $n$ is the refractive index.

Now we are ready to explain the range of interaction. The term $e^{i\mathbf{k}\cdot\mathbf{R}}$ is a fast-oscillating term. The integration of this fast-oscillating term $e^{i\mathbf{k}\cdot\mathbf{R}}$ cancels when $\mathbf{R}$, the distance between the two transitions, is large. This is why the interaction vanishes for large $R$. In addition to $R$, the photonic environment also plays an important role through the shape of the isosurface $\Sigma(\omega)$. To see this impact, we fix a distance $\mathbf{R}$ between two transitions and consider photonic environments with different $\Sigma(\omega)$. If we have a large isosurface, the integration of $e^{i\mathbf{k}\cdot\mathbf{R}}$ cancels more compared to a small isosurface. As a result, a large isosurface leads to a shorter-range interaction than a small isosurface. This is what we observed in Fig. 2(b) and 2(c). The isosurface, which is an iso-contour in this two-dimensional space, is smaller around Dirac point, leading to a long interaction range. Away from the Dirac point, the isosurface is larger, resulting in a short interaction range.

Now we understand one aspect of the interaction-the range. We can examine the other side of the trade-off. The size of isosurface has a quite different impact on interaction strength. Larger isosurface generally leads to a greater interaction strength. The maximum interaction strength is obtained at $R=0$ when $e^{i\mathbf{k}\cdot\mathbf{R}}$ becomes unity. A large isosurface has a large integration area, and thus a greater integration value. This relationship can also be understood using the optical density of states (DOS), which is proportional to the size of isosurface. The larger the DOS, the greater the spontaneous emission rate, which is directly related to the interaction strength.

Now the trade-off in Fig. 2(d) can be easily understood. Close to the Dirac point, the isosurfaces become smaller and smaller (Fig. 2(b)). Correspondingly, the range of interaction increases. However, smaller isosurfaces reduce the density of states, which decreases the spontaneous emission rate and the interaction strength.

The understanding of interaction above is general for a photonic environment with an explicit dispersion relationship. The trade-off caused by isosurfaces is built into the mathematic integration Eq. (1) that determines the interaction. To mitigate this trade-off, we have to look into other physical mechanisms that contribute to the interaction: the first is the local field effect in the term $p_{\mathbf{k}}$, and the other is the group velocity in the term $v_{\mathbf{k}}$. By carefully balancing the isosurface, local field, and group velocity in judiciously designed photonic environments, we can overcome the apparent trade-off to realize simultaneous enhancement of interaction strength and range.

Our strategy starts by considering a photonic environment with a small isosurface because this is necessary to obtain a long range. However, it comes with a weak interaction strength. We can then use strong local fields and small group velocity to offset the negative impact from the small isosurface. The result is simultaneous enhancement of strength and range by many orders of magnitude. This strategy can be applied to many different types of engineered photonic media. Here we choose one of the simplest implementations in a Fabry-Pérot cavity [39, 40] to illustrate this method.

The cavity structure consisting two mirrors is shown in Fig. 3(a). The mirrors are extremely large and restrict the available modes and reduces the spherical isosurface in vacuum to iso-contours such as those rings and points shown in Fig. 3(c)-(f). Here we are particularly interested in the frequency region around the fundamental mode $\omega=\pi c/n_{0}D$, where $D$ is the cavity length and $n_{0}$ is the refractive index of material inside the cavity. The isosurface of the fundamental mode reduces to two points (Fig. 3(f)). In comparison, high-order resonant modes consist of ring-shape isosurfaces (Fig. 3(d,e)). The diminishing isosurface at the fundamental mode is ideal for extending the interaction range. Unlike Dirac points and Weyl points, here the group velocity is zero in the $x$-$y$ plane, which helps to off-set the negative impact of small isosurface on interaction strength.

We perform direct calculation of the resonant dipole-dipole interaction inside a Fabry-Pérot cavity (see more details in Appendix A). Fig. 3(b) shows the strength and range comparing the fundamental Fabry-Pérot modes to two other modes in the cavity. We can see the interaction range is longest for the fundamental mode because its smallest isosurface. Here we managed to maintain the strength of interaction because of the low group velocity.

The fundamental mode in a Fabry-Pérot cavity provides a long interaction range while maintaining an interaction strength similar to that in vacuum. Next, we explore a second mechanism to greatly enhance the strength beyond that of vacuum using the local field effect [41]. This effect occurs at the interface between two materials of different refractive indices when a small volume of the lower index material is embedded in the higher index material. Polarization charge accumulated at the interfaces creates enhanced local electric field [41, 42, 43], which can be used to enhance the field $\mathbf{E}_{\mathbf{k}}(\mathbf{r}_{1,2})$ in Eq. (1). To illustrate this effect, we consider an ellipsoidal defect of low index $n_{\mathrm{d}}$ in a material of high index $n_{0}$. The dimension of the ellipsoid is much smaller than wavelength (typically smaller than $\lambda_{0}/10$). Fig. 4(a) shows the field distribution under plane wave illumination with electric field polarized in $x$ direction. The intensity of the field inside is enhanced by $F^{2}=n_{0}^{4}/[(1-r)n_{0}^{2}+rn_{\mathrm{d}}^{2}]^{2}$, where $r=(l^{2}s/2)\int_{0}^{\infty}dq/[(q+l^{2})^{2}(q+s^{2})^{1/2}]$ is the geometrical factor that depends on the major radius $l$ and minor radius $s$ of the ellipsoid (see details in Appendix B). Figure 4(b) shows the enhancement of the field intensity defects with different index contrast and geometrical ratio. With an index contrast around $3$, $90$ times enhancement can be realized for elongated defect shapes. Since it is small with a deep sub wavelength size, the defect does not significantly affect the modes in the continuum which are mostly determined by larger and wavelength-scale structures. The local field effect explores a completely different physics of enhancement than the group velocity and isosurface. It does not interfere with other enhancement mechanisms.

Next, we combine the effects of local field enhancement and the fundamental mode of a Fabry-Pérot cavity to demonstrate a case of long and strong interaction. The cavity is filled with a high-index material $n_{0}=3.67$. Inside the cavity, we have two small air holes as the low-index defects. We consider one resonant dipole inside each defect as shown in Fig. 4(c). The two dipoles are aligned to the short axis of the ellipsoidal defect. The radiative interaction between two quantum transitions is given by $\Gamma=(2\omega^{2}/\epsilon_{0}c^{2})\mathrm{Im}[\mathbf{d}_{2}\cdot\bar{\bar{\mathbf{G}}}(\mathbf{r}_{2},\mathbf{r}_{1};\omega)\cdot\mathbf{d}_{1}]$, where $\bar{\bar{\mathbf{G}}}(\mathbf{r}_{2},\mathbf{r}_{1};\omega)$ is the electric dyadic Green’s function of the photonic environment. Then dyadic Green’s function can be obtained numerically by a full-wave simulation shown in Appendix D. The numerical results are marked as the red diamonds in Fig. 4 (d), which agree with our theoretical estimation in Appendix B. For the cavity with a quality factor of $3.14\times 10^{4}$, the same as the cavity in Fig. 3, the interaction strength can reach up to about $500\Gamma_{0}$. Meanwhile, the interaction range is $8$ times the range in vacuum. We also examine the results from non-Markovian calculation, which shows fairly consistent conclusions as the Markovian approximation used here. The details can be found in Appendix E. The retardation effect is included in the full electrodynamic treatment [44].

To conclude, we discuss the origin of the apparent trade-off between the interaction strength and interaction range. We analyze three main factors that contribute to radiative interaction: the isosurface of photonic environment, the local field, and the group velocity. The isosurface leads to the trade-off between the strength and the range of interaction. The local field and group velocity can be modified relatively independently. With this recognition, we developed a recipe for long and strong range interaction. We start with a small isosurface to guarantee a slow decay with a long range. Then we engineer the photonic environment to exploit the local field and the group velocity. It leads to long and strong interaction between two resonant transitions. It would be interesting to study a network of quantum transitions where they interact with each other strongly over extended distance, which could lead to quite different results from networks that only consider nearest neighbor interactions.

This work was supported by the National Science Foundation (NSF) through the University of Wisconsin Materials Research Science and Engineering Center DMR-1720415. L. Y. was also supported by the National Key Research and Development Program of China 2019YFA0308100.