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Extended range of dipole-dipole interactions in periodically structured photonic media

Extended range of dipole-dipole interactions in periodically structured photonic media 1 1 2 1 3 2 1, ∗ Lei Ying, Ming Zhou, Michael Mattei, Boyuan Liu, Paul Campagnola, Randall H. Goldsmith, and Zongfu Yu Department of Electrical and Computer Engineering, University of Wisconsin, Madison, Wisconsin 53706, USA Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706, USA Department of Biomedical Engineering, University of Wisconsin, Madison, Wisconsin 53706, USA (Dated: October 25, 2019) The interaction between quantum two-level systems is typically short-range in free space and most photonic environments. Here we show that diminishing momentum isosurfaces with equal frequen- cies can create a significantly extended range of interaction between distant quantum systems. The extended range is robust and does not rely on a specific location or orientation of the transition dipoles. A general relation between the interaction range and properties of the isosurface is described for structured photonic media. It provides a new way to mediate long-range quantum behavior. (a) The resonant dipole-dipole interaction between two (c) quantum two-level systems (TLS) is typically short- range. There has been strong interest in realizing long- range interactions to exploit collective physics such as superradiance [1, 2], collective frequency shift [3], (b) Fo¨rster resonance energy transfer [4, 5], and quantum entanglement [6–12]. The ability to modulate the dis- tance dependence of these processes could have po- tential applications in quantum information process- ing [8, 13] and energy conversion [14]. Two components contribute to the interaction: the evanescent near fields and the propagating far fields (Fig. 1a&b). To enable Figure 1. Schematics of interactions between two TLSs medi- long-range interaction from the evanescent fields, one ated by (a) evanescent near-field modes, (b) propagating far- could use evanescent fields with a long tail, such as de- field modes. (c) Momentum isosurface S with equal ω(k)=ω fect modes in the photonic bandgap [15–17]. However, frequencies ω and dS is a small surface element. 0 k it is less obvious how to engineer propagating far fields to enable long-range interaction. It is the goal of this of physical systems and can reveal new systems capable letter to provide a new perspective to understand the of realizing long-range interactions. general physical mechanism that is responsible for long- We begin by examining the interaction between two range interaction induced by propagating far fields, and TLSs over a long distance. The TLSs are embedded in identify photonic structures that are capable of extend- a photonic environment that can be described by a dis- ing the interaction range. persion relation ω = ω (k). For example, in free space, In free space, the range of far-field interaction is lim- ω = c|k| = ck, where c is the speed of light. Other dis- ited to the wavelength scale. When the wavelength is persion relations can be seen in metamaterials, photonic long, such as in index-near-zero materials [18–22], the crystals or waveguides. In general, the Hamiltonian of interaction range can increase proportionally. However, the TLSs and the photonic modes is given by [33] there are a few intriguing examples where the inter- X X action range extends beyond the effective wavelength. † † H = ω σˆ σˆ + ω aˆ aˆ 0 k k These include low-dimensional spaces, such as photonic i k i=1,2 k crystal waveguides and fibers [2, 23–30], or hyperbolic (1) X X materials in selected directions [31, 32]. These interest- † ik·r +i ig (r ) σˆ + σˆ aˆ e + H.c. , k i i k ing but isolated examples heavily rely on very specific i=1,2 k configurations. Thus, it is difficult to generalize the the- oretical treatments to identify the underlying physics, where ω is the resonant transition frequency of TLSs. which unfortunately remains elusive. In this letter, we σˆ (σˆ ) is the raising (lowering) operator of ith TLS. ω i k show the deep connection between the interaction range † and aˆ (aˆ ) are the frequency and creation (annihilation) and the size and shape of the isofrequency surface in operator of photons, respectively. g (r ) = ω /2ε V µ · k i k 0 momentum space. It can be generalized to a broad range ǫ is the coupling between the ith TLS and the photonic mode k, where µ is the transition dipole moment of the ith TLS and ǫ is the polarization direction of the photonic mode k . One can derive the radiative interac- zyu54@wisc.edu tion Γ = Γ + iΓ between two TLSs based on the above Re Im arXiv:1906.08389v3 [physics.optics] 24 Oct 2019 2 Hamiltonian. The real and imaginary parts describe the cooperative decay rate and cooperative energy shift, re- spectively. The focus of this letter will be the cooper- ative decay rate. Similar conclusions can be drawn for the cooperative energy shift. We first provide a graphic illustration of why the interaction between TLSs is short-range in free space. Unlike most theoretical treatments used in the litera- ture [32], we do not use the Green’s function method to describe the radiative environment. Instead, we try to keep all radiative modes in their explicit forms in order to gain a more intuitive picture. As shown in Section I of Supplementary Material (SM), the real part of the radiative interaction between TLSs can be expressed in the following form: ik·R Γ = ρ e dS . (2) Re k k ω (k) The integral is performed on an isosurface in mo- mentum space, i.e. all wavevectors k that satisfy ω (k) = ω . The integrand includes two terms. The first term is simply a polarization factor ρ = µ · ǫ µ · ǫ , which describes the relative 2 k k 1 2 16ε π v (k) 0 g orientation of the transition dipole µ and the polariza- tion of the electric field ǫ. Here v (k) is the group ve- locity of mode k. For degenerate polarization states, the integration should also include all polarizations. Since the polarization factor ρ is independent of the inter- TLS distance, it does not affect the interaction range. It ik·R Figure 2. (a) Two dipolar quantum transitions spaced by a dis- is the second term, e , that plays the critical role in tance R = 10λ in free space, where λ = 2πc/ω. The right panel the physics of the interaction range. Here R = r − r 1 2 shows the isosurface for the transition frequency in momen- is the distance vector between the two TLSs. The inte- ik·R ik·R tum space. The real part of the integrand ρ e is plotted on grand ρ e is a fast oscillating function, which gener- k the isosurface. Red and blue colors indicate positive and neg- ally results in cancellation of the integration when the ative maximum,respectively. A long R leads to fast oscillation inter-TLS distance R is large. Therefore, the interaction and cancellation of the integral over the isosurface. (b) Simi- is always short-range. We can see this effect in Fig. 2a. lar to (a) but with a shorter distance R = 0.3λ and thus slow Here we consider two TLSs in free space. The spherical oscillation on the isosurface. (c) The situation can change sig- isosurface has a radius of k = |k| = ω /c. The real part nificantly if two quantum transitions are placed in a general ik·R of ρ e is plotted on the isosurface. When R = 10λ, photonic environment, such as Weyl photonic crystal, where there are rapid oscillations as k varies on the isosurface. the isosurface can be very small. Here R = 10λ. The isosurface has a radius of q = |k − k |. The inset in the right panel shows The resulting value of the integral is small, and there- the zoom-in view of the small isosurface, showing that even fore the interaction is weak at this long distance. When a large R may not result significant cancellation due to small the inter-TLS distance is small, for example R = 0.3λ, isosurface size. R in (a-c) is fixed as (1,0,1)/ 2. (d) & (e) The the oscillation is slow (Figure 2c), leading to a sizeable real part of radiative interaction, normalized by Γ (R = 0), as Re value of the integral and thus a strong interaction. The a function of distance between two TLSs in free space and the interaction decays as the distance R grows (Fig. 2d). Weyl photonic crystal, respectively. Red dots correspond to The graphic illustration also indicates that the inter- the cases in (a), (b), and (c), respectively. action range is inversely proportional to the size of the isosurface in momentum space. A large inter-TLS dis- tance R on a large isosurface leads to a fast oscillating integrand on the isosurface that results in a small value times that of the free-space isosurface. While the os- of the integral. One way to counteract this effect is to cillation is still fast, the small isosurface cannot accom- substantially reduce the isosurface size. Small isosur- modate many oscillations, yielding a sizable value of faces can save the integral from cancellation even for a the integral. Figure 2e shows that this strong interac- fast-oscillating function. Figure 2c shows the real part tion is sustained over a long distance if the isosurface is ik·R of the integrand ρ e with a long inter-TLS distance small. Specifically, for an isosurface with a radius of q, R = 10λ on an isosurface that has a radius that is 0.03 the real part of interaction Γ scales as sin(qR)/qR. As Re 3 Figure 3. (a) Structure of Weyl photonic crystal. The locations of four air spheres with a radius of 0.07a in the double-gyroid unit cell are same with Ref. [34]. (b) Dispersion relation on the plane of k = 0. The momentum k is normalized by 2π/a. (c) The z x,y real part of the radiative interaction Γ (normalized by Γ (R=0)) as a function of distance for TLS transition frequencies (upper) Re Re ω = 0.5545, (middle) 0.5520 and (lower) 0.5512[2πc/a], which are marked with white contours i, ii, and iii in (b),respectively. The √ √ inter-TLS direction is R = (−1,1,1)/ 3. The dipole orientations are µˆ = (−1,1,1)/ 3 and µ is fixed at central point of the unit 1,2 1 cell. Green dashed curves are the envelops of the solid curves. Inset (i-iii) are the isosurfaces in momentum space. (d) The linear relationship between decay length ℓ and inverse size of isosurfaces 1/ q. the isosurface radius approaches zero q → 0, the range MPB software package [37]. The details of the calcu- becomes infinite. Here, we use a polarization factor ρ lation are shown in SM. Figure 3c shows the interaction based on plane waves, which, although a simplification, as a function of the inter-TLS distance for three different is sufficient for estimating the scaling. transition frequencies, which are also labeled by white lines in Fig. 3b. The isosurfaces have four lobes because The size of isosurface is fixed in free space. But there are four Weyl points, as shown in Fig. 3c (i-iii). there are many structured photonic environments that As the TLS transition frequency approaches the Weyl offer smaller isosurfaces. Here, we use Weyl photonic point, the isosurface size decreases, causing the interac- crystals as an example to demonstrate the inverse rela- tion extends to extend to a significantly greater range. tionship between the interaction range and the isosur- When the transition frequency is 0.00024[2πc/a] away face. Weyl photonic crystals [34, 35] exhibit a conic from the Weyl point (panel iii in Fig. 3c), the interaction dispersion relation in three-dimensional space, similar shows a negligible decay even at 180 wavelengths (Fig. to Dirac dispersion relations in two-dimensional space. 3c bottom). The isosurface gradually reduces to a point around the apex of the conic dispersion, i.e. the Weyl point. The decaying and oscillating patterns in these curves Observation of this small isosurface suggests that we are attributed to a few different origins. At the largest could expect long-range interactions around isolated scale, the envelop scales as sin(q¯R)/q¯R, where we use q¯ Weyl points. Specifically, we consider a double gyroid to roughly characterize the size of the isosurface (we will structure described by g (r)=sin(2πx/a) cos(2πy/a) + discuss the impact of the shape of isosurface later). The sin(2πy/a) cos(2πz/a) + sin(2πz/a) cos(2πx/a) , where a medium-range oscillation is due to the interplay of four is the lattice constant. A material with a dielectric con- Weyl points at the same frequency. The fastest oscilla- stant ε = 13 fills the regions defined by |g (r)| > 1.1. tion is due to the modulation of the nonuniform field Four air spheres are placed in the dielectric material within a unit cell of the photonic crystal. The long- as defects to break parity symmetry yielding two pairs range interaction observed here is robust in that it does of Weyl points at identical frequencies [36]. The unit not rely on the orientation of the dipole direction or the structure is shown Fig. 3(a). The dispersion relation spatial placement of TLSs (See more discussion in SM). on the momentum plane of k = 0 is shown in Fig. 3b We can quantitatively characterize the interaction with two pairs of Weyl points at the frequency ω = wp range by numerically fitting the envelope. These en- 0.55096[2πc/a]. The isosurface becomes infinitesimally velops are shown by the dashed line in Fig. 3c. We fur- small at the Weyl point. ther define a range ℓ as the distance when the envelop Using these isosurfaces, we numerically calculate the drops to half of its maximum value. We calculate this interaction between two TLSs placed inside the Weyl range for TLSs at different transition frequencies near crystal. The photonic modes are simulated using the the Weyl points, corresponding to different isosurface 4 (a) sizes. The results are shown in Fig. 3d. A clear linear re- lationship is demonstrated between ℓ and the inverse of the isosurface size 1/q¯. Because the isosurfaces are not spherical, we use q¯ = S /4π to define the isosur- face size, where S is the surface area of isosurfaces. ω (b) Thus far, we have shown that the size of the isosur- face plays a critical role in the interaction range. Next, we will discuss the role of the shape of the isosurface. A spherical isosurface leads to an isotropic interaction (c) range. On the other hand, a non-spherical isosurface generally creates an anisotropic interaction range: the interaction range depends on the direction of the inter- TLS distance vector R. There is a general reciprocal re- lationship between the interaction range and the size of 0 5 10 15 the isosurface when projected along R. Let us take the example of an ellipsoidal isosurface Figure 4. (a) Real part of the integrand in Eq. (2) on an el- in an anisotropic media. The interaction range is longer liptical isosurface with (left) R=(0,1,0), (middle) (0,1,1)/ 2, when the two TLSs are placed along the direction of the and (right) (0,0,1). Unit vectors sˆ and l represent short short axis of the ellipsoid sˆ, than when they are along and long axis of the anisotropic isosurface. The dipole ori- the long axis l. We can easily see this effect by observing entation is fixed as µˆ = (0,0,1). (b) Same as (a), but the ik·R 1,2 the oscillation pattern of ρ e on an ellipsoidal iso- isosurface is in the Weyl photonic crystal in Fig. 3a at fre- surface as shown in Fig. 4a. When R is parallel to the quency ω = 0.555[2πc/a] and the dipole orientation is fixed long axis l, we have many oscillations and strong can- as µˆ = (0,1,0). (c) The absolute value of Γ as a function Re 1,2 ˆ of distance R. Light green, blue and red curves, respectively, cellation of the integration. On the other hand, when R correspond to R in left, middle, and right cases of (b). is parallel to the short axis sˆ, we have fewer oscillations and weaker cancellation. To demonstrate this effect in Weyl photonic crys- tals, we plot the isosurface at frequency ω = ω + first kind. For a two-dimensional photonic crystal, a wp spherical isosurface with a radius of q creates a differ- 0.00404[2πc/a], where the isosurface has a flat edge- softened rectangular geometry (Fig. 4b). We plot the ent scaling law that follows J (qR). More examples are discussed in Sec. II of SM. real part of the integrand in Eq. (2) on the isosurface for three different R. Here the magnitude of R is fixed, We have discussed that the interaction range. An- but its direction R varies from the short axis s to the other important aspect is the strength of the interac- long axis l. The cancellation effect is weaker when R is tion. We chose the linear dispersion near Weyl points aligned with the short axis and stronger along the long because it makes it easy to separate the effect of the iso- axis. We also calculate the interaction as a function of surface from other effects such as group velocity and the distance for the three directions shown in Fig. 4b. density of states. However, the shrinking isosurface The range is conspicuously longer for TLSs placed along combined with a finite group velocity also decreases the short axis of the isosurface than that for the long axis the interaction strength. At the Weyl point, the inter- as shown in Fig. 4c. In the case shown in Fig. 4, the action strength is zero. The linear dispersion near a frequency is greatly detuned from the Weyl point, and Weyl point results in a trade-off between the interaction thus, the interaction range is not as long as those shown range and strength. Such a trade-off can be alleviated in in Fig. 3. two-dimensional crystals and with a high order disper- The extended range of the dipole-dipole interaction sion relation. We discussed the scaling of the interaction strength in Sec. II in SM. extends beyond quantum systems. In the microwave regime, where Weyl photonic crystals have been exper- Visual inspection of the isosurface provides a conve- imentally realized on a printed circuit board [34], the nient tool to understand a broad class of long-range in- resonant dipole-dipole interaction range can also be ex- teraction phenomena. We now comment on the connec- tended. The range will also be limited by the propaga- tion between our approach and the existing literature. tion length of the waves inside such systems due to finite The behavior of index-near-zero materials [18] was ex- absorption by metallic materials. plained by a long effective wavelength. Alternatively, it We also emphasize that the relation between the in- can also be conveniently explained by our method: the teraction range and the isosurface is not unique to index-near-zero material also has an ultra-small isosur- Weyl photonic crystals. It is generally applicable to face. In addition to these examples, we can envision that periodically structured media. For example, in two- Dirac points in two-dimensional photonic crystals also dimensional space, the scaling of the interaction range provide small ‘isosurfaces’ (isofrequency contours) for follows J (kR), where J is the Bessel function of the long-range interaction. Ref. [16] shows that inside the 0 0 5 photonic bandgap, long tails of evanescent fields can in- the interaction range. The method introduced here pro- duce long-range interaction. Here we can also see that vides an intuitive understanding of underlying physics outside the photonic bandgap but near the band edge, that is somewhat buried in traditional treatments, and the propagating far fields have small isosurfaces, offer- we were able to use our method to help understand sev- ing a different mechanism for long-range interaction. A eral photonic systems from the existing literature. It hyperbolic material, where long-range interactions were also provides a general recipe to search for new pho- allowed along specific directions, was treated using the tonic systems that support long-range interactions. Green’s function method [32]. Using our graphic inter- This work was supported by the National Science pretation allows one to intuitively see that only special Foundation (NSF) through the University of Wiscon- directions allow long-range interactions (see the visual- sin Materials Research Science and Engineering Center ization in SM). DMR-1720415. 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THEORY OF LONG-RANGE INTERACTION as [15]M = iΓ , where Γ is the radiative inter- FI action. Utilizing the relation [f (x)/ (x − x )]dx = A. General resonant interaction theory between two quantum two-level systems [P(f (x)/ (x − x )) + iπδ(x − x )f (x)]dx, we can write 0 0 the radiative interaction as The Hamiltonian of quantum two-level systems Γ = Γ + iΓ Re Im (TLSs) in an arbitrary photonic environment is given by H = H + H + H . (3) ph tls int v (k)  3 ik·R = d k πδ (ω − ω )ρ e k 0 k,η  (7) They are explicitly written as [33] (~ = 1) X  † ik·R −ik·R e e H = ω aˆ aˆ ph k k,η ∗ k,η  + iP ρ + ρ , k,η k,η  k,η ω − ω ω + ω k 0 k 0 ˆ ˆ H = ω σ σ tls 0 i where P denotes the Cauchy principal value and the po- (4) i=1,2 larization factor is X X † ik·r H = ig (r ) σˆ + σˆ aˆ e + H.c. , ρ = µ · ǫ µ · ǫ . (8) int k,η i i k,η i k,η k,η k,η 1 2 16π ε v (k) 0 g i=1,2 k,η The real part of Γ is the cooperative decay rate and its where ω and σˆ (σˆ ) are the transition frequency and 0 i explicit expression is raising (lowering) operator of ith two-level system (TLS). ω and aˆ (aˆ ) are the frequency and cre- ik·R k k,η k,η Γ (ω ) = ρ (ω )e dS , (9) Re 0 k,η 0 k ation (annihilation) operator of photon. g (r ) = η ω (k) k,η i ω /2ε V µ · ǫ is the coupling between ith TLS and k 0 k,η where S is the isosurface of ω = ω in momentum ω (k) 0 photonic mode k. µ is the transition dipole moment of space and v (k) = |∇ ω | is the group velocity of mode g k k ith quantum TLS and ǫ is the polarization of photonic k,η k. The cooperative energy shift is mode k with polarization index η. Z " ∞ X The transition probability from initial to final states is 1 Γ (ω ) = P dω dS Im 0 k given by the Fermi’s Golden rule 2π/~|M | δ (E − E ), FI F I π 0 S η ω(k) where the transition matrix element M can describe FI   the resonant dipole-dipole interaction between two  ρ  k,η k,η   ik·R −ik·R   ×  e + e    TLSs. For the weak light-matter interaction, it can be (10) ω − ω ω + ω 0 0 written as the second-order form: hF|H |R ihR |H |Ii Z ! int α α int ∞ ∗ M = hF|H |Ii + + ··· . (5) 1 Γ (ω) Γ (ω) FI int Re Re E − E = P dω + . I R π ω − ω ω + ω 0 0 Here, |Ii = |e ,g ;0i and |Fi = |g ,e ;0i denote initial and 1 2 1 2 final states, where‘e’ and ‘g’ in the Dirac bracket notions B. Interaction in free space vacuum represent excited and ground states, respectively, and the number ‘0’ or ‘1’ is the photon number in the pho- In the free space, the dispersion relation is given by tonic environment. The intermediate state |R i has two (1) (2) options: |g ,g ;1 i with energy E = E + E + ~ω ω = c|k| = ck, (11) 1 2 k,η R g g k (1) (2) and |e ,e ;1 i with energy E = E + E + ~ω . e e 1 2 k,η R k where c is the speed of light. Because the isosur- ! ! (1) (2) face is isotropic, we have dS = k dΩ = The energy of the initial state is E = E + E + ~ω . k k I g g k ω(k) (1,2) R R 2π π Since two identical TLSs are considered, we have E − k dθ dϕsinϕ and group velocity v (k) = c. As- 0 0 (1,2) E = ~ω . Then, Eq. (3) can be explicitly given by suming k·R=ξ kR = cosϕkR, the cooperative decay rate g 0 in free space is written as (c = 1) X ik·R Z Z 3 X 2π π M = g (r ) g (r ) FI k,η 1 k,η 2 µ µ k 1 2 ω − ω k 0 Γ = dθ dϕ sinϕ k,η Re (6) 16π ε 0 0 0 η (12) −ik·R +g (r )g (r ) , ikRξ k,η 1 k,η 2 × µ · ǫ µ · ǫ e . k,η k,η ω + ω 1 2 k 0 7 The polarization sum rule is given by components of the group velocity. For simplicity, we as- sume the isosurface is isotropic, i.e. v = v = v = v. x y z (η) (η) ˆ ˆ The dispersion relation near the Weyl point is given by ǫ ǫ = δ −k k (13) 12 1 2 k,i k,j ω = ω ± v|q|. (18) q wp (η) ˆ ˆ with ǫ = µˆ · ǫ , δ = µˆ · µˆ , and k = k · µˆ . Then, we k,η 12 i i 1 2 i k,i Then, the cooperative decay rate is given by have µ µ ω 1 2 Z Z ik ·R 3 2π π Γ (ω) = e Re µ µ k 1 2 ikRξ ˆ ˆ k 16π ε v Γ = dθ dϕ δ − k k sinϕe Re 12 1 2 (19) 16π ε 0 0 0 iq·R Z Z × dS µˆ · ǫ µˆ · ǫ e . 2π π q k,η k,η 1 2 µ µ k 1 2 2 ikRξ ω(q) = −∇ δ + ∇ ∇ dθ dϕ sinϕe 12 1 2 16π ε 0 0 0 Here, we use a polarization based on plane waves. Al- µ µ k sinkR 1 2 = −∇ δ + ∇ ∇ 12 1 2 though this approximation is simplified, it is sufficient 4πε R  for calculating the scaling. At frequencies near the Weyl µ µ k sinkR 1 2  point, q ≪ k and the polarization factor term is a con-  ˆ ˆ = δ − R R 12 i j 4πε kR stant µ µ ω ∗ 1 2 coskR sinkR ρ = µˆ · ǫ µˆ · ǫ ≃ ρ. (20) ˆ ˆ  k,η k,η k,η 1 2 + δ − 3R R + , 2 12 i j  16π ε v 2 3 (kR) (kR) (14) Consequently, the cooperative decay rate is ˆ ˆ where R = µˆ · R. Also, with the relation in Eq. (13), 1,2 1,2 ik ·R iq·R Γ ≃ρe dS e Re q the cooperative energy shift is given by ω (q) (21) Z Z 2π π sinqR 2 ik ·R µ µ c 1 2 2 =4πq ρe . Γ = −∇ δ + ∇ ∇ dθ dϕ sinϕ Im 12 1 2 qR 16π ε 0 0 0 Z ! ikRξ −ikRξ k k e e The imaginary part of the radiative interaction (the × P dkk + . k − k k + k cooperative energy shift) is an integral over frequencies 0 0 0 from zero to infinity, as shown in Eq. (10). For Weyl (15) photonic crystals, the dispersion relation is quite dif- After calculating the Cauchy principal integral and in- ferent from Eq. (18) at frequencies far from the Weyl tegral over isosurface [? ], we have point [35]. Thus, we do not show an analytic estima- tion of the cooperative energy shift here, but numerical µ µ k cosk R 1 2 0 2 0 details will be discussed in Sec. III. Γ = −∇ δ + ∇ ∇ Im 12 1 2 4πε R µ µ k  cosk R 1 2  0 0  ˆ ˆ = δ − R R − II. THE SCALING OF THE INTERACTION STRENGTH  12 i j (16) 4πε k R 0 0 IN 3D AND 2D PHOTONIC ENVIRONMENTS sink R cosk R 0  ˆ ˆ  + δ − 3R R + . 12 i j The interaction range increases as the isosurface de- 2 3 (k R) (k R) 0 0 creases. However, at the same time, the density of states also decreases, particularly when the group ve- locity does not scale to zero. The consequence is that C. Interaction near Weyl points the interaction strength reduces while the range extends unless the group velocity scales in a way to cancel the ef- At first, we only consider a single Weyl point, as fect. Here below, we discuss how the strength scales in shown in Fig. 2c of the main text. The Hamiltonian for different photonic environments. the continuum around the Weyl point is given by [35] H (k) = v q σ , (17) A. Three-dimensional photonic media wp i i i i=x,y,z In the 3D case, the derivation starts from the defini- where σ are Pauli matrices and q = q , q , q = k − x,y,z x y z tion of the real part of Γ as shown in Eq. (9). Here, we k is the distance to the Weyl point in momentum space. will show the real part of radiative interaction in differ- The Weyl point is at k when q = 0. v are the x,y,z ent 3D photonic media. c x,y,z 8 1. 3D free space For the free space case, the dispersion relation is ω = c|k| = ck, where c is the speed of light. The radiative interaction strength is given by (see detailed in Sec. I B) k sinkR (3D)  ˆ ˆ Γ =B δ − R R + Re 12 1 2 c kR (22) coskR sinkR ˆ ˆ  δ − 3R R + , 12 1 2 2 3 (kR) (kR) (3D) where B = µ µ ω/4πε . In the long-distance regime, 1 2 0 we have k sinkR (3D) ˆ ˆ Γ (R > λ)  B δ − R R . (23) Re 12 1 2 c kR 2. 3D Weyl photonic environment The linear dispersion relation near a Weyl point is given by Eq. (18). Assuming µˆ · ǫ µˆ · ǫ ≈ 1, we k,η k,η 1 2 Figure S1. (a) Double-gyroid dielectric structure in have the radiative interaction (also see details in Sec. IC) a body-centered cubic unit cell with a set of basis vec- tors a = (−1/2,1/2,1/2)a, a = (1/2,−1/2,1/2)a, and a = µ µ ω 1 1 2 3 1 2 ik ·R iq·R Γ  e dS e Re q (1/2,1/2,−1/2)a. Four air spheres with a radius r = 0.07a 16π ε v 0 S ω(q) are located at (1/4,−1/8,1/2)a, (1/4,1/8,0)a, (5/8,0,1/4)a and (24) (3/8,1/2,1/4)a, respectively. The dielectric constant of solid q sinqR (3D) ik ·R = B e . gyroid structure is 13. (b) Dispersion relation on k = 0 plane. v qR The white curves correspond to isosurface in (c). (c) Isosur- face at ω = 0.5545 [2πc/a] normalized by the momentum in As the radius of the small isosurface q shrinks to zero, free space k . (d) The radiative interaction Γ in vacuum the interaction strength accordingly diminishes to zero. 0 Re and in the Weyl photonic crystal. The distance direction is However, if the group velocity v and the radius of iso- R = [−1,1,1] and the dipole orientation is µˆ = [0,0,1]. γ is 1,2 surface q are finite and small, the interaction can still be the spontaneous decay rate in free space and λ is the wave- much larger and longer than that in vacuum (see com- length in free space. parison in Fig. S1(d)). 3. 3D quadratic photonic environment The interaction strength reduces to zero when q = 0, which is similar to Weyl point. However, when the For a quadratic dispersion relation such as group velocity v and the radius of isosurface q are finite and small, the radiative interaction can be significantly 2 2 ω = β|k−k | + ω = βq + ω , (25) k c c c larger than that in vacuum. the group velocity is v = 2βq. Supposing k = 0, we g c have ZZ iq·R ω q ∗ Γ = dS µ · ǫ µ · ǫ e Re q k k 1 2 16π ε 2β 0 S ω(q) B. Two-dimensional photonic media q sinqR (3D)  ˆ ˆ =B δ − R R + 12 1 2 (26) 2β qR In 2D case, the real part of the radiative interaction is written as cosqR sinqR ˆ ˆ  δ − 3R R + . 12 1 2 2 3 (qR) (qR) ik·R ω ∗ e Γ (ω) = dℓ µ · ǫ µ · ǫ , (28) In the long-distance regime, the radiative interaction is Re k k k 1 2 8πε ℏ v (k) 0 ℓ g ω(k) written as q sinqR (3D) ˆ ˆ Γ (R > λ)  B δ − R R . (27) Re 12 1 2 where ℓ is the isofrequency contour. 2β qR ω(k) 9 Figure S2. The real part of radiative interaction Γ as Re a function of distance R with dipole orientations (a) µˆ = 1,2 (1, 0, 0), (b) (0, 1, 0), (c) (0, 0, 1), (d) (−1, 1, 1)/ 3, and (e) 6 at ω = 0.5512[2πc/a]. The wavelength is given (2, 1, 1)/ by λ = 2πc/ω. Γ is normalized by µ µ ω /16π ε . Re 1 2 0 0 Figure S3. The real part of radiative interaction Γ as a Re function of distance R with distance vectors (a) R = (0, 1, 0), √ √ √ (b) (0, 1, 1)/ 2, (c) (1, 0, 0), (d) (1, 0, 1)/ 2, (e) (1, 1, 0)/ 2, 1. 2D free space and (f) (1, 1, 1)/ 3 at ω = 0.5512[2πc/a]. The wavelength is given by λ = 2πc/ω. Γ is normalized by µ µ ω /16π ε . Re 1 2 0 0 We assume the dipole direction is normal to the 2D plane, the radiative decay rate is written as Then the group velocity is written as v = v. The radia- 2π tive interaction is given by µ µ ωk 1 2 ikRcosθ Γ (ω) = dθe Re 4ε c 0 0 2π (29) µ µ ωq 1 2 ik ·R iqRcosθ c q k Γ (ω) = e dθ e Re q (2D) = B J (kR), 4ε v 0 0 (31) (2D) ik ·R = B J (qR)e . where J (x) is the zero-order of the Bessel functions of (2D) the first kind and B = µ µ ω/4ε is a frequency- 1 2 0 The 2D Dirac points provide stronger and longer-range related parameter. interaction than Weyl points because of the reduced di- mensionality. 2. 2D linear dispersion near Dirac point 3. 2D quadratic dispersion near band edges. The dispersion relation near a Dirac point is written as If the 2D dispersion relation is quadratic such as 2 2 ω = v |k−k | + ω = vq + ω . (30) ω = β|k−k | + ω = βq + ω (32) k c c c k c c c 10 and the group velocity is v = 2βq. Then, 2π µ µ ω 1 2 ik ·R iqRcosθ c q Γ (ω) = e dθ e Re q 8ε ℏβ (33) (2D) ik ·R = B J (qR)e . 2β In this case, the strength do not be limited by small q. III. THE WEYL PHOTONIC CRYSTAL A. Numerical method and details The index of polarization η is replaced by the index of band n. If the transition frequency of TLSs ω is in the n th band, the cooperative decay rate is given by Γ (ω ) = dS Re 0 k 16π ε 0 S ω (k) (34) ik·R × µ · ǫ µ · ǫ . n ,k n ,k 1 0 2 0 v (k,n ) g 0 The cooperative energy shift is written as ik·R Figure S4. (a) & (b) The real part of integrand ρ e on the ◦ ◦ Z max " X ω hyperbolic isosurface with θ = 45 and 80 , respectively. The 1 ω dipole orientations are µˆ = (0, 1, 0). (c) Γ as a function of Γ (ω ) = P dω dS Re Im 0 n k 1,2 min 16π ε v 0 ω S n,k θ with a fixed distance R = 10λ. Red dots correspond to the n n ω (k) cases in (a) and (b). (d) Γ as a function of distance for the hy- Re ik·R ∗ e perbolic (red) and vacuum (gray) cases. γ is the spontaneous ×  µ · ǫ µ · ǫ n,k n,k  1 2 ω − ω decay rate in free space. The angle is fixed as θ = 45 . n 0 −ik·R ∗  + µ · ǫ µ · ǫ n,k n,k 1 2 ω + ω arbitrary dipole orientations. Although the the dipole n 0 orientations affect the oscillation patterns of Γ curves, Re the envelopes of all curves show negligible decay even Z max X ω 1 at 30 wavelengths. In addition, the amplitudes of Γ Re = P dω with different dipole orientations stay on a same order min n n of magnitudes. 1 1 ∗ To show the effect of distance vector, the dipole ori- × Γ (ω ) + Γ (ω ) . Re n n Re ω − ω ω + ω entations are fixed as µˆ = (0, 1, 0). Fig. S3 shows the n 0 n 0 1,2 real part of radiative interaction Γ as a function of dis- (35) Re tance R with different distance vectors. Similarly to the Numerically, we use the MPB software package [37] effect of dipole orientations, the variation of R only in- to calculate eigen-modes of the Weyl photonic crystal in fluences the oscillation patterns of Γ curves. However, Re Fig. 3 of the main text. We set the resolution in the unit the envelopes of all curves exhibit negligible decay at cell as 30 × 30 × 30. Then, the frequency of the Weyl 30 wavelengths. Similary, the variation of the first TLS points is ω = 0.55096[2πc/a], which falls in between location r in a unit cell does not affect the interaction wp 4th and 5th bands. range. B. Dipole orientations and spatial placements of TLSs IV. INTERACTIONS IN OTHER PHOTONIC ENVIRONMENTS The details of the gyroid photonic crystal are shown in Fig. S1. To demonstrate the effect of TLS dipole ori- Our theory is also applicable for understanding the entations, we fix the direction of the distance vector as dipole-dipole interactions in other photonic environ- R = (−1, 1, 1)/ 3. Fig. S2 shows the real part of ra- ments, such as near the bandedge of photonic crys- diative interaction Γ as a function of distance R with tals, index-near-zero materials, hyperbolic materials, Re 11 ◦ ◦ etc. Here, we describe the interaction in hyperbolic me- θ = 45 and 80 , respectively. Here, θ is the included ˆ ˆ dia as an especially interesting case. Its isosurface is in- angle between R and x-axis. When R is normal to the finitely large, which normally leads to a very short inter- isosurface, i.e. θ = 45 ,there is a thick red (positive) action range. This is indeed the case. However, because strip in the oscillating pattern and it results in a large of its unique shape, for the direction R that is normal integral value. If R is aligned with a different direc- ◦ ik·R to the isosurface, the interaction range can be very long. tion, say θ = 80 , the fast oscillation of e results in This can also be clearly seen in the illustration of the in- cancellation of the integral and thus a weak interaction tegrand on the surface as we will show now. The disper- strength. In Fig. S4(c), we plot the cooperative decay sion relation of hyperbolic materials can be described rate at R = 10λ as a function of θ. The left- and right- by hand red dots correspond to the case in Fig. S4 (a) and (b), respectively. Our results agree with the exact nu- 2 2 2 2 k + k x y k ω merical result in Ref. [32]. + = . (36) ε ε z x Furthermore, we also plot the radiative interaction Here, we choose the second type hyperbolic material Γ as a function of distance for the hyperbolic material Re with ε = −ε = 1. The isosurface and the real part of compared to the case in free space, as shown in Fig. S4 z x ik·R integrand ρ e are shown in Fig. S4 (a) and (b) with (d). It greatly agrees with the result in Ref. [32]. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Condensed Matter arXiv (Cornell University)

Extended range of dipole-dipole interactions in periodically structured photonic media

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0031-9007
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ARCH-3331
DOI
10.1103/PhysRevLett.123.173901
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Abstract

1 1 2 1 3 2 1, ∗ Lei Ying, Ming Zhou, Michael Mattei, Boyuan Liu, Paul Campagnola, Randall H. Goldsmith, and Zongfu Yu Department of Electrical and Computer Engineering, University of Wisconsin, Madison, Wisconsin 53706, USA Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706, USA Department of Biomedical Engineering, University of Wisconsin, Madison, Wisconsin 53706, USA (Dated: October 25, 2019) The interaction between quantum two-level systems is typically short-range in free space and most photonic environments. Here we show that diminishing momentum isosurfaces with equal frequen- cies can create a significantly extended range of interaction between distant quantum systems. The extended range is robust and does not rely on a specific location or orientation of the transition dipoles. A general relation between the interaction range and properties of the isosurface is described for structured photonic media. It provides a new way to mediate long-range quantum behavior. (a) The resonant dipole-dipole interaction between two (c) quantum two-level systems (TLS) is typically short- range. There has been strong interest in realizing long- range interactions to exploit collective physics such as superradiance [1, 2], collective frequency shift [3], (b) Fo¨rster resonance energy transfer [4, 5], and quantum entanglement [6–12]. The ability to modulate the dis- tance dependence of these processes could have po- tential applications in quantum information process- ing [8, 13] and energy conversion [14]. Two components contribute to the interaction: the evanescent near fields and the propagating far fields (Fig. 1a&b). To enable Figure 1. Schematics of interactions between two TLSs medi- long-range interaction from the evanescent fields, one ated by (a) evanescent near-field modes, (b) propagating far- could use evanescent fields with a long tail, such as de- field modes. (c) Momentum isosurface S with equal ω(k)=ω fect modes in the photonic bandgap [15–17]. However, frequencies ω and dS is a small surface element. 0 k it is less obvious how to engineer propagating far fields to enable long-range interaction. It is the goal of this of physical systems and can reveal new systems capable letter to provide a new perspective to understand the of realizing long-range interactions. general physical mechanism that is responsible for long- We begin by examining the interaction between two range interaction induced by propagating far fields, and TLSs over a long distance. The TLSs are embedded in identify photonic structures that are capable of extend- a photonic environment that can be described by a dis- ing the interaction range. persion relation ω = ω (k). For example, in free space, In free space, the range of far-field interaction is lim- ω = c|k| = ck, where c is the speed of light. Other dis- ited to the wavelength scale. When the wavelength is persion relations can be seen in metamaterials, photonic long, such as in index-near-zero materials [18–22], the crystals or waveguides. In general, the Hamiltonian of interaction range can increase proportionally. However, the TLSs and the photonic modes is given by [33] there are a few intriguing examples where the inter- X X action range extends beyond the effective wavelength. † † H = ω σˆ σˆ + ω aˆ aˆ 0 k k These include low-dimensional spaces, such as photonic i k i=1,2 k crystal waveguides and fibers [2, 23–30], or hyperbolic (1) X X materials in selected directions [31, 32]. These interest- † ik·r +i ig (r ) σˆ + σˆ aˆ e + H.c. , k i i k ing but isolated examples heavily rely on very specific i=1,2 k configurations. Thus, it is difficult to generalize the the- oretical treatments to identify the underlying physics, where ω is the resonant transition frequency of TLSs. which unfortunately remains elusive. In this letter, we σˆ (σˆ ) is the raising (lowering) operator of ith TLS. ω i k show the deep connection between the interaction range † and aˆ (aˆ ) are the frequency and creation (annihilation) and the size and shape of the isofrequency surface in operator of photons, respectively. g (r ) = ω /2ε V µ · k i k 0 momentum space. It can be generalized to a broad range ǫ is the coupling between the ith TLS and the photonic mode k, where µ is the transition dipole moment of the ith TLS and ǫ is the polarization direction of the photonic mode k . One can derive the radiative interac- zyu54@wisc.edu tion Γ = Γ + iΓ between two TLSs based on the above Re Im arXiv:1906.08389v3 [physics.optics] 24 Oct 2019 2 Hamiltonian. The real and imaginary parts describe the cooperative decay rate and cooperative energy shift, re- spectively. The focus of this letter will be the cooper- ative decay rate. Similar conclusions can be drawn for the cooperative energy shift. We first provide a graphic illustration of why the interaction between TLSs is short-range in free space. Unlike most theoretical treatments used in the litera- ture [32], we do not use the Green’s function method to describe the radiative environment. Instead, we try to keep all radiative modes in their explicit forms in order to gain a more intuitive picture. As shown in Section I of Supplementary Material (SM), the real part of the radiative interaction between TLSs can be expressed in the following form: ik·R Γ = ρ e dS . (2) Re k k ω (k) The integral is performed on an isosurface in mo- mentum space, i.e. all wavevectors k that satisfy ω (k) = ω . The integrand includes two terms. The first term is simply a polarization factor ρ = µ · ǫ µ · ǫ , which describes the relative 2 k k 1 2 16ε π v (k) 0 g orientation of the transition dipole µ and the polariza- tion of the electric field ǫ. Here v (k) is the group ve- locity of mode k. For degenerate polarization states, the integration should also include all polarizations. Since the polarization factor ρ is independent of the inter- TLS distance, it does not affect the interaction range. It ik·R Figure 2. (a) Two dipolar quantum transitions spaced by a dis- is the second term, e , that plays the critical role in tance R = 10λ in free space, where λ = 2πc/ω. The right panel the physics of the interaction range. Here R = r − r 1 2 shows the isosurface for the transition frequency in momen- is the distance vector between the two TLSs. The inte- ik·R ik·R tum space. The real part of the integrand ρ e is plotted on grand ρ e is a fast oscillating function, which gener- k the isosurface. Red and blue colors indicate positive and neg- ally results in cancellation of the integration when the ative maximum,respectively. A long R leads to fast oscillation inter-TLS distance R is large. Therefore, the interaction and cancellation of the integral over the isosurface. (b) Simi- is always short-range. We can see this effect in Fig. 2a. lar to (a) but with a shorter distance R = 0.3λ and thus slow Here we consider two TLSs in free space. The spherical oscillation on the isosurface. (c) The situation can change sig- isosurface has a radius of k = |k| = ω /c. The real part nificantly if two quantum transitions are placed in a general ik·R of ρ e is plotted on the isosurface. When R = 10λ, photonic environment, such as Weyl photonic crystal, where there are rapid oscillations as k varies on the isosurface. the isosurface can be very small. Here R = 10λ. The isosurface has a radius of q = |k − k |. The inset in the right panel shows The resulting value of the integral is small, and there- the zoom-in view of the small isosurface, showing that even fore the interaction is weak at this long distance. When a large R may not result significant cancellation due to small the inter-TLS distance is small, for example R = 0.3λ, isosurface size. R in (a-c) is fixed as (1,0,1)/ 2. (d) & (e) The the oscillation is slow (Figure 2c), leading to a sizeable real part of radiative interaction, normalized by Γ (R = 0), as Re value of the integral and thus a strong interaction. The a function of distance between two TLSs in free space and the interaction decays as the distance R grows (Fig. 2d). Weyl photonic crystal, respectively. Red dots correspond to The graphic illustration also indicates that the inter- the cases in (a), (b), and (c), respectively. action range is inversely proportional to the size of the isosurface in momentum space. A large inter-TLS dis- tance R on a large isosurface leads to a fast oscillating integrand on the isosurface that results in a small value times that of the free-space isosurface. While the os- of the integral. One way to counteract this effect is to cillation is still fast, the small isosurface cannot accom- substantially reduce the isosurface size. Small isosur- modate many oscillations, yielding a sizable value of faces can save the integral from cancellation even for a the integral. Figure 2e shows that this strong interac- fast-oscillating function. Figure 2c shows the real part tion is sustained over a long distance if the isosurface is ik·R of the integrand ρ e with a long inter-TLS distance small. Specifically, for an isosurface with a radius of q, R = 10λ on an isosurface that has a radius that is 0.03 the real part of interaction Γ scales as sin(qR)/qR. As Re 3 Figure 3. (a) Structure of Weyl photonic crystal. The locations of four air spheres with a radius of 0.07a in the double-gyroid unit cell are same with Ref. [34]. (b) Dispersion relation on the plane of k = 0. The momentum k is normalized by 2π/a. (c) The z x,y real part of the radiative interaction Γ (normalized by Γ (R=0)) as a function of distance for TLS transition frequencies (upper) Re Re ω = 0.5545, (middle) 0.5520 and (lower) 0.5512[2πc/a], which are marked with white contours i, ii, and iii in (b),respectively. The √ √ inter-TLS direction is R = (−1,1,1)/ 3. The dipole orientations are µˆ = (−1,1,1)/ 3 and µ is fixed at central point of the unit 1,2 1 cell. Green dashed curves are the envelops of the solid curves. Inset (i-iii) are the isosurfaces in momentum space. (d) The linear relationship between decay length ℓ and inverse size of isosurfaces 1/ q. the isosurface radius approaches zero q → 0, the range MPB software package [37]. The details of the calcu- becomes infinite. Here, we use a polarization factor ρ lation are shown in SM. Figure 3c shows the interaction based on plane waves, which, although a simplification, as a function of the inter-TLS distance for three different is sufficient for estimating the scaling. transition frequencies, which are also labeled by white lines in Fig. 3b. The isosurfaces have four lobes because The size of isosurface is fixed in free space. But there are four Weyl points, as shown in Fig. 3c (i-iii). there are many structured photonic environments that As the TLS transition frequency approaches the Weyl offer smaller isosurfaces. Here, we use Weyl photonic point, the isosurface size decreases, causing the interac- crystals as an example to demonstrate the inverse rela- tion extends to extend to a significantly greater range. tionship between the interaction range and the isosur- When the transition frequency is 0.00024[2πc/a] away face. Weyl photonic crystals [34, 35] exhibit a conic from the Weyl point (panel iii in Fig. 3c), the interaction dispersion relation in three-dimensional space, similar shows a negligible decay even at 180 wavelengths (Fig. to Dirac dispersion relations in two-dimensional space. 3c bottom). The isosurface gradually reduces to a point around the apex of the conic dispersion, i.e. the Weyl point. The decaying and oscillating patterns in these curves Observation of this small isosurface suggests that we are attributed to a few different origins. At the largest could expect long-range interactions around isolated scale, the envelop scales as sin(q¯R)/q¯R, where we use q¯ Weyl points. Specifically, we consider a double gyroid to roughly characterize the size of the isosurface (we will structure described by g (r)=sin(2πx/a) cos(2πy/a) + discuss the impact of the shape of isosurface later). The sin(2πy/a) cos(2πz/a) + sin(2πz/a) cos(2πx/a) , where a medium-range oscillation is due to the interplay of four is the lattice constant. A material with a dielectric con- Weyl points at the same frequency. The fastest oscilla- stant ε = 13 fills the regions defined by |g (r)| > 1.1. tion is due to the modulation of the nonuniform field Four air spheres are placed in the dielectric material within a unit cell of the photonic crystal. The long- as defects to break parity symmetry yielding two pairs range interaction observed here is robust in that it does of Weyl points at identical frequencies [36]. The unit not rely on the orientation of the dipole direction or the structure is shown Fig. 3(a). The dispersion relation spatial placement of TLSs (See more discussion in SM). on the momentum plane of k = 0 is shown in Fig. 3b We can quantitatively characterize the interaction with two pairs of Weyl points at the frequency ω = wp range by numerically fitting the envelope. These en- 0.55096[2πc/a]. The isosurface becomes infinitesimally velops are shown by the dashed line in Fig. 3c. We fur- small at the Weyl point. ther define a range ℓ as the distance when the envelop Using these isosurfaces, we numerically calculate the drops to half of its maximum value. We calculate this interaction between two TLSs placed inside the Weyl range for TLSs at different transition frequencies near crystal. The photonic modes are simulated using the the Weyl points, corresponding to different isosurface 4 (a) sizes. The results are shown in Fig. 3d. A clear linear re- lationship is demonstrated between ℓ and the inverse of the isosurface size 1/q¯. Because the isosurfaces are not spherical, we use q¯ = S /4π to define the isosur- face size, where S is the surface area of isosurfaces. ω (b) Thus far, we have shown that the size of the isosur- face plays a critical role in the interaction range. Next, we will discuss the role of the shape of the isosurface. A spherical isosurface leads to an isotropic interaction (c) range. On the other hand, a non-spherical isosurface generally creates an anisotropic interaction range: the interaction range depends on the direction of the inter- TLS distance vector R. There is a general reciprocal re- lationship between the interaction range and the size of 0 5 10 15 the isosurface when projected along R. Let us take the example of an ellipsoidal isosurface Figure 4. (a) Real part of the integrand in Eq. (2) on an el- in an anisotropic media. The interaction range is longer liptical isosurface with (left) R=(0,1,0), (middle) (0,1,1)/ 2, when the two TLSs are placed along the direction of the and (right) (0,0,1). Unit vectors sˆ and l represent short short axis of the ellipsoid sˆ, than when they are along and long axis of the anisotropic isosurface. The dipole ori- the long axis l. We can easily see this effect by observing entation is fixed as µˆ = (0,0,1). (b) Same as (a), but the ik·R 1,2 the oscillation pattern of ρ e on an ellipsoidal iso- isosurface is in the Weyl photonic crystal in Fig. 3a at fre- surface as shown in Fig. 4a. When R is parallel to the quency ω = 0.555[2πc/a] and the dipole orientation is fixed long axis l, we have many oscillations and strong can- as µˆ = (0,1,0). (c) The absolute value of Γ as a function Re 1,2 ˆ of distance R. Light green, blue and red curves, respectively, cellation of the integration. On the other hand, when R correspond to R in left, middle, and right cases of (b). is parallel to the short axis sˆ, we have fewer oscillations and weaker cancellation. To demonstrate this effect in Weyl photonic crys- tals, we plot the isosurface at frequency ω = ω + first kind. For a two-dimensional photonic crystal, a wp spherical isosurface with a radius of q creates a differ- 0.00404[2πc/a], where the isosurface has a flat edge- softened rectangular geometry (Fig. 4b). We plot the ent scaling law that follows J (qR). More examples are discussed in Sec. II of SM. real part of the integrand in Eq. (2) on the isosurface for three different R. Here the magnitude of R is fixed, We have discussed that the interaction range. An- but its direction R varies from the short axis s to the other important aspect is the strength of the interac- long axis l. The cancellation effect is weaker when R is tion. We chose the linear dispersion near Weyl points aligned with the short axis and stronger along the long because it makes it easy to separate the effect of the iso- axis. We also calculate the interaction as a function of surface from other effects such as group velocity and the distance for the three directions shown in Fig. 4b. density of states. However, the shrinking isosurface The range is conspicuously longer for TLSs placed along combined with a finite group velocity also decreases the short axis of the isosurface than that for the long axis the interaction strength. At the Weyl point, the inter- as shown in Fig. 4c. In the case shown in Fig. 4, the action strength is zero. The linear dispersion near a frequency is greatly detuned from the Weyl point, and Weyl point results in a trade-off between the interaction thus, the interaction range is not as long as those shown range and strength. Such a trade-off can be alleviated in in Fig. 3. two-dimensional crystals and with a high order disper- The extended range of the dipole-dipole interaction sion relation. We discussed the scaling of the interaction strength in Sec. II in SM. extends beyond quantum systems. In the microwave regime, where Weyl photonic crystals have been exper- Visual inspection of the isosurface provides a conve- imentally realized on a printed circuit board [34], the nient tool to understand a broad class of long-range in- resonant dipole-dipole interaction range can also be ex- teraction phenomena. We now comment on the connec- tended. The range will also be limited by the propaga- tion between our approach and the existing literature. tion length of the waves inside such systems due to finite The behavior of index-near-zero materials [18] was ex- absorption by metallic materials. plained by a long effective wavelength. Alternatively, it We also emphasize that the relation between the in- can also be conveniently explained by our method: the teraction range and the isosurface is not unique to index-near-zero material also has an ultra-small isosur- Weyl photonic crystals. It is generally applicable to face. In addition to these examples, we can envision that periodically structured media. For example, in two- Dirac points in two-dimensional photonic crystals also dimensional space, the scaling of the interaction range provide small ‘isosurfaces’ (isofrequency contours) for follows J (kR), where J is the Bessel function of the long-range interaction. Ref. [16] shows that inside the 0 0 5 photonic bandgap, long tails of evanescent fields can in- the interaction range. The method introduced here pro- duce long-range interaction. Here we can also see that vides an intuitive understanding of underlying physics outside the photonic bandgap but near the band edge, that is somewhat buried in traditional treatments, and the propagating far fields have small isosurfaces, offer- we were able to use our method to help understand sev- ing a different mechanism for long-range interaction. A eral photonic systems from the existing literature. It hyperbolic material, where long-range interactions were also provides a general recipe to search for new pho- allowed along specific directions, was treated using the tonic systems that support long-range interactions. Green’s function method [32]. Using our graphic inter- This work was supported by the National Science pretation allows one to intuitively see that only special Foundation (NSF) through the University of Wiscon- directions allow long-range interactions (see the visual- sin Materials Research Science and Engineering Center ization in SM). DMR-1720415. L.Y. and Z.Y. were also supported by the To conclude, we show the deep connection between Defense Advanced Research Projects Agency (DARPA) the interaction range and the isosurface in momentum (DETECT program). L. Y. also acknowledges the finan- space. Both the size and shape of the isosurface affect cial support from NSF EFRI Award-1641109. [1] M. O. Scully and A. A. Svidzinsky, Science 325, 1510 and E. 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General resonant interaction theory between two quantum two-level systems [P(f (x)/ (x − x )) + iπδ(x − x )f (x)]dx, we can write 0 0 the radiative interaction as The Hamiltonian of quantum two-level systems Γ = Γ + iΓ Re Im (TLSs) in an arbitrary photonic environment is given by H = H + H + H . (3) ph tls int v (k)  3 ik·R = d k πδ (ω − ω )ρ e k 0 k,η  (7) They are explicitly written as [33] (~ = 1) X  † ik·R −ik·R e e H = ω aˆ aˆ ph k k,η ∗ k,η  + iP ρ + ρ , k,η k,η  k,η ω − ω ω + ω k 0 k 0 ˆ ˆ H = ω σ σ tls 0 i where P denotes the Cauchy principal value and the po- (4) i=1,2 larization factor is X X † ik·r H = ig (r ) σˆ + σˆ aˆ e + H.c. , ρ = µ · ǫ µ · ǫ . (8) int k,η i i k,η i k,η k,η k,η 1 2 16π ε v (k) 0 g i=1,2 k,η The real part of Γ is the cooperative decay rate and its where ω and σˆ (σˆ ) are the transition frequency and 0 i explicit expression is raising (lowering) operator of ith two-level system (TLS). ω and aˆ (aˆ ) are the frequency and cre- ik·R k k,η k,η Γ (ω ) = ρ (ω )e dS , (9) Re 0 k,η 0 k ation (annihilation) operator of photon. g (r ) = η ω (k) k,η i ω /2ε V µ · ǫ is the coupling between ith TLS and k 0 k,η where S is the isosurface of ω = ω in momentum ω (k) 0 photonic mode k. µ is the transition dipole moment of space and v (k) = |∇ ω | is the group velocity of mode g k k ith quantum TLS and ǫ is the polarization of photonic k,η k. The cooperative energy shift is mode k with polarization index η. Z " ∞ X The transition probability from initial to final states is 1 Γ (ω ) = P dω dS Im 0 k given by the Fermi’s Golden rule 2π/~|M | δ (E − E ), FI F I π 0 S η ω(k) where the transition matrix element M can describe FI   the resonant dipole-dipole interaction between two  ρ  k,η k,η   ik·R −ik·R   ×  e + e    TLSs. For the weak light-matter interaction, it can be (10) ω − ω ω + ω 0 0 written as the second-order form: hF|H |R ihR |H |Ii Z ! int α α int ∞ ∗ M = hF|H |Ii + + ··· . (5) 1 Γ (ω) Γ (ω) FI int Re Re E − E = P dω + . I R π ω − ω ω + ω 0 0 Here, |Ii = |e ,g ;0i and |Fi = |g ,e ;0i denote initial and 1 2 1 2 final states, where‘e’ and ‘g’ in the Dirac bracket notions B. Interaction in free space vacuum represent excited and ground states, respectively, and the number ‘0’ or ‘1’ is the photon number in the pho- In the free space, the dispersion relation is given by tonic environment. The intermediate state |R i has two (1) (2) options: |g ,g ;1 i with energy E = E + E + ~ω ω = c|k| = ck, (11) 1 2 k,η R g g k (1) (2) and |e ,e ;1 i with energy E = E + E + ~ω . e e 1 2 k,η R k where c is the speed of light. Because the isosur- ! ! (1) (2) face is isotropic, we have dS = k dΩ = The energy of the initial state is E = E + E + ~ω . k k I g g k ω(k) (1,2) R R 2π π Since two identical TLSs are considered, we have E − k dθ dϕsinϕ and group velocity v (k) = c. As- 0 0 (1,2) E = ~ω . Then, Eq. (3) can be explicitly given by suming k·R=ξ kR = cosϕkR, the cooperative decay rate g 0 in free space is written as (c = 1) X ik·R Z Z 3 X 2π π M = g (r ) g (r ) FI k,η 1 k,η 2 µ µ k 1 2 ω − ω k 0 Γ = dθ dϕ sinϕ k,η Re (6) 16π ε 0 0 0 η (12) −ik·R +g (r )g (r ) , ikRξ k,η 1 k,η 2 × µ · ǫ µ · ǫ e . k,η k,η ω + ω 1 2 k 0 7 The polarization sum rule is given by components of the group velocity. For simplicity, we as- sume the isosurface is isotropic, i.e. v = v = v = v. x y z (η) (η) ˆ ˆ The dispersion relation near the Weyl point is given by ǫ ǫ = δ −k k (13) 12 1 2 k,i k,j ω = ω ± v|q|. (18) q wp (η) ˆ ˆ with ǫ = µˆ · ǫ , δ = µˆ · µˆ , and k = k · µˆ . Then, we k,η 12 i i 1 2 i k,i Then, the cooperative decay rate is given by have µ µ ω 1 2 Z Z ik ·R 3 2π π Γ (ω) = e Re µ µ k 1 2 ikRξ ˆ ˆ k 16π ε v Γ = dθ dϕ δ − k k sinϕe Re 12 1 2 (19) 16π ε 0 0 0 iq·R Z Z × dS µˆ · ǫ µˆ · ǫ e . 2π π q k,η k,η 1 2 µ µ k 1 2 2 ikRξ ω(q) = −∇ δ + ∇ ∇ dθ dϕ sinϕe 12 1 2 16π ε 0 0 0 Here, we use a polarization based on plane waves. Al- µ µ k sinkR 1 2 = −∇ δ + ∇ ∇ 12 1 2 though this approximation is simplified, it is sufficient 4πε R  for calculating the scaling. At frequencies near the Weyl µ µ k sinkR 1 2  point, q ≪ k and the polarization factor term is a con-  ˆ ˆ = δ − R R 12 i j 4πε kR stant µ µ ω ∗ 1 2 coskR sinkR ρ = µˆ · ǫ µˆ · ǫ ≃ ρ. (20) ˆ ˆ  k,η k,η k,η 1 2 + δ − 3R R + , 2 12 i j  16π ε v 2 3 (kR) (kR) (14) Consequently, the cooperative decay rate is ˆ ˆ where R = µˆ · R. Also, with the relation in Eq. (13), 1,2 1,2 ik ·R iq·R Γ ≃ρe dS e Re q the cooperative energy shift is given by ω (q) (21) Z Z 2π π sinqR 2 ik ·R µ µ c 1 2 2 =4πq ρe . Γ = −∇ δ + ∇ ∇ dθ dϕ sinϕ Im 12 1 2 qR 16π ε 0 0 0 Z ! ikRξ −ikRξ k k e e The imaginary part of the radiative interaction (the × P dkk + . k − k k + k cooperative energy shift) is an integral over frequencies 0 0 0 from zero to infinity, as shown in Eq. (10). For Weyl (15) photonic crystals, the dispersion relation is quite dif- After calculating the Cauchy principal integral and in- ferent from Eq. (18) at frequencies far from the Weyl tegral over isosurface [? ], we have point [35]. Thus, we do not show an analytic estima- tion of the cooperative energy shift here, but numerical µ µ k cosk R 1 2 0 2 0 details will be discussed in Sec. III. Γ = −∇ δ + ∇ ∇ Im 12 1 2 4πε R µ µ k  cosk R 1 2  0 0  ˆ ˆ = δ − R R − II. THE SCALING OF THE INTERACTION STRENGTH  12 i j (16) 4πε k R 0 0 IN 3D AND 2D PHOTONIC ENVIRONMENTS sink R cosk R 0  ˆ ˆ  + δ − 3R R + . 12 i j The interaction range increases as the isosurface de- 2 3 (k R) (k R) 0 0 creases. However, at the same time, the density of states also decreases, particularly when the group ve- locity does not scale to zero. The consequence is that C. Interaction near Weyl points the interaction strength reduces while the range extends unless the group velocity scales in a way to cancel the ef- At first, we only consider a single Weyl point, as fect. Here below, we discuss how the strength scales in shown in Fig. 2c of the main text. The Hamiltonian for different photonic environments. the continuum around the Weyl point is given by [35] H (k) = v q σ , (17) A. Three-dimensional photonic media wp i i i i=x,y,z In the 3D case, the derivation starts from the defini- where σ are Pauli matrices and q = q , q , q = k − x,y,z x y z tion of the real part of Γ as shown in Eq. (9). Here, we k is the distance to the Weyl point in momentum space. will show the real part of radiative interaction in differ- The Weyl point is at k when q = 0. v are the x,y,z ent 3D photonic media. c x,y,z 8 1. 3D free space For the free space case, the dispersion relation is ω = c|k| = ck, where c is the speed of light. The radiative interaction strength is given by (see detailed in Sec. I B) k sinkR (3D)  ˆ ˆ Γ =B δ − R R + Re 12 1 2 c kR (22) coskR sinkR ˆ ˆ  δ − 3R R + , 12 1 2 2 3 (kR) (kR) (3D) where B = µ µ ω/4πε . In the long-distance regime, 1 2 0 we have k sinkR (3D) ˆ ˆ Γ (R > λ)  B δ − R R . (23) Re 12 1 2 c kR 2. 3D Weyl photonic environment The linear dispersion relation near a Weyl point is given by Eq. (18). Assuming µˆ · ǫ µˆ · ǫ ≈ 1, we k,η k,η 1 2 Figure S1. (a) Double-gyroid dielectric structure in have the radiative interaction (also see details in Sec. IC) a body-centered cubic unit cell with a set of basis vec- tors a = (−1/2,1/2,1/2)a, a = (1/2,−1/2,1/2)a, and a = µ µ ω 1 1 2 3 1 2 ik ·R iq·R Γ  e dS e Re q (1/2,1/2,−1/2)a. Four air spheres with a radius r = 0.07a 16π ε v 0 S ω(q) are located at (1/4,−1/8,1/2)a, (1/4,1/8,0)a, (5/8,0,1/4)a and (24) (3/8,1/2,1/4)a, respectively. The dielectric constant of solid q sinqR (3D) ik ·R = B e . gyroid structure is 13. (b) Dispersion relation on k = 0 plane. v qR The white curves correspond to isosurface in (c). (c) Isosur- face at ω = 0.5545 [2πc/a] normalized by the momentum in As the radius of the small isosurface q shrinks to zero, free space k . (d) The radiative interaction Γ in vacuum the interaction strength accordingly diminishes to zero. 0 Re and in the Weyl photonic crystal. The distance direction is However, if the group velocity v and the radius of iso- R = [−1,1,1] and the dipole orientation is µˆ = [0,0,1]. γ is 1,2 surface q are finite and small, the interaction can still be the spontaneous decay rate in free space and λ is the wave- much larger and longer than that in vacuum (see com- length in free space. parison in Fig. S1(d)). 3. 3D quadratic photonic environment The interaction strength reduces to zero when q = 0, which is similar to Weyl point. However, when the For a quadratic dispersion relation such as group velocity v and the radius of isosurface q are finite and small, the radiative interaction can be significantly 2 2 ω = β|k−k | + ω = βq + ω , (25) k c c c larger than that in vacuum. the group velocity is v = 2βq. Supposing k = 0, we g c have ZZ iq·R ω q ∗ Γ = dS µ · ǫ µ · ǫ e Re q k k 1 2 16π ε 2β 0 S ω(q) B. Two-dimensional photonic media q sinqR (3D)  ˆ ˆ =B δ − R R + 12 1 2 (26) 2β qR In 2D case, the real part of the radiative interaction is written as cosqR sinqR ˆ ˆ  δ − 3R R + . 12 1 2 2 3 (qR) (qR) ik·R ω ∗ e Γ (ω) = dℓ µ · ǫ µ · ǫ , (28) In the long-distance regime, the radiative interaction is Re k k k 1 2 8πε ℏ v (k) 0 ℓ g ω(k) written as q sinqR (3D) ˆ ˆ Γ (R > λ)  B δ − R R . (27) Re 12 1 2 where ℓ is the isofrequency contour. 2β qR ω(k) 9 Figure S2. The real part of radiative interaction Γ as Re a function of distance R with dipole orientations (a) µˆ = 1,2 (1, 0, 0), (b) (0, 1, 0), (c) (0, 0, 1), (d) (−1, 1, 1)/ 3, and (e) 6 at ω = 0.5512[2πc/a]. The wavelength is given (2, 1, 1)/ by λ = 2πc/ω. Γ is normalized by µ µ ω /16π ε . Re 1 2 0 0 Figure S3. The real part of radiative interaction Γ as a Re function of distance R with distance vectors (a) R = (0, 1, 0), √ √ √ (b) (0, 1, 1)/ 2, (c) (1, 0, 0), (d) (1, 0, 1)/ 2, (e) (1, 1, 0)/ 2, 1. 2D free space and (f) (1, 1, 1)/ 3 at ω = 0.5512[2πc/a]. The wavelength is given by λ = 2πc/ω. Γ is normalized by µ µ ω /16π ε . Re 1 2 0 0 We assume the dipole direction is normal to the 2D plane, the radiative decay rate is written as Then the group velocity is written as v = v. The radia- 2π tive interaction is given by µ µ ωk 1 2 ikRcosθ Γ (ω) = dθe Re 4ε c 0 0 2π (29) µ µ ωq 1 2 ik ·R iqRcosθ c q k Γ (ω) = e dθ e Re q (2D) = B J (kR), 4ε v 0 0 (31) (2D) ik ·R = B J (qR)e . where J (x) is the zero-order of the Bessel functions of (2D) the first kind and B = µ µ ω/4ε is a frequency- 1 2 0 The 2D Dirac points provide stronger and longer-range related parameter. interaction than Weyl points because of the reduced di- mensionality. 2. 2D linear dispersion near Dirac point 3. 2D quadratic dispersion near band edges. The dispersion relation near a Dirac point is written as If the 2D dispersion relation is quadratic such as 2 2 ω = v |k−k | + ω = vq + ω . (30) ω = β|k−k | + ω = βq + ω (32) k c c c k c c c 10 and the group velocity is v = 2βq. Then, 2π µ µ ω 1 2 ik ·R iqRcosθ c q Γ (ω) = e dθ e Re q 8ε ℏβ (33) (2D) ik ·R = B J (qR)e . 2β In this case, the strength do not be limited by small q. III. THE WEYL PHOTONIC CRYSTAL A. Numerical method and details The index of polarization η is replaced by the index of band n. If the transition frequency of TLSs ω is in the n th band, the cooperative decay rate is given by Γ (ω ) = dS Re 0 k 16π ε 0 S ω (k) (34) ik·R × µ · ǫ µ · ǫ . n ,k n ,k 1 0 2 0 v (k,n ) g 0 The cooperative energy shift is written as ik·R Figure S4. (a) & (b) The real part of integrand ρ e on the ◦ ◦ Z max " X ω hyperbolic isosurface with θ = 45 and 80 , respectively. The 1 ω dipole orientations are µˆ = (0, 1, 0). (c) Γ as a function of Γ (ω ) = P dω dS Re Im 0 n k 1,2 min 16π ε v 0 ω S n,k θ with a fixed distance R = 10λ. Red dots correspond to the n n ω (k) cases in (a) and (b). (d) Γ as a function of distance for the hy- Re ik·R ∗ e perbolic (red) and vacuum (gray) cases. γ is the spontaneous ×  µ · ǫ µ · ǫ n,k n,k  1 2 ω − ω decay rate in free space. The angle is fixed as θ = 45 . n 0 −ik·R ∗  + µ · ǫ µ · ǫ n,k n,k 1 2 ω + ω arbitrary dipole orientations. Although the the dipole n 0 orientations affect the oscillation patterns of Γ curves, Re the envelopes of all curves show negligible decay even Z max X ω 1 at 30 wavelengths. In addition, the amplitudes of Γ Re = P dω with different dipole orientations stay on a same order min n n of magnitudes. 1 1 ∗ To show the effect of distance vector, the dipole ori- × Γ (ω ) + Γ (ω ) . Re n n Re ω − ω ω + ω entations are fixed as µˆ = (0, 1, 0). Fig. S3 shows the n 0 n 0 1,2 real part of radiative interaction Γ as a function of dis- (35) Re tance R with different distance vectors. Similarly to the Numerically, we use the MPB software package [37] effect of dipole orientations, the variation of R only in- to calculate eigen-modes of the Weyl photonic crystal in fluences the oscillation patterns of Γ curves. However, Re Fig. 3 of the main text. We set the resolution in the unit the envelopes of all curves exhibit negligible decay at cell as 30 × 30 × 30. Then, the frequency of the Weyl 30 wavelengths. Similary, the variation of the first TLS points is ω = 0.55096[2πc/a], which falls in between location r in a unit cell does not affect the interaction wp 4th and 5th bands. range. B. Dipole orientations and spatial placements of TLSs IV. INTERACTIONS IN OTHER PHOTONIC ENVIRONMENTS The details of the gyroid photonic crystal are shown in Fig. S1. To demonstrate the effect of TLS dipole ori- Our theory is also applicable for understanding the entations, we fix the direction of the distance vector as dipole-dipole interactions in other photonic environ- R = (−1, 1, 1)/ 3. Fig. S2 shows the real part of ra- ments, such as near the bandedge of photonic crys- diative interaction Γ as a function of distance R with tals, index-near-zero materials, hyperbolic materials, Re 11 ◦ ◦ etc. Here, we describe the interaction in hyperbolic me- θ = 45 and 80 , respectively. Here, θ is the included ˆ ˆ dia as an especially interesting case. Its isosurface is in- angle between R and x-axis. When R is normal to the finitely large, which normally leads to a very short inter- isosurface, i.e. θ = 45 ,there is a thick red (positive) action range. This is indeed the case. However, because strip in the oscillating pattern and it results in a large of its unique shape, for the direction R that is normal integral value. If R is aligned with a different direc- ◦ ik·R to the isosurface, the interaction range can be very long. tion, say θ = 80 , the fast oscillation of e results in This can also be clearly seen in the illustration of the in- cancellation of the integral and thus a weak interaction tegrand on the surface as we will show now. The disper- strength. In Fig. S4(c), we plot the cooperative decay sion relation of hyperbolic materials can be described rate at R = 10λ as a function of θ. The left- and right- by hand red dots correspond to the case in Fig. S4 (a) and (b), respectively. Our results agree with the exact nu- 2 2 2 2 k + k x y k ω merical result in Ref. [32]. + = . (36) ε ε z x Furthermore, we also plot the radiative interaction Here, we choose the second type hyperbolic material Γ as a function of distance for the hyperbolic material Re with ε = −ε = 1. The isosurface and the real part of compared to the case in free space, as shown in Fig. S4 z x ik·R integrand ρ e are shown in Fig. S4 (a) and (b) with (d). It greatly agrees with the result in Ref. [32].

Journal

Condensed MatterarXiv (Cornell University)

Published: Jun 19, 2019

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