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Abstract
The chiral interaction between light and matter is mainly caused by the spin-momentum locking and makes the chiral quantum optics enter a vigorous development stage. Here, we explore the condition of the perfect chiral interaction between an atom possessing circular dipole and the surface plasmon polariton (SPP) mode. The realization of the perfect chiral interaction must satisfy the following two conditions at the same time. First, the SPP mode should possess the transverse circular polarization; and second, the atom decays mainly into the SPP mode, while the decay through other channel can be ignored. In this paper, we adopt a simple but effective structure to satisfy both of requirements, which is the sandwiched waveguide made of metal. We found that the transverse circular polarization of SPP mode might be achieved within the structure possessing multiple interfaces instead of the interface separating two semi-infinite materials. In our model, the decay rate into SPP mode overwhelms that through traveling wave, which provides higher quantum efficiency. What’s more, we found that only the symmetric TM-polarized SPP mode might get the transverse circular polarization. For the sandwiched structure containing metal, the existence of two SPP modes weakens the overall chiral interaction. However, the structure containing left-handed materials (LHMs), which can only support one symmetric TM-polarized SPP mode, can get the nearly perfect chiral interaction. We measure the chiral interaction through the decay rate, radiation field distribution and the unidirectional rate through the energy flux. Our work provides a reference for exploring the perfect chiral interaction in more complex structures and has potential and wide applicability to other optical processes.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
One of the central goals of quantum optics is to fully control the interaction between light and matter on quantum level [1–3]. It is known that the spontaneous emission of an atom depends not only on the property of the atom itself, but also on the environment surrounding it [4]. Atoms are usually quantified by transition frequency and dipole moment. There are two kinds of dipole moments; i.e., linear dipole and circular dipole. Previous studies mainly focused on linear dipoles, because the circular dipole can be seen as the superposition of two orthogonal linear dipoles, and the phase ±π/2 of circular dipole has no contribution to the spontaneous emission rate [5,6]. Such treatment can be extended to the researches of quantum interference [7,8] and Casimir-Polder force [9], in which rare attention was drawn on the different behavior between the linear dipole and the circular dipole.
However, the situation changed recently due to the discovery of chiral interaction between light and matter. The chiral interaction originates from the photonic spin Hall Effect (PSHE) [10,11], which refers to the fact that photons with opposite transverse spin (clockwise and counter clockwise) may move into different directions. It usually happens when the light is strongly transversely limited, which results in a nontrivial longitudinal component of the electric field, and the polarization of the electric field becomes elliptical. As a proof, considering a transversely limited wave propagating along the ±$x$-axis with complex wave vector $\boldsymbol{K} ={\pm} h{\boldsymbol{e}_x} + i\boldsymbol{\kappa }$, its electric field can be written as ${\boldsymbol{E}_ \pm }({\boldsymbol{r},t} )= ({{E_ \pm }(\boldsymbol{r} )/2} ){e^{ - i({\omega t \mp hx} )}} + c .c.$, where ${E_ \pm }$ is the complex amplitude vector, κ is the imagery wave vector in transverse $y$-$z$ plane, and c.c. refers to the complex conjugation. Assuming that the amplitude vector varies slowly along the $x$-axis. Due to the transverse condition $\boldsymbol{E} \cdot \boldsymbol{K} = 0$, the complex amplitude vector ${E_ \pm }$ produces an imaginary longitudinal component as ${E_{ {\pm} ,x}} ={\mp} {i / k}({{{\partial {E_{ {\pm} ,z}}} / {\partial z}} + {{\partial {E_{ {\pm} ,y}}} / {\partial y}}} )$, where the index “i” makes the electric field elliptically polarized and generate an unusual transverse spin. Such “spin” depends on the direction of propagation ${\pm} h{\boldsymbol{e}_x}$, and flips its sign when the propagating direction is opposite, which shows the spin-momentum locking property.
The chiral interaction provides a new method for quantum control, in which the coupling of light and atom depends on the propagating direction of light and the polarization of the atomic dipole moment. Several abnormal phenomena, such as the directional emission, scattering and absorption of photons, can be obtained [12]. This inherent spin-momentum locking has been demonstrated for evanescent waves in total internal reflection, surface state and optical fibre/waveguide [13]. Some experiments were also performed to achieve directional excitation of waveguide models, evansecent waves [14] and topological photonic crystals [3] recently. For instance, the polarization of the guided mode is nearly circular close to and outside the surface of an optical nanofiber, of which the direction is opposite on left and right part [1]; the one-way waveguide mode can be excited by a circularly polarized emitter near the interface of two single-negative metamaterials, where the propagation direction was determined by the chirality of the emitter [10]; in the near-field of the dipole source near the dielectric or plasma waveguide, the unidirectional excitation of the waveguide mode can both be realized [15,16]; the circularly polarized incident light can excite the electromagnetic surface wave near the metal-dielectric interface unidirectionally [17]; spin-dependent directional emission can be achieved even in dielectric nanobeam waveguides with the symmetric structure and non-chiral dielectric materials [18]; in topological photonic metamaterials, the propagating directions of the topological interfacial electromagnetic waves emitted by a linearly polarized dipole source depend on the sign of orbital angular momentum [19]; the efficient near-field photonic directional routing is experimentally achieved in waveguides composed of two kinds of single-negative metamaterials [20].
However, except for Ref. [12], most of the above works just focus on the phenomena of unidirectional emission, and rarely explore how to achieve the perfect chiral interaction. As the atomic dipole moment can be rigorous circularly polarized according to Zeeman splitting, the effects of the chiral interaction are determined by the ellipticity of the transversely limited mode and the quantum efficiency coupled into such mode. In other words, only the mode with transverse circular polarization can achieve the perfect chiral interaction between the atom with this mode, while the quantum efficiency coupled in such mode refers to the ratio of decay rate into such mode to all modes.
The chiral interaction was initially demonstrated by dielectric nanofibers [1], which is very enlightening. On the one hand, the fundamental waveguide mode of nanofibers is not exactly circularly polarized on the surface of it, on the other hand, the quantum efficiency of atoms coupling into this fundamental mode is not high, which were confirmed by P. Solano et al. in 2019 [21]. They analyzed the modification of the atomic decay rate of 87Rb in the vicinity of an optical nanofiber (∼500-nm-diameter) both theoretically and experimentally, and calculated theoretically the coupling efficiency into the guided mode which is about 0.2 when atom is at the surface of nanofiber with $n = 1.45367$ at $\lambda = 780.241\textrm{ nm}$ [21]. Therefore, to improve the effect of chiral interaction, most of the previous works used more complex electromagnetic structures [12,15–19,22], the ellipticity of the mode can only be checked in numerical simulation, but cannot be discussed with analytic solution. For example, in the glide-plane photonic crystal waveguide; i.e., a photonic-crystal waveguide that breaks mirror symmetry, when the magnetic field is approximately 1 T, the directionality is about 90% and the coupling efficiency can reach 98% experimentally [12]; with the similar glide-plane photonic crystal waveguide, which is composed of a photonic crystal waveguide where two top and bottom mirrors are shifted by a half period, when there’s no external magnetic field, the coupling efficiency can be around 30% and the directionality gets as high as 0.35 experimentally [22]. Therefore, this prevented finding the general law to achieve the transverse circularly polarized modes and high quantum efficiency.
In this paper, we focus on a simple sandwiched structure, in which two semi-infinite metal separated by vacuum with a finite thickness. And the atom with circular dipole is embedded in vacuum. The surface plasmon polariton (SPP) mode of such structure can be analyzed analytically. We found that only the symmetric SPP mode can meet the transverse circular polarization, and the position of such circular polarization is related to permittivity, permeability and vacuum width. In addition, such structure is similar to the Fabry-Pérot cavity, the decay rate of atom into the travelling mode can be totally inhibited for narrow thickness of vacuum layer. In Sec. 2, we analyze the SPP mode in a sandwiched structure and give the general law to get the transverse circular polarization. In Sec. 3, we check the chiral interaction through the spontaneous emission field of an atom with the circular dipole. In Sec. 4, we replace the metal by left-handed materials, and find that it can support only one symmetric SPP mode and can get the perfect chiral interaction. We draw the conclusion in Sec. 5.
2. Model and TM SPP mode with circular polarization
Unlike the complicated models in above references, we consider the surface plasmon polariton (SPP) mode in a simple sandwiched structure shown in Fig. 1. The two semi-infinite regions ($z \le 0$ and $z > d$) are homogeneous metal with the permittivity ${\varepsilon _\textrm{L}} = {\varepsilon _R} < 0$ and permeability ${\mu _\textrm{L}} = {\mu _R} = 1$. The middle region with width d is a vacuum with ${\varepsilon _M} = {\mu _M} = 1$. Although the model requires the semi-infinite metal to be infinitely extended in the x-y plane, in future experiments, as long as the scale of x and y direction is more than 10 times of the thickness d, the experimental model is consistent with the theoretical one. This half open structure also facilitates the manipulation of atoms.
Fig. 1. Sketch of the sandwiched structure. Two semi-infinite metals separated by a vacuum, and an atom is placed at ${\boldsymbol{r}_A} = ({0,0,{z_A}} )$ in the vacuum denoted by a black solid circle.
Such sandwiched structure only supports the TM-polarized SPP modes. We take the magnetic field along $y$-axis and the electric field in the $x$-$z$ plane as example. The spatial distribution of TM-polarized SPP mode can be written as follows.
Here h is the $x$-component of wave vector, and is named as the propagation constant of SPP mode. Meanwhile, ${k_L}$, ${k_M}$ and ${k_R}$ are the imaginary part of $z$-component of wave vectors in the left, middle and right regions, respectively. h and ${k_j}$ satisfy the dispersion relation
${k_0} = {\omega / c}$ is the wave number in vacuum.Actually, the above equations were shown in the text book and well known, but few attentions were drawn on the chiral interaction before. From Eq. (2), the electric field propagates along the $x$-axis, marked by h, and is localized near the interfaces due to ${e^{ {\pm} {k_j}z}}$. Its polarization is in the $z$-$x$ plane, and the “i” of the $x$-component makes the polarization elliptical, which created a transverse spin along $y$-axis. If h flips its sign, the transverse spin reverses, which is the inherent spin-momentum locking.
The crux of the problem is that the polarization of electric field is elliptical in general rather than circular. For example, the electric fields in the left and right regions are proportional to $({i{k_{L/R}}{\boldsymbol{e}_x} + h{\boldsymbol{e}_z}} )$, but h never equals to ${k_L}$ and ${k_R}$ according to Eq. (3). Therefore, the surface modes near a single interface separating two semi-infinite materials never possess circular polarization; meanwhile, the surface modes outside the structure cannot produce circular polarization.
However, for our sandwiched structure, the electric field in middle region is complex. We can rewrite it in the following formation
There is a symmetric and an antisymmetric mode in our model as the result of the interference between the surface modes of two interfaces. The propagation constant h of the symmetric and the antisymmetric mode should satisfy the following dispersion relations [23]
For the dispersion relation, it is easy to find that the propagation constant of symmetric mode is obtained in the range ${k_0} < h < {k_0}\sqrt {{{{\varepsilon _L}} / {({{\varepsilon_L} + {\varepsilon_M}} )}}} $, while that of antisymmetric mode is in the range $h > {k_0}\sqrt {{{{\varepsilon _L}} / {({{\varepsilon_L} + {\varepsilon_M}} )}}} $.
We then take the position on the right of the vacuum region; i.e., d/2 < z < d, the circular polarization requires α=β. From the expressions of α and β in Eq. (5), the equation α=β can be transformed into
As a result, the symmetric SPP mode may possess the circular polarization but the antisymmetric mode does not.
To judge the elliptical polarizability, the ratio β⁄α is considered. If β⁄α=±1, it refers to circular polarization; other values of β⁄α refer to elliptical polarization. There are two extreme cases: β⁄α=0 refers to z linear polarization, while β⁄α→±∞ refers to x linear polarization.
The structure with ${\varepsilon _{L/R}} ={-} 2.31$, ${\mu _{L/R}} = 1$, and d=0.5λ is taken as an example. This metal waveguide supports two TM-polarized SPP modes with propagation constants ${h_1} = 1.1835{k_0}$ (symmetric mode) and ${h_2} = 1.4007{k_0}$ (antisymmetric mode). The elliptical polarizability β⁄α as a function of the position in the vacuum region is shown in Fig. 2, in which the solid curve refers to the symmetric mode, while the dashed one refers to the asymmetric mode. It is clear that |β⁄α| is always smaller than 1 for the antisymmetric mode, while |β⁄α| can be 1 at the position z=0.1λ and 0.4λ for symmetric modes, which confirms the existence of circular polarization. At the center of the vacuum; i.e., z=0.25λ, there are only linear polarization for both symmetric and antisymmetric modes.
Fig. 2. Elliptical polarizability rate as a function of position z in the vacuum region for the sandwiched structure. The structure parameters are ${\varepsilon _{L/R}} ={-} 2.31$, ${\mu _{L/R}} = 1$, and d=0.5λ. The solid (dashed) curve refers to the symmetric (antisymmetric) SPP mode.
In fact, the symmetric SPP mode is the necessary but insufficient conditions to achieve circular polarization. The existence of circular polarization is related to the permittivity of metal, the thickness of the vacuum and the position of concern. We fix the relative position of concern at z = d/5, and show the combination of parameters $({{\varepsilon_{L/R}},d} )$ that can obtain circular polarization in Fig. 3(a). And then, we fix the relative position of concern at the interface; i.e., z=0, the corresponding parameters $({{\varepsilon_{L/R}},d} )$ are shown in Fig. 3(b).
Fig. 3. The parameter combination of ${\varepsilon _{L/R}}$ and d to get the circular polarization for TM symmetric SPP mode (a) at fixed relative position z = d/5 and (b) at the interface z=0.
From Fig. 3(a), to get the circular polarization at the relative position z = d/5 for TM symmetric SPP mode, the region of metal permittivity is limited from −1.3 to −3.5, while the vacuum thickness d is limited in a narrow interval from 0.41λ to 0.53λ which is about half wavelength. We also show the parameter combination of ${\varepsilon _{L/R}}$ and d to get the circular polarization for TM symmetric SPP mode at the interface z=0 in Fig. 3(b), in which the region of permittivity narrows down to −1.1–−1.6 and the corresponding thickness d also shrinks to an interval of 0.24λ to 0.32λ.
When the thickness of the vacuum layer is very large, the circular polarization will disappear. This is because the surface mode of this case is similar to that of single semi-infinite metal, which cannot achieve circular polarization as mentioned above.
When the parameters are out of curve in Fig. 3, there is no circular polarization at these relative positions. We take the structure with ${\varepsilon _{L/R}} ={-} 1.3$, ${\mu _{L/R}} = 1$, and d=0.273λ as example, shown in Fig. 4. In this case, the elliptical polarizability β⁄α for both symmetric and antisymmetric modes deviates from ±1. For all positions, the absolute elliptical polarizability |β⁄α| of antisymmetric mode is always smaller than 1, while that of symmetric mode is larger than 1.
Fig. 4. Elliptical polarizability as a function of position z in the vacuum region for the sandwiched structure. The structure parameters are ${\varepsilon _{L/R}} ={-} 1.3$, ${\mu _{L/R}} = 1$, and d=0.273λ. The solid (dashed) curve refers to the symmetric (antisymmetric) SPP mode.
3. Chiral interaction between an atom and SPP modes
As the degree of perfection of chiral interaction depends on both the circular polarization of SPP mode and the quantum efficiency coupled in such mode, we will discuss the spontaneous emission of an atom with the circular dipole embedded in such structure in detail. The quantum efficiency can be measured by the decay rate, while the unidirectional emission can be judged by the spatial distribution of radiative field.
The circular dipoles can be obtained by applying a magnetic field to the atom according to Zeeman splitting. So, it is possible to get a two-level atom with a linear or circular dipole as will. Here we define $|e \rangle $ and $|g \rangle $ as the atomic excited and ground states, respectively, and $\boldsymbol{P} = \left\langle e \right|\hat{\boldsymbol{P}}|g \rangle $ is the transition dipole moment. Such a two-level atom located in the vacuum region at the position ${\boldsymbol{r}_A} = ({0,0,{z_a}} )$ ($0 < {z_a} < d$) is shown in Fig. 1.
When the atom is prepared in the excited state initially, it will decay to the ground state spontaneously due to the vacuum fluctuation, and emit radiation field. Under the Markovian approximation [4], the probability amplitude of atom in the excited state C(t) decays exponentially as
Following the discussion in above section, we check the structure with ${\varepsilon _{L/R}} ={-} 2.31$, ${\mu _{L/R}} = 1$, and d=0.5λ, which produces circular polarization at z=0.1λ/0.4λ for the symmetric SPP mode according to Fig. 3.
It should be noticed that the atom with the linear dipole of $\boldsymbol{P} = P{{({{\boldsymbol{e}_x} + {\boldsymbol{e}_z}} )} / {\sqrt 2 }}$ has the same decay rate to that with circular dipole $\boldsymbol{P} = P{{({ \pm i{\boldsymbol{e}_x} + {\boldsymbol{e}_z}} )} / {\sqrt 2 }}$, because their decay rates can be rewritten into the same formula $\gamma = ({{{\omega_0^2} / {\hbar {\varepsilon_0}{c^2}}}} ){P^2}[{{G_{xx}}({{z_A},{z_A},{\omega_0}} )+ {G_{zz}}({{z_A},{z_A},{\omega_0}} )} ]$ according to the definition of decay rate in Eq. (9). This is the reason why rare attention was paid on the atomic decay rate of circular dipole.
To check the quantum efficiency, we plot the atomic decay rate as a function of atomic position in Fig. 5. Since the structure considered here is relatively simple, it is easy to distinguish the decay rates through different modes. Besides the decay through SPP modes, there is also decay through travelling wave, which does not possess circular polarization, namely, does not contribute to the chiral interaction. The quantum efficiency is defined by ${\beta _r} = {{{\gamma _{SPP }}} / {{\gamma _{total }}}}$ here, and its relationship with atom position is shown in Fig. 5(b). Due to the large thickness 0.5λ, symmetric and antisymmetric modes are almost degenerate, so their decay rates are almost the same. Meanwhile, the decay rate through the traveling wave is tiny, and can be omitted compared with the decay rate through SPP mode. The reason is the structure can be seen as a cavity with half wavelength length, and no traveling modes exist [8]. It means that the quantum efficiency is near 1 (the specific value is about 95% at ${z_A} = 0.1\lambda $), higher than the case of dielectric nanofiber. Therefore, we can only discuss the contribution of the surface modes to the emitted electric field here.
Fig. 5. (a) Spontaneous decay rate $\gamma $ as a function of atom position ${z_A}$. (b) Quantum efficiency ${\beta _\textrm{r}}$ as a function of atom position ${z_A}$. The sandwiched structure possesses parameters with ${\varepsilon _{L/R}} ={-} 2.31$, ${\mu _{L/R}} = 1$, and d=0.5λ. The atomic dipole moment is circularly polarized with $\boldsymbol{P} = P{{({ \pm i{\boldsymbol{e}_x} + {\boldsymbol{e}_z}} )} / {\sqrt 2 }}$. ${\gamma _0} = {{{P^2}\omega _0^3} / {(3\pi {\varepsilon _2}}}\hbar {c^3})$ is the decay rate in free space.
Considering the spontaneous radiation field, i.e., Eq. (11), there were obvious differences for the linear dipole and the circular dipole. We check the case when the atom is placed at the position ${z_A} = 0.1\lambda $ in which shows the circular polarization for TM symmetric SPP mode.
Figure 6(a) illustrates the distribution of symmetric SPP field intensity |E| excited by the atom at ${z_A} = 0.1\lambda $ with $\boldsymbol{P} = P{{({i{\boldsymbol{e}_x} + {\boldsymbol{e}_z}} )} / {\sqrt 2 }}$. It should be noticed that the electric field is normalized by ${E_0} = {{P\omega _0^3|{{C_u}(t)} |} / {(8\pi {c^3})}}$. In this case, the excited symmetric SPP field shows the unidirectional propagation along -x-axis near the interface. The electric fields on both left and right interfaces are excited simultaneously. If we change the dipole into left-handed circular polarized; i.e., $\boldsymbol{P} = P{{({ - i{\boldsymbol{e}_x} + {\boldsymbol{e}_z}} )} / {\sqrt 2 }}$, the propagating direction of excited field reverses to x-axis accordingly, showing the spin-momentum locking and the chiral interaction. For the atom with linearly polarized dipole, the emitted symmetric SPP electric field loses the unidirectional property as displayed in Fig. 6(b), where the excited electric field simultaneously emits along both + x-axis and -x-axis. Comparing Figs. 6(a) with 6(b), the maximum intensity of the electric field excited by the circularly polarized dipole is stronger than that excited by the linearly polarized dipole. This is because the unidirectional emission focuses more energy in one direction. Notice that though the structure considered is one dimensional, the radiative field is calculated in three dimensions. We just show the distribution of field in x-z plane, which clearly illustrate the unidirectional emission.
Fig. 6. Distribution of the symmetric SPP electric field $|{\boldsymbol {E}} |$ in x-z plane generated by an atom at ${z_A} = 0.1\lambda $ with diploe moment (a) $\boldsymbol{P} = P{{({i{\boldsymbol{e}_x} + {\boldsymbol{e}_z}} )} / {\sqrt 2 }}$ and (b) $\boldsymbol{P} = P{{({{\boldsymbol{e}_x} + {\boldsymbol{e}_z}} )} / {\sqrt 2 }}$. Distribution of total electric field $|{{\boldsymbol{E}_{Total}}} |$ in x-z plane generated by (c) an atom with $\boldsymbol{P} = P{{({ - i{\boldsymbol{e}_x} + {\boldsymbol{e}_z}} )} / {\sqrt 2 }}$ and (d) a classical circular antenna $\boldsymbol{P} = P{{({ - i{\boldsymbol{e}_x} + {\boldsymbol{e}_z}} )} / {\sqrt 2 }}$ at ${z_A} = 0.1\lambda $. The sandwiched structure possesses parameters with ${\varepsilon _{L/R}} ={-} 2.31$, ${\mu _{L/R}} = 1$, and d=0.5λ. The black circle in figures means the atom.
In Figs. 6(a) and 6(b), we merely consider the symmetric SPP field, but the atomic spontaneous emission field includes the contribution of all modes actually. For the sandwiched structure containing metal, there are always both symmetric and antisymmetric TM SPP modes. According to Fig. 2, the symmetric SPP mode possesses the circular polarization at $z = 0.1\lambda $, but the antisymmetric SPP mode does not, so the unidirectionality of total field should be weakened. We give the distribution of total electric field $|{{\boldsymbol{E}_{Total}}} |$ excited by an atom with $\boldsymbol{P} = P{{({ - i{\boldsymbol{e}_x} + {\boldsymbol{e}_z}} )} / {\sqrt 2 }}$ at ${z_A} = 0.1\lambda $ in Fig. 6(c). The contribution of traveling wave has also been included, but this contribution is too tiny to be noticed. It shows that the total emitted field also has the phenomenon of unidirectional emission, but it seems only left side of the electric field is stimulated. This is because the destructive interference between symmetric and antisymmetric modes on right interface makes the energy concentrate on the left interface. In addition, when the atom is placed at ${z_A} = 0.4\lambda $, it excites the right side of the electric field. Still, when the dipole moment changes its circular direction, it emits electric field into opposite directions. Hence the position and polarization of the atom can be inferred from the distribution of the total electric field, and verse vise. Although the excited electric field in Fig. 6(c) is still unidirectional, the degree of unidirectionality is weaker than that in Fig. 6(a). We also perform the simulation of a classical circular antenna by commercial electromagnetic software (Comsol) in between two semi-infinite metal which has the same relative size with that in Fig. 6(c). Comparing Figs. 6(d) with 6(c), it can be seen that the main features of the two pictures are the same. This fully proves that our calculation is reasonable and credible. Since the numerical simulation can only get the total field and cannot distinguish the contribution of different electromagnetic modes, our analysis is more instructive. However, there is an obvious dividing line in the x=0 plane in Figs. 6(a) and 6(c), which is due to the point dipole model used in the calculation. In the actual electromagnetic environment, the antenna has a limited size, shown in Fig. 6(d) and such dividing line disappears. It should be noticed that, in the simulation of Fig. 6(d), we choose the wavelength as λ=1 µm.
In order to measure the degree of unidirectionality, we define the unidirectional rate U as
U=1 indicates the perfect unidirectional emission, that is, the perfect chiral interaction; while U=0 declares the disappearance of chiral interaction. In the following, we set ${x_a} = 0.2\lambda $.
If we just discuss the contribution of the symmetric SPP mode; i.e., the case of Fig. 6(a), the unidirectional rate as a function of the position of atom with circular dipole is given by the solid line in Fig. 7. When ${z_A} = 0.1\lambda /0.4\lambda $, the case of Fig. 6(a), the unidirectional rate of the symmetric SPP field is the largest, which is U=0.992, that is, the circular polarization of SPP mode leads to the perfect chiral interaction and unidirectional emission. However, the unidirectional rate for the atom at the middle position ${z_A} = 0.25\lambda $ is 0, which shows the bidirectional emission phenomenon. Besides, the atom deviating from the center shows unidirectional emission in varying degrees.
Fig. 7. The unidirectional rate of the symmetric mode electric field (solid curve) and the total electric field (dashed curve) as function of the position of atom with $\boldsymbol{P} = P{{({ - i{\boldsymbol{e}_x} + {\boldsymbol{e}_z}} )} / {\sqrt 2 }}$. The sandwiched structure possesses parameters with ${\varepsilon _{L/R}} ={-} 2.31$, ${\mu _{L/R}} = 1$, and d=0.5λ.
Then we consider the contribution of all modes, the unidirectional rate of the total field as a function of the position of atom with circular dipole is given at the dashed curve in Fig. 7. The unidirectional rate of total emission is lower than that of the symmetric mode electric field, namely, antisymmetric modes destroy the chirality and the unidirectionality of the whole. For example, when ${z_A} = 0.1\lambda /0.4\lambda $, the case of Fig. 6(c), the unidirectional rate of the total field decreases to U=0.867.
However, the antisymmetric mode does not always weaken the chiral interaction. For the case of the symmetric mode without circular polarization, just as Fig. 4, the elliptical polarizability |β⁄α| of the symmetric mode is always greater than 1 while that of the antisymmetric modes is less than 1. Compared with the situation that only symmetric SPP mode being considered, the superposing of both symmetric and antisymmetric emitted field will improve the chiral interaction. For example, metal material in Fig. 4 supports two TM-polarized SPP modes with propagation constants ${h_1} = 1.0096{k_0}$ (symmetric mode) and ${h_2} = 2.3332{k_0}$ (antisymmetric mode). Since the symmetric mode does not produce optimum circular polarization, we locate the atom at position ${z_A} = 0.01\lambda $. In this case, the total electric field excited by a left-handed circularly polarized dipole is almost unidirectional as shown in Fig. 8(a). Because of large difference between the propagation constants of the symmetric and antisymmetric modes, contributions of these two modes are not equal, so the electric field cannot be completely offset on the right side. Besides, Fig. 8(b) shows that the unidirectional rate of total electric field is larger than that of symmetric mode field. Therefore, the superposition of the symmetric and the antisymmetric SPP field can improve the degree of unidirectionality in the case that the symmetric mode doesn’t have optimum transverse circular polarization, but the maximum unidirectional rate is still smaller than the case of Fig. 7.
Fig. 8. (a) Distribution of the total electric field intensity $|{{\boldsymbol{E}_{Total}}} |$ in x-z plane generated by an atom with $\boldsymbol{P} = P{{({ - i{\boldsymbol{e}_x} + {\boldsymbol{e}_z}} )} / {\sqrt 2 }}$ at ${z_A} = 0.01\lambda $. The black circle in figures means the atom. (b) The unidirectional rate of the symmetric mode electric field (solid curve) and the total electric field (dashed curve) as function of the position of atom with $\boldsymbol{P} = P{{({ - i{\boldsymbol{e}_x} + {\boldsymbol{e}_z}} )} / {\sqrt 2 }}$. The sandwiched structure possesses parameters with ${\varepsilon _{L/R}} ={-} 1.3$, ${\mu _{L/R}} = 1$, and d=0.273λ.
4. Chiral interaction in the structure containing LHMs
In above section, there are always two TM SPP modes in the sandwiched structure containing metal. The circular polarization condition can only be satisfied by symmetric SPP mode, while the antisymmetric SPP mode cannot. Also, this limits the effect of chiral interaction when considering all modes. A natural generalization is that whether we can find a special material in which only the TM symmetry SPP exists.
Left-handed materials (LHMs) were first proposed by Veselago in 1968 [25]. It refers to the materials with negative effective permittivity, permeability and refractive index at a certain frequency. In 2001, LHMs was experimentally proven in the microwave range using arrays of metallic split-ring resonators by Shelby et al. [26]. Up to now, LHMs had been fabricated in the infrared [27] and visible light frequencies [28]. Different from metal, it had been proved that some LHM layer can support only one SPP mode [29]. In the following, we perform the detail analysis of SPP modes in the sandwiched structure of two infinite LHM separated by a vacuum. The scheme is the same with that in Fig. 1, except that both ${\varepsilon _{L/R}}$ and ${\mu _{L/R}}$ are negative here.
4.1 LHMs with ${n_{L/R}} < - 1$ (${n_{L/R}} = \sqrt {{\varepsilon _{L/R}}} \sqrt {{\mu _{L/R}}} $)
When ${\varepsilon _{L/R}} < - 1$, the two SPP modes are both TE-polarized for large d. If the width d is small, there is a TE-polarized SPP mode and a TM-polarized antisymmetric SPP mode. So there does not have any position with circular polarization. Therefore, in the case of ${\varepsilon _{L/R}} < - 1$, the chiral interaction and the unidirectional emission are not good.
When $- 1 < {\varepsilon _{L/R}} < 0$, there are two TM SPP modes. We choose the LHM with ${\varepsilon _{L/R}} ={-} 0.67$ and ${\mu _{L/R}} ={-} 2$ as an example and set the width of vacuum is as 0.69λ. Such sandwiched structure supports two TM-polarized SPP modes with propagation constants ${h_1} = 1.2971{k_0}$ (symmetric mode) and ${h_2} = 1.2452{k_0}$ (antisymmetric mode), and the symmetric mode possesses left-handed (right-handed) circular polarization at z=0.2λ (0.49λ), but the antisymmetric mode does not. Therefore, in the case of $- 1 < {\varepsilon _{L/R}} < 0$, the total electric field has the quasi-unidirectional emission phenomenon, which is similar to the case of sandwiched structure containing metal.
4.2 LHMs with $- 1 < {n_{L/R}} < 0$
Next, we discuss the case of $- 1 < {n_{L/R}} < 0$. The permittivity and permeability of LHM are chosen as ${\varepsilon _{L/R}} ={-} 0.96$ and ${\mu _{L/R}} ={-} 0.98$ for example. Its dispersion relationship [Fig. 9(a)] shows that, if the width of the vacuum d is large, there are two SPP modes (one TE and one TM mode); if the width is small, there is only one TM mode. We focus on the case of only one TM mode, that is the thickness region is fall into d∈[0.052λ, 0.060λ]. We plot the elliptical polarizability β⁄α at the interface z=0 of the TM SPP mode as a function of thickness d in Fig. 9(b). It shows that the elliptical polarizability tends to −1 within the thickness region.
Fig. 9. (a) Dispersion relation of the LHM; i.e., propagation constant h varies with d, in which the solid line indicates TM-polarized mode and the dashed line indicates TE-polarized mode. No transverse circular polarization case with dipole $\boldsymbol{P} = P{{({ - i{\boldsymbol{e}_x} + {\boldsymbol{e}_z}} )} / {\sqrt 2 }}$. (b) Elliptical polarizability of the symmetric TM-polarized mode at z=0 varies with d. The sandwiched structure possesses the parameters of ${\varepsilon _{L/R}} ={-} 0.96$, ${\mu _{L/R}} ={-} 0.98$, and d=0.06λ.
To check the effect of chiral interaction, we take the case of d=0.06λ as example. It supports only one symmetric TM-polarized mode with ${h_1} = 10.3538{k_0}$. In Fig. 10(a), the decay rate of atom with $\boldsymbol{P} = P{{({ - i{\boldsymbol{e}_x} + {\boldsymbol{e}_z}} )} / {\sqrt 2 }}$ was calculated as a function of position ${z_A}$. Actually, the decay rate through traveling wave is about $0.667{\gamma _0}$ for all position as shown in the inset, which is close to the decay rate in free space. However, the decay rate through SPP mode is much larger than that through traveling wave, which leads to nearly unitary quantum efficiency (the actual value of ${\beta _r}$ is around 99.9%). This is because the LHM with ${\varepsilon _{L/R}} = {\mu _{L/R}} ={-} 1$ is a special case which supports infinite number of SPP mode [30]. Here we take ${\varepsilon _{L/R}} ={-} 0.96$, ${\mu _{L/R}} ={-} 0.98$ will tend to the case of ${\varepsilon _{L/R}} = {\mu _{L/R}} ={-} 1$ and leads to huge decay rate through SPP mode. Therefore, the structure with the present index possesses high quantum efficiency.
Fig. 10. (a) Spontaneous decay rate γ of an atom with $\boldsymbol{P} = P{{({ - i{\boldsymbol{e}_x} + {\boldsymbol{e}_z}} )} / {\sqrt 2 }}$ as a function of atom position ${z_A}$. (b) The unidirectional rate U of the electric field as a function of the atomic position ${z_A}$. Distribution of total electric field intensity $|{{\boldsymbol{E}_{Total}}} |$ in x-z plane generated by an atom with dipole $\boldsymbol{P} = P{{({ - i{\boldsymbol{e}_x} + {\boldsymbol{e}_z}} )} / {\sqrt 2 }}$ at (c) ${z_A} = 0$ and (d) ${z_A} = 0.012\lambda $. The common parameters are d=0.06λ, ${\varepsilon _{L/R}} ={-} 0.96$, ${\mu _{L/R}} ={-} 0.98$. The black circle in figures means the atom.
In Fig. 10(c), the distribution of total electric field $|{{\boldsymbol{E}_{Total}}} |$ excited by the atom at ${z_A} = 0$ with $\boldsymbol{P} = P{{({ - i{\boldsymbol{e}_x} + {\boldsymbol{e}_z}} )} / {\sqrt 2 }}$ is given. The electric field generates unidirectional emission phenomenon, and the left and right sides are simultaneously excited and symmetrical, which is because there is only one symmetric mode and the elliptical polarizability on the boundaries is closest to −1. As revealed in Fig. 10(b), we find that the unidirectional rate near boundaries is not monotonous, the unidirectional rate reaches the maximum value at ${z_A} = 0.012\lambda $, rather than at ${z_A} = 0$. This is because the electric field generated by x and z components of the dipole completely destructive interference in the opposite direction of the total field at this position. We plot the electric field excited by the atom at ${z_A} = 0.012\lambda $ in Fig. 10(d). Comparing Fig. 10(d) with 10(c), the total electric field shows perfect unidirectional emission. Combined with the high quantum efficiency, in the case of $- 1 < {\varepsilon _{L/R}} < 0$, $- 1 < {\mu _{L/R}} < 0$, the sandwiched structure with the small thickness can overcome the mutual interference between the two SPP modes in metal case, and achieve nearly perfect chiral interaction.
5. Conclusion
In this paper, we take a simple structure to explore the condition of perfect chiral interaction. An atom with circularly polarized dipole in a sandwiched waveguide is analyzed in detail. Different from usual sandwiched structure, we take the middle layer as vacuum and the outside semi-infinite materials as metal or left-handed materials. When the atom is placed in such structure, the decay rate through SPP mode can overwhelm that through traveling wave, and provide high quantum efficiency. When the waveguide is made of metal, its symmetric SPP modes may possess the transverse circular polarization at certain position, but antisymmetric SPP mode cannot. Therefore, for the atom at the optimum position with circular dipole, its excited symmetric SPP field gets the perfect unidirectional rate, but the antisymmetric SPP field weakens the unidirectional rate. When the waveguide is made of LHMs with $- 1 < {\varepsilon _{L/R}} < 0$ and $- 1 < {\mu _{L/R}} < 0$, it may support only one TM symmetric SPP mode, and may get the nearly transverse circular polarization at the interface. Therefore, it can realize the nearly perfect chiral interaction and unidirectional emission. The dissipation of materials obviously affects the results, but a small dissipation will only make some corrections to the position of transverse circular polarization, and will not change our results qualitatively. In addition, the degree of unidirectionality of SPP emission is adjustable, and the position and polarization state of atoms can be known by the distribution of total electric field. These effects can be used to orientate SPP emission and absorption, and to study the emission and scattering processes near the chiral interface.
With the development of the fabrication technology of artificial microstructure materials, people have been able to prepare effective media with various permittivity and permeability from microwave to infrared. Corresponding to these bands, there are quantum dots, Rydberg atoms, NV color centers and other quantum emitters. Therefore, it is possible to implement the parameters described in our article.
However, the significance of our work is not to let experiments verify our results in the future. Noticed that we just take a simple structure as example to analyze the requirement to get the perfect chiral interaction. We find that the atom in a structure possessing multiple interfaces is necessary to reduce the number of SPP modes and inhibit the decay through traveling mode. Therefore, some more complex structures are worthy of further analysis.
Finally, this fundamental physics has potential and wide applicability to other optical processes. Our work is an important step in the realization of cascaded quantum optical networks. Chiral interaction between photons and atom can be assembled into complex integrated circuits by controlling the flow of light in the model, which has broad application prospects in quantum information processing and the construction of complex quantum networks. In addition, the chiral interface enables the realization of a special non-equilibrium quantum multibody photonic system and a one-dimensional emitter.
Funding
National Natural Science Foundation of China (11874287, 61975154); Science and Technology Commission of Shanghai Municipality (18JC1410900); Shanghai Education Commission Foundation.
Acknowledgments
We sincerely thank Prof. Hong Chen for constructive suggestion, and Prof. Jie Ren and Dr. Yang Long for their help in simulation of electromagnetic field.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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