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Abstract
We study the excitation conditions of the supported field modes, as well as the spontaneous decay property of a two-level quantum emitter coupled to photonic structures containing topological insulators (TIs) and left-handed materials. Within the proper field quantization scheme, the spontaneous decay rates of dipoles with different polarizations are expressed in forms of the Green’s functions. We find that in the proposed structure, the variation in the topological magnetoelectric polarizability (TMP) has a deterministic effect on the excitation of different field modes. As the result, the spontaneous decay property of the quantum emitter can be engineered. For a dipole placed in different spatial regions, the spontaneous decay feature indicates a dominant contribution from the waveguide modes, the surface plasmon modes or the free vacuum modes. Moreover, a special kind of the surface plasmon modes displaying asymmetric density of states at the interfaces, becomes legal in the presence of nontrivial TIs. These phenomena manifest the feasibility in controlling dipole emissions via manipulations of the topological magnetoelectric (TME) effect. Our results have potential applications in quantum technologies relied on the accurate control over light-matter interactions.
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1. Introduction
The research on the cavity quantum electrodynamics in different optical systems is a cornerstone of contemporary physics, which paves the way for the development of various quantum technologies. According to the Fermi’s golden rule, the control over the spontaneous emission of quantum emitters (QEs) can be realized by engineering the photonic modes [1]. In 1946, Purcell claimed that the spontaneous decay of a quantum emitter can be effectively modified by coupling it to a resonant cavity [2]. Nowadays, a rich variety of photonic systems have been developed to control the spontaneous emission, including the photonic media [3–5], multilayer metamaterials [6], plasmonic [7–9] and the integrated structures [10]. Among them, the hyperbolic metamaterials with subwavelength multilayer structures support the excitation of bulk plasmon modes [11]. Owing to the divergent photonic density of states [12] determined by the iso-frequency surfaces, they allow broadband controls over the light-matter interaction. Besides, it has been reported that in two-dimensional graphene, the surface conductivity can be anisotropic under the application of strains [13]. In this case, the dynamic modification of the spontaneous emission for dipoles with different polarizations becomes feasible. The structures mentioned above indeedly demonstrate the possibility in tailoring the electromagnetic environment by engineering the optical boundaries, which in turn plays a key role in determining the spontaneous decay property of QEs. The modification of the dipole emission leads to a diverse range of intriguing phenomena, such as spontaneously generating quantum interference [14,15], vacuum Casimir force enhancement [16,17], squeezing in the resonance fluorescence [18,19] and atom localization [20].
Benefit from the development of nanotechnology, now it becomes possible to fabricate artificial materials exhibiting different optical properties in the lab. Among them, TIs [21,22] are novel kinds of materials that behave as normal insulators in the bulk and possess gapless surface states protected by the time-reversal symmetry [23]. On the other hand, they can be classified into a new class of quantum state, which defines the insulators with nontrivial topological order [24]. Due to the presence of quantum Hall effect, the electromagnetic response of the TIs is dominated by the modified constitutive relations and thus different from normal dielectrics [25]. When the time-reversal symmetry of the surface state is broken, the TME effect [26] dramatically changes the reflection and transmission behaviors of the incident light at the surface. It had been reported that in the structures containing TIs, the optical phenomena like Goos-H$\ddot {a}$nchen shift [27] or Kerr and Faraday [28] effects can exhibit peculiar properties.
Another kind of metamaterials that has attracted much attention are left-handed materials [29,30], where the electric component, the magnetic component and the wave vector form a left-handed triad for the propagating electromagnetic field in the medium. Based on this special property, left-handed materials exhibit both the negative index of refraction and opposite group velocities for an incident light [31]. They have been extensively investigated in 1990s [32,33] since the theoretical prediction [29], accompanied with the experimental realizations of the negative refraction index [34] and the reversed Doppler effect [35] in the early 2000s. Despite the structures working at the microwave region [34,36,37], recent researches demonstrate the possibility of realizing left-handed materials in the infrared [38,39], visible [40,41] and ultraviolet [42,43] frequency regions. Inspired by the success in fabricating the metamaterials, a series of applications have been reported, such as quantum entanglement [44], imaging [45], light absorbers [46] and cloaking [47].
In this paper, we propose a system composed of a two-level QE and a complex optical structure constructed by a left-handed slab (LHS) mounted on an insulator substrate. Applying the light scattering theory in multi-layered planar structures, the Green functions and the spontaneous decay rates for dipoles with orthogonal polarizations are derived. It is known that the TME effect strongly modifies the electromagnetic field at the surface of TIs [48]. Thus if the insulator in the proposed optical structure becomes topologically nontrivial, one can expect prominent modifications on the excitation of the electromagnetic modes supported by the structure. From this point of view, we give a comprehensive discussion on the spontaneous decay property of the QE under different conditions.
The paper is organized as follows. In Sec. 2, starting from the Schrödinger equation, the dipole decay rates are expressed in forms of the Green’s functions. In Sec. 3, the attention is focused on the excitation conditions of different field modes and their contributions to the spontaneous decay of the QE. In Sec. 4, we numerically investigate the influences of the topological effect and the spatial position on the spontaneous decay behavior of the QE, by considering different dipole polarizations. Finally we present our conclusions in Sec. 5.
2. Quantum scheme of the model and the derivation of the dipole decay rates
Consider a two-level QE placed near a complex structure constructed by the LHS and the TI, as shown in Fig. 1. The excited and ground states of the QE are $\left |e\right \rangle$ and $\left |g\right \rangle$, where the transition dipole is denoted by $\mathbf {p}_{a}$ and the resonant frequency is $\omega _{a}$. The TI is assumed to be semi-infinite in the space, with permittivity and permeability denoted by $\varepsilon _{t}$ and $\mu _{t}$. The LHS is mounted on the TI, with thickness $d_{l}$, permittivity $\varepsilon _{l}$ and permeability $\mu _{l}$. It is also highlighted that when the time-reversal symmetry of the surface states is broken, nontrivial topological magnetoelectric polarizability (denoted by $\Phi$) can be induced owing to the existence of the TME effect [49]. Notice that the strength of the TMP can be experimentally controlled by covering the TI with a thin magnetic membrance and applying a gate voltage [50], which provides the tunability of the proposed structures.
Fig. 1. Schematic representation of an initially excited two-level QE interacting with the photonic structures consist of the LHS and TI. The $z$ axis is perpendicular to the interfaces and the spatial position of the QE is denoted by $\mathbf {r}_{a}=\left (0,0,z_{a}\right )$.
Here we adopt the quantization scheme mentioned in the work [51,52], which is especially convenient in describing the modified photonic reservoir in the presence of metamaterials. Under the dipole and rotating-wave approximations, the Hamiltonian of the system can be expressed as
In the above equation, $\hat {\mathbf {J}}_{N}\left (\mathbf {r}',\omega \right )$ is the noise current operator related to the bosonic field operator $\hat {\mathbf {f}}_{\sigma }\left (\mathbf {r}',\omega \right )$, $\overset {\leftrightarrow }{\mathbf {G}}\left (\mathbf {r},\mathbf {r}',\omega \right )$ is the dyadic Green’s tensor of the electromagnetic field satisfying the wave equation [52]
Here the systematic state $\left |e,0\right \rangle$ demonstrates that the QE stays in its excited state, and there is no excitation in the media-assisted reservoir. In contrast, the state vector $\left |g,\mathbf {1}_{\sigma }\left (\mathbf {r},\omega \right )\right \rangle$ describes the QE in its ground state and there is an excitation in the field degrees $\left (\left |\mathbf {1}_{\sigma }\left (\mathbf {r},\omega \right )\right \rangle =\hat {\mathbf {f}}_{\sigma }^{\dagger }\left (\mathbf {r},\omega \right )\left |0\right \rangle \right )$. The probability amplitudes of the states are denoted by $C_{e}\left (t\right )$ and $\mathbf {C}_{\sigma g}\left (\mathbf {r},\omega,t\right )$, respectively. We assume that the QE is initially prepared in the excited state, thus by inserting Eqs. (1) and (4) into the Schrödinger equation, after formally eliminating the probability amplitude $\mathbf {C}_{\sigma g}\left (\mathbf {r},\omega,t\right )$ in the time evolution of the probability amplitude $C_{e}\left (t\right )$ and applying the Markovian approximation, the spontaneous decay rate of the quantum emitter can be extracted from the following equation
It should be stressed that the property $\omega ^{2}c^{-2}\int d\mathbf {s}\,\textrm {Im}\left [\varepsilon \left (\mathbf {s},\omega \right )\right ]\overset {\leftrightarrow }{\mathbf {G}}_{ln}\left (\mathbf {r},\mathbf {s},\omega \right )\overset {\leftrightarrow }{\mathbf {G}}_{mn}^{\ast }\left (\mathbf {r}',\mathbf {s},\omega \right )=\textrm {Im}\left [\overset {\leftrightarrow }{\mathbf {G}}_{lm}\left (\mathbf {r},\mathbf {r}',\omega \right )\right ]$ has been adopted in the derivation of Eq. (5), where $\textrm {Im}\left [\overset {\leftrightarrow }{\mathbf {G}}\left (\mathbf {r}_{a},\mathbf {r}_{a},\omega _{a}\right )\right ]$ is the imaginary part of the Green’s tensor evaluated at the transition frequency $\omega _{a}$ and the spatial position $\mathbf {r}_{a}$ (QE’s position). Also we notice that the spontaneous decay rate $\Gamma$ is proportional to the imaginary part of the classical Green function, thus could be modified in the presence of the materials. For two-level QEs, the polarization of the transition dipole could be arbitrary, here we mainly focus on two typical cases: the dipole moment parallel and normal to the surface of the complex optical structure. After expanding the field into the plane wave forms and imposing the boundary conditions [53], the Green’s tensor for the proposed structure are proved to be
It is known that the TMP (denoted by $\Phi$ in our notation) is often introduced to describe the strength of TME effect, which can induce mixed polarizations between the electric and magnetic fields at the interface. According to Chern-Simons theory, one can categorize all time-reversal invariant insulators into two groups: $\Phi =0$ for the $Z_{2}$ trivial insulators and $\Phi =\pi$ for the $Z_{2}$ nontrivial TIs. In this case, all physically measurable quantities are invariant when TMP suffers $2\pi$ shifts. However, when a magnetic coating is applied on the surface of the TI, time-reversal symmetry of the surface states will be broken, which also accompanied with the destruction of the degenerate surface states. In this scenario, the presence of the quantized Hall effect leads to the discrete TMP with odd values of $\pi$, which can be denoted by $\Phi =\left (2n+1\right )\pi$, with $n\in Z$ [54]. Moreover, the modified constituent relations under the influence of the TME effect can be expressed as $\mathbf {D}=\varepsilon \mathbf {E}+\left (\alpha \Phi /\pi \right )\mathbf {B}$ and $\mathbf {H}=\mathbf {B}/\mu -\left (\alpha \Phi /\pi \right )\mathbf {E}$ [49]. Starting from the constituent relations, after combining the Maxwell’s equations and applying the transfer matrix method [55], the reflection coefficients at the TI’s surface are of the forms
In the above expressions, $r_{pp}$ and $r_{ss}$ are reflection coefficients of the TM and TE modes, respectively. The other two cross terms in Eq. (10) refer to the ratio of the field amplitudes describing the polarization conversion before and after reflections by the interface, which actually indicate the polarization conversions of light between the reflected and incident waves owing to the existence of the TME effect. Thus apart from the bulk refractive indices of the TI and the medium mounted on it (denoted by $n_{t}$ and $n_{m}$), light reflections at the interface also strongly depends on the TMP-dependent coefficient $\bar {\Phi }=\alpha \Phi \left (\pi \varepsilon _{0}c\right )^{-1}$. Other parameters are defined as $\zeta _{\pm }=\beta _{m}k_{t}\left (\beta _{t}k_{m}\right )^{-1}\pm \beta _{t}k_{m}\left (\beta _{m}k_{t}\right )^{-1}$ and $\eta =n_{m}^{2}+n_{t}^{2}+\bar {\Phi }^{2}+n_{m}n_{t}\zeta _{+}$, with $\beta _{t\left (m\right )}$ and $k_{t\left (m\right )}=n_{t\left (m\right )}\omega _{a}/c$ representing the vertical and total wave numbers in the TI (or in the mounted medium, denoted by the index $m$).
Now the Green’s tensor in Eqs. (6) and (7) can be well understood with the definition of the complex reflection coefficients $R_{\sigma _{1}\sigma _{2}}$, which have the form
Here $r_{p}=\left (\varepsilon _{0}\beta _{l}-\varepsilon _{l}\beta _{0}\right )/\left (\varepsilon _{l}\beta _{0}+\varepsilon _{0}\beta _{l}\right )$ and $r_{s}=\left (\mu _{0}\beta _{l}-\mu _{l}\beta _{0}\right )/\left (\mu _{l}\beta _{0}+\mu _{0}\beta _{l}\right )$ are reflection coefficients of the field reflected by the LHS-vacuum interface with different polarizations, where $r_{\sigma _{1}\sigma _{2}}\left (\sigma _{1}\neq \sigma _{2}=s,p\right )$ represent the cross terms related to polarization conversions defined in Eq. (10). The vertical component of the wave numbers in the LHS and the vacuum are $\beta _{l}=-\sqrt {\varepsilon _{l}\mu _{l}k_{0}^{2}-k^{2}}$ and $\beta _{0}=\sqrt {k_{0}^{2}-k^{2}}$, where $\varepsilon _{l\left (0\right )}$ and $\mu _{l\left (0\right )}$ are permittivity and permeability of the left-handed material (vacuum), $k_{0}=\omega _{a}/c$ is the vacuum wave number and $k=\left |\mathbf {k}\right |$ is the amplitude of the in-plane wave vector. Combining the results given in Eqs. (6)–(11), it is clear that for topologically nontrivial insulators, the field reflected by the complex optical structure will contain both the TM and TE modes for arbitrary polarizations of the incident field. However, since polarization conversions has no influence on the spontaneous decay of the QE, only the influence of the reflection coefficients $R_{pp}$ and $R_{ss}$ in the Green’s tensor should be taken into account. Thus according to the Eqs. (5) and (6), a straightforward calculation of the spontaneous decay rate for the QE with dipole momentum parallel or normal to the interfaces reveals to be
3. Spontaneous decay of the QE via different field modes
3.1 Decay through the radiation modes
The spontaneous decay through radiation modes refers to the case that the QE decays from its excited state to the ground state by emitting a propagating photon in the whole space [56]. Under this circumstance, the refractive indices of the LHS and the TI should be larger than the vacuum value ($\left |n_{l}\right |,n_{t}>n_{0}$), the contribution of the radiation modes to the total decay rate can be obtained by performing the integrals in Eqs. (12) and (13) over the parallel wave number $k$ from $0$ to $k_{0}$.
3.2 Decay through the substrate modes
The substrate modes refer to the electromagnetic modes which are evanescent waves in the vacuum and standing waves in both the LHS and the TI. In other words, this kind of field modes can be supported when the in-plane wave number falls into the region $k\in \left (k_{0},n_{t}k_{0}\right )$ for $n_{t}<\left |n_{l}\right |$, or into the region $k\in \left (k_{0},\left |n_{l}\right |k_{0}\right )$ for $n_{t}>\left |n_{l}\right |$, where both the vertical components $\beta _{t}$ and $\beta _{l}$ are real and $\beta _{0}$ becomes imaginary. Thus in this case, the contribution of the substrate modes to the spontaneous decay can be numerically evaluated by performing the integrals in Eqs. (12)–(15) over the wave number $k$ from $k_{0}$ to $\left |n_{t\left (l\right )}\right |k_{0}$.
3.3 Decay through the tunneling modes
The contribution of the tunneling modes to the spontaneous decay of the QE should be taken into account when $n_{t}>\left |n_{l}\right |$, ${\it i.}e.$ the refractive index of the TI becomes larger than that of the LHS. This kind of the electromagnetic modes behaves as standing waves in the TI and evanescent waves in both the LHS and vacuum. Thus when the QE is placed near the optical structure (with $z_{a}>0$), the spontaneous decay can also take place via field tunneling in the LHS [30,57]. The relevant in-plane wave number of this field modes falls into the region $k\in \left (\left |n_{l}\right |k_{0},n_{t}k_{0}\right )$ and the spontaneous decay rate can be evaluated by performing the integrals in Eqs. (12)–(15) over $k$ from $\left |n_{l}\right |k_{0}$ to $n_{t}k_{0}$.
3.4 Decay through the WGM
The WGM [58] in the proposed structure corresponds to the resonant standing wave modes propagating in the LHS with finite thickness, where the field becomes evanescent in the two semi-infinite spatial regions filled with the TI and the vacuum. Thus it is clear that the WGM are legal when the refractive indices of the materials satisfy $n_{0}<n_{t}<\left |n_{l}\right |$, with the in-plane wave number $k$ falls into the region $k\in (n_{t}k_{0},\left |n_{l}\right |k_{0})$. If the QE is placed outside the LHS waveguide, the spontaneous decay mainly takes place through the evanescent modes near the LHS-vacuum interface. The exact solutions of the WGM often require the complex reflection coefficients with modulus $1$. Under this circumstance, the real parts of the integrands in Eqs. (12)–(15) vanish except for the cases that $D_{p\left (s\right )}=0$. In other words, the WGM can be excited at discrete wave numbers $k$ leading to $D_{p\left (s\right )}=0$ for different field polarizations [59]. However, one can also notice that the integrands become divergent owing to the presence of the WGM. In order to evaluate the spontaneous decay rate of the QE via coupling to this kind of field modes, we adopt the residue theorem for the poles and obtain
In Fig. 2 we investigate the properties of the WGM in the proposed structure, where both the trivial ($\Phi =0$) and nontrivial ($\Phi =\pi$) cases are taken into account. Figure 2(a) illustrates that the number of WGM exhibits a ladder-type increase with the growth in the waveguide thickness $d_{l}$ for the trivial insulator. Also we notice the WGM may exist even for ultrathin waveguide with sub-wavelength thickness, which can be attributed to the enhancement in the phase change of the optical path induced by the Goos-Hanchen shifts at the interfaces formed by positive and negative index materials [60]. Moreover, discontinuities (none WGM can be supported for both field polarizations when the thickness of the LHS locates in the regions $\left [0,0.07\lambda _{a}\right ]$ and $\left [0.2\lambda _{a},0.4\lambda _{a}\right ]$) and fluctuations (the sudden increase of the WGM numbers $N_{p}^{wg}$ at $d_{l}=0.45\lambda _{a}$ and $d_{l}=0.75\lambda _{a}$) in the mode numbers of the WGM can be observed. These behaviors can be well explained by checking the resonant conditions $D_{p\left (s\right )}=0$ for the WGM. If we rewrite the reflection coefficients with unit modulus in forms of $r_{p}=\exp \left (2i\theta _{0}\right )$ and $r_{pp}=\exp \left (2i\theta _{i}\right )$ (their definitions can be found in the Eqs. (8) and (11)), the resonant conditions for the TM-polarized WGM are equivalent to $\beta _{l}d_{l}+\theta _{0}+\theta _{i}=m\pi \left (m=0,1,2\cdots,N_{p}^{wg}\right )$. It is easy to verify that with the increase of the parallel wave number $k$ in the region $\left (n_{t}k_{0},\left |n_{l}\right |k_{0}\right )$, both the phases $\theta _{0}$ and $\theta _{i}$ suffer monotonic decreases, where the geometrical phase term $\beta _{l}d_{l}$ exhibits monotonic growths according to the changes. As the result, the sum of all the phase terms mentioned above becomes uncertain (neither monotonic decrease nor monotonic increase) with the growth of the waveguide thickness $d_{l}$, which is the main reason for the appearance of the discontinuities and fluctuations in the mode numbers of the supported WGM.
Fig. 2. (a) Mode numbers of the WGM as a function of the thickness $d_{l}$ for the trivial insulator with $\Phi =0$, where the thickness step is $0.05\lambda _{a}$. The material parameters are $\varepsilon _{t}=3$, $\mu _{t}=1$, $\varepsilon _{l}=-3$ and $\mu _{l}=-2$, the mode numbers $N_{p}^{wg}$ and $N_{s}^{wg}$ are denoted by the blue circle and the red dot, respectively. (b) Modulus square of the reflection coefficients $r_{pp}$ (blue solid line) and $r_{ps}$ (red dashed-dotted line) defined in Eqs. (8) and (10) versus the in-plane wave number $k$, also the summation is given (black dashed line). Here the nontrivial case of $\Phi =\pi$ is considered, where other parameters are the same as in the subplot (a).
Different from the case that the LHS waveguide is mounted on the trivial insulator (the insulator behaves as a normal dielectric), for nontrivial TI cases the complex optical structure can not support the excitation of the WGM. To better understand the physical mechanism lies behind this phenomenon, we investigate the modulus square of the reflection coefficients $r_{pp}$ and $r_{ps}$. The results are shown in Fig. 2(b), where the value of the TMP is set to be $\Phi =\pi$ and other material parameters are the same as in Fig. 2(a). It is known that the reflection coefficient measures the ratio between the complex amplitudes of the reflected and incident fields, thus the modulus square of them in fact indicates the energy reflected by the interface compared to the incident part. As illustrated, there is a dip in the region $k\in \left (\sqrt {3}k_{0},\sqrt {6}k_{0}\right )$ for $\left |r_{pp}\right |^{2}$, which demonstrates a deviation from the resonant conditions of the WGM (which requires $\left |r_{pp}\right |=1$). In this scenario, the incident TM modes partially transfers into the TE modes after reflected by the interface (indicated by the red dashed-dotted line). The polarization conversion effect of the field modes has no contributions on the spontaneous decay, thus although total reflection still takes place at the interface (the black dashed line) the destruction of the WGM is inevitable.
3.5 Decay through the SPM
In positive-index waveguides formed by multilayer dielectric structures, evanescent modes cannot propagate out and thus do not contribute to the spontaneous decay of the QE. However, the circumstance changes when one of the layers is replaced by the LHS, where the excited SPM can be well guided along the interfaces [61,62]. In the proposed structure, the SPM refers to the field modes that exponentially decay away from the interfaces in the regions filled with the semi-infinite TI and the vacuum. This kind of modes often accompany with large mode density, whose in-plane wave numbers should satisfy $k>\left |n_{l}\right |k_{0}$. Different from the metallic structure that only the TM-polarized SPM can be excited, the LHS supports the excitation of both the TM- and TE-polarized SPM near the surface [63] owing to the simultaneous negativity of the permittivity and permeability in the working frequencies.
Similar to the WGM, the excitation of the SPM requires the vanishment of the multireflection factors, ${\it i.e.}$ the condition determined by $D_{p}=0$ or $D_{s}=0$. In this case, the spontaneous decay rates of the QE via coupling to the SPM can be simply evaluated through the expressions given in Eqs. (16)–(19), by only replacing the mode numbers ($N_{p}^{wg}$ and $N_{s}^{wg}$) and the supported wave numbers ($k_{pj}^{wg}$ and $k_{sj}^{wg}$) for the WGM with the mode numbers ($N_{p}^{sp}$ and $N_{s}^{sp}$) and propagation constant ($k_{pj}^{sp}$ and $k_{sj}^{sp}$) for the SPM. It should be noticed that the in-plane wave numbers of the supporting SPM satisfy the relation $k_{p\left (s\right )j}^{sp}>\left |n_{l}\right |k_{0}$, which differs from the excitation condition of the WGM $k_{p\left (s\right )j}^{wg}\in \left (n_{t}k_{0},\left |n_{l}\right |k_{0}\right )$.
In Fig. 3, the reflection coefficients of the TI-LHS interface, as well as the supported wave numbers of the SPM versus the slab thickness, are investigated to illustrate the influence of the TME effect on the excitation of the SPM. In the trivial case ($\Phi =0$ plane), the reflection coefficients $r_{pp\left (ss\right )}$ and $r_{p\left (s\right )}$ are negative in the plasmon region, where $r_{ps}$ and $r_{sp}$ (denoted by the dashed-dotted line) are vanished due to the absence of the TME effect. Under this circumstance, the product of $r_{\sigma _{1}}r_{\sigma _{1}\sigma _{2}}$ becomes positive for both field modes. The wave numbers (denoted by $k_{p}^{sp}$ and $k_{s}^{sp}$) of the SPM can be obtained through the excitation condition $D_{p\left (s\right )}=0$ for the LHS with the thickness in the subwavelength region. This property is clearly demonstrated in Fig. 3(b), where the SPM with different polarizations can be excited if the thickness of the LHS ranges in $\left [0.026\lambda _{a},0.067\lambda _{a}\right ]$ or $\left [0.07\lambda _{a},0.126\lambda _{a}\right ]$. However, the situation changes quite a lot when the TME effect becomes dominated in determining the reflection behavior of light at the interface. For nontrivial TIs, the reflection coefficients exhibit different features, which can be recognized through the $\Phi =\pi$ plane in Fig. 3(a). In this case, the polarization conversion effect (dashed-dotted line) of the reflected light becomes prominent, where the reflection coefficient of the TM waves (blue solid line) changes its sign with the increase of the wave number $k$ in the plasmon region. As we would see later, this is the main reason for the appearance of the super modes [64]$-$a special kind of the SPM that can be always supported by the structure regardless of the slab thickness. Further investigations show that the reflectivity of the interface ($r_{pp}$ and $r_{ss}$) suffers sufficient increases as the TMP grows, which is accompanied with the enhancement in the phase difference between the incident and reflected TE waves.
Fig. 3. Influence of the TMP on the excitation conditions of the SPM under different circumstances, other material parameters are the same as in Fig. 2. (a) Variation of the reflection coefficients $r_{pp}$, $r_{ss}$ and $r_{ps(sp)}$ versus the scaled in-plane wave number and TMP in the plasmon region ($k>n_{l}k_{0}$). (b) and (c): Wave numbers of the supported SPM as a function of the normalized slab thickness $d_{l}$ for trivial ($\Phi =0$) and nontrivial ($\Phi =\pi, 5\pi$) insulators.
In Fig. 3(c) we present the wave number of the SPM as a function of the LHS thickness, where the nontrivial insulator cases with $\Phi =\pi$ and $\Phi =5\pi$ are taken into account. It is clear that in the former case, both the TM- and TE-polarized SPM can be supported by the structure. Meanwhile, owing to the different phases of the reflected waves, the super TM modes can also be excited for certain slab thicknesses. Different from the normal SPM that the fields distributed at both the interfaces, the super modes do not require a strong coupling between the interfaces (its excitation relies on the vanishment of the reflection at the TI-LHS interface in the plasmon region). For slab thickness varies from $0.144\lambda _{a}$ to $0.169\lambda _{a}$, strong coupling takes place between the interfaces. As the result, both the normal and super TM modes can be simultaneously supported by the structure. Then with the increase of the slab thickness, only the super TM modes are legal, which can be attributed to the decoupling of the interfaces. It is shown that an increase in the TMP further limits the slab thickness in order to support the TE modes. Specifically, the reflection coefficient $r_{pp}$ turns out to be positive in the region $k>\left |n_{l}\right |k_{0}$ for $\Phi =5\pi$. In this case the condition $D_{p}=0$ can not be fulfilled for real $k$, which is the reason for the absence of the TM modes. Here it should be pointed out that the SPM can also be supported under different circumstances for the proposed structure. Nevertheless, due to the similarity of the SPM in the modification of the QE’s spontaneous decay, we do not give more discussions.
4. Spontaneous decay properties of the QE in proximity to the complex optical structure
Fig. 4. Illustrations of the supported WGM and the spontaneous decay properties of the QE with different dipole polarizations, where the material parameters are $\varepsilon _{t}=3$, $\mu _{t}=1$, $\varepsilon _{l}=-3$, $\mu _{l}=-2$ and $d_{l}=0.15\lambda _{a}$. (a) and (b): Wave numbers of the WGM supported by the structure, which are determined by the conditions $D_{p(s)}=0$. (c) and (d): Normalized spontaneous decay rates(scaled by $\Gamma _{0}$) for dipoles with $x$ and $z$ polarizations, as a function of the QE’s position. Here $\Gamma _{\textrm {x}}$ and $\Gamma _{\textrm {z}}$ are total decay rates, where $\Gamma _{x(z)}^{\textrm {rad}}$, $\Gamma _{x(z)}^{\textrm {sub}}$ and $\Gamma _{x(z)}^{\textrm {wg}}$ represent the spontaneous decay rates via the radiation, substrate and wave guide modes, respectively.
In this part we will focus on discussing the spontaneous decay properties of a QE near the proposed structure, where different dipole polarizations and material parameters are taken into account. Firstly, we neglect the dissipation of the materials without loss of generality, where the branch points of the integrals in Eqs. (12)–(15) located on the real axis of the complex $k$ plane under this circumstance. Then in comparison, the dissipative cases are also considered. As we would see later, the TME effect dramatically modifies the electromagnetic modes, which plays a key role in determining the spontaneous emission features of the QE.
4.1 Refractive indices $n_{t}<\left |n_{l}\right |$
According to the discussions in Sec. 3, it is clear that for $n_{t}<\left |n_{l}\right |$ the spontaneous decay can take place through the radiation, the substrate, the waveguide and the surface plasmon modes. Here we consider the LHS with thickness $d_{l}=0.15\lambda _{a}$ and assume that the TME effect is absent, where other material parameters are the same as in Fig. 2. Obviously, according to the results shown in Figs. 2 and 3, only the TM- and TE-polarized WGM can be supported under this circumstance. This feature is well illustrated by Figs. 4(a) and 4(b), where the vanishment of the multireflection factor $D_{p\left (s\right )}$ in the region $\left (n_{t}k_{0},\left |n_{l}\right |k_{0}\right )$ can be observed. Due to the presence of the boundaries, there are also branch points (also zeros of the multireflection factors) in the $k$-domain, which do not contribute to the spontaneous decay.
Figures 4(c) and 4(d) demonstrate that the spontaneous decay property of the QE inside the LHS is mainly determined by the WGM. As the result, the spontaneous decay rates is position-dependent and exhibit antinodes for both the $x$- and $z$- polarized dipoles. When the QE is close to the LHS-vacuum interface, both the waveguide and substrate modes have prominent contributions to the spontaneous decay. Outside the LHS, the QE gradually decouples from these two modes with the increasing distance, which in turn leads to sharp decreases in the decay rates (dashed and dashed-dotted lines in the region $z_{a}\in \left [0,2\lambda _{a}\right ]$). Notice that owing to the smaller imaginary part of the $\beta _{0}$, the attenuation length of the substrate modes is larger than that of the WGM. When the QE is far away from the surface, the influence of the complex optical structure is negligible and thus the spontaneous decay mainly takes place via the free vacuum modes.
Fig. 5. Illustrations of the supported SPM and the spontaneous decay properties of the QE for the nontrivial TI case ($\Phi =\pi$), other material parameters are the same as in Fig. 4. (a) and (b): Wave numbers of the SPM determined by the conditions $D_{p(s)}=0$. (c) and (d): Normalized decay rates as a function of the QE’s position, logarithmic values are taken in the inset panels. The decay rates denoted by $\Gamma _{x(z)}^{\textrm {bou}}$, $\Gamma _{x(z)}^{\textrm {nor}}$ and $\Gamma _{x(z)}^{\textrm {sup}}$ refer to the bounded modes (propagating in the LHS and evanescent in other spatial regions, which differs from the WGM), normal and super SPM, respectively.
Then we consider the nontrivial TI case with $\Phi =\pi$, where both the normal and super surface plasmon modes with TM polarization are supported by the structure. It is more evident by combining the results shown in Figs. 5(a) and 5(b), where two solutions can be found for the excitation of the SPM. In this case, the super SPM possesses larger wave number and mode density. Thus as Figs. 5(c) and 5(d) demonstrate, the super SPM has a prominent contribution to the spontaneous decay when the QE is placed near the TI-LHS interface. Besides, we notice that for dipole with $z$ polarization, the spontaneous decay is deeply suppressed near $z_{a}=-0.02\lambda _{a}$ owing to the destructive interference of the TM modes. Similar behaviors can also be observed for other modes, which finally forms a dip in the spontaneous decay rate $\Gamma _{z}$. Based on this property, the dipole decay rates becomes highly anisotropic near this position (illustrated by the black solid curves in the insets). Outside the waveguide, the spontaneous decay property is mainly determined by the SPM when the QE is close to the interface. For larger QE-interface distances, the spontaneous decay rates display oscillatory behaviors around the vacuum value.
Fig. 6. The multireflection factors and spontaneous decay properties of the QE for nontrivial TI case ($\Phi =9\pi$), other material parameters are the same as in Fig. 4. (a) and (b): The multireflection factors of different field modes. (c) and (d): Normalized decay rates as a function of the QE’s position.
As indicated by Figs. 6(a) and 6(b), when the TME effect is strong ($\Phi =9\pi$), neither the WGM nor the SPM can be excited in the proposed structure (the zero(s) of the multireflection factors can not be found in the waveguide ($k\in \left [n_{t}k_{0},\left |n_{l}\right |k_{0}\right ]$) or plasmon ($k>\left |n_{l}\right |k_{0}$) region). Consequently, under this circumstance the spontaneous decay of the QE exhibits different properties from the former cases. It is clear that in the LHS, the spontaneous decay rates are small due to low density of states of the relevant field modes. Near the TI-LHS interface, the spontaneous decay of the dipole with $x$ polarization is largely inhibited. This can be attributed to the destructive interference of the incident and reflected waves, where the phase of the reflected field changes the sign in comparison with the incident field for the TE modes. Outside the waveguide, the substrate modes and the bounded modes contribute to the spontaneous decay only for small distances between the QE and the surface.
Fig. 7. Illustrations of the supported WGM and the spontaneous decay properties of the QE for trivial insulator case ($\Phi =0$), the thickness of the LHS is $d_{l}=3\lambda _{a}$ and other material parameters are kept unchanged in accordance with Fig. 4. (a) and (b): Wave numbers of the supported WGM determined by the conditions $D_{p(s)}=0$. (c) and (d): Normalized decay rates with different dipole polarizations as a function of the QE’s position.
The above discussions indicate that the interfaces between negative and positive index materials can lead to the enhancement in the phase change of the electromagnetic field in the waveguide, which is the main reason for the presence of the WGM in the LHS with subwavelength thickness. As illustrated by Fig. 2(a), the WGM can also be formed for waveguides with thickness comparable or larger than the vacuum wavelength. In this case, the number of the WGM suffers increases with the growth of the waveguide thickness $d_{l}$. In the following we consider the case that the waveguide thickness is $3\lambda _{a}$, where other parameters are the same as in Fig. 4. The relevant results are shown in Fig. 7 and as demonstrated, there are totally $9$ distinct WGM supported by the LHS. Under this condition, the spontaneous decay via the WGM contains the contributions of all the eigenmodes when the QE is placed in the LHS. Thus position-dependent oscillatory behaviors of the decay rates can be observed, which implies the coupling between the QE and the propagating WGM. Moreover, notice that the oscillatory behavior of the total decay rates becomes weak when the QE is placed near the center of the LHS. This can be well understood by combining the contributions of different field modes on the spontaneous decay of the QE. In one aspect, destruction interferences of the WGM take place in the LHS. On the other hand, superpositions among the WGM and other field modes also takes place when the QE is away from the surface, which further weakens the influence of the QE’s position on the spontaneous decay in this region. In Fig. 8, we investigate the nontrivial TI case ($\Phi =\pi$) to see how the TME effect influences the field modes and the spontaneous decay in the LHS. Obviously, the existence of the nonvanished TMP effectevely modifies the field modes inside the LHS, where the structure can not support the excitation of the WGM anymore. However, as indicated by the inset panel in Fig. 8(a), a novel kind of the SPM - super TM modes can be supported owing to the presence of the TME effect. In detail, it is clear that near the supported wave number of the super TM mode, the multireflection factor $D_{p}$ changes rapidly and becomes extremely large. Combining Eqs. (18), (19) and the results in Fig. 3(a), the weak reflection feature indicated by the coefficient $r_{pp}$ near the eigenmode $k_{p}^{sp}$ finally leads to the decoupling of the interfaces, with large density of states distributing only near the TI-LHS interface. Thus the distribution of the SPM turns out to be highly asymmetric in the waveguide, which has prominent contributions to the spontaneous decay only when the QE is close to the TI-LHS interface.
Fig. 8. Illustrations of the supported SPM and the spontaneous decay properties of the QE for nontrivial TI case ($\Phi =\pi$), other material parameters are the same as in Fig. 7. (a) and (b): Wave numbers of the super SPM supported by the structure. (c) and (d): Normalized decay rates as a function of the QE’s position.
Interestingly, as demonstrated by Figs. 9(a) and 9(b), quasi-resonant modes appear in the LHS waveguide when the TME effect is strong. The reason lies in the prominent reflection behavior of the TI-LHS interface, which can be easily verified through Eqs. (8)–(10) and checking the reflection coefficients by following the method mentioned in Fig. 2(b). Owing to the nearly vanished multireflection factors in the WGM region (clearly showed by the inset panels), these modes are recognized as quasi-waveguide modes (QWGM). As predicted, the QWGM has a prominent contribution on the spontaneous decay in the LHS, which exhibits similar oscillatory features as the WGM holds. Also we notice that large phase differences between the incident and the reflected fields can be induced at the TI-LHS interface for the TE modes, where for TM modes this phase differences are negligible. Thus when the QE is close to the interface, the spontaneous emissions for dipoles with $x$ and $z$ polarizations are inhibited and enhanced, respectively. Generally, the spontaneous decay rates through the radiation modes have larger oscillating amplitudes compared with the trivial case, which can be attributed to the dramatic changes in the reflection behaviors of the TI-LHS interface when the TME effect is prominent.
Fig. 9. The multireflection factors and spontaneous decay properties for nontrivial TI case ($\Phi =9\pi$), other material parameters are the same as in Fig. 7. (a) and (b): Illustrations of the QWGM supported by the structure. (c) and (d): Normalized spontaneous decay rates as a function of the QE’s position, here $\Gamma _{x(z)}^{q-wg}$ represents the decay rate via the QWGM.
4.2 Refractive indices $n_{t}>\left |n_{l}\right |$
From the discussions in Sec. 3, it is known that the spontaneous decay in the presence of the complex structure can take place via the radiation, the substrate, the tunneling and the surface plasmon modes when the refractive indices of the materials satisfy $n_{t}>\left |n_{l}\right |$. Here we focus on the cases that the refractive index of the LHS is close to $-1$. With this restriction the reflection of the vacuum-LHS interface is nearly vanished, thus the zero(s) of the multireflection factors can never be found. As the result, the proposed structure do not support the excitation of the SPM and only the contribution of the other field modes on the spontaneous decay needs to be taken into account.
In Fig. 10 we study the spontaneous decay properties of the QE under the conditions: the permittivity of the TI is $\varepsilon _{t}=30$, where the refractive index of the LHS is close to $-1$ (i.e. the LHS nearly acts as a perfect lens). As indicated, the spontaneous decay behavior of the QE is largely modified by the electromagnetic property of the TI-LHS interface. Moreover, the spontaneous decay through the radiation modes exhibit a large dependence on the optical property of the TI near the spatial position $z_{a}=0.15\lambda _{a}$. This feature can be attributed to the focus effect of the LHS, where the distance between the QE and the surface of the TI is optically cancelled. For nontrivial insulator with large TMP, it is clear that the spontaneous decay through the radiation modes is strongly modified near the focus. Outside the LHS, the decay rate suffers a deep suppression or enhancement for dipoles with $x$ and $z$ (dashed-dotted lines in Fig. 10(c) and Fig. 10(d)) polarizations, respectively. We also notice that owing to the amplification of the evanescent waves in the LHS, the tunneling modes have a prominent contribution on the spontaneous decay. In this case, the spontaneous decay of the QE is dominated by the tunneling modes near the LHS-vacuum interface.
Fig. 10. Normalized decay rates with different dipole polarizations as a function of the QE’s position, the material parameters are $\varepsilon _{t}=30$, $\mu _{t}=1$, $\varepsilon _{l}=-1.001$, $\mu _{l}=-1.001$ and the thickness of the LHS is $d_{l}=0.15\lambda _{a}$. (a) and (b): Trivial insulator case with $\Phi =0$. (c) and (d): Nontrivial TI case with $\Phi =9\pi$.
Then consider the LHS with thickness comparable to the radiation wavelength ($d_{l}=3\lambda _{a}$), where other material parameters are kept unchanged. Under this circumstance, strong modifications on the spontaneous decay via the radiation modes can be observed when the QE is placed near the focus. In detail, it is clear that the phenomenon is mainly due to the topological effect of the TI, which plays an important role in tailoring the electromagnetic environment. As the result, strong suppression and enhancement of the spontaneous decay through the radiation modes (dashed dotted line in Figs. 11(c) and 11(d)) near the focus can be realized, where the contributions of the other modes are negligible. That is to say, the spontaneous decay with different dipole polarizations becomes highly anisotropic in this spatial region.
Fig. 11. Normalized decay rates with different dipole polarizations as a function of the QE’s position, the LHS thickness is $d_{l}=3\lambda _{a}$ and other material parameters are the same as in Fig. 10. (a) and (b): Trivial insulator case with $\Phi =0$. (c) and (d): Nontrivial TI case with $\Phi =9\pi$.
4.3 Influence of the material dissipation on the spontaneous decay
It has been reported in both the theoretical and experimental works that the optical response of the TIs and metamaterials can be described by the Drude-Lorentz model [65,66], where the material dissipation is included in this scenario. Usually, materials become dissipative when its permittivity or permeability contains an imaginary part, which is often related to the absorption of the incident light. To investigate its influence on the spontaneous decay of the QE, in the following the material parameters are assumed to be complex.
In Fig. 12, we investigate the spontaneous decay of the QE by considering the material dissipations. As illustrated by the inset panels, material dissipation has dominant contributions to the spontaneous decay when the QE is embedded in the LHS (dashed curves). Moreover, the relevant decay rates display obvious increases when the QE is close to the interfaces, which is mainly caused by the energy loss through the dissipation [67]. Also we notice that compared to the lossless cases, the contributions of other modes to the spontaneous decay do not change a lot. For a QE placed inside the dissipative LHS, as indicated by the insets, the spontaneous decay property is largely modified by the material dissipation. When the QE is placed near the complex structure (i.e., outside the LHS), material dissipation plays an important role on the spontaneous decay only when the QE is close to the LHS-vacuum interface. Interestingly, the enhancement and suppression of the spontaneous decay rates with different dipole polarizations near the focus is immune from the material dissipation. This novel feature, makes the proposed structure to be a good candidate in engineering the radiation properties of QEs with large distances to the interfaces.
Fig. 12. Normalized decay rates with different dipole polarizations as a function of the QE’s position when the material dissipation is included. (a) and (b): Trivial insulator case ($\Phi =0$) with the parameters $\varepsilon _{t}=3+0.01i$, $\mu _{t}=1$, $\varepsilon _{l}=-3+0.01i$, $\mu _{l}=-2+0.01i$ and $d_{l}=0.15\lambda _{a}$. (c) and (d): Nontrivial TI case ($\Phi =9\pi$) with the parameters $\varepsilon _{t}=30+0.01i$, $\mu _{t}=1$, $\varepsilon _{l}=-1.001+0.01i$, $\mu _{l}=-1.001+0.01i$ and $d_{l}=3\lambda _{a}$.
5. Conclusions
In conclusion, we have investigated the spontaneous decay property of a two-level QE in the presence of photonic structures constructed by the TI and LHS. We have shown that under certain conditions, the proposed structure can support the propagation of the WGM and the SPM. In one aspect, the spontaneous decay of the QE via the WGM exhibits a strong dependence on the spatial position owing to the standing wave feature of the field modes. On the other hand, if the SPM are supported by the proposed structure, the spontaneous decay can take place mainly through the SPM when the QE is close to the interfaces. It has been shown that for LHS with subwavelength thickness, dipole radiation in the slab can be largely suppressed owing to the destructive interference of the electromagnetic modes. As the result, sharp declines in the spontaneous decay rates can be observed for both the WGM and SPM. Moreover, we have proved that super SPM can be excited in the LHS by manipulating the TMP of the TI, which is mainly caused by the decoupling of the interfaces. In the extreme cases, the structure can only support the excitation of the SPM at the TI-LHS interface. Apart from the WGM and SPM, other field modes such as the radiation, substrate, bounded and tunneling modes have been proved to be contributed to the spontaneous decay under certain conditions. The circumstance becomes more interesting when the LHS nearly behaves as a perfect lens. We have shown that the electromagnetic environment can be engineered at distances far away from the optical structure, where the spontaneous decay rates for dipoles with different polarizations become highly anisotropic near the focus. Furthermore, for LHS with thickness comparable to the vacuum wavelength, the spontaneous decay property of the QE placed near the focus is immune from the material dissipation. Based on the features mentioned above, our proposal has potential applications in controlling the spontaneous decay property of quantum transition systems, which is important in the realization of many quantum effects.
Funding
National Natural Science Foundation of China (11874287, 11947044); Huaqiao University (605-50Y19046), National Key Research and Development Program of China (2021YFA1400602)
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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