Topological hybrid nanocavity for coupling phase transition

Topological photonics provides new insights for manipulating light-matter interactions and becomes a hotspot in recent years. Novel optical cavities based on topological photonic states attract much attention due to their advantages of robustness. In recent years, several types of topological photonic cavities have been proposed. For example, topological photonic cavities based on gyromagnetic materials have been demonstrated by Bahari et al in 2017. Since time-reversal symmetry of the system is broken under a static external magnetic field, the cavities of arbitrary geometrical shape were realized, which provide opportunities to develop the topological devices with arbitrary geometries [1]. Considering there are no ideal gyromagnetic materials at optical frequencies, topological photonic cavities based on all-dielectric materials have also been explored. In 2018, topological photonic cavities composed of coupled ring-resonator array has been proposed and realized by Harari et al, which produced topologically protected edge-mode lasing in nonmagnetic materials [2, 3]. Recently, topological photonic cavities based on band-inversion-induced reflection [4] and valley edge states [5, 6] have been put forward, and the robustness of the cavities' mode has been verified. These studies reflect that topological photonic cavities have fruitful physics and applications [7, 8].

The quality factor Q and the mode volume V play key roles in designing photonic cavities, since they represent how long time the cavity can store light and how small space the cavity can localize light, where the figure of merit Q/V is introduced as a critical index to evaluate the property of photonic cavities [9–12]. As for topological photonic cavities, the effective way to improve the value of Q/V is to obtain higher quality factor Q and smaller mode volume V simultaneously. The quality factor of topological photonic cavities previously reported is not relatively high until topological photonic crystal (PhC) nanocavities with high quality factor Q being realized [13, 14]. However, the figure of merit Q/V is still limited, since the mode volume V of topological PhC nanocavities is hard to further decrease due to the light diffraction limit. As a result, the figure of merit Q/V of above topological photonic cavities is not high, which mainly support to realize weak coupling, such as Purcell enhancement [15]. Till now, it is still a great challenge to realize high Q/V topological photonic cavities, which has restricted the applications of topological photonic cavities in certain areas, such as cavity quantum electrodynamics [16–18], nonlinear optics [19, 20], topological lasers [4–7], etc.

In this work, we propose a topological photonic-plasmonic hybrid nanocavity for the first time, which consists of a topological PhC nanocavity and a plasmonic nano-antenna. Combining the advantage of the high quality factor of the topological PhC nanocavity and the ultra-small volume of plasmonic nano-antennas, the highest figure of merit Q/V of $1.5 \times {10^6}{(\lambda /n)^{ - 3}}$ is achieved in the proposed hybrid regime, and it is two orders of magnitude more than the bare topological PhC nanocavity. An ultra-high single-atom cooperativity parameter of $1.1 \times {10^5}$ is obtained and it is enhanced more than 60 times with respect to the bare topological PhC nanocavity, which makes the coupling between light and a single emitter enter a strong coupling region in topological photonic realm for the first time. Meanwhile, weak coupling and strong coupling can be easily switched in the topological hybrid system by adjusting plasmonic nano-antennas parameters. This work demonstrates that topological photonic-plasmonic hybrid nanocavity possesses higher Q/V compared with that of the bare topological PhC nanocavity. The combination of the research of topological photonics and surface plasmon resonance will open a new research direction for enhancing light–matter interaction. Before this work, only weak coupling can be realized in topological photonic cavities [13–15, 21]. The topological photonic-plasmonic hybrid nanocavity provides a robust platform to realize strong coupling and control the coupling phase transition between light and a single emitter, which will play key roles in quantum optics, quantum information and topological lasers.

A schematic of topological photonic-plasmonic hybrid nanocavity is shown in figure 1(a). The left is an overall diagram of proposed hybrid structure, which consists of a topological PhC nanocavity and a gold (Au) plasmonic bowtie nano-antenna. The corresponding enlarged view in x–y plane is shown on the right. The topological PhC nanocavity is based on a topological corner state [22–24]. It is composed of two kinds of PhCs with different topological properties. The brown part represents the topological PhC with a Zak phase of ${\theta ^{zak}} = \left( {\pi ,{\text{ }}\pi } \right)$ and the blue part represents the trivial PhC with ${\theta ^{zak}} = \left( {0,{\text{ }}0} \right)$. When the two PhCs with different Zak phase join together, the edge polarizations generate along the both interface in x and y direction and the 0D corner state is induced as a convergence [22]. The cavity mode produced by the 0D corner state in 2D PhC is demonstrated to be topologically protected and immune to disorders [22]. The quality factor Q can be improved by adjusting the gap W between the two kinds of PhCs [15, 21]. As is shown in the enlarged view of figure 1(a), the two kinds of the PhCs mentioned above are both based on square lattice with same lattice constant, but have different ways in defining unit cells. The unit cell of trivial PhC (blue part) has one air hole with length of side D (green dashed frame), and the topological PhC (brown part) has four air holes with length of side d (black dashed frame). The relation between D and d is d = 2/D. Since the mode volume V of topological PhC cavity is on the order of a cubic wavelength and hard to decrease due to the light diffraction limit, an Au plasmonic bowtie nano-antenna is introduced to trap the light into an ultra-small space and break the light diffraction limit [25–27]. The Au plasmonic bowtie nano-antenna is placed close to the corner of the air hole and the reason will explain in following contents. A quantum dot (red spot) is inserted in the gap of the Au plasmonic bowtie nano-antenna. As is shown in figure 1(b), the gap between the two Au triangle is G. The length and thickness of the Au plasmonic bowtie nano-antenna are L and T. When appropriate parameters of Au plasmonic bowtie nano-antenna is selected, the high quality factor Q and the ultra-small mode volume V can be obtained simultaneously in the hybrid structure, which makes the topological photonic-plasmonic hybrid nanocavity possess ultra-high figure of merit Q/V.

Zoom In Zoom Out Reset image size

The properties of the topological PhC nanocavity and topological photonic-plamonic hybrid nanocavity are investigated respectively by using the three-dimensional (3D) finite-difference time-domain method. The topological PhC nanocavity is designed based on the silicon-on-insulator substrate with 220 nm thickness of Silicon, which is a standard commercial film widely used in silicon photonics. Considering both the simulation accuracy and calculation time, the calculation area of topological PhC nanocavity is set to be 33 a × 33 a in x–y plane and 3 μm in the z-direction. The two-dimensional (2D) PhC slab has a square lattice of air holes with lattice constant of 420 nm. In order to guarantee near infrared (NIR) communication wavelength 1310 nm is located in the bandgap, the air hole length sides of trivial and topological unit cells are 300 nm for D and 150 nm for d, as is shown in figure 1(a). The refractive index of Silicon and air are 3.45 and 1 in NIR communication wavelength. High quality factor topological PhC cavity is calculated through adjusting the gap W between the two kinds of PhC [15, 21]. The quality factor Q is firstly increased with the increase of gap W between the two kinds of PhC and then decrease when the value of gap W larger than 50 nm, so the largest quality factor Q is obtained when the value of gap W is 50 nm for our configuration. The value of quality factor Q is $1.6 \times {10^4}$ and corresponding mode volume V is 0.68 ${(\lambda /n)^3}$, so the figure of merit Q/V is $2.3 \times {10^4}{(\lambda /n)^{ - 3}}$.

The electric field distribution of a topological PhC nanocavity and a topological photonic-plamonic hybrid nanocavity are calculated, which are shown in figure 2. For a bare topological PhC nanocavity, the electric field distribution of |E| in x–y plane is concentrated around the central air hole, as is shown in the left of figure 2(a) and the corresponding enlarged view is shown in the right. The electric field of topological PhC nanocavity is not localized in a small space, which leads to a relatively large mode volume V. In order to realize strong electric field confinement, the Au plasmonic bowtie nano-antenna is chosen to be situated near the corner of air holes of the topological PhC nanocavity, which possesses the maximum of electric field. The refractive index of gold used in the simulations is from the experimental data by Johnson and Christy in 1972 [28]. As is shown in figure 2(b), the left picture shows the electric field distribution of |E| of topological photonic-plamonic hybrid nanocavity in x–y plane and the right one is the corresponding enlarged view. The electric field is concentrated in the gap of the bowtie nano-antennas and the length of the Au plasmonic bowtie nano-antennas is 45 nm. A hotspot is generated when the gap of Au plasmonic bowtie nano-antenna is 5 nm. The electric field intensity of the proposed hybrid structure in figure 2(b) is enhanced more than two orders of magnitude compared with the bare topological PhC nanocavity in figure 2(a). Thus, the topological photonic-plasmonic hybrid nanocavity provides a robust platform to enhance the light-emitter interaction.

Zoom In Zoom Out Reset image size

Next, we quantitatively investigate the intrinsic properties of the topological photonic-plasmonic hybrid nanocavity and light-emitter coupling in the hybrid structure. As is shown in figure 3(a), the variation of the ${Q_{c,m}}$ and ${V_{c,m}}$ with the Au plasmonic bowtie nano-antenna length are calculated, where ${Q_{c,m}}$ and ${V_{c,m}}$ represent the quality factor and mode volume of topological photonic-plasmonic hybrid nanocavity (the subscript c denotes cavity, and m denotes metal). The value of ${Q_{c,m}}$ decreases when the nano-antenna length increases. This mainly arises from the more mental losses in larger nano-antennas [29] and much energy dissipates into the free space. The value of mode volume ${V_{c,m}}$ behaves a downward trend with the increase of Au plasmonic nano-antennas length, since larger nano-antennas have stronger ability to localize light [30]. The coupling between a single emitter and hybrid cavity mode is also calculated. The quantum emitter we used here is a typical PbS quantum dot, because the resonant wavelength of the PbS quantum dot matches the resonant wavelength of the topological photonic-plamonic hybrid nanocavity [31]. The corresponding dipole moment of the PbS quantum dot is $\mu = 5.8 \times {10^{ - 29}}{\text{ C}} \cdot {\text{m}}$ [32]. Considering the PbS quantum dot is located in the maximum of electric field and the dipole is parallel to the electric field, the variation of ${g_{c,m}}$ and ${\kappa _{c,m}}$with Au plasmonic nano-antennas length are calculated, as is shown in figure 3(b). The ${g_{c,m}} = { }\mu \sqrt {{\omega _{c,m}}/2\hbar {\varepsilon _0}{\varepsilon _c}{V_{c,m}}}$ and ${\kappa _{c,m}} = { }{\omega _{c,m}}/{Q_{c,m}}$ are coupling strength and decay rate of topological photonic-plasmonic hybrid nanocavity respectively. Here, ${\varepsilon _c}$ and ${\omega _{c,m}}$ are ${3.45^2}$ and $2{{\pi }} \times 229$ THz (corresponding to cavity resonant wavelength of 1310 nm). The value of coupling strength ${g_{c,m}}$ behaves a upward trend with the increase of nano-antennas length because the value of ${V_{c,m}}$ decreases with the increase of nano-antennas length. The value of decay rate ${\kappa _{c,m}}$ increases with the increase of nano-antenna because the value of ${Q_{c,m}}$ decreases with the increase of nano-antennas length. As is shown in figure 3(b), when the nano-antenna length side is below 75 nm, the value of coupling strength ${g_{c,m}}$ is always higher than the value of the decay rate ${\kappa _{c,m}}$. It indicates that strong coupling can be achieved in our topological photonic-plamonic hybrid cavity, since the condition of strong coupling ${g_{c,m}}/{\kappa _{c,m}} > 1$ is satisfied [33]. The maximum value of the coupling strength ${g_{c,m}}$ is three times larger than the decay rate ${\kappa _{c,m}}$, so the strong coupling is still realized when the PbS quantum dot is not in the maximum position of electric field. However, in the case of bare topological PhC cavity, coupling strength ${g_c}$ (black dash line) is always smaller than the decay rate ${\kappa _c}$ (red dot line), which indicates the strong coupling cannot be realized.

Zoom In Zoom Out Reset image size

In order to evaluate the ability of topological photonic-plasmonic hybrid nanocavity in enhancing light–matter interaction and the light-emitter coupling strength in the hybrid structure, the figure of merit ${Q_{c,m}}/{V_{c,m}}$ and single-atom cooperativity parameter ${C_{c,m}}$ (defined as ${C_{c,m}} = 4g_{c,m}^2/{\kappa _{c,m}}\gamma$) are both calculated, and γ is the spontaneous emission rate of the PbS quantum dot. As is shown in figure 3(c), the value of ${Q_{c,m}}/{V_{c,m}}$ and ${C_{c,m}}$ behave fluctuant since both of them are related to the mode volume ${V_{c,m}}$. The complex behavior of the mode volume ${V_{c,m}}$ in figure 3(a) might result from complicated interaction effect of light confinement by the total internal reflection and the photonic bandgap. The value of ${Q_{c,m}}/{V_{c,m}}$ and ${C_{c,m}}$ of topological photonic-plasmonic hybrid nanocavity is always greater than the value of ${Q_c}/{V_c}$ and ${C_c}$ of bare topological PhC cavity. The largest value of ${Q_{c,m}}/{V_{c,m}}$ and ${C_{c,m}}$ are obtained when the nano-antenna length side is 50 nm. The same parameters mentioned above are calculated with variation of the nano-antennas gap when the nano-antenna length maintains 50 nm. As is shown in figure 3(d), when the gap distance is decreased, the value of ${Q_{c,m}}$ and ${V_{c,m}}$ have two peaks at the position of 7.5 nm and 15 nm. It is indicates that metal loss does not have a significant effect on the value of ${Q_{c,m}}$ and the value of ${V_{c,m}}$ in these positions. As is shown in figure 3(e), the variation of the value of ${g_{c,m}}$ and the value of ${\kappa _{c,m}}$ with the decrease of gap are also calculated. The maximum of ${g_{c,m}}$ is at the position of 5 nm since ultra-small ${V_{c,m}}$ is obtained at this positon (see figure 3(d)). Besides, the value of ${g_{c,m}}$ is always larger than ${\kappa _{c,m}}$, it indicates that strong coupling can be realized in the topological photonic-plasmonic hybrid nanocavity. The variation of the value of ${Q_{c,m}}/{V_{c,m}}$ and the value of ${C_{c,m}}$ with the decrease of gap are calculated in figure 3(f), The largest figure of merit ${Q_{c,m}}/{V_{c,m}}$ of $1.5 \times {10^6}{(\lambda /n)^{ - 3}}$ is obtained when the gap is 5 nm, which is two orders more than that of bare topological PhC nanocavity. The largest single-atom cooperativity parameter ${C_{c,m}}$ of $1.1 \times {10^5}$ is also obtained when the gap is 5 nm, which is over 60 times than that of bare topological PhC nanocavity.

In addition to realizing strong coupling between light and a single emitter, topological photonic-plasmonic hybrid nano-cavity provides a robust platform to control coupling phase transition, where the strong coupling and weak coupling can be easily switched by adjusting the length and gap of the Au plasmonic bowtie nano-antennas. As is shown in figure 4(a), a color map of coupling phase transition shows the strong coupling area (${g_{c,m}}/{\kappa _{c,m}} > 1$, marked as 'S') and the weak coupling area (${g_{c,m}}/{\kappa _{c,m}} < 1$, marked as 'W') clearly [33, 34]. The black dotted line represents the value of ${g_{c,m}}/{\kappa _{c,m}}$ is 1. Since the quantum effects cannot be ignored when the gap is smaller than 5 nm and the size of Au bowtie nano-antennas is smaller than 25 nm, the minimum of gap and size are set to be 5 and 25 nm here. The value of ${g_{c,m}}/{\kappa _{c,m}}$ is increased firstly and then decreased with the increase of length when the gap G is 5, 15 and 20 nm. It becomes more difficult to realize strong coupling with the increase of antennas length, because much metal loss is introduced. However, when the length and gap are 85 and 5 nm respectively, the strong coupling is realized again because large size and small gap size Au plasmonic bowtie nano-antennas can realize ultra-small volume [35]. It can be seen that when the gap G is 10 nm, the value of ${g_{c,m}}/{\kappa _{c,m}}$ experience four period with the increase of length: increase, decrease, increase and decrease. The sudden decrease in the second periods is caused by smaller quality factor ${Q_{c,m}}$ and larger mode volume ${V_{c,m}}$, which leading to the figure of merit ${Q_{c,m}}/{V_{c,m}}$ is smaller than the around parameters. According to figure 4(a), it indicates that we can realize strong coupling or weak coupling by tuning the gap or length of nano-antennas in our topological hybrid nanocavity. If a weak coupling is expected, we can choose larger size and gap of nano-antennas. If strong coupling is needed, we can decrease the gap between nano-antennas or choose smaller nano-antennas. It provides an efficient method to manipulate the coupling phase transition between strong coupling and weak coupling. The variation of the value of ${Q_{c,m}}/{V_{c,m}}$ of topological hybrid cavity is also calculated with the variation of length and gap of Au plasmonic bowtie nano-antennas. A color map is shown in figure 4(b), when the length is invariant, the value of Q/V is increased with the decrease of the gap, because smaller mode volume can be achieved when the gap decreases [35]. When the gap is invariant, the value of Q/V is firstly increased and then decreased with the increase of the nano-antennas length. As is shown in figure 4(b), high figure of merit Q/V of the topological hybrid cavity can be obtained when we use the middle size Au plasmonic bowtie nano-antennas.

Zoom In Zoom Out Reset image size

In order to validate the coupling between light and a single emitter in our proposed hybrid structure has entered strong coupling region, we calculated the emission power spectrum of the single emitter (PbS quantum dot) based on Jaynes–Cummings model [36].

The Hamiltonian of the system is defined as:

$$\begin{align}{H_{JC}} = {\omega _c}{a^\dagger }a + {\omega _e}{\sigma _ + }{\sigma _ - } + g\left( {{a^\dagger }{\sigma _ - } + a{\sigma _ + }} \right),\end{align} \tag{ 1 }$$

where ${\omega _c}$ and ${\omega _e}$ are cavity resonance and quantum emitter emission resonance, ${a^\dagger }$ and $a$ are creation and annihilation operators of cavity mode, ${\sigma _ + }$ and ${\sigma _ - }$ are Pauli operator, the emission spectral function S(ω) of the emitter is defined as follows:

$$\begin{align}{\text{S}}\left( {{\omega }} \right) = \mathop \int \langle {\tilde \sigma _ + }\left( {\omega^{^{\prime}} } \right){\tilde \sigma _ - }\left( \omega \right)\rangle d\omega^{^{\prime}}, \end{align} \tag{ 2 }$$

where ${\tilde \sigma _ + }\left( {\omega^{^{\prime}} } \right)$ and ${\tilde \sigma _ - }\left( \omega \right)$ can be resolved by the frequency domain quantum Langevin equation:

$$\begin{align}\begin{array}{l} - i\omega \tilde a\left( \omega \right) = \left( { - i{\omega _c} - \kappa /2} \right)\tilde a\left( \omega \right) - ig{{\tilde \sigma }_ - }\left( \omega \right) - \sqrt {\kappa {{\tilde a}_{in,1}}\left( \omega \right)} \\ - i\omega {{\tilde \sigma }_ - }\left( \omega \right) = \left( { - i{\omega _e} - {\gamma _e}/2} \right){{\tilde \sigma }_ - }\left( \omega \right) - ig\tilde a\left( \omega \right)\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \sqrt {{\gamma _{in}}{{\tilde a}_{in, - }}\left( \omega \right)} , \end{array}\end{align} \tag{ 3 }$$

where the ${\gamma _e}$ is assumed be equal to ${\gamma _s}$. According to the definition ${\gamma _s} = \varepsilon _b^{\frac{1}{2}}{\mu ^2}\omega _e^3/\left( {3\pi {\varepsilon _0}\hbar {c^3}} \right)$ [33], the value of decay rate of emitter is 42.72 MHz. As is shown in figure 4(c), the orange solid curve represents the emission power spectrum of a single quantum emitter in topological photonic-plasmonic hybrid nanocavity and Rabi splitting is produced in the spectrum. For comparison, the emission power spectrum of a single quantum emitter in bare PhC nanocavity is also calculated, which is denoted by a blue dash curve. It is obvious that there no Rabi splitting happened. Thus, strong coupling can be realized in our proposed topological photonic-plasmonic hybrid nanocavity, but cannot be realized in bare topological PhC nanocavity.

Finally, we discuss the robustness of corner states under the presence of the bowtie nano-antennas. For 2D PhCs system, the corner states is characterized by quantized topological corner charge ${Q_c}$, which is determined by the edge polarizations ($p_x^{{v _y}},{\text{ }}p_y^{{v _x}}$) and bulk polarizations P = (${P_x}$, ${P_y}$) as:

$$\begin{equation}{Q_c} = p_x^{{v _y}} + p_y^{{v _x}} = 4{P_x}{P_y}.\end{equation} \tag{ 4 }$$

Joining trivial PhCs and nontrivial PhCs together, the corner states are protected existence when the quantized topological corner charge ${Q_c} = 1$, as well as corresponding edge polarizations $p_x^{{v _y}} = {\text{ }}p_y^{{v _x}} = 1/2$ [22]. The bowtie nano-antennas on the surface of the topological corner state nanocavity here can be seemed as a metal scatter, which has not affect the value of topological corner charge although it is lossy. The robustness of the corner sates preserved under the presence of the bowtie nano-antenna. In order to further improve the robustness of the topological photonic-plasmonic hybrid nanocavity, we introduce perturbations and defects around the cavity mode. As is shown in figures 5(a) and (b), we have moved and deformed the air holes randomly around the corner. The perturbations of displacement and deformation are denoted as $\Delta x$, $\Delta y$, $\Delta \delta$ and $\Delta \eta$. The perturbations to the proposed topological hybrid structures are Δx = Δy = 0.05a, $\Delta \delta$ = $-$0.07a and $\Delta \eta$ = 0.07a. In the case of perturbation, the value of Q/V of topological hybrid structures are shown as figures 5(a) and (b), which show that they are in the same order of magnitude compared with the value of Q/V of perfect topological hybrid structures. It indicates that the topological photonic-plasmonic hybrid nanocavity is almost immune to external perturbations and has higher tolerance of errors in fabrication. In comparison, the value of Q/V of trivial hybrid nanocavity is decreased more rapidly when introducing disorders and it is not in the same order of magnitude compared with the value of Q/V of perfect trivial hybrid structures. Thus, the combination of topological corner state nanocavity with plasmonic nanoantennas shows great advantages in robustness when compared with conventional defected-based PhC nanocavity with plasmonic nano-antennas, which is vulnerable to structure parameters due to without topological protection.

Zoom In Zoom Out Reset image size

In conclusion, we have proposed a topological photonic-plasmonic hybrid nanocavity which consists of a topological PhC nanocavity and a plasmonic nano-antenna. Since high quality factor of $2 \times {10^3}$ and ultra-small mode volume of $1.3 \times {10^{ - 3}}{(\lambda /n)^3}$ are realized simultaneously in the proposed hybrid structure, the highest figure of merit Q/V of $1.5 \times {10^6}{(\lambda /n)^{ - 3}}$ is obtained and it is two orders more than that of the bare topological PhC nanocavity. An ultra-high single-atom cooperativity parameter of $1.1 \times {10^5}$ is achieved and it is enhanced more than 60 times with respect to the bare topological PhC nanocavity, which makes the coupling between light and a single emitter enter a strong coupling region in topological photonic realm for the first time. Besides, strong coupling and weak coupling can be easily switched in the topological hybrid system by tuning the structure dimension of plasmonic nano-antennas. Through combining topological photonics with surface plasmon resonance, the work opens a new research direction for enhancing light–matter interaction. The topological photonic-plasmonic hybrid nanocavity provides a robust platform to realize strong coupling and control the coupling phase transition between light and a single emitter, which will play a key role in quantum optics, quantum information and topological lasers.

The authors acknowledge the financial support from the National Natural Science Foundation of China (Grant Nos. 91850117, 11654003, 61775003, 11734001, 91950204, 92050110, and 61775244), Beijing Institute of Technology Research Fund Program for Young Scholars, National Key Research and Development Program of China under Grant No. 2018YFB2200403, and Beijing Municipal Science & Technology Commission No. Z191100007219001.

All data that support the findings of this study are included within the article (and any supplementary files).

The authors declare that there are no relevant financial interests in the manuscript and no other potential conflicts of interest to disclose.