1. Introduction

Optical nanostructures are efficient architectures in tailoring the electromagnetic vacuum and controlling light-matter interactions. Among them, a novel kind of deep subwavelength polariton modes-surface plasmon modes (SPM), has been reported and observed in various structures [1–3]. It is known that the excitation of the SPM can lead to dramatically large density of the states near the supported boundaries, which often accompanied by prominent enhancements in the Purcell factor [4,5]. In this aspect, systems consist of quantum emitters coupled to plasmonic platforms are of both fundamental and practical interest due to their ability in modifying dipole radiations. They also give rise to potential applications in quantum domains such as superradiance [6], single photon sources [7], quantum entanglement [8], resonance fluorescence [9] and quantum interference [10]. The two-dimensional metasurfaces are uniaxial structures with either opposite or the same sign in the effective surface conductivities. So far, surface polariton modes with both the elliptical and hyperbolic dispersion topologies have been reported in metasurfaces composed of silver [11], graphene [12], black phosphorus (BP) [13,14], van der Waals materials [15] and hexagonal boron nitride [16]. Among them, two-dimensional anisotropic metasurfaces like BP and graphene heterostructures possess actively tunable electronic and optical properties. Based on these novel features, they have found numerous applications in plasmonics and optoelectronics [17–20].

As a building block of understanding the mechanism lies behind the light scattering in the steady-state regime, the resonance fluorescence of driven systems has attracted attentions since it was proposed [21–23]. Many intriguing phenomena demonstrating the quantum features of dipole radiations via coherent pumpings have been reported, such as squeezing in resonance fluorescence [24], photon antibunching [25] and ultra-coherent photons generation [26]. Squeezed light-a nonclassical state of radiation, now becomes an important resource for applications in quantum information areas [27,28]. The squeezed states of light can be generated by reducing the variance of the electric field below the shot-noise limit, which are useful in the quantum metrology [29,30], gravitational wave detections [31], quantum key distributions [32,33], continuous variable quantum computings [34,35] and quantum sensing [36]. There are various theoretical methods in producing the squeezed states of light, and a wide range of experiments have been accomplished to explore their practical applications. These include the Gaussian squeezed states as well as non-Gaussian types, where the latter often behaves more ’non-classically’ and can be generated via non-Gaussian operations on the Gaussian coherent states [37–39]. Moreover, squeezed state generations via multiphoton processes containing four-wave mixing in atomic and solid-state systems [40–42], optical parametric amplifiers [43,44], cavity optomechanical or cavity quantum electrodynamical systems [45–48], and Bose-Einstein condensations [49] are also intensely investigated by scientists.

It was first predicted by Walls and Zoller in 1981 that non-Gaussian quadrature squeezing can be generated in the resonance fluorescent field of a driven two-level quantum emitter, which is originated from the buildup and survival of the steady-state coherence in the weak excitation regime [50]. Recently, this nonclassical phenomenon was also observed in the semiconductor quantum dot platform [51], which possesses higher photon collection efficiency compared to the atomic approach [52]. In the theoretical work, Grünwald and Vogel demonstrated the possibility in increasing the squeezing via atomic-state purifications in atom-cavity coupled systems. It was found that the atomic coherence is important in optimizing the squeezing in resonance fluorescence [53]. The results also indicate the enhancement of squeezing in the radiation field from a coherent driven quantum emitter by coupling dipoles to an optimal environment, which is no longer limited to weakly driven systems. Besides, they predicted the large enhancement of squeezing by coupling quantum emitters to the surface plasmon modes supported by the gold nanospheres [54]. The antenna effect [55] boosts the squeezing in the far field, which is robust against the dephasing of the solid-state system owing to the strong light-matter interactions at the plasmon resonances. However, few works focus on the generation of squeezed light with high tunability through exciton-environment couplings in driven QD systems [56], where the surface plasmon modes [57,58] serve as the tunable photonic reservoir.

In this work we investigate the anisotropic Purcell effect as well as the resonance fluorescence properties of a driven cascaded exciton-biexciton quantum dot coupled to anisotropic metasurfaces. The central frequency of the drive is tuned to half of the biexciton energy such that the two-photon resonance condition is satisfied. Starting from the Green’s functions, the influence of the carrier concentration on the Purcell factor [59] is investigated. We show that the plasmon resonances are highly tunable, and the Purcell factors of the orthogonal in-plane dipoles exhibit large differences in the frequency domain. Based on these intriguing properties, we then study the steady-state populations of the driven QD. It is shown that different energy levels can be selectively populated by manipulating the carrier concentration of the metasurfaces [60]. As the result, the resonance fluorescent field exhibits asymmetric Mollow peaks. Finally we find that two-mode squeezing of the photon pairs with different resonances and polarizations can be generated and further enhanced according to the variations in the carrier concentration.

Our paper is organized as follows. In Sec. 2, the model is introduced and the Green functions of the electromagnetic field are derived by applying the boundary conditions. Then the influence of the carrier concentration on the Purcell factor is discussed. In Sec. 3. the Bloch equations of the dressed QD are obtained according to the master equation, where the steady-state populations, the resonance fluorescence and the two-mode noise spectra at different carrier concentrations are studied. Finally we present our conclusions in Sec. 4. and an appendix is given to gain insights into the main results of this work.

2. Anisotropic Purcell effect mediated by anisotropic metasurfaces

The system under consideration can be well described by a biexciton QD coupled to the BP metasurfaces, where the QD is driven by a laser and the conductivity of the metasurfaces takes the form $\overset {\leftrightarrow }{\boldsymbol {\kappa }}=diag\left (\kappa _{xx},\kappa _{yy}\right )$. As illustrated by Fig. 1, the QD can be effectively treated as a four-level transition system, with bare states defined as the biexciton state $\left |B\right \rangle$, the $X-$exciton state $\left |X\right\rangle$, the $Y-$exciton state $\left |Y\right \rangle$ and the ground state $\left |G\right \rangle$. Here the small energy splitting between two excitonic states is neglected [61] and the resonant frequency is taken as $\omega _{e}=400\,\textrm {THz}$, the energy of the biexciton state is equal to $2\hbar \omega _{e}-\chi$ owing to the existence of the binding energy $\chi$ [62]. For linearly polarized excitations, it is possible to induce the transitions from the ground state to the biexciton state through two orthogonal channels, with photonic polarizations along the $x-$ or $y-$direction. The $y-$polarized transition dipoles $\boldsymbol {p}_{y1}$ and $\boldsymbol {p}_{y2}$ couple to the drive with Rabi frequencies $\Omega _{1}$ and $\Omega _{2}$, where the undressed transition dipoles are denoted by $\boldsymbol {p}_{x1}$ and $\boldsymbol {p}_{x2}$. All the dipoles are modeled in the point dipole approximation and we take $\boldsymbol {p}_{x1}=\boldsymbol {p}_{y1}=60\,\textrm {D}$, $\boldsymbol {p}_{x2}=\boldsymbol {p}_{y2}$ for further calculations. In this configuration, the spontaneous emission of the QD takes place via both the $x$- and $y$-polarized transitions, where the relevant strength is modified by the metasurfaces owing to the strong coupling with the SPM. Then within the rotating wave approximation, after rotating with the laser’s frequency the unperturbed and the interaction Hamiltonians appear to be

 

Fig. 1. Interaction between a driven QD and the anisotropic metasurfaces (here we consider BP structures): the QD couples to the anisotropic SPM supported by the surfaces. The pump frequency is reasonably chosen such that two-photon resonance condition is satisfied in the transition channel $\left |G\right \rangle \leftrightarrow \left |Y\right \rangle \leftrightarrow \left |B\right \rangle$. In the energy level diagram, the spontaneous emission of the exciton and biexciton states are modified by the anisotropy of the surfaces, the corresponding decay rates are denoted by $\gamma _{jj}^{y}$ ($y-$polarized transitions) and $\gamma _{jj}^{x}$ ($x-$polarized transitions).

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The Green’s functions $\overset {\leftrightarrow }{\mathbf {G}}_{yy}\left (\mathbf {r},\mathbf {r}'\right )$ and $\overset {\leftrightarrow }{\mathbf {G}}_{xx}\left (\mathbf {r},\mathbf {r}'\right )$, which are diagonal components of the Green’s dyadic, plays an important role in determining the spontaneous decay property of the driven QD. Both of them can be separated into the principle and the scattered parts (denoted by the superscript $P$ and $R$, respectively), where the former part originates from the dipole radiation in the free space and the later part reflects the influence of the metasurfaces on light-matter couplings owing to the excitation of the SPM. Notice that the principle part is determined by the isotropic radial wave number $k$ and the scattered part depends unequally on the wave numbers along two optical axes ($\mathbf {k}=k_{x}\mathbf {e}_{x}+k_{y}\mathbf {e}_{y}$, $\mathbf {e}_{x(y)}$ is unit vectors in the $x(y)$ direction), which can be clearly explained by exploring the dispersion of the SPM. Other parameters are: $\eta =\sqrt {\mu _{0}/\varepsilon _{0}}$ ($\varepsilon _{0}$ and $\mu _{0}$ are vacuum permittivity and permeability), $k_{0}$ is the vacuum wave number and $p=\sqrt {k_{0}^{2}-k_{x}^{2}-k_{y}^{2}}$ is the $z-$component of the wave vector. The denominator of the reflection coefficient $g_{yy(xx)}^{R}\left (\mathbf {k}\right )$ is denoted by $D(\mathbf {k})=2\kappa _{xx}\left (k_{x}^{2}-k_{0}^{2}\right )+2\kappa _{yy}\left (k_{y}^{2}-k_{0}^{2}\right )+4pk_{0}\eta ^{-1}\left (1+\kappa _{xx}\kappa _{yy}\eta ^{2}/4\right )$, which determines the isofrequency contour of the SPM. It is known that in heterostructures containing BP or graphene [12–14], the intraband excitations can be well described by the Drude model [65] and the interband transitions at high frequencies are defined by the step absorption function. In the following we consider the dispersive BP layers with the conductivity components [66]

It has been demonstrated that two-dimensional structures are good candidates in supporting anisotropic excitations of the SPM [69–71]. In most cases, the increase in the carrier concentration will lead to declines in the real part of the conductivity due to Pauli blocking, whereas the imaginary part suffers prominent growths at the same time. The imaginary part of the conductivity often ranges from $\mu \textrm {S}$ to $m\textrm {S}$ for different doping levels, where the real part is negligibly small [14,72] for the strong excitation cases of the SPM. Owing to this reason, metasurfaces with high carrier concentrations can exhibit strong electromagnetic responses and low dissipations. In the proposed systems, it is instructive to inspect the Purcell factor [59] in order to understand the behavior of light-matter coupling in the presence of the BP metasurfaces. In the proper scheme of the field quantization, the spontaneous decay of dipoles with different polarizations is often related to the imaginary part of the diagonal Green’s functions [73]. Based on this property, the Purcell factors can be defined as $\Gamma _{y(x)}/\Gamma _{0}=6\pi \left (c\,\omega _{e}^{-1}\right )^{3}\textrm {Im}[\overset {\leftrightarrow }{\mathbf {G}}_{yy(xx)}\left (\mathbf {r}_{Q},\mathbf {r}_{Q},\omega _{e}\right )]$ ($\Gamma _{y(x)}$ denotes the spontaneous decay rate of the $y(x)$-polarized dipole), where the vacuum decay rate takes the form $\Gamma _{0}=\omega _{e}^{3}p_{y(x)}^{2}/3\varepsilon _{0}\pi \hbar c^{3}$. In the previous work, we found that the total coupling between quantum emitters and the SPM supported by anisotropic metasurfaces can be realized under certain conditions [78], where the radiations into both the free vacuum modes and the absorption of materials are negligible.

In Figs. 2(a) and 2(b) we inspect the Purcell factor of dipoles with different polarizations as a function of the excitation frequency and the carrier concentration, where the distance between the QD and the metasurface is assumed to be $20\,\textrm {nm}$ such that the SPM has a dominant contribution on the dipole radiations. The results demonstrate strong enhancements in the Purcell factor of the $y-$oriented dipole, which is mainly due to the Drude-like behavior of the conductivity components for excitation frequencies below the threshold of the interband transitions (see Fig. S1 for more details). Also we notice that although the spontaneous emission of the $x-$polarized dipoles is enhanced, the corresponding Purcell factor is much smaller than that of the $y-$polarized dipoles. However, the situation changes quite a lot at higher frequencies. In this regime, the Purcell factor of the $x-$polarized dipoles becomes much larger for small carrier concentrations, which indicates a predominant contribution of the $x-$polarized dipoles on the spontaneous decay. The unequal enhancements in the Purcell factors can be attributed to the deformation of the isofrequency contours toward one of the optical axes (see Fig. S1 for more details). For example, at low excitation frequencies the isofrequency contours of the surface plasmon modes become quasi-elliptical, which are elongated along the $y$ axis. As the result, the SPM are mostly guided in the directions with small angles respect to the $x$ axis, which is the main reason for the prominent enhancement in the Purcell factor of the $y-$polarized dipole. On the other hand, it is shown that the variations in the carrier concentration can effectively modify the Purcell factors. A novel feature is that the the plasmon resonance suffers blueshifts with the increase in the carrier concentration, where declines in the strength of the Purcell factors take place at the same time. That is to say, the in-plane coupling between the dipoles and the BP metasurfaces turns out to be highly anisotropic, where the coupling strength is highly tunable via the gate doping.

 

Fig. 2. Purcell factors of dipoles with different polarizations coupled to the BP metasurfaces, different carrier concentrations $n_{c}=\left (0.2,0.5,1,1.5,2\right )\times 10^{18}/\textrm {m}^{2}$ are considered to stress the modification of the metasurfaces on the dipole radiation. The spatial coordination of the QD is $\mathbf {r}_{Q}=\left (0,0,20\right )\textrm {nm}$, the material parameters are $m_{x}=0.2m_{0}$, $m_{y}=0.7m_{0}$, $\omega _{r}=2.5\textrm {THz}$, $\varpi _{x}/2\pi =240\textrm {THz}$, $\varpi _{y}/2\pi =80\textrm {THz}$, $\mathcal {I_{\mathit {x}}}=1.5\kappa _{0}$ and $\mathcal {I_{\mathit {y}}}=3.5\kappa _{0}$, here $\kappa _{0}=e^{2}/4\hbar$ is the conductance quantum and $m_{0}$ is the free electron mass. (a) The Purcell factor of $y-$polarized dipoles. (b) The Purcell factor of $x-$polarized dipoles.

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3. Tunable resonance fluorescence and noise spectra in the steady-state regime

Based on the above discussions, the application of the anisotropic metasurfaces indeedly opens a new avenue for controlling light-matter interactions. In this section, we will mainly focus on its influence on the radiation properties of the driven QD. It is known that the dynamical and spectral features of the driven systems can be well understood by working in the dressed state picture. After diagonalizing the unperturbed Hamiltonian defined in Eq. (1), the dressed state basis [74] of the system under consideration can be expressed in forms of the combinations of the bare QD operators as

In order to investigate the influence of the anisotropic Purcell effect on the population distributions of the QD, in Fig. 3 we plot the steady-state populations versus the Rabi detuning $\Delta _{R}$ by considering different carrier concentrations, where other parameters are the same as in Fig. 2. It can be seen from Fig. 3(a) that for low carrier concentrations, there are prominent population differences between the state $\left |+\right \rangle$ and other energy states when $\Delta _{R}<0$. When the carrier concentration increases to $n_c=2\times 10^{17}/m^{2}$, the steady-state population of the dressed QD is rapidly accumulated to the state $\left |X\right\rangle$ with the increase of the Rabi amplitude $\Omega _{2}$, which is shown in Fig. 3(b).

 

Fig. 3. Steady-state populations of the energy states $\left |+\right \rangle$, $\left |-\right \rangle$, $\left |M\right \rangle$ and $\left |X\right \rangle$ (denoted by $\sigma _{++}^{ss}$, $\sigma _{--}^{ss}$, $\sigma _{MM}^{ss}$ and $\sigma _{XX}^{ss}$), as a function of the Rabi detuning ($\Delta _{R}=\Omega _{2}-\Omega _{1}$, the Rabi energy $\Omega _{1}=20\;\textrm {meV}$). The influence of the carrier concentration on the population distributions are studied: (a) $n_{c}=10^{17}/\textrm {m}^{2}$, (b) $n_{c}=2\times 10^{17}/\textrm {m}^{2}$, (c) $n_{c}=1.1\times 10^{18}/\textrm {m}^{2}$ and (d) $n_{c}=2.2\times 10^{18}/\textrm {m}^{2}$, other parameters are the same as in Fig. 2.

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Figures 3(c) and 3(d) demonstrate the behavior of the steady-state populations at large carrier concentrations. Obviously, the increases in the carrier concentration can effectively redistribute the populations of the energy states. This can be well understood by combining the results shown in Fig. 2, where the growth of the carrier concentration leads to blueshifts in the resonant peak of the Purcell factor. As the result, different transition channels can be selectively enhanced without changing the driving conditions. In other words, the anisotropic metasurfaces act as a tunable reservoir coupled to the dressed QD, which provides unique capabilities to control the steady-state properties of the driven system. For example, as indicated by Fig. 3(c), the dressed state $\left |+\right \rangle$ is largely populated for the Rabi detuning $\Delta _{R}=-8.3\;\textrm {meV}$. However, under the same driving conditions a prominent population inversion between the state $\left |-\right \rangle$ and other states can be achieved by only increasing the carrier concentration, with steady-state population $\rho _{--}^{ss}\approx 0.97$ (see Fig. 3(d)). Thus it is clear that the steady-state populations of the dressed levels can be well controlled by manipulating the material parameters or driving conditions.

For the system under consideration, the steady-state spectrum of the resonance fluorescence is proportional to the Fourier transformation of the correlation function $\lim _{t\rightarrow \infty }\langle \hat {\mathbf {E}}_{-}\left (\boldsymbol {r},t+\tau \right )\cdot \hat {\mathbf {E}}_{+}(\boldsymbol {r},t) \rangle$, where $\hat {\mathbf {E}}_{+}(\boldsymbol {r},t)$ and $\hat {\mathbf {E}}_{-}(\boldsymbol {r},t)$ are positive and negative parts of the dipole field in the far zone [75], respectively. In our notation, the resonance fluorescence originates from the sideband transitions can be generally written as

We now focus on how the BP metasurfaces modify the resonance fluorescence of the driven QD. The analysis will be carried out in the dressed state picture, where the function $\textrm {S}_{jl}(\delta _{L})$ corresponds to the spectrum of the two-photon emission via the transition $\left |j\right \rangle \rightarrow \left |l\right \rangle \rightarrow \left |j\right \rangle$). It is known that the two-time correlation functions can be evaluated by applying the quantum regression theory [77] on the Bloch equations given in Eqs. (14) and (15). Within the secular approximation, the couplings between the populations and coherences are neglected, thus the resonance fluorescence spectra can be decomposed as a sum of Lorentzian peaks. Numerical results are displayed in Fig. 4, where the influence of the carrier concentrations is taken into account. The resonance fluorescence spectra of different transitions are plotted separately to clearly illustrate the modification the metasurfaces. Notice that the peak locations of the spectra are solely determined by the driving condition and thus are insensitive to the carrier concentration. However, the peak values are strongly modified as the carrier concentration changes. As the result, the peak signals can exhibit tunable asymmetries. For example, as depicted in Fig. 4(f), distinct Mollow peaks of the spectrum $\textrm {S}_{+-}(\delta _{L})$ can be observed when $n_{c}$ varies from $10^{17}/\textrm {m}^2$ to $2.2\times 10^{18}/\textrm {m}^2$. This fact is exactly a signature of the selective enhancement in the dipole transitions. More detailed, from Fig. 3(a) it is clear that the state $\left |+\right \rangle$ is largely populated under the conditions $\hbar \Delta _{R}=-3\;\textrm {meV}$ and $n_{c}=10^{17}/\textrm {m}^2$, which is the reason for the appearance of the strong fluorescence peak at the right sideband. On the contrary, when $n_{c}$ increases to $2.2\times 10^{18}/\textrm {m}^2$ the population of the state $\left |-\right \rangle$ becomes prominent, thus a strong peak at the left sideband occurs. Similar modifications on the resonance fluorescence spectra corresponds to other sideband transitions can also be found under certain conditions, which are illustrated by Figs. (4(a))–(4(e)).

 

Fig. 4. Steady-state resonance fluorescence spectra versus the photon-laser detuning ($\delta _{L}=\omega -\omega _{L}$) and the carrier concentration of the metasurfaces, the material parameters are the same as used in Fig. 2. The resonance fluorescence spectra denoting distinct two-photon emissions are plotted: (a) $\left |M\right \rangle \rightarrow \left |-\right \rangle \rightarrow \left |M\right \rangle$, $\hbar \Delta _{R}=-1\;\textrm {meV}$, (b) $\left |M\right \rangle \rightarrow \left |+\right \rangle \rightarrow \left |M\right \rangle$, $\hbar \Delta _{R}=-3\;\textrm {meV}$, (c) $\left |M\right \rangle \rightarrow \left |X\right \rangle \rightarrow \left |M\right \rangle$, $\hbar \Delta _{R}=3\;\textrm {meV}$, (d) $\left |X\right \rangle \rightarrow \left |+\right \rangle \rightarrow \left |X\right \rangle$, $\hbar \Delta _{R}=3\;\textrm {meV}$, (e) $\left |X\right \rangle \rightarrow \left |-\right \rangle \rightarrow \left |X\right \rangle$, $\hbar \Delta _{R}=-1\;\textrm {meV}$, and (f) $\left |+\right \rangle \rightarrow \left |-\right \rangle \rightarrow \left |+\right \rangle$, $\hbar \Delta _{R}=-3\;\textrm {meV}$. The carrier concentrations are taken as $n_{c}=(0.2,1.1,2.2)\times 10^{18}/\textrm {m}^2$ for subfigures (a), (c), (d), (e) and $n_{c}=(0.1,1.1,2.2)\times 10^{18}/\textrm {m}^2$ for subfigures (b), (f).

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In the driven systems, quadrature squeezing occurs when the noise spectra exhibit negative peaks. Notice that the photon pairs generated via distinct two-photon transitions are correlated in the frequency, thus by transforming the spectral functions into the frequency domain, the two-mode noise spectra can be defined through

To provide a general scheme for the generation of two-mode squeezing, we plot the noise spectra as the function of the photon-sideband detunings under different carrier concentrations, where the leapfrog arrows denote different two-photon transitions and the transition frequencies are also given (the level energies are $E_{+}$, $\chi /2$, $0$ and $E_{-}$ for the states $\left |+\right \rangle$, $\left |X\right \rangle$, $\left |M\right \rangle$ and $\left |-\right \rangle$). As indicated by the results, the carrier concentration of the BP metasurfaces plays a key role in the generation and the enhancement of squeezing. In Fig. 5(a), the noise spectrum $\textrm {S}_{M-}^{in}$ exhibits remarkable squeezing for $n_{c}=2.2\times 10^{18}/\textrm {m}^2$ at the Rabi detuning $\Delta _{R}=-1\;\textrm {meV}$. Recalling the result of Fig. 3(d), it is clear that in this case strong population inversion can be observed, where the steady-state population of the state $\left |-\right \rangle$ is much larger than that of the state $\left |M\right \rangle$. Thus the term $\Omega _{1}\sigma _{MM}^{ss}-\Omega _{2}\sigma _{--}^{ss}$ is effectively modified and leads to the enhancement of squeezing in the noise spectrum $\textrm {S}_{M-}^{in}$. In addition, the result also indicates a photon-sideband detuning of $0.25\;\textrm {meV}$ in order to obtain half of the maximum squeezing.

 

Fig. 5. Two-mode noise spectra of the in-phase and out-of-phase quadratures versus the photon-sideband detunings under different carrier concentrations, where the diagram of the related two-photon transitions are also drawn (including the transition channels and the resonant frequencies, the emissions with orthogonal polarizations can be distinguished from the arrows marked in different colors). The material parameters are the same as in Fig. 3, with the Rabi detunings taken as: (a) $\Delta _{R}=-1\;\textrm {meV}$, (b) $\Delta _{R}=-2\;\textrm {meV}$, (c) $\Delta _{R}=1\;\textrm {meV}$, (d) $\Delta _{R}=10\;\textrm {meV}$, (e) $\Delta _{R}=6\;\textrm {meV}$ and (f) $\Delta _{R}=-12\;\textrm {meV}$.

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The production of squeezing via other two-photon transitions are demonstrated in Figs. 5(b)–5(f). From previous discussions one may notice that the coupling between the QD and the metasurfaces is important in controlling the steady-state populations. As the result, two-mode squeezing can be produced and largely enhanced by manipulating the carrier concentration, where the photon-sideband detuning ranges from $0.014\;\textrm {meV}$ (Fig. 5(f)) to $0.57\;\textrm {meV}$ (Fig. 5(e)) in order to achieve half degree of the maximum squeezing. Moreover, Figs. 5(b) and 5(d) clearly illustrate the enhancements in the squeezing via manipulations on the carrier concentration. This behavior can be attributed to the prominent population accumulations in one of the energy states corresponds to the two-photon emissions. For example, in Fig. 5(d) the photon-sideband detuning of the noise spectrum strongly depends on the real part of the polarization coefficient $f_{X+}=i\Omega _{-}-\left (\Gamma _{++}+\sum _{m=+,M,-}\Gamma _{{X}\scriptstyle m}+\sum _{n=X,M,-}\Gamma _{{+}\scriptstyle n}\right )$. The decrease in the carrier concentration then results in the accumulation of the population into the state $\left |+\right \rangle$ (see Fig. 3), which indicates a decline in the decay rate from the state $\left |+\right \rangle$ to other energy states, i.e., the decay term $\sum _{n=X,M,-}\Gamma _{{+}\scriptstyle n}$ turns out to be smaller according to the steady-state solutions given in Eq. (15). Thus in one aspect, gains in the steady-state population $\sigma _{++}^{ss}$ prominently enhances the squeezing. On the other hand, the declines in the level decay rates reduces the tolerance of the photon-sideband detunings in the generation of squeezing. As demonstrated by the green and blue solid lines in Fig. 5, squeezing can be largely enhanced at the cost of declines in the tolerance of the photon-sideband detunings.

4. Conclusions

In conclusion, we have investigated the Purcell effect and the properties of the resonance fluorescent field in hybrid systems consist of a driven four-level biexciton QD coupled to the BP metasurfaces. The dispersive model of the material is introduced [66] to describe the optical response of the metasurfaces at different frequencies. Within the framework of the quantization scheme in the presence of inhomogeneous dissipative media [63], the dipole decay rates are expressed in terms of the Green’s functions [73]. We have shown that the Purcell factors of the orthogonal in-plane dipoles display unequal enhancements in different frequency regions for certain material parameters. In the meantime, the plasmon resonance exhibits prominent shifts according to the variation of the carrier concentration,which is experimentally achievable by the methods of biasing [79] or chemical doping [80]. Further studies indicate that the energy levels of the driven QD can be selectively populated by engineering the anisotropy of the metasurfaces. Based on this novel feature, we have explored the steady-state properties of the resonance fluorescence and the noise spectra under the two-photon resonance condition. As demonstrated, the resonance fluorescence of the driven QD are strongly modified by the metasurfaces owing to the couplings between transition dipoles and the SPM. Moreover, the generation of two-mode squeezing in the resonance fluorescence relies on the control of the steady-state populations of the driven system, which is feasible by manipulating the carrier concentration of the materials. We have also found that, although at the cost of declines in the tolerance of photon-sideband detunings, the degrees of squeezing can be largely enhanced according to the variation in the carrier concentration.

Beyond the BP metasurfaces considered in this work, it has been reported that anisotropic polariton modes can also be supported by two-dimensional or nanostructured metasurfaces like arrays of graphene nanoribbons [12], carbon phosphide [81] and van der Waals layers [15,82]. From this point of view, our proposal is promising in developing other tunable hybrid systems containing the metasurfaces mentioned above. After considering all the merits, we believe that our proposal is useful for the realization of integrated photonic platforms with high tunability, which may have potential applications in the atomic state manipulation, squeezed light generation and other quantum technologies.