Controllable radiation properties of a driven exciton-biexciton quantum dot couples to a graphene sheet

Wei Fang; Gao-Xiang Li; Yaping Yang

doi:10.1364/OE.26.029561

1. Introduction

The optical properties of quantum emitters have been intensively investigated and proved to be dramatically modified when they are located in close proximity to plasmonic structures [1–11]. Different from the vacuum case, the strong local field restricted near the surface of the plasmonic structures leads to an enhancement of the Purcell factor [12, 13]. Graphene, a two-dimensional version of graphite, which can be treated as a semiconductor with zero bandgap, possesses tunable conductivity [14] and has promising applications in terahertz science [15, 16]. The tunability of its conductivity lies in the high mobility of the carrier concentration [17], where the Fermi energy (or chemical potential) can be modulated by either applying a gate voltage [18, 19] or different chemical dopings [20, 21]. In far infared region and low temperature limits, the intraband transition has a prominent contribution to the conductivity according to the Kubo formula [22]. Meanwhile, as predicted by the theoretical results [23, 24], strong plasmon absorption peaks have been successfully measured in laboratories [25, 26].

Resonant interaction between light and matter have attracted significant interest in the fundamental researches [27, 28] and plays a crucial role in a plethora of applications [29–31]. Basically, even the coherent interaction between an optical field and a two-level system leads to a diverse range of intriguing phenomena, such as nonclassical and ultra-coherent light emission for driven quantum dots [32], the observation of the Mollow triplet [33] and the quadrature squeezing in the resonance fluorescence [34]. In large amount of the quantum optical experiments, the semiconductor quantum dots (QD) exhibits narrow emission linewidth [35] and atomlike density of states [36], thus can be described as quantum systems with discrete energy levels. State-manipulation through coherent optical excitations, which have been illustrated by Rabi oscillations of the exciton occupation and conditional control of the biexciton system (consisting of two electron-hole pairs) [37, 38], promotes the applications of multi-level QD in quantum optics and quantum information [39, 40].

As a key point to study the optical properties of the photons scattered by coherently driving QDs, the studies on the resonance fluorescence have been carried out both theoretically and experimentally [30, 41–47]. In the domain of strong field excitation, the resonance fluorescence spectrum often consists of several distinct spectral resonances, which denote different transitions of the dressed QD. Meanwhile, the strengths of sidebands are determined by the local density of states from different spectral regions of the photonic reservoir [48]. Furthermore, the influence of the pump power and phonon-QD interaction on the Mollow triplet sideband of the QD-cavity system has been reported in the experiment [49]. The scattered spectrum modified by the plasmonexciton coupling [50], and multi-Mollow triplets modulation by bichromatic drives [51] have also been investigated for QDs with different configurations. Besides, an important nonclassical feature of the resonance fluorescence-the quadrature squeezing effect [52], which has been widely studied owing to its wide applications in gravitational wave detection [53], quantum computing [54], quantum teleportation [55] and other quantum information fields. It has been shown that in strong cavity-atom coupling regime, maximum squeezing can be realized around the cavity frequency with little loss [56]. Recently, the quadrature squeezing of the resonance fluorescence field generated by large optical dipole has been successfully observed by Schulte et al, where the squeezed photons also exhibits antibunching property [34]. Although both theoretical and experimental researches on the generation of the squeezed light have been proposed and illustrated in the last four decades, few of them concerns the generation of two-mode squeezed state in the driven QD systems.

In this work, we study both the resonance fluorescence and noise spectra of an excitonbiexciton QD couples to the graphene-assisted reservoir. The drive couples to the horizontal polarized dipoles with different Rabi amplitudes, where the two-photon resonance condition is satisfied. The strong coupling between quantum emitters and the plasmonic modes supported by monolayer graphene sheet has been intensively studied owing to the tunability of graphene’s optical properties [18,57–60], where the modification in the energy structure leads to the changes of the collective modes of the electron gas i.e., the surface plasmon modes (SPM) [26]. Thus the influence of the chemical potentials on the Purcell factors are analysed by using the Green tensor method [61], which is important in illustrating the mode distribution of the electromagnetic field around the graphene sheet. We then show that the population imbalance in the steady-state regions can be achieved by tuning the chemical potential. As the result, the enhancement of the resonance fluorescence sidebands without population inversion can be observed. Finally we derive both the in-phase and out-of-phase quadratures of the noise spectra within the resonant two-photon emission scheme, and prove that even at room temperature, simultaneous squeezing of the photon pairs can be realized.

Our paper is organized as follows. In Sec. 2, the Purcell factors are given in forms of the Green functions, where the influences of the dipole polarization and the environment temperature are considered. Then we derive the master equation and discuss the influence of the chemical potential and the Rabi amplitudes on the steady-state occupations, simplified Bloch equations are given to better understand the transition processes. In Sec. 3 the resonance fluorescence and absorption spectra are studied for different chemical potentials. We investigate both the in-phase and out-of-phase quadratures of the two-mode noise spectra in Sec. 4, and present our conclusions in Sec. 5. Finally, appendixes are added to clarify some essential concepts of the work.

2. The system and its steady-state properties

The system under consideration consists of a semiconductor QD driven by a monochromatic laser and couples to a graphene sheet, where vacuum-exciton and exciton-biexciton transitions can both take place in the QD subsystem. As Fig. 1(a) indicates, the graphene sheet is placed in the x − y plane and the QD has a distance l above from the graphene layer, in the following part of the text we use ${\vec{r}}_{o} = (0, 0, l)$ to denote the spacial position of the QD.

Fig. 1 (a) Geometry of the system. The QD is placed above the graphene sheet and coherently driven by a laser field. Both the upper and lower half spaces are assumed to be vacuum. (b) The diagram of bare QD transitions. Two horizontal transition dipoles ${\vec{p}}_{1}$ and ${\vec{p}}_{2}$ are coherently driven by a linearly polarized laser with central frequency ω_L, which has an energy difference Δ from the single exciton state. Two undressed vertical polarized dipoles are denoted by ${\vec{p}}_{3}$ and ${\vec{p}}_{4}$ . The energy of excitonic states is ħω₀ and owing to the existence of the binding energy χ, the energy of biexcitonic state is denoted by 2ħω₀ − χ.

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In solid-state systems, the atomic transitions can be replaced by the excitonic resonance of a quantum dot. According to Pauli’s principle of exclusion, only single excitons can be created for a certain circularly polarized excitation. For linearly polarized excitations, which is an equal combination of two circular (σ⁺ and σ⁻) excitation components, it is possible to induce the transitions from the ground state to the biexciton state through two orthogonal channels. Thus it is advantageous to introduce the horizontal and vertical single-exciton states denoted as $| H 〉 = (| σ^{-} 〉 + | σ^{+} 〉) / \sqrt{2}$ and $| V 〉 = i (| σ^{-} 〉 - | σ^{+} 〉) / \sqrt{2}$ , where the photons generated by two orthogonal decay paths are distinguishable in polarizations. In this configuration, the biexciton levels is modeled by a four-level “diamond” structure, as shown in Fig. 1(b). The excitation frequency of the excitonic state is denoted by ω₀ and locates in the terahertz region [62, 63], where we take ω₀ = 206 meV. The horizontal transition dipoles ${\vec{p}}_{1}$ and ${\vec{p}}_{2}$ , couple to the driving laser with Rabi amplitudes Ω₁ and Ω_2. The other two undressed dipoles, which are both vertically polarized, are denoted by ${\vec{p}}_{3}$ and ${\vec{p}}_{4}$ . All dipoles are modeled in the point dipole approximation, and we take ${\vec{p}}_{1} = {\vec{p}}_{3} = 50 D$ , ${\vec{p}}_{2} = {\vec{p}}_{4}$ for further calculations. The spontaneous decay of high energy levels then generates both horizontal and vertical polarized spectra, which are modified by coupling to the electromagnetic modes on the graphene surface [24]. Obviously two-photon resonance condition is satisfied when Δ = χ/2, where the detuning between the driving field and excitonic state is equal to half of the biexcitonic binding energy [64]. Within the dipole and rotating wave approximations, the Hamiltonian of the driven QD-reservoir system described above has the form

\begin{matrix} \hat{H} = ℏ \int d \vec{r} \int ω {\hat{f}}^{†} (\vec{r}, ω) \hat{f} (\vec{r}, ω) d ω + ℏ ω_{0} (σ_{H H} + σ_{V V}) + (2 ℏ ω_{0} - χ) σ_{B B} \\ [Ω_{1} (σ_{H G} e^{- i ω_{L} t} + σ_{G H} e^{i ω_{L} t}) + Ω_{2} (σ_{B H} e^{- i ω_{L} t} + σ_{H B} e^{i ω_{L} t})] \\ - [(σ_{H G} {\vec{p}}_{1} + σ_{B H} {\vec{p}}_{2} + σ_{V G} {\vec{p}}_{3} + σ_{B V} {\vec{p}}_{4}) \int d ω {\hat{E}}^{(+)} ({\vec{r}}_{o}, ω) e^{i (ω_{L} - ω) t} + H . c .], \end{matrix}

where the bosonic operators

{\hat{f}}^{†} (\vec{r}, ω)

and

\hat{f} (\vec{r}, ω)

represent the creation and annihilation of an excitation in the graphene-assisted reservoir [65], σ_ij = |i〉〈j| (i, j = G, H, V and B) are Pauli operators of the bare QD, ħω₀ and 2ħω₀ − χ are level energies of the exciton and biexciton states, where a small energy splitting between two excitonic states has been neglected [66]. In the presence of the boundary conditions, the positive frequency part of the field operator can be written in the following form [67]

{\hat{E}}^{(+)} ({\vec{r}}_{o}, ω) = \frac{i ω^{2}}{c^{2}} \int d \vec{r} \sqrt{\frac{ε^{″} (\vec{r}, ω) ℏ}{π ε_{0}}} \overset{\leftrightarrow}{G} ({\vec{r}}_{o}, \vec{r}, ω) \cdot \hat{f} (\vec{r}, ω),

with

ε^{″} (\vec{r}, ω)

denoting the imaginary part of the permittivity that bounded material holds. Here the Green tensor

\overset{\leftrightarrow}{G} ({\vec{r}}_{o}, \vec{r}, ω)

is introduced to depict the spacial distribution of the electromagnetic field in the upper and lower half spaces separated by the graphene sheet, where the explicit form has been given in Appendix A.

Before investigating the steady-state populations of the driving QD, firstly we focus on the spontaneous decay properties of the QD under the influence of the graphene-supported electromagnetic modes at different environment temperatures. It has been proved that graphene can be treated as an infinitesimally thin, local two-side surface characterized by the surface conductivity [22]

\begin{matrix} σ (ω, μ_{f}, T) = \frac{i e^{2} k_{B} T}{π ℏ^{2} (ω + i γ_{a})} [\frac{μ_{f}}{k_{B} T} + 2 \ln (e^{- \frac{μ_{f}}{k_{B} T}} + 1)] \\ + \frac{i e^{2} (ω + i γ_{e})}{π} \int_{0}^{\infty} \frac{d_{f} (ζ) - d_{f} (- ζ)}{4 ζ^{2} - ℏ^{2} {(ω + i γ_{e})}^{2}} d ζ . \end{matrix}

In the above expression, T represents the environment temperature, e is the charge of the electron, k_B is the Boltzmann constant and ħ is the reduced Plank constant. Obviously, the conductivity can be affected by experimental conditions. The Fermi energy (also called chemical potential) of the graphene µ_f can be simply modulated by changing the bias voltage, where the Fermi-Dirac distribution function $d_{f} (ζ) = {[e^{(ζ - μ_{f}) / k_{B} T} - 1]}^{- 1}$ varies according to the change of µ_f. The scattering rates of the intraband and interband transitions are denoted by γ_a and γ_e, which depend on the impurities of graphene [71]. However, the interband scattering only broadens the transition width and does not affect the main physics, thus in the following calculations we take γ_e = 0 for simplicity [72]. Meanwhile, the intraband transitions of graphene exhibit Drude-like dispersion response for the incident photons in the terahertz region [73], which indicates a metal-like behavior. Notice that in the first term of Eq. (3), the change of Fermi energy has a prominent influence on the intraband transition, which in turn affects the plasmonic absorption. This property provides the possibility in the realization of tunable SPM on the graphene sheet. For tailored electromagnetic environment due to the presence of boundaries, it is convenient to introduce the Purcell factor [12] to investigate the spontaneous emission behavior of the atomic system. According to the decay rates defined in Eq. (8), the simplified graphene-assisted Purcell factors are

R_{H (V)} (ω) = \frac{6 π c^{3}}{ω^{3}} Im [{\overset{\leftrightarrow}{G}}_{x x (z z)} ({\vec{r}}_{o}, {\vec{r}}_{o}, ω)] .

In Fig. 2 we investigate the Purcell factor of two orthogonal transitions in zero environment temperature case. It can be clearly seen in Figs. 2(a) and 2(b) that for a specific chemical potential, as the sharp peak indicates, the Purcell factor exhibits a Lorentz-like distribution over the frequency for both horizontal and vertical polarized transition dipoles. This metal-like behavior of the photon-field coupling can be well understood by analysing the plasmon contribution on QD’s decay, where the relevant results have been given in Eqs. (30) and (31) in Appendix A. It can be easily verified through the analytical expressions that the Lorentzian shape absorption originates from the interaction between the transverse magnetic (TM) surface modes and the radiated photons, where the coupling between the QD and transverse electric (TE) modes is weak in low terahertz region. This weak coupling between the QD and the TE modes then results in a steep decrease of the Purcell factor [23]. Notice that the increase in the chemical potential leads to a blue shift of the resonant absorption peak. Meanwhile, although the Lorentzian absorption of the plasmon field still holds, the Purcell factor suffers a decrease.

Fig. 2 Purcell factor of two orthogonal polarized dipoles. Subplots (a) and (b) are Purcell factors varies with the frequency and chemical potential, both the surface plot and its projection have been given. In subplots (c) and (d), the curves of the purcell factors at both zero and room temperatures have been considered, where the chemical potential is tuned to µ_f = 80 meV and µ_f = 150 meV, respectively.

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Figures 2(c) and 2(d) are curves of the Purcell factor with chemical potential µ_f = 80 meV and µ_f = 150 meV, where both horizontal/vertical polarized dipoles and different environment temperature have been considered. It is clear that for either horizontal or vertical dipoles, the resonant peaks of the Purcell factors have blueshifts for large chemical potential, and the Lorentzian type absorption becomes weak due to the broadening of linewidth. The reason lies in the varying of the intraband transition region, that the graphene-supported TM surface mode becomes able to couple to high frequency photons with the increase of the chemical potential. Meanwhile, the increase in the environment temperature has a destructive influence on the photon-field coupling, where the peaks of the Purcell factors suffer decreases and the curves appear to be distributed averagely over the frequency. Additionally, for large chemical potential the Purcell factors become insensitive to the variation of the environment temperature, as illustrated by Fig. 2(d). Thus by reasonable tuning the chemical potential of the graphene and controlling the environment temperature, one can selectively tailored the electromagnetic environment around the QD.

The systematic Hamiltonian in Eq. (1) contains the unperturbed and interacting terms, where the first eight terms characterize the unperturbed Hamiltonian of the whole system, denoting the total energy of the graphene-assisted reservoir, the bare QD and the coupling between the QD and driving laser. Besides, the left part in Eq. (1) describes the interaction between different transition dipoles and the electromagnetic modes supported by the reservoir, which has a prominent influence on the spontaneous decay process. By applying a canonical transformation on the unperturbed and interaction Hamiltonians, where the transforming operator is

U^{†} = \exp {i [\int d \vec{r} \int d ω ℏ ω {\hat{f}}^{†} (\vec{r}, ω) \hat{f} (\vec{r}, ω) + ℏ ω_{L} (σ_{H H} + σ_{V V}) + 2 ℏ ω_{L} σ_{B B}] t}

under the two-photon resonance condition, we can acquire the systematic Hamiltonian in the frame rotating with both the laser and the reservoir frequencies. Then continue with the transformed unperturbed Hamiltonian, after introducing dressed state basis and entering the interaction picture, the interaction Hamiltonian appears to be

{\tilde{V}}_{I} (t) = [σ_{H G} (t) {\vec{p}}_{1} + σ_{B H} (t) {\vec{p}}_{2} + σ_{V G} (t) {\vec{p}}_{3} + σ_{B V} (t) {\vec{p}}_{4}] \cdot \int d ω {\hat{E}}^{(+)} ({\vec{r}}_{o}, ω) e^{i (ω_{L} - ω) t} + H . c .,

where the time dependent Pauli operators σ_HG(t), σ_BH(t), σ_VG(t) and σ_BV(t) can be expressed in combinations of the Pauli operators in the dressed state representation scaled by the transformation coefficients, which are given in the Appendix B. Starting from the interaction Hamiltonian in the dressed state picture, it is convenient to perform the second-order perturbation calculation [68] to derive the master equation with the reservoir interaction included. We can then trace over the reservoir degrees of freedom to obtain the master equation for the reduced density matrix operator

\tilde{ρ}

of the QD through

\frac{\partial \tilde{ρ}}{\partial t} = - \frac{1}{ℏ^{2}} \int_{0}^{t} d τ {Tr}_{R} {[{\tilde{V}}_{I} (t), [{\tilde{V}}_{I} (t - τ), \tilde{ρ} (t) \otimes ρ_{R}]]},

where Born-Markovian approximation has been applied in the above equation. Under the approximation, operator

\tilde{ρ} (t - τ)

is replaced by

\tilde{ρ} (t) \otimes ρ_{R}

, that ρ_R is the initial reservoir operator and Tr_R denotes tracing over the reservoir variables. The bath is described by

{Tr}_{R} [{\hat{f}}^{†} (\vec{r}, ω) \hat{f} ({\vec{r}}^{'}, ω^{'})] = \bar{n} (ω) δ (\vec{r} - {\vec{r}}^{'}) δ (ω - ω^{'})

and

{Tr}_{R} [\hat{f} (\vec{r}, ω) {\hat{f}}^{†} ({\vec{r}}^{'}, ω^{'})] = [\bar{n} (ω) + 1] δ (\vec{r} - {\vec{r}}^{'}) δ (ω - ω^{'})

, where the distribution function

\bar{n} (ω) = {(e^{ℏ ω / k_{B} T} - 1)}^{- 1}

represents the average excitation number of the reservoir. After tedious deduction the master equation can be rearranged in the following form

\begin{matrix} \frac{\partial ρ}{\partial t} = - \frac{i}{ℏ} [{\hat{H}}_{0}^{'}, ρ] + \int_{0}^{\infty} d ω \int_{0}^{t} d τ e^{- i (ω - ω_{L}) τ} \\ \times {γ_{11}^{\bar{n} + 1} (ω) D_{G H, H G} + γ_{22}^{\bar{n} + 1} (ω) D_{H B, B H} + γ_{12}^{\bar{n} + 1} (ω) [D_{G H, B H} + D_{H B, H G}]} \\ + \int_{0}^{\infty} d ω \int_{0}^{t} d τ e^{- i (ω - ω_{L}) τ} {γ_{33}^{\bar{n} + 1} (ω) D_{G V, V G} + γ_{44}^{\bar{n} + 1} (ω) D_{V B, B V}} \\ + \int_{0}^{\infty} d ω \int_{0}^{t} d τ e^{i (ω - ω_{L}) τ} \\ \times {γ_{11}^{\bar{n}} (ω) D_{H G, G H} + γ_{22}^{\bar{n}} (ω) D_{B H, H B} + γ_{12}^{\bar{n}} (ω) [D_{B H, G H} + D_{H G, H B}]} \\ + \int_{0}^{\infty} d ω \int_{0}^{t} d τ e^{i (ω - ω_{L}) τ} {γ_{33}^{\bar{n}} (ω) D_{V G, G V} + γ_{44}^{\bar{n}} (ω) D_{B V, V B}} + H . c ., \end{matrix}

where the operators

D_{j l, m n}

are defined as

D_{j l, m n} = {\tilde{σ}}_{j l} (- τ) ρ {\tilde{σ}}_{m n} - {\tilde{σ}}_{m n} {\tilde{σ}}_{j l} (- τ) ρ .

The coefficients $γ_{i j}^{\bar{n} + 1} (ω)$ and $γ_{i j}^{\bar{n}} (ω)$ characterize the decay rates of the different transition dipoles owing to the interaction with the reservoir, where the influence of the bath temperature has been included through the thermal distribution function $\bar{n} (ω)$ . The explicit forms of the decay coefficients are

γ_{i j}^{M} (ω) = \frac{M ω^{2}}{ℏ π g ε_{0} c^{2}} {\vec{p}}_{i} \cdot Im [\overset{\leftrightarrow}{G} ({\vec{r}}_{o}, {\vec{r}}_{o}, ω)] \cdot {\vec{p}}_{j}^{*} .

In the above expression, the index M is used to denote the mean excitation number of the reservoir. Notice in the derivation of Eq. (8), the contributions of the cross-terms, which describe the interaction between two orthogonal dipoles, are vanished at QD’s position (the off-diagonal terms of the Green tensor are equal to zero according to Appendix A). As we will see later, the absence of this interactions then restrains the simultaneous squeezing of orthogonal polarized photon pairs. In real cases the effective dipoles depended on the experimental conditions and the growing methods, where in our further calculations the dipoles ${\vec{p}}_{2}$ and ${\vec{p}}_{4}$ are assumed to be equal and variable.

It is clear that when the environment is at zero temperature, all the reservoir-assisted decay rates with $\bar{n}$ mean excitations vanished and only the terms described by $γ_{i j}^{\bar{n} + 1} (ω)$ contribute to the QD’s decay. After ignoring the principle part of the integrals that describe small energy shifts of the levels [70], the simplified master equation can be derived within secular approximation, as shown in Eq. (37). The relevant Bloch equations can be rearranged in the form of diagonal terms

(\begin{matrix} 〈 {\dot{\tilde{σ}}}_{+ +} 〉 \\ 〈 {\dot{\tilde{σ}}}_{- -} 〉 \\ 〈 {\dot{\tilde{σ}}}_{00} 〉 \\ 〈 {\dot{σ}}_{V V} 〉 \end{matrix}) = (\begin{matrix} D_{+} & Γ_{- +} & Γ_{0 +} & Γ_{V +} \\ Γ_{+ -} & D_{-} & Γ_{0 -} & Γ_{V -} \\ Γ_{+ 0} & Γ_{- 0} & D_{0} & Γ_{V 0} \\ Γ_{+ V} & Γ_{- V} & Γ_{0 V} & D_{V} \end{matrix}) (\begin{matrix} 〈 {\tilde{σ}}_{+ +} 〉 \\ 〈 {\tilde{σ}}_{- -} 〉 \\ 〈 {\tilde{σ}}_{00} 〉 \\ 〈 σ_{V V} 〉 \end{matrix})

and the evolution of off-diagonal terms

〈 {\dot{\tilde{σ}}}_{j l} 〉 = C_{j l} 〈 {\tilde{σ}}_{j l} 〉 (j \neq l) .

Obviously, the motion equations illustrate that the populations of different QD states couple to each other during the evolution, where the coefficients D_l (l =+, 0, −, V) represent the total decay rates from level |l) to all other possible final states, and the corresponding physical scheme can be well understood through their definitions

\begin{array}{l} D_{+} = - \sum_{j = -, 0, V} Γ_{+ j}, D_{-} = - \sum_{j = +, 0, V} Γ_{- j}, \\ D_{0} = - \sum_{j = +, -, V} Γ_{0 j}, D_{V} = - \sum_{j = +, -, 0} Γ_{V j} . \end{array}

The reservoir-assisted decay coefficients, marked as Γ_mn(m ≠ n =+, 0, −, V), representing the transition rates of the dressed QD from state |m〉 to the state |n〉, their expressions can be found in Eq. (38).

The off-diagonal terms, which denote the horizontal and vertical polarizations ${\tilde{σ}}_{j l} (j \neq l = +, 0, -, V)$ , plays a key role in the determination of the quadrature squeezing properties of the resonance fluorescence field. It should be pointed out that the coefficients, which indicate the frequency shifts of different transition channels under the influence of the drive, are thus complex (the relations $C_{j l} = C_{l j}^{*}$ are satisfied) and appear to be

\begin{array}{l} C_{V - (+ 0)} = i {\tilde{Ω}}_{+} - \frac{1}{2} [Γ_{+ +} - D_{V (+)} - D_{- (0)}], C_{+ -} = i \tilde{Ω} - \frac{1}{2} (4 Γ_{+ +} - D_{+} - D_{-}), \\ C_{+ V (0 -)} = i {\tilde{Ω}}_{-} - \frac{1}{2} [Γ_{+ +} - D_{+ (0)} - D_{V (-)}], C_{V 0} = i Δ + \frac{1}{2} (D_{0} + D_{V}) . \end{array}

As shown in Fig. 3, the resonance fluorescence spectra generated through all possible transitions exhibit eight different frequencies, which is coincident with the results given in Eq. (14). Among these transitions, four of them are degenerate in frequency, but distinguishable in polarization. The transition rates of different decay channels, which are frequency dependent and described by Γ_mn in Eq. (11), will be influenced by both the driving conditions and the couplings to the reservoir at different spectral regions.

Fig. 3 The diagram of the energy levels and all possible transitions in the frame rotating with the laser frequency, where the blue arrows denote the horizontal transitions and red arrows denote the vertical transitions. Either the photons generated by the horizontal or the vertical transitions are distinguishable in frequency, which symmetrically distributed with respect to the drive.

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The steady-state solution of the driving system can be obtained through the time-independent formations of the Eq. (11), where we use ${〈 {\tilde{σ}}_{+ +} 〉}_{s s}$ , ${〈 {\tilde{σ}}_{- -} 〉}_{s s}$ , ${〈 {\tilde{σ}}_{00} 〉}_{s s}$ and ${〈 {\tilde{σ}}_{V V} 〉}_{s s}$ to denote the population distribution of different states. In Fig. 4, we start from Eq. (8) and numerically study the steady state populations of the dressed QD versus the chemical potential and the difference of the Rabi amplitudes (DRA) denoted by ΔΩ = Ω₂ − Ω_1, where the laser-exciton detuning Δ = 2 meV and the Rabi amplitude Ω₁ = 10 meV. Generally, it can be seen through the contours that when DRA takes positive values and the chemical potential is tuned below 90 meV, the populations of the dressed QD are mostly distributed in the states |0〉 and |V〉. Then with the increase of the chemical potential, the population transfers from other states to the state |−〉 are enhanced due to the variation of resonant photon-reservoir coupling region, which can be understood with the combination of the analyses given in Fig. 2 and Appendix B. As the result, large population occupation can be observed in the state |−〉. It should be pointed out that the rapid increases in the populations are mainly due to the steep decreases of the Purcell factor at the critical frequencies, whereas the imaginary part of graphene’s conductivity changes its sign by crossing these points. The variation in the conductivity then leads to the change of the graphene-supported electric modes, where the vanish of TM modes finally results in the weak photon-reservoir coupling [23]. For chemical potentials ranging from 110 meV to 140 meV, the populations are mainly distributed either in state |+〉 or state |V〉, which also depends on the values of DRA. However, for most chemical potentials and DRA values, the transitions to the state |+〉 become dominant in all possible decay channels. Thus a relative high population occupation in the upper dressed level |+〉 can be observed, as shown in Fig. 4(a). Finally, large population distributions in the state |V〉 for positive DRA values can be observed when chemical potential exceeds 160 meV.

Fig. 4 Steady-state populations of the dressed QD under the two-photon resonance driving condition. The environment temperature is 0K, where subplots (a), (b), (c) and (d) depicts the population of the state |+〉, |−〉, |0〉 and |V〉 versus the chemical potentials (µ_f) and the difference between two Rabi amplitudes (Ω₂ − Ω_1, which proportional to the drive’s power and the transition dipoles), respectively.

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Similarly, unbalanced population distribution could be achieved when DRA takes negative values, where large population occupations in the states |+〉 and |−〉 are obtained for different chemical potentials. It should be mentioned that although we have taken Δ = 2 meV in achieving these results, the populations distributions are insensitive to the small variation of the laser-exciton detuning (determined by the biexciton binding energy) for small ratios of Δ/ω_L, which can be well understood through the analytical expressions given in the Eq. (38).

To investigate the influence of the environment temperature on the steady-state population occupations, we start from Eq. (8) and derive the exact Bloch equations with the assumption that the reservoir is at room temperature (T = 300 K). As shown in Fig. 5, for different values of DRA, states |+〉, |−〉 and |V〉 exhibits relatively high population occupations by tuning graphene’s chemical potential, respectively. Moreover, the major population distribution patterns are similar to what has been discussed in the zero temperature case. Nevertheless, the shifts of the resonant peaks and the broadening of the coupling curves at room temperature redefine the light-matter interaction and bring modifications to the population occupations. Meanwhile, the discrepancies in transition rates among different transition channels become less prominent, which indicates the probable destruction influences on the unbalanced population distributions. As a result, the maximum population occupations suffer decreases compared with the zero temperature case, thus for some chemical potentials and DRA values the decline of population imbalance can be observed with the increase of the environment temperature.

Fig. 5 Steady-state populations of the dressed QD under the two-photon resonance driving condition. All the parameters are the same as we used in Fig.4 except the environment temperature is 300K, where subplots (a), (b), (c) and (d) depicts the variation in the population of the state |+〉, |−〉, |0〉 and |V〉, respectively.

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In a short summary, we may conclude that in our system, the electromagnetic environment around the driving transition system has been tailored mainly through the excitation of the surface modes on the graphene sheet. Then by applying a voltage bias on the graphene sheet the chemical potential can be tuned, which in turn influences the spontaneous emission of different transition dipoles and plays a key role in the generation of the unbalanced steady-state population occupations. Furthermore, the increase in the environment temperature might have a destructive effect in achieving unbalanced population distributions, which can be attributed to the declines in the maximum values of the population occupations. However, the tunability of the coupling between the graphene-assisted reservoir and the driving system, makes it possible to selectively enhance the spontaneous emission of transition dipoles. As we will show later, this tunability is important in reshaping the resonance fluorescence spectrum and realizing quadrature squeezing of the radiation field.

3. Resonance fluorescence and probe absorption spectra with different polarizations

In this section we focus on the spectral properties of the driving system under the influence of the graphene-assisted reservoir. The steady-state spectrum of the resonance fluorescence is proportional to the Fourier transformation of the correlation function [74] $\lim_{t \to \infty} 〈 {\vec{E}}_{-} (\vec{r}, t + τ) \cdot {\vec{E}}_{+} (\vec{r}, t) 〉$ , where ${\vec{E}}_{+} (\vec{r}, t)$ and ${\vec{E}}_{-} (\vec{r}, t)$ are the positive and negative part of the dipole radiated field in the far zone [68]. The source field part, which has dominant contribution to the radiation field and influenced by the spectral property of the reservoir, is proportional to the polarization operator of the dressed QD. The resonance fluorescence spectrum with different polarizations can be generally written as

\begin{array}{l} S^{P} (ω) = \lim_{t \to \infty} Re \int_{0}^{\infty} d τ [〈 σ_{P G} (t + τ), σ_{G P} (t) 〉 + 〈 σ_{B P} (t + τ), σ_{P B} (t) 〉 \\ + 〈 σ_{P G} (t + τ), σ_{P B} (t) 〉 + 〈 σ_{B P} (t + τ), σ_{G P} (t) 〉] e^{- i (ω - ω_{L}) τ} (P = H, V), \end{array}

where the index P represents the polarization of the dipole transitions, the spectrum functions denoted by S^H (ω) and S^V (ω) represent the resonance fluorescence field generated through horizontal and vertical decay channels, respectively. In the above expression the terms of the form

\lim_{t \to \infty} Re \int_{0}^{\infty} d τ 〈 σ_{i j} (t + τ), σ_{k l} (t) 〉 e^{- i (ω - ω_{L}) τ}

correspond to the Fourier transformation of the two-time correlation functions, which contain the spectral information of the radiation field and can be obtained by applying the quantum regression theorem [75] (the results are given in Eqs. (39)–(41) in Appendix C). With the help of their formations, the dipole-reservoir interaction process can be easily understood.

We consider the resonance fluorescence spectra of the dressed system with small binding energy (the laser-exciton detuning has been chosen as Δ = χ/2 = 2 meV to satisfy the two-photon resonance condition), where other parameters are Ω₁ = 10 meV, Ω₂ = 7 meV and µ_f = 350 meV, the results are given in Fig. 6. As it can be seen through the previous analysis according to the Fig. 4, with these parameters the steady-state population largely distributes in the state |−〉, where the occupations of the other states are negligible. This population imbalance indicates that the transitions from other states to the state |−〉, which are denoted by diagrams |j〉 → |−〉(j ≠ −) near the resonant peaks in the figure, have been enhanced owing to the strong couplings with the reservoir. Consequently, the sidebands of the relevant resonance fluorescence spectrum are much stronger than the sidebands describing the transitions through the state |−〉 to other QD states, which can be clearly identified by comparing the resonant peaks in subplots (a) and (b).

Fig. 6 Resonance fluorescence spectra of the driving system under the conditions: T = 0 K, Ω₁ = 10 meV, ΔΩ = −3 meV, Δ = 2 meV and µ_f = 350 meV. (a) The fluorescence spectrum of the horizontal transitions. (b) The fluorescence spectrum of the vertical transitions.

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Then the case of µ_f = 125 meV has been investigated to reveal the role of graphene’s chemical potential in controlling the spectral property, where the driving conditions are the same as we used in Fig. 6. Under this circumstance, the bulk of the steady-state population distributes in the state |+〉 according to the contour plots given in Fig. 4. It is not difficult to see from Fig. 7 that the lower-frequency outer sidebands correspond to the transitions |−〉 → |+〉 and |V〉→|+〉 are largely enhanced, where the higher-frequency ones are suppressed for both the horizontal and vertical radiations. Furthermore, owing to small population occupation of other states, the radiation strength of other transitions are weak compared with the transitions to the state |+〉. It should be noticed that owing to the extremely large population occupation in the highest excited state, the sidebands depict the transitions to the state |+〉 are much stronger than other resonant peaks.

Fig. 7 Resonance fluorescence spectra of the driven QD under the conditions: T = 0 K, Ω₁ = 10 meV, ΔΩ = −3 meV, Δ = 2 meV and µ_f = 125 meV. (a) The fluorescence spectrum of the horizontal transitions. (b) The fluorescence spectrum of the vertical transitions.

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To clearly demonstrate the enhancement of the transition probabilities between non-inverted states, we then numerically study the absorption spectra in the weak probe limit. It is known that the steady-state absorption spectrum of a weak probe field is proportional to the Fourier transform of the stationary mean value of the two-time commutator of the transition operators [42, 69]:

\begin{array}{l} A^{P} (v) = \lim_{t \to \infty} Re \int_{0}^{\infty} d τ {〈 [σ_{P G} (t + τ), σ_{G P} (t)] 〉 + 〈 [σ_{B P} (t + τ), σ_{P B} (t)] 〉 \\ + 〈 [σ_{P G} (t + τ), σ_{P B} (t)] 〉 + 〈 [σ_{B P} (t + τ), σ_{G P} (t)] 〉} e^{i δ^{'} τ} (P = H, V), \end{array}

where δ^′ = ν − ω_L denotes the frequency deviation of the probe field from the driving laser. The analytical form of the absorption spectra are formally the same as the resonance fluorescence spectra, which can be acquired by simply replacing the populations

{〈 {\tilde{σ}}_{j j} 〉}_{s s}

in the spectral functions (as shown in Eq. (C3)) with population differences

{〈 {\tilde{σ}}_{j j} 〉}_{s s} - {〈 {\tilde{σ}}_{l l} 〉}_{s s}

, the transformation parameter s with −iδ^′ and neglects the contribution from the central line.

The probe absorption spectra with different polarizations are then investigated for the case of T = 0 K, Ω₁ = 10 meV, ΔΩ = −3 meV, Δ = 2 meV and µ_f = 350 meV, where the relevant resonance fluorescence spectra have been studied in Fig. 6. As Fig. 8 shows, neither the probe absorptions nor probe gains at the driving’s frequency can be observed, and the sidebands exhibit a Lorentzian line shape. Moreover, the sidebands result from the transitions |0〉 → |−〉, |+〉 → |−〉 and |V〉 → |−〉 exhibit negative values, which indicates the amplification of the probing field at the relevant frequencies. Notice that the population of the dressed QD mainly distributes in the lower state |−〉 in this case, that is to say, the amplification of the probe has been achieved without the help of population inversion. Additionally, although the discrepancy is small, the population in the state |0〉 exceeds the population in the state |+〉 or |V〉. Thus the probe gains observed at the frequencies corresponds to the transitions |+〉→|0〉 and |V〉→|0〉 also demonstrates the inversionless enhancement of the transition probabilities.

Fig. 8 Absorption spectra with the same parameters used in Fig. 6. (a) The absorption spectrum of the horizontal transitions. (b) The absorption spectrum of the vertical transitions.

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So far, the influences of the chemical potential on the resonance fluorescence spectrum have been investigated. It has been discussed previously that by tuning graphene’s chemical potential, the population of the driving system can selectively distribute on different states. This unbalanced occupations lead to an enhancement in the dipole transitions to the largely populated state, and as a result, the relevant resonance fluorescence sidebands has prominent increases in strength. A further study on the absorption spectra displays probe gains at these frequencies, which clearly demonstrates the enhancement in the transition rates without population inversion. In room temperature case, the coupling between the transition dipoles and the reservoir also exhibits Lorentzian-like dispersion relations. Although we do not give more discussion here, similar role of the chemical potential in controlling the resonance fluorescence spectrum can be expected.

4. Two-mode noise spectrum of the resonance fluorescence field

The normal-order noise spectrum of the in-phase and out-of-phase quadratures of the resonance fluorescence field can be expressed in forms of the two-time correlation functions of the transition operators, where quantum regression theory has been used according to the master equation given in Eq. (8) previously. Starting from the definition, under two-photon resonance condition the photon pairs generated through transitions |j〉→|l〉 and |l〉→|j〉 are correlated in frequency. Then by transforming the time dependent functions into the spectral region we obtain the two-mode noise spectra of the photon pairs. The relevant spectrum functions have been given in Eqs. (55)–(60), which indicate that the squeezing property of the photon pairs strongly depends on the driving conditions, the steady-state population occupations and the frequency deviation of the photons to their sidebands.

To provide a general scheme for the influence of the driving conditions and the photon-sideband detuning on two-mode noise spectrum, firstly we investigate the in-phase quadrature noise spectrum, as shown in Fig. 9. In obtaining these plots, we have assumed that the laser-exciton detuning is 2 meV and the Rabi amplitude Ω₁ = 10 meV. According to Eq. (56), the squeezing of the noise spectrum can be generated when there exists a sufficient steady-state population difference between the states |0〉 and |−〉. As discussed in Sec. 2, if the drive couples to the dipoles with close Rabi amplitudes (i.e. |ΔΩ| is small) and the chemical potential is huge (above 200 meV), a relative large population imbalance could be achieved. In Fig. 9(a) the chemical potential has been chosen to be 350 meV to realize sufficient population imbalance between states |0〉 and |−〉. The result shows that two-mode squeezing occurs when Rabi amplitude Ω₁ is larger than Ω_2, where the squeezing achieves its maximum around ΔΩ = −5 meV and finally disappears for smaller DRA values. Meanwhile, squeezing maintains for small frequency deviations of the photon pairs to their sidebands. Only for deviation larger than three times of the halfwidth, squeezing strength then decreases to less than 10% compared with the maximum value and can be neglected.

Fig. 9 Contour plots of the in-phase quadrature two-mode noise spectrum, where the parameters have been chosen as T = 0 K, Δ = 2 meV and Ω₁ = 10 meV. The squeezing spectrum under investigation varies with the DRA and the photon-sideband detunings, which are denoted by δ₋₀ and δ_0+. (a) The noise spectrum of the photon pairs generated by transitions |−〉 → |0〉 → |−〉. (b) The noise spectrum of the photon pairs generated by transitions |0〉→|+〉→|0〉.

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In Fig. 9(b) we investigate the squeezing property of the photon pairs produced by transitions |0〉→|+〉→|0〉, where the chemical potential is taken to be 125 meV. As illustrated by Eq. (55), an unbalanced population distribution between states |0〉 and |+〉 is required in the generation of the squeezing. Similar to the case that has been studied in Fig. 9(a), squeezing occurs when DRA takes negative values, and can be observed for a wide range of the DRA. However, although the squeezing also maintains for detunings smaller than three times of the halfwidth, the bandwidth has been narrowed owing to the increase in the transition rates, which in turn only allows an observable two-mode squeezing for sideband-detuning up to 0.07 meV.

Similarly, as Eqs. (58)–(60) illustrates, two-mode squeezing in the out-of-phase quadrature can also be realized by reasonably tuning the chemical potential of graphene. In Fig. 10 we investigate the out-of-phase quadrature noise spectrum produced by different transitions. For photon pairs generated through transitions |−〉 → |+〉 → |−〉, according to the analytical form of the noise spectrum given in Eq. (58), a chemical potential equal to 350 meV has been chosen to achieve large population difference between states |−〉 and |+〉 when Rabi amplitude Ω₂ is smaller than half of the Rabi amplitude Ω_1. As shown in Fig. 10(a), the squeezing of the photon pairs occurs when Rabi amplitude Ω₂ is smaller than 4 meV. However, the decrease in the transition rates broadens the halfwidth of the spectrum and the squeezing can be achieved for sideband-detuning up to 0.5 meV.

Fig. 10 Two-mode noise spectrum versus the DRA and the photon-sideband detunings denoted by δ_−+, δ₋_V and δ_V+. Subplots (a), (b) and (c) depict the noise spectra of the photon pairs generated by transitions |−〉 → |+〉 → |−〉, |−〉 → |V〉 → |−〉 and |+〉→|V〉→|+〉, respectively.

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Then in Fig. 10(b) the case of the chemical potential tuned to 350 meV has been investigated, where large population distributes in the states |−〉 or |V〉. According to Eq. (59), for negative DRA values the population largely populated in the state |−〉 rather than state |V〉, thus squeezing can be generated due to the prominent population imbalance. When DRA takes positive values, the population occupation of the state |−〉 gradually decreases while the occupation of the state |V〉 gains with the increase of the DRA, thus the population imbalance becomes sharper and finally leads to the appearance of squeezing for DRA value around 5 meV. Finally, the vertical polarized noise spectrum generated via transitions |+〉→|V〉→|+〉 have been given in Fig. 10(c), where the chemical potential has been chosen as 125 meV to achieve a prominent population occupation in the state |+〉. As shown in the figure, the squeezing of the photon pairs occurs when Rabi amplitude Ω₁ is larger than Ω_2, and its strength is gradually enhanced with the decrease of the Rabi amplitude Ω_2. Also we notice that in order to observe notable squeezing effect, the photon-sideband detunings should not be larger than three times of the halfwidth.

From the discussions in Sec. 2, it is clear that the variation in the environment temperature has an influence on the dipole-reservoir coupling. For this reason, we investigate the two-mode noise spectrum at room temperature, as shown in Fig. 11. It should be pointed out that the noise spectrum $S_{x}^{0, +}$ does not show any squeezing, which mainly due to the decline of the maximum population occupation of the state |+〉 around µ_f = 125 meV with the increase of the environment temperature. In the following we focus on the noise spectra $S_{y}^{-, +}$ and $S_{y}^{+, V}$ , whose squeezing properties might have notable changing owing to the decrease in the population imbalance. In Fig. 11(a) the noise spectrum of transition |−〉 → |+〉 → |−〉 has been studied. By comparing with the noise spectrum at zero environment temperature, it can be found that the redistribution of the population occupation in the highest excited state has an obvious influence on the production of squeezing, where smaller DRA (negative values) are needed to realize quadrature squeezing. Meanwhile, the destruction of the population imbalance leads to the decline of maximum squeezing. Finally, the noise spectrum $S_{y}^{+, V}$ at room temperature has also been investigated. Similar to the zero temperature case shown in Fig. 10(c), squeezing occurs for negative values of DRA. However, the circumstance is different when Rabi amplitude Ω₂ exceeds Ω_1. At room temperature, squeezing can also be observed for large values of Ω_2, where not squeezing has been indicated in the zero temperature case. This phenomenon can be clearly illustrated with the combination of the results given in Fig. 5 and Eq. (60), that for large DRA values the population tends to distribute in the state |V〉 rather than the state |+〉 at room temperature. The population imbalance between the two QD states becomes outstanding with the increase of the DRA, which finally leads to the generation of squeezing.

Fig. 11 Contour plots of the out-of-phase quadrature two-mode noise spectrum at room temperature. (a) The noise spectrum of the photon pairs generated by transitions |−〉 → |+〉 → |−〉. (b) The noise spectrum of the photon pairs generated by transitions |+〉→ |V〉→|+〉.

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5. Conclusions

In conclusion, we have demonstrated the resonance fluorescence and the noise spectra of a dressed QD under two-photon resonance condition. It is known that in most plasmonic structure including metal slabs [76], nanorods [77] or nanoparticles [78], light-matter interaction mainly depends on the optical response of the bulk, where the coupling curve would be totally determined for a certain boundary condition. We have shown the dipole-reservoir coupling in our model exhibits Lorentz-like distributions for both the zero and room temperature cases, and the resonant peak can be tuned in accordance with the variation of the graphene’s chemical potential. Owing to the enhancement of different dipole transitions, unbalanced steady-state population occupations are then achieved by modulating the optical response of graphene sheet. Meanwhile, both the spectral asymmetries in the resonance fluorescence spectrum and probe gains in the absorption spectrum can be observed, which indicate the enhancement in the probabilities of different transition channels.

The study of squeezing in the resonance fluorescence has been mostly focused in the cavity-QED [79–81] system, where the dipole transitions are modified by the cavity mode(s). Owing to the striking electronic property [14] and the tunability in its conductivity by electrostatic doping [57, 82], graphene has been recognized as a suitable material in plasmon photonics [83], which enables fast switch and on-chip integration [84, 85]. In our work, we have proved that with the flexibility in tuning the SPM, the two-mode noise spectrum of different transition channels can be squeezed below the vacuum fluctuation limit. Moreover, the squeezing maintains for small frequency deviations of the photon pairs to their sidebands. Finally it has been shown owing to the strong dipole-plasmon interaction, although there are decreases in the strength compared with the zero environment cases, quadrature squeezing of the fluorescence fields can be observed even at room temperature. Our study provides a promising method to prepare two-mode squeezed states, which may have potential applications in other quantum research areas.

Appendix A Green tensor of a dipole placed above the graphene sheet

The Green tensor that depicts the transmission of the electromagnetic field in our model can be obtained by imposing the boundary conditions on the electric Hertzian potential, and can be generally written in the form

\overset{\leftrightarrow}{G} ({\vec{r}}_{o}, {\vec{r}}^{'}, ω) = (k_{0}^{2} \overset{\leftrightarrow}{I} + \vec{\nabla} \vec{\nabla}) [{\overset{\leftrightarrow}{g}}_{p} ({\vec{r}}_{o}, {\vec{r}}^{'}, ω) + {\overset{\leftrightarrow}{g}}_{s} ({\vec{r}}_{o}, {\vec{r}}^{'}, ω)],

with

\overset{\leftrightarrow}{I}

denoting the unit dyadic,

{\vec{r}}^{'}

and

{\vec{r}}_{o}

denoting the source and observe point, and k₀ representing the wave vector in the vacuum. In Eq. (17), the first term denotes the contribution of the free space dipole on the total field that observed at the spacial point

{\vec{r}}_{o}

, and the second term describes the field distribution in the presence of the boundary conditions. Both of them can be expanded as Sommerfeld integrals

{\overset{\leftrightarrow}{g}}_{p} ({\vec{r}}_{o}, {\vec{r}}^{'}, ω) = \overset{\leftrightarrow}{I} g_{p}^{o} ({\vec{r}}_{o}, {\vec{r}}^{'}, ω),

\begin{matrix} {\overset{\leftrightarrow}{g}}_{s} ({\vec{r}}_{o}, {\vec{r}}^{'}, ω) = \hat{z} \hat{z} g_{s}^{n} ({\vec{r}}_{o}, {\vec{r}}^{'}, ω) + (\hat{x} \hat{x} + \hat{y} \hat{y}) g_{s}^{t} ({\vec{r}}_{o}, {\vec{r}}^{'}, ω) \\ + (\hat{z} \hat{x} \frac{\partial}{\partial x} + \hat{z} \hat{y} \frac{\partial}{\partial y}) g_{s}^{c} ({\vec{r}}_{o}, {\vec{r}}^{'}, ω) . \end{matrix}

In Eqs. (18) and (19), k_║ the projection of the wave vector in the x−y plane, $β_{0} = {(k_{0}^{2} - k_{| |}^{2})}^{1 / 2}$ is the z component of the wave vector in the vacuum, $\hat{j} (j = x, y, z)$ is unit vector and J₀ (k) is the Bessel function. Moreover, z_o = l denotes the distance between the QD and graphene sheet, where we have taken z_o = 10 nm in our calculations. The general forms of both the principle and scattering parts are [86]

g_{p}^{o} ({\vec{r}}_{o}, {\vec{r}}^{'}, ω) = \frac{i}{2 π} \int_{0}^{\infty} e^{i β_{0} | z_{o} - z^{'} |} \frac{J_{0} (k_{∥} ρ)}{2 β_{0}} k_{∥} d k_{∥},

g_{s}^{j} ({\vec{r}}_{o}, {\vec{r}}^{'}, ω) = \frac{i}{2 π} \int_{0}^{\infty} R_{s}^{j} (k_{∥}, z, z_{o}) \frac{J_{0} (k_{∥} ρ)}{2 β_{0}} k_{∥} d k_{∥} (j = n, t, c),

where the reflection coefficients can be derived through the boundary conditions and appear to be

R_{s}^{n} (k_{∥}, z, z_{o}) = \frac{σ β_{0} e^{i β_{0} (z + z_{o})}}{2 ω ε_{0} + σ β_{0}},

R_{s}^{t} (k_{∥}, z, z_{o}) = \frac{σ ω μ_{0} e^{i β_{0} (z + z_{o})}}{σ ω μ_{0} + 2 β_{0}},

R_{s}^{c} (k_{∥}, z, z_{o}) = \frac{2 i σ β_{0} e^{i β_{0} (z + z_{o})}}{(σ ω μ_{0} + 2 β_{0}) (2 ω ε_{0} + σ β_{0})} .

In the above equations, σ is the conductivity of graphene whose explicit form is given in Eq. (3). It can be seen from Eqs. (22) and (23) that $g_{s}^{n} ({\vec{r}}_{o}, {\vec{r}}^{'}, ω)$ and $g_{s}^{t} ({\vec{r}}_{o}, {\vec{r}}^{'}, ω)$ characterize the interaction between graphene and different polarized photons, where in detail they illustrates the process of the QD emitting photons into the TM (transverse magnetic field) and TE (transverse electric field) surface modes, respectively. The other reflection coefficient, which has been given in Eq. (24), describes the possible interaction between the radiation photons and either the TM or TE surface modes that mentioned above. Then replace the principle part and scattering part in Eq. (17) by the expressions given in Eqs. (18) and (19), we can acquire the components of the Green function

\begin{matrix} \overset{\leftrightarrow}{G} {({\vec{r}}_{o}, {\vec{r}}_{o}, ω)}_{z z} = (k_{0}^{2} + \frac{\partial^{2}}{\partial z^{2}}) {[g_{p}^{o} ({\vec{r}}_{o}, {\vec{r}}^{'}, ω) + g_{s}^{n} ({\vec{r}}_{o}, {\vec{r}}^{'}, ω)]}_{{\vec{r}}^{'} = {\vec{r}}_{o}} \\ = \frac{i}{2 π} \int_{0}^{\infty} \frac{k_{∥}^{3}}{2 β_{0}} d k_{∥} + \frac{i}{2 π} \int_{0}^{\infty} R_{s}^{n} (k_{∥}, z_{o}, z_{o}) \frac{k_{∥}^{3}}{2 β_{0}} d k_{∥}, \end{matrix}

\begin{matrix} \overset{\leftrightarrow}{G} {({\vec{r}}_{o}, {\vec{r}}_{o}, ω)}_{x x} = (k_{0}^{2} + \frac{\partial^{2}}{\partial x^{2}}) [g_{p}^{o} ({\vec{r}}_{o}, {\vec{r}}^{'}, ω) + g_{s}^{t} ({\vec{r}}_{o}, {\vec{r}}^{'}, ω)] \\ {+ \frac{\partial^{3}}{\partial x^{2} \partial z} g_{s}^{c} ({\vec{r}}_{o}, {\vec{r}}^{'}, ω) |}_{{\vec{r}}^{'} = {\vec{r}}_{o}} \\ = \frac{i}{2 π} \int_{0}^{\infty} \frac{2 k_{0}^{2} - k_{∥}^{2}}{4 β_{0}} k_{∥} d k_{∥} + \frac{i}{2 π} \int_{0}^{\infty} R_{s}^{t} (k_{∥}, z_{o}, z_{o}) \frac{2 k_{0}^{2} - k_{∥}^{2}}{4 β_{0}} k_{∥} d k_{∥} \\ + \frac{1}{2 π} \int_{0}^{\infty} R_{s}^{c} (k_{∥}, z_{o}, z_{o}) \frac{k_{∥}^{2}}{4} k_{∥} d k_{∥}, \end{matrix}

\overset{\leftrightarrow}{G} {({\vec{r}}_{o}, {\vec{r}}_{o}, ω)}_{z x} = \overset{\leftrightarrow}{G} {({\vec{r}}_{o}, {\vec{r}}_{o}, ω)}_{x z} = 0 .

In the derivation of these results, the properties of Bessel functions $J_{v - 1} (z) + J_{v + 1} (z) = 2 v J_{v} (z) / z$ , $J_{v - 1} (z) - J_{v + 1} (z) = 2 J_{v}^{'} (z)$ and $J_{0}^{'} (z) = - J_{1} (z)$ have been used. It is clear through Eqs. (25)–(27) that when the graphene is removed, the Green tensor is coincident with its vacuum form. Meanwhile, the off-diagonal terms illustrate that different polarized dipoles can not interact with each other. The diagonal elements, which contain the influence of graphene due to the appearance of the scattering terms, illustrate the interactions between horizontal/vertical polarized dipoles and the surface modes. Commonly, the dispersion of the surface modes supported by planar boundaries can be given by finding the zeros of the scattering coefficients [87]. For vertical polarized dipoles, the poles of the reflection coefficient $R_{s}^{n} (k_{∥}, z, z_{o})$ requires 2ωε₀ + σβ₀ = 0, thus after deductions the wave vector of the supported TM surface modes is found to be

k_{V s}^{2} = k_{0}^{2} [1 - \frac{4 ε_{0}}{σ^{2} μ_{0}}] .

If the conductivity of graphene is assumed to be in the form σ = σ_r + iσ_i, where σ_r and σ_i represents the real and imaginary part of the conductivity, then the normal component of the wave vector can be acquired through the Eq. (28) and has the form

β_{V s} = - \frac{2 ω ε (σ_{r} - j σ_{i})}{| σ |^{2}} .

Obviously, Eq. (29) indicates that the TM surface modes supported by the graphene sheet can be excited only when the imaginary part of the conductivity is positive, i.e. σ_i > 0. Meanwhile, the contribution of the SPM on the spontaneous emission of the vertical QD transitions can be solved by applying residue theorem on the second term in Eq. (25), where the imaginary part of the Green function can be analytical given as

\frac{i}{2 π} \int_{0}^{\infty} R_{s}^{n} (k_{∥}, z_{o}, z_{o}) \frac{k_{∥}^{3}}{2 β_{0}} d k_{∥} \approx \frac{1}{4} β_{V s} k_{V s}^{2} e^{2 i β_{V s} z_{o}} .

Similarly, for horizontal polarized dipoles, the contribution of the TM surface modes on the spontaneous emission is proportional to

\frac{i}{2 π} \int_{0}^{\infty} R_{s}^{c} (k_{∥}, z_{o}, z_{o}) \frac{k_{∥}^{2}}{4} k_{∥} d k_{∥} \approx \frac{1}{8} \frac{β_{V s} k_{V s}^{2} (ω ε_{0} - σ β_{V s})}{(2 ω ε_{0} + σ β_{V s})} e^{2 i β_{V s} z_{o}} .

Appendix B Dressed state basis under two-photon resonance condition and the master equations

By applying a canonical transformation of the form U^† AU on both the unperturbed and interaction Hamiltonians, we then acquire the transformed Hamiltonians that appear to be

{\hat{H}}_{0}^{'} = Δ (σ_{H H} + σ_{V V}) + [Ω_{1} σ_{H G} + Ω_{2} σ_{B H} + H . c .],

{\hat{H}}_{I}^{'} (t) = - [σ_{H G} {\vec{p}}_{1} + σ_{B H} {\vec{p}}_{2} + σ_{V G} {\vec{p}}_{3} + σ_{B V} {\vec{p}}_{4}] \int d ω {\hat{E}}^{(+)} ({\vec{r}}_{o}, ω) e^{i (ω_{L} - ω) t},

with

Δ = ℏ (ω_{0} - ω_{L}) = χ / 2

denoting the energy difference between the driving photons and the excitonic levels. For a driven system, it is convenient to introduce the dressed state picture [88] in order to better understand the spectral features. By diagonalizing the unperturbed Hamiltonian, the eigenstates then forms a new set of basis that transforms the Pauli operators of the bare QD into the dressed picture, which are

\begin{matrix} | + 〉 = \frac{1}{Ω_{e}} \sqrt{\frac{{\tilde{Ω}}_{-}}{\tilde{Ω}}} (Ω_{1} | G 〉 + {\tilde{Ω}}_{+} | H 〉 + Ω_{2} | B 〉), \\ | 0 〉 = \frac{1}{Ω_{e}} (- Ω_{2} | G 〉 + Ω_{1} | B 〉), \\ | - 〉 = \frac{1}{Ω_{e}} \sqrt{\frac{{\tilde{Ω}}_{+}}{\tilde{Ω}}} (Ω_{1} | G 〉 - {\tilde{Ω}}_{-} | H 〉 + Ω_{2} | B 〉) \end{matrix}

for different eigenvalues

λ_{+} = {\tilde{Ω}}_{+} = (Δ + \tilde{Ω}) / 2

, λ₀ = 0 and

λ_{-} = - {\tilde{Ω}}_{-} = (Δ - \tilde{Ω}) / 2

. The effective Rabi amplitudes are

Ω_{e} = {(Ω_{1}^{2} + Ω_{2}^{2})}^{1 / 2}

and

\tilde{Ω} = {(4 Ω_{e}^{2} + Δ^{2})}^{1 / 2}

. In dressed basis, the interaction Hamiltonian can be expressed in time-dependent form, as Eq. (6) indicates. The time-dependent Pauli operators are the combination of the dressed state operators multiplied by the transforming parameters, which have the forms

\begin{array}{l} σ_{H G} (t) = R_{1} {\tilde{σ}}_{+ +} + R_{21} e^{i {\tilde{Ω}}_{+} t} {\tilde{σ}}_{+ 0} + R_{13} e^{i \tilde{Ω} t} {\tilde{σ}}_{+ -} + R_{14} e^{- i \tilde{Ω} t} {\tilde{σ}}_{- +} + R_{22} e^{- i {\tilde{Ω}}_{-} t} {\tilde{σ}}_{- 0} - R_{1} {\tilde{σ}}_{- -}, \\ σ_{B H} (t) = R_{2} {\tilde{σ}}_{+ +} + R_{11} e^{- i {\tilde{Ω}}_{+} t} {\tilde{σ}}_{0 +} + R_{23} e^{- i \tilde{Ω} t} {\tilde{σ}}_{- +} + R_{24} e^{i \tilde{Ω} t} {\tilde{σ}}_{+ -} + R_{12} e^{i {\tilde{Ω}}_{-} t} {\tilde{σ}}_{0 -} - R_{2} {\tilde{σ}}_{- -}, \\ σ_{V G} (t) = - R_{21} e^{- i {\tilde{Ω}}_{-} t} {\tilde{σ}}_{V +} - R_{2} e^{i Δ t} {\tilde{σ}}_{V 0} + R_{11} e^{i {\tilde{Ω}}_{+} t} {\tilde{σ}}_{V -}, \\ σ_{B V} (t) = R_{22} e^{i {\tilde{Ω}}_{-} t} {\tilde{σ}}_{+ V} + R_{1} e^{- i Δ t} {\tilde{σ}}_{0 V} - R_{21} e^{- i {\tilde{Ω}}_{+} t} {\tilde{σ}}_{- V}, \end{array}

where the transforming coefficients are explicitly defined by

\begin{array}{l} R_{2} = \frac{Ω_{2}}{Ω_{1}} R_{1} = - \frac{Ω_{2}}{\tilde{Ω}}, \\ R_{21} = \frac{Ω_{2}}{Ω_{1}} R_{11} = \frac{Ω_{2}}{Ω_{e}} {(\frac{{\tilde{Ω}}_{+}}{\tilde{Ω}})}^{1 / 2}, R_{22} = - \frac{Ω_{2}}{Ω_{1}} R_{12} = - \frac{Ω_{2}}{Ω_{e}} {(\frac{{\tilde{Ω}}_{-}}{\tilde{Ω}})}^{1 / 2}, \\ R_{23} = \frac{Ω_{2}}{Ω_{1}} R_{13} = - \frac{Ω_{2}}{Ω_{e}} (\frac{{\tilde{Ω}}_{+}}{\tilde{Ω}}), R_{24} = \frac{Ω_{2}}{Ω_{1}} R_{14} = \frac{Ω_{2}}{Ω_{e}} (\frac{{\tilde{Ω}}_{-}}{\tilde{Ω}}) . \end{array}

For strong driving field, secular approximation is valid in describing the evolution of the system. The fluorescence spectrum can be well explained with the simplified Bloch equations in the dressed state basis, where the strengths of the sidebands are related to the relaxations of the dipole operators (off-diagonal terms of the reduced density matrix). In what follows we give the master equation in the zero environment temperature case through Eq. (8)(The non-zero temperature master equation can also be directly derived through the same equation. However, the expression is complicated thus we do not list here, the relevant steady-state populations has been discussed in Sec. 2.), where the principle part of the integrals has been neglected because these terms only cause small Lamb shifts and do not affect the main physics. After straightforward deduction, the master equation can be arranged in the following form

\begin{matrix} \dot{ρ} = - \frac{i}{ℏ} [{\hat{H}}_{0}^{'}, ρ] + \frac{1}{2} \sum_{j = +, 0, -} (Γ_{+ j} ℒ_{+ j} + Γ_{- j} ℒ_{- j} + Γ_{0 j} ℒ_{0 j} + Γ_{V j} ℒ_{V j}) \\ + \frac{1}{2} \sum_{j = +, 0, -} Γ_{j V} ℒ_{j V} - Γ_{+ +} ({\tilde{σ}}_{+ +} \tilde{ρ} {\tilde{σ}}_{- -} + {\tilde{σ}}_{- -} \tilde{ρ} {\tilde{σ}}_{+ +}), \end{matrix}

where the operator

ℒ_{j k} = 2 {\tilde{σ}}_{k j} ρ {\tilde{σ}}_{j k} - {\tilde{σ}}_{j k} {\tilde{σ}}_{k j} ρ - ρ {\tilde{σ}}_{j k} {\tilde{σ}}_{k j}

and the decay rates in the summations are

\begin{array}{l} Γ_{+ +} = Γ_{- -} = 2 [R_{1}^{2} γ_{11} (ω_{L}) + 2 R_{1} R_{2} γ_{12} (ω_{L}) + R_{2}^{2} γ_{22} (ω_{L})], \\ Γ_{+ -} = 2 [R_{13}^{2} γ_{11} (ω_{L} + \tilde{Ω}) + 2 R_{13} R_{24} γ_{12} (ω_{L} + \tilde{Ω}) + R_{24}^{2} γ_{22} (ω_{L} + \tilde{Ω})], \\ Γ_{- +} = 2 [R_{14}^{2} γ_{11} (ω_{L} - \tilde{Ω}) + 2 R_{14} R_{23} γ_{12} (ω_{L} - \tilde{Ω}) + R_{23}^{2} γ_{22} (ω_{L} - \tilde{Ω})], \\ \begin{matrix} Γ_{+ 0} = 2 R_{21}^{2} γ_{11} (ω_{L} + {\tilde{Ω}}_{+}), & Γ_{0 +} = 2 R_{11}^{2} γ_{22} (ω_{L} - {\tilde{Ω}}_{+}), \\ Γ_{V -} = 2 R_{11}^{2} γ_{33} (ω_{L} + {\tilde{Ω}}_{+}), & Γ_{- V} = 2 R_{21}^{2} γ_{44} (ω_{L} - {\tilde{Ω}}_{+}), \\ Γ_{- 0} = 2 R_{22}^{2} γ_{11} (ω_{L} - {\tilde{Ω}}_{-}), & Γ_{0 -} = 2 R_{12}^{2} γ_{22} (ω_{L} + {\tilde{Ω}}_{-}), \\ Γ_{V +} = 2 R_{12}^{2} γ_{33} (ω_{L} - {\tilde{Ω}}_{-}), & Γ_{+ V} = 2 R_{22}^{2} γ_{44} (ω_{L} + {\tilde{Ω}}_{-}), \\ Γ_{V 0} = 2 R_{2}^{2} γ_{33} (ω_{L} + Δ), & Γ_{0 V} = 2 R_{1}^{2} γ_{44} (ω_{L} - Δ) . \end{matrix} \end{array}

In the above expressions, $γ_{i j} (ω) = ω^{2} {\vec{p}}_{i} \cdot Im [\overset{\leftrightarrow}{G} ({\vec{r}}_{o}, {\vec{r}}_{o}, ω)] \cdot {\vec{p}}_{j}^{*} / ℏ π ε_{0} c^{2}$ denote spontaneous decay rates of different transition dipoles in zero temperature case. The total decay rates between different transition channels in the dressed state basis, which have been denoted by Γ_jk (j ≠ k = +, −, 0 and V), denotes the strength of the spontaneous decay of the dressed QD jumps from state |j〉 to state |k〉.

Appendix C Resonance fluorescence and two-mode squeezing spectra expressed in the dressed state basis

The resonance fluorescence spectra of the horizontal and vertical polarized transitions can be expressed by use of the dressed Pauli operators, which have the forms

\begin{matrix} S^{H} (ω) = Re [{(R_{1} + R_{2})}^{2} L_{z z, z z} + (R_{14}^{2} + R_{23}^{2}) L_{- +, + -} + (R_{13}^{2} + R_{24}^{2}) L_{+ -, - +} + R_{11}^{2} L_{0 +, + 0} \\ + R_{21}^{2} L_{+ 0, 0 +} + R_{12}^{2} L_{0 -, - 0} + R_{22}^{2} L_{- 0, 0 -}] \end{matrix}

and

S^{V} (ω) = Re [R_{12}^{2} L_{V +, + V} + R_{22}^{2} L_{+ V, V +} + R_{11}^{2} L_{V -, - V} + R_{21}^{2} L_{- V, V -} + R_{2}^{2} L_{V 0, 0 V} + R_{1}^{2} L_{0 V, V 0}],

where in our notation we use the symbol

L_{j l, m n} = \int_{0}^{\infty} d τ 〈 {\tilde{σ}}_{j l} (τ), {\tilde{σ}}_{m n} (0) 〉 e^{- i (ω - ω_{L}) τ}

to denote the Laplace transformation of the two-time correlation functions. Meanwhile, the population difference operator

〈 {\tilde{σ}}_{z z} 〉 = 〈 {\tilde{σ}}_{+ +} 〉 - 〈 {\tilde{σ}}_{- -} 〉

has been introduced to simplify the formations of the resonance fluorescence spectra, where the spectrum functions are

\begin{array}{l} L_{z z, z z} = \frac{1}{d} {n_{11} [{〈 {\tilde{σ}}_{+ +} 〉}_{s s} + {〈 {\tilde{σ}}_{- -} 〉}_{s s} - {〈 {\tilde{σ}}_{z z} 〉}_{s s}^{2}] - n_{12} {〈 {\tilde{σ}}_{z z} 〉}_{s s} {〈 {\tilde{σ}}_{00} 〉}_{s s} - n_{13} {〈 {\tilde{σ}}_{z z} 〉}_{s s} {〈 σ_{V V} 〉}_{s s}}, \\ L_{j l, l j} = \frac{{〈 {\tilde{σ}}_{j j} 〉}_{s s}}{s - C_{l j}} (j \neq l = +, 0, -, V), \end{array}

with s = i(ω − ω_L₎ and the other parameters can be obtain through the Bloch equations. It can be seen from Eq. (41) that the spectral function L_jl_,_lj depicts the transition from state |m〉 to the state |n〉, and the related spectrum exhibits a Lorentz-like shape with specific central frequency and halfwidth, which depends on the imaginary and real part of the coefficient C_mn (the definitions are given in Eq. (14)). Furthermore, the strength of the spectrum is proportional to the steady-state populations of the final state. Thus by increasing the population occupations of the QD states, the corresponding resonance fluorescence sidebands can be largely enhanced.

In the dressed state basis, the in-phase quadrature of the normally-order noise spectrum of the resonance fluorescence field can be expanded in the following forms

S_{x}^{H} (ω) = Re \int_{0}^{\infty} d τ \cos ω τ [T_{x}^{z z} 〈 {\tilde{σ}}_{z z} (τ), {\tilde{σ}}_{z z} (0) 〉 + \sum_{j \neq l = +, 0, -} T_{x}^{j l} 〈 {\tilde{σ}}_{j l} (τ), {\tilde{σ}}_{l j} (0) 〉]

and

S_{x}^{V} (ω) = Re \int_{0}^{\infty} d τ \cos ω τ [\sum_{m = +, 0, -} T_{x}^{V m} 〈 {\tilde{σ}}_{V m} (τ), {\tilde{σ}}_{m V} (0) 〉 + \sum_{n = +, 0, -} T_{x}^{n V} 〈 {\tilde{σ}}_{n V} (τ), {\tilde{σ}}_{V n} (0) 〉],

where subscript x denotes the in-phase quadrature and H (V) indicates the horizontal (vertical) polarized resonance fluorescence field. Similarly, the out-of-phase quadrature squeezing spectra evaluated by a π/2 phase shift of the fluorescence field can be written in the following forms

S_{y}^{H} (ω) = Re \int_{0}^{\infty} d τ \cos ω τ \sum_{j \neq l = +, 0, -} T_{y}^{j l} 〈 {\tilde{σ}}_{j l} (τ), {\tilde{σ}}_{l j} (0) 〉

and

S_{y}^{V} (ω) = Re \int_{0}^{\infty} d τ \cos ω τ [\sum_{m = +, 0, -} T_{y}^{V m} 〈 {\tilde{σ}}_{V m} (τ), {\tilde{σ}}_{m V} (0) 〉 + \sum_{n = +, 0, -} T_{y}^{n V} 〈 {\tilde{σ}}_{n V} (τ), {\tilde{σ}}_{V n} (0) 〉] .

The coefficients appear in the above spectrum functions are

\begin{array}{l} T_{x}^{z z} = 2 {(R_{1} + R_{2})}^{2}, \\ T_{x (y)}^{+ -} = R_{13}^{2} + R_{24}^{2} \pm R_{14} R_{13} \pm R_{14} R_{24} \pm R_{13} R_{23}, \\ T_{x (y)}^{- +} = R_{14}^{2} + R_{23}^{2} \pm R_{24} R_{23} \pm R_{13} R_{23} \pm R_{14} R_{24}, \\ \begin{matrix} T_{x (y)}^{+ 0} = R_{21}^{2} \pm R_{21} R_{11}, & T_{x (y)}^{0 +} = R_{11}^{2} \pm R_{21} R_{11}, \\ T_{x (y)}^{- 0} = R_{22}^{2} \pm R_{22} R_{12}, & T_{x (y)}^{0 -} = R_{12}^{2} \pm R_{22} R_{12}, \\ T_{x (y)}^{+ V} = R_{22}^{2} \mp R_{12} R_{22}, & T_{x (y)}^{V +} = R_{12}^{2} \mp R_{12} R_{22}, \\ T_{x (y)}^{0 V} = R_{1}^{2} \mp R_{1} R_{2}, & T_{x (y)}^{V 0} = R_{2}^{2} \mp R_{1} R_{2}, \\ T_{x (y)}^{- V} = R_{21}^{2} \mp R_{11} R_{21}, & T_{x (y)}^{V -} = R_{11}^{2} \mp R_{11} R_{21} . \end{matrix} \end{array}

Under two-photon resonance condition, the two-mode noise spectrum of the resonance fluorescence field can also be acquired through the definitions. For example, the in-phase quadrature squeezing spectrum originating from the transitions between states |+〉 and |−〉 is

S_{x}^{+, -} (ω) = Re \int_{0}^{\infty} d τ \cos ω τ [T_{x}^{+ -} 〈 {\tilde{σ}}_{+ -} (τ), {\tilde{σ}}_{- +} (0) 〉 + T_{x}^{- +} 〈 {\tilde{σ}}_{- +} (τ), {\tilde{σ}}_{+ -} (0) 〉] .

In the frame rotating with the laser’s frequency, the transition operators of the dressed QD can be given as

{\tilde{σ}}_{+ -} (t) = {\tilde{σ}}_{+ -}^{'} (t) e^{- i (ω_{L} - \tilde{Ω}) t}, {\tilde{σ}}_{- +} (t) = {\tilde{σ}}_{- +}^{'} (t) e^{- i (ω_{L} + \tilde{Ω}) t},

that

{\tilde{σ}}_{+ -}^{'} (t)

and

{\tilde{σ}}_{- +}^{'} (t)

are slowly varied operators obey the following equations

〈 {\dot{\tilde{σ}}}_{+ -}^{'} 〉 = Re [C_{+ -}] 〈 {\tilde{σ}}_{+ -}^{'} 〉, 〈 {\dot{\tilde{σ}}}_{- +}^{'} 〉 = Re [C_{- +}] 〈 {\tilde{σ}}_{- +}^{'} 〉 .

It is not difficult to see through Eqs. (38) that the photons generated by transitions |+〉 → |−〉 and |−〉 → |+〉, with frequencies $ω_{L} - \tilde{Ω}$ and $ω_{L} + \tilde{Ω}$ , satisfy the two-photon resonance condition. Thus by considering the frequency connection of the photon pair, we can obtain the following noise spectrum functions

T_{x}^{+ -} \int_{0}^{\infty} d τ e^{i ω_{1} τ} 〈 {\tilde{σ}}_{+ -} (τ), {\tilde{σ}}_{- +} (0) 〉 = T_{x}^{+ -} \int_{0}^{\infty} d τ e^{i δ_{- +} τ} 〈 {\tilde{σ}}_{+ -}^{'} (τ), {\tilde{σ}}_{- +}^{'} (0) 〉

and

T_{x}^{- +} \int_{0}^{\infty} d τ e^{i ω_{2} τ} 〈 {\tilde{σ}}_{- +} (τ), {\tilde{σ}}_{+ -} (0) 〉 = T_{x}^{- +} \int_{0}^{\infty} d τ e^{- i δ_{- +} τ} 〈 {\tilde{σ}}_{- +}^{'} (τ), {\tilde{σ}}_{+ -}^{'} (0) 〉 .

In the above expressions, the two-photon resonance condition reveals the relation ω₁ + ω₂ = 2ω_L should be satisfied, where $δ_{- +} = ω_{1} - (ω_{L} - \tilde{Ω})$ denotes the frequency deviation to the photons generated by the dipole transition |−〉 → |+〉. The positive and negative values of the δ₋₊ indicates a blue shift and red shift in the frequency of the generated photon, where its counterpart undergoes an opposite frequency shift from the corresponding sideband (in this case a positive value of δ₋₊ indicates a red shift of its paired photons from the central frequency $ω_{L} + \tilde{Ω}$ ). Combining Eqs. (47) and (49)–(51), the in-phase (out-of-phase) quadrature of the two-mode noise spectrum, which describes the squeezing property of the photon pairs generated via two-photon process, appears to be

S_{x (y)}^{-, +} (δ_{- +}) = - \frac{C_{- +}^{'}}{C_{- +}^{'^{2}} + δ_{- +}^{2}} [T_{x (y)}^{+ -} {〈 {\tilde{σ}}_{+ +} 〉}_{s s} + T_{x (y)}^{- +} {〈 {\tilde{σ}}_{- -} 〉}_{s s}] .

In the above expression, $C_{- +}^{'} = C_{+ -}^{'}$ represent real parts of the coefficients C₋₊ and C₊₋. In fact, the two-mode noise spectra of the resonance fluorescence field generated through other transition channels can be acquired and have similar forms

S_{x (y)}^{j, l} (δ_{j l}) = - \frac{C_{j l}^{'}}{C_{j l}^{' 2} + δ_{j l}^{2}} [T_{x (y)}^{l j} {〈 {\tilde{σ}}_{l l} 〉}_{s s} + T_{x (y)}^{j l} {〈 {\tilde{σ}}_{j j} 〉}_{s s}], (j \neq l = +, 0, -, V)

where the subscripts x and y represent the in-phase and out-of-phase quadratures of the resonance fluorescence field. The detunings denoted by δ_jl, illustrate the frequency deviation of the photon pairs to the corresponding sidebands, and

C_{j l}^{'}

is the real part of C_jl. The detunings are defined as

\begin{array}{l} δ_{- +} = ω_{1} - (ω_{L} - \tilde{Ω}), δ_{0 +} = δ_{- V} = ω_{1} - (ω_{L} - {\tilde{Ω}}_{+}), \\ δ_{0 V} = ω_{1} - (ω_{L} - Δ), δ_{- 0} = δ_{V +} = ω_{1} - (ω_{L} - {\tilde{Ω}}_{-}) . \end{array}

According to Eqs. (52) and (53), we can expand the two-mode squeezing spectra in more detail forms

S_{x}^{0, +} (δ_{0 +}) = K_{0 +} \frac{Δ Ω {\tilde{Ω}}_{+}}{\tilde{Ω} Ω_{e}^{2}} [Ω_{1} {〈 {\tilde{σ}}_{00} 〉}_{s s} - Ω_{2} {〈 {\tilde{σ}}_{+ +} 〉}_{s s}],

S_{x}^{-, 0} (δ_{- 0}) = K_{- 0} \frac{Δ Ω {\tilde{Ω}}_{-}}{\tilde{Ω} Ω_{e}^{2}} [Ω_{1} {〈 {\tilde{σ}}_{00} 〉}_{s s} - Ω_{2} {〈 {\tilde{σ}}_{- -} 〉}_{s s}],

S_{x}^{0, V} (δ_{0 V}) = K_{0 V} \frac{Δ Ω}{{\tilde{Ω}}^{2}} [Ω_{1} {〈 {\tilde{σ}}_{00} 〉}_{s s} - Ω_{2} {〈 {\tilde{σ}}_{V V} 〉}_{s s}],

\begin{matrix} S_{x}^{-, +} (δ_{- +}) = - K_{- +} \frac{{〈 {\tilde{σ}}_{+ +} 〉}_{s s}}{{\tilde{Ω}}^{2}} {\frac{Δ Ω}{Ω_{e}^{2}} [Ω_{1} {\tilde{Ω}}_{+}^{2} - Ω_{2} {\tilde{Ω}}_{-}^{2}] + Ω_{1}^{2}} \\ - K_{- +} \frac{{〈 {\tilde{σ}}_{- -} 〉}_{s s}}{{\tilde{Ω}}^{2}} {\frac{Δ Ω}{Ω_{e}^{2}} [Ω_{1} {\tilde{Ω}}_{-}^{2} - Ω_{2} {\tilde{Ω}}_{+}^{2}] + Ω_{2}^{2}}, \end{matrix}

S_{y}^{-, V} (δ_{- V}) = K_{- V} \frac{Δ Ω}{2 Ω_{e}^{2}} [Ω_{1} {〈 σ_{V V} 〉}_{s s} - Ω_{2} {〈 {\tilde{σ}}_{- -} 〉}_{s s}],

S_{y}^{V, +} (δ_{V +}) = K_{V +} \frac{Δ Ω}{2 Ω_{e}^{2}} [Ω_{1} {〈 σ_{V V} 〉}_{s s} - Ω_{2} {〈 {\tilde{σ}}_{+ +} 〉}_{s s}],

where the coefficients

K_{j k} = C_{j k}^{'} / (C_{j k}^{' 2} + δ_{j k}^{2})

. Obviously, both the in-phase and out-of-phase quadratures of the noise spectra strongly depend on the driving conditions and the steady-state populations.

Funding

National Natural Science Foundation of China (NSFC) (11474221); Shanghai Science and Technology Committee (18JC1410900); Shanghai Education Commission Foundation.

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Controllable radiation properties of a driven exciton-biexciton quantum dot couples to a graphene sheet

Abstract

1. Introduction

2. The system and its steady-state properties

3. Resonance fluorescence and probe absorption spectra with different polarizations

4. Two-mode noise spectrum of the resonance fluorescence field

5. Conclusions

Appendix A Green tensor of a dipole placed above the graphene sheet

Appendix B Dressed state basis under two-photon resonance condition and the master equations

Appendix C Resonance fluorescence and two-mode squeezing spectra expressed in the dressed state basis

Funding

References

Cited By

Figures (11)

Equations (60)

Optics Express