Considering a single quantum dot (QD) that is modeled as a TLS, with ground-state | 0〉 and an excited-state | x〉 , which is coupled to the photon bath (vacuum fluctuation) and phonon bath, both represented by sets of harmonic oscillators, and a square laser pulse pumps the single emitter from electronic ground state | 0〉 to the excited state | x〉 . We make a semiclassical treatment of the coupling between the square laser pulse and SPSL. The Rabi frequency Ω is not so large that it dresses the states leading to Mollow’ s triplet states. With the rotating-wave approximation (RWA) on both the driving laser and system– radiation coupling terms, the Hamiltonian is

Here and represent the annihilation (creation) operators for photons and phonons respectively, the subscripts k and q stand for the wave vector of photon and phonon respectively, ħ ω ₀ is the total exciton energy, ω _k (ω _q) denotes the photon (phonon) frequencies, g_k (f_q) represents the exciton– photon (exciton– phonon) coupling factor, and Rabi frequency Ω describes the amplitude of classical laser pulse with frequency ω _L, coupling to the QD by dipole approximation Ω = E(t)· | μ | /ħ , E(t) is the time-domain envelope of the laser pulse, μ is the dipole moment. A period time-dependent Rabi frequency Ω (t) is

and Ω (t + n(T + T₀)) = Ω (t) where n = 1, 2, ..., (see Fig.  1).

Fig.  1.

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	Fig.  1. The periodic square laser pulses is described by Eqs.  (2a) and (2b). The starting point of the laser pulse is at t = 0, and T is the width of the pulse. T0 is the time interval between two successive pluses with conditions T0 ≫ T and T0 ≫ 1/Γ .

The exciton population is described by Pauli operator σ _z = | x〉 〈 x| – | 0〉 〈 0| , the lowering operator σ ⁻ = | 0〉 〈 x| , and the raising operator σ ^† = | x〉 〈 0| . Since the fundamental band gap ħ ω ₀ is much larger than κ _BT, the photon field can be taken as remaining at zero temperature, only the pure dephasing process includes how the environment temperature affects the PSPE and how it destroys the coherent property of photons. Presuming the phonon bath to be in thermal equilibrium, by taking trace over the phonon modes, the spectral density

contains the entire information about the phonon bath which completely describes the interaction between the QD and phonon bath.^[25] For arsenide QDs, the coupling to acoustic phonons by means of deformation potential causes the pure dephasing process that has been found to be the dominant decoherence mechanism at elevated temperature.^{[21– 23]} An appropriate choice for the spectral function and its related cut-off frequency, can be specified by the QD environment. We adopt the model in Ref.  [26] and get the spectral density function as follows:

In the model, the wave-function is determined by the spherically symmetric harmonic confinement potential for the QD, the structure constant is

where the cut-off frequency , D_h and D_e are the corresponding bulk deformation-potential constants, ρ is the sample density, u is the velocity of phonon, and d is the electron (hole) ground-state localization length. The explicit form for 𝛾 _deph within Markov limit^[26] reads

The formula manifests the influence of temperature T and Rabi frequency Ω to the pure dephasing rate. We are more interested in the high temperature situation, since the realization of SPS in high temperature will be more practical. With the temperature increasing, the first-order approximation with respect to ħ Ω /κ _BT is derived by Tayler series expanding, here we consider the cutoff frequency ω _c ∼ 10¹²  Hz which is far larger than the Rabi frequency Ω ∼ 10⁹  Hz, then we find an approximation formula at high temperature

The functions of the pure dephasing rate with respect to the environment temperature in Eqs.  (5) and (6) are shown in Fig.  2. When the κ _BT is large enough in comparison with ħ Ω , the dephasing rate displays a linear relationship with the environment temperature T, which is in agreement with the result of Ref.  [27]. For simplicity, we just considered the resonance situation, in the rotating frame of reference with frequency ω _L, within Born– Markov approximation and RWA, the dynamic evolution of the density operator is given by^[28]

The first term describes the evolution of the coupling of the driving laser to the system, δ (t) = ω _o − ω _L, the second term describes the influence of radiation field, ^[11] 𝛾 _sp is the half width of spontaneous radiation rate Γ in reference, ^{[8, 9]} the third term describes the pure dephasing process caused by the coupling of phonon bath to the QD, this term includes the temperature effects into the master equations. In the resonance situation, the frequency shift caused by the phonon bath has been neglected, we obtain the stochastic Liouville equations by taking the pure dephasing process into account.^[29] The main difference between the isolated system and the solid-state system is that the dynamic equation contains the third term and 𝛾 _deph is temperature-dependent.

Fig.  2.

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Fig.  2. The dephasing rate function with respect to temperature. We calculated the parameters as those in Ref.  [26]: De = − 14.6  eV, Dh = − 4.8  eV, d = 4.5  nm, u = 5110  ms− 1, ρ = 5370  kg· m− 3, then obtained α = 0.032  ps2. Here Ω = 10Γ , Γ = 100  MHz, the crossing and the circle represent the results of the modeled function Eq.  (5) and the first-order approximation function of temperature factor Eq.  (6) respectively.

By utilizing state vectors | 0〉 and | x〉 , we expand the operator equation Eq.  (7) into the following density matrix elements differential equations

By solving the above linear coupling differential equations with initial condition (ρ ₀₀, ρ _xx, ρ _0x, ρ _x0) = (1, 0, 0, 0), we obtain the time-dependent analytical expression of density matrix elements.

The definition of the second-order correlation function of a photon is

Here τ = t − T represents the time delay between the first photon and the second one, T is the time that the first photon has been recorded, g⁽²⁾(τ ) is proportional to the probability of detecting two emitted photons, separated by τ and normalized by the probability of independent detection. Since we get ρ _xx(t), then utilizing the quantum regression theorem, ^{[30, 31]} we find the analytical expression of g⁽²⁾(τ ) as follows:

Here , which is a measurement to characterize the quantum coherent properties of SPS.^[32] When g⁽²⁾(τ ) is smaller than zero, the anti-bunching phenomena occurr, this is a quantum light phenomenon which cannot be explained by classical theory of light. Compared with the formula in Ref.  [30], we include the 𝛾 _deph term. By using Cook’ s conditional density operator , ^[33] where n is the number of photons radiated in the time interval t. We now transform the dynamic evolution of the emitter described by Eqs.  (8) into the following set of equations

Note that σ ^{(− 1)} ≡ 0, the distinct difference compared to Eqs.  (8) is the term, this indicates that one other photon must emit after the emitter has been excited; it is not possible that two photons emit simultaneously by the SPS. With the generating function method, ^[9] we define the generating functions as follows:

where s is an auxiliary variable which can be specified on needs. We define the generating Bloch variables as

and define the vector V = (𝒰 (s, t), 𝒱 (s, t), 𝒲 (s, t), 𝒴 (s, t))^T, we then get the modified generalized optical Bloch equations as

where

We solve the modified generalized optical Bloch equations in two separate time intervals with two different Rabi frequencies. At first, we consider the solution when T < t < T + T₀, Ω = 0, and zero detuning frequency δ (t) = ω ₀ − ω _L = 0, ω ₀ = ω _x − ω ₀. ω _i(i = 0, x) represents the energy of QDs’ electrons in state | i〉 (i = 0, x). In this case, the modified generalized optical Bloch equations become

Taking the generalized Bloch vector V(s, T) = (𝒰 (s, T), 𝒱 (s, T), 𝒲 (s, T), 𝒴 (s, T))^T as the initial condition, after making Laplace-transformation, and performing simple algebra operations, we derive the time-dependent generating function solution as follows:

At time t = 0, the SPS is at the ground state, and the vector of the reduced density matrix elements σ (t = 0) = (1, 0, 0, 0)^T. Thus, we have the initial condition of the generalized Bloch vector (𝒰 (s, 0), 𝒱 (s, 0), 𝒲 (s, 0), 𝒴 (s, 0) = (0, 0, − 1/2, 1/2, ), Ω = const., and zero detuning frequency δ _L = ω _L − ω ₀ = 0. Now the modified generalized optical Bloch equations become

By making Laplace-transform t → z on both sides of Eqs.  (18) with initial conditions given above, then transforming the differential equations to algebraic equations. The solution of the algebraic equations is given below

We could derive the initial condition of Eqs.  (16) by making the inverse Laplace transform of this solution with z → T.

Here we are interested in the probability of single photon emission and Mandel’ s Q parameter. Once we get the generating function, under the condition T₀ ≫ 1/𝛾 _sp, we only need the solution when t → ∞

the probabilities of emitting n photons after time t (T < t < T + T₀) could be obtained by SPS interacting with laser pulse in time interval T, and then we get the fractional moments of N(t). The formulas to extraction probability and fractional moments are presented below

where 〈 N⁽ⁿ⁾〉 = 〈 N(N − 1)(N − 2)… (N − n + 1)〉 , (n < N). Through Eq.  (21) and Eq.  (22) we can derive the PSPE, the average number of N photon emission in time t, and the second-order fractional moment of N as given below

The definition of Mandel’ s Q parameter is

it also could be written as

By Eqs.  (20) and (23), we can derive the zero-order approximation solution of the PSPE with respect to spontaneous radiation rate 𝛾 _sp as follows:

Within zero approximation, once the emitter is in the fixed pure dephasing rate, i.e., the environment temperature is small enough such that the coherence still exists, the probability will oscillate without damping, this is the unreasonable case caused by neglecting the spontaneous radiation rate 𝛾 _sp. For the sake of physical reasonability, the spontaneous radiation rate 𝛾 _sp should be included into the exponential part of the above formula, we thus correct Eq.  (28) by including the spontaneous radiation rate 𝛾 _sp. The corrected formula of the PSPE is

By making use of Eqs.  (20), (19), (24), and (25), we can derive the analytical expressions of the first-order fractional moment and the second-order fractional moment as follows:

where the coefficients A_i, B_i, C_i (i = 1, 2) are given in Appendix  A. By substituting Eqs.  (30) and (31) into Eq.  (27) we obtain the analytical expression of Q as follows: