The deterministic single photon source and photon pair source are important for quantum information, quantum computing, and quantum communications.^{[1– 5]} In quantum communications, most of the communication protocols are based on entangled photon pairs.^{[6– 9]} Generally, there are two ways to generate entangled photon pairs: one is to use parametric down conversion in nonlinear optical media, and the emission photons satisfy the Poissonian statistics; ^[10] the other way is to use the radiative recombination of the bi-exciton state in single quantum dots.^[10] The last method cannot avoid the effect of the single photon process.

We suggest that two single quantum emitters are coupled as a stable photon pairs source. Because the coupled two single quantum emitters show the same energy structure as that of the single quantum dots, and the coupling between the single quantum emitters demonstrates the two-photon resonance peak at , with , and being the transition frequencies of the single quantum emitters A and B, respectively. At the two photon resonance peak position, the coupled quantum emitters emit two photons (more than one photon) at a time. The interaction strength between the single quantum emitters can be conveniently modified by changing the separated distance between the two quantum emitters.

In this paper, we consider the two coupled single quantum emitters as a photon pair source. The coupling of the two single quantum emitters provides a way to excite the two single quantum emitters to their excited states (the bi-exciton states) by a single external field. The external field excites the emitter A into its excited state, then the emitters A and B exchange their status, and the external field then excites the emitter A into its excited state again, which combines into a bi-exciton state. When the external field satisfies , the external field can excite the two emitters into their excited states. On this condition, the coupled two single quantum emitters can be considered as a photon pair source. The probability of emitting two photons P₂ and the correlation coefficient R_AB are employed to describe the photon pair source quality. R_AB ∼ 1 corresponds to the fact that the emitters A and B emit photons at the same time, while R_AB ∼ 0 corresponds to the fact that the emitters A and B randomly emit photons, while Mandel’ s Q parameters are used to describe the self correlations of the photons emitted from the molecule A, molecule B, and the coupling system.

The rest of this paper is organized as follows. In Section  2 the model of the coupled two single quantum emitters is introduced, and the generating function approach is briefly reviewed. In Section  3, we demonstrate the main numerical results, and a short discussion is also presented. Some conclusions are drawn in Section  4.

We consider two separated single quantum emitters A and B, which are coupled through the dipole– dipole interaction. The emitters A and B can be described as a two-level system. The energy states of the emitters A and B are denoted by | g_A〉 and | g_B〉 (ground states), respectively, | e_A〉 and | e_B〉 (excited states) respectively. The transition frequencies of emitters A and B are and , respectively. The coupled emitters can be described in the direct product space of the eigenstates space of emitters A and B. Then the Hamiltonian can be written as^{[11, 12]}

where and are the “ bare” Hamiltonians of emitters A and B, and are the Hamiltonians of emitters A and B interacting with an external light field, and are the unity operators in the Hilbert space of emitters A and B, and is the Hamiltonian of the dipole– dipole interaction between the emitters.

The “ bare” Hamiltonians, in the rotating wave approximation (RWA), can be written as

where are the detuning frequencies, and ω _L is the angular frequency of the external field.

The Hamiltonians of the interaction between emitters and external field in the RWA can be written as

where h.c. represents the Hermitian conjugate, are the Rabi frequencies, r_n is the position of emitter n, μ _n is the transition dipole moments of the emitter n (n = A, B), and is the amplitude of the external field at the position r_n.

The coupling Hamiltonian of the two emitters, the dipole– dipole interaction, can be written as^{[11, 13, 14]}

with

where and are the unit vectors of the dipole moments of molecules A and B, and is the unit vector of the displacement of emitters A and B. The z = k₀r_AB, , r_AB = | r_AB| ≡ | r_A − r_B| is the distance between the emitters A and B, and c is the speed of light in a vacuum.

The evolution of the system can be described by the Liouville-von Neumann equation^{[15, 16]}

where describes the disspative part of evolution, which can be expressed as^[13]

where Γ _A and Γ _B are the spontaneous emission rates of emitters A and B, Γ _AB is the cross damping rate, which can be written as^{[11, 13, 14]}

The spontaneous emission rates Γ _n, (n = A, B) read

The generating function satisfies

where , with ρ ^{(k+ l)} being the density matrix part corresponding to the case that the emitter A has emitted k photons and emitter B has emitted l photons; and are the spontaneous emission operators; ^[17]s_A and s_B are the auxiliary parameters for photon counting statistics.

The working generating function is defined as

where | m〉 = | g_Ag_B〉 , | g_Ae_B〉 , | e_Ag_B〉 , and | e_Ae_B〉 .

The moments of emission photons from emitters A and B can be written as

the corresponding Mandel’ s Q parameters can be expressed as

The moments of the total emission photons can also be expressed as

and the corresponding Mandel’ s Q parameter Q_tot can be written as

The correlation of the photons, emission from emitters A and B, is^[18]

and the correlation coefficent can be written as^{[19, 20]}

where

with n = A and B.

The probability of emitting m photons in time interval [0, t] can be written as

In this section, we consider the single terrylene molecules in a paraterphenyl crystal.^{[11, 12]} The system has been used to demonstrate the high spatial resolution and two-photon resonance, ^[12] and implemented some quantum gates in the small quantum network.^[11] The spontaneous emission rates of molecule A and molecule B are Γ _A = Γ _B = 50  MHz.^{[11, 12]} The cross damping rate Γ _AB = 9  MHz, and the interaction between molecules A and B is , and the difference transition frequency between the molecules is .^[12]

The probabilities of emitting one, two, and more photons at Δ _A = Δ ω /2 are shown in Fig.  4, while the central peak (Δ _A = Δ ω /2) is known as the two-photon resonance peak. Figures  1(a)– 1(c) correspond to the Rabi frequencies Ω _A = Ω _B = 50, 100, and 150  MHz, respectively. The solid lines correspond to the probabilities of emitting one photon [P₁(t)], the dashed lines correspond to the probabilities of emitting two photons [P₂(t)], and the dash-dotted lines correspond to the probabilities of emitting multi- photons [P_multi(t)]. From the figures, one could know that the maximum value of the probability of emitting two photons is two times greater than that of the probability of emitting one photon, for all the driving field strengths.

Fig.  1.

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Fig.  1. Probabilities of emitting one, two, and more photons versus evolution time t, with the detuning frequency Δ A = Δ ω /2. Panels (a), (b), and (c) correspond to the Rabi frequencies Ω A = Ω B = 50, 100, and 150  MHz, respectively. The other parameters are the same as those in Fig.  1. The solid lines correspond to the probabilities of emitting one photon P1(t), dashed lines correspond to the probabilities of emitting two photons P2(t), and the dash-dotted lines correspond to the probabilities of emitting multi- photon Pmulti(t) = 1 − P0(t) − P1(t) − P2(t).

The values of correlation coefficent R_AB of the photons emitted from molecule A and molecule B are shown in Fig.  2. Figures  2(a)– 2(c) correspond to the weak, medium, and strong driving fields, respectively. The correlation coefficent R_AB shows a similar behavior to Mandel’ s Q parameter Q_tot. At the left and right resonance peak positions, the correlation coefficient R_AB is negative, the negative correlation, of the photons emitted from molecules A and B, means the molecules A and B cannot emit photons at the same time. On the weak driving field condition, the photons emitted from molecule A and molecule B show positive correlation at the peak position (Δ _A = Δ ω /2), which means that molecule A emits a photon at time interval [t, t + dt], while molecule B has a big probability to emit a photon at the same time interval [t, t + dt]. The photons emitted from molecule A and molecule B are bunching together, which makes Mandel’ s Q parameter greater than zero. The correlation coefficient R_AB first increases and then decreases with the driving field strength inceasing, which means that the strong driving field destroys the correlation of the photons emitted from molecules A and B.

Fig.  2.

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	Fig.  2. The correlation coefficent RAB versus the detuning frequency Δ A. (a), (b), and (c) correspond to the Rabi frequencies Ω A = Ω B = 50, 100, and 150  MHz, respectively. The other parameters are the same as those in Fig.  3.

The emission photons 〈 N〉 and Mandel’ s Q parameters versus the detuning frequency are plotted in Fig.  1. The first column [Figs.  3(a) and 3(d)] corresponds to the Rabi frequencies Ω _A = Ω _B = 50  MHz, the second column [Figs.  3(b) and 3(e)] corresponds to the Rabi frequencies Ω _A = Ω _B = 100  MHz, and the last column [Figs.  3(c) and 3(f)] corresponds to the Rabi frequencies Ω _A = Ω _B = 150  MHz. The red solid lines correspond to the total photons and Mandel’ s Q parameters of molecules A and B, the dashed-dotted lines correspond to the photons and Mandel’ s Q parameters of molecule A, and the dashed lines correspond to the photons and Mandel’ s Q parameters of molecule B.

Fig.  3.

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Fig.  3. The emission photons number 〈 N〉 versus the detuning frequency Δ A for different Mandel’ s parameters, with spontaneous emission rates Γ A = Γ B = 50  MHz, cross damping rate Γ AB = 9  MHz, and the interaction . Panels  (a) and (d) correspond to the Rabi frequencies Ω A = Ω B = 50  MHz, panels  (b) and (e) correspond to the Rabi frequencies Ω A = Ω B = 100  MHz, and panels  (c) and (f) correspond to the Rabi frequencies Ω A = Ω B = 150  MHz. The red lines correspond to the total emission photons, dashed– dotted lines correspond to the photons emitted from molecule A, and the dashed lines correspond to the photons emitted from molecule B.

From Fig.  3, one can know that the total emission photon number 〈 N_tot〉 emitted from molecules A and B is a little smaller than the photon number 〈 N_B〉 emitted from molecule B at the left peak, and a little greater than the photon number 〈 N_A〉 emitted from molecule A at the right peak, for all the driving field strengths Ω _A = Ω _B = 50  MHz (weak condition), 100  MHz (medium condition), and 150  MHz (strong condition). The differences between the total emission photons 〈 N_tot〉 and the emission photons 〈 N_A〉 and 〈 N_B〉 are caused by the coherence of the states | e_Ag_B〉 and | g_Ae_B〉 , and the virtual photons emission and absorption between the states.^[21] The dipole– dipole interaction causes the resonance peak to shift.^[13] The Mandel’ s Q parameters Q_tot, Q_A, and Q_B are negative at the positions of left and right resonance peaks for all driving field strength conditions. That means that the photons emitted from molecule A, from molecule B, and from molecules A and B are anti-bunching at the resonance peak positions. On the weak driving condition, the emission photon number 〈 N_tot〉 is very small, and the corresponding Mandel’ s Q parameter Q_tot is positive at the position Δ _A = Δ ω /2 (central peak), which corresponds to the emission photons showing bunching behavior, the result is the same as that in Ref.  [12]. The emission photon number 〈 N_tot〉 increases, and the corresponding Mandel’ s Q parameter Q_tot first increases and then decreases with the driving field strength increasing. On the strong driving field condition, Mandel’ s Q parameter Q_tot decreases to a negative value, corresponding to the fact that the emission photons show anti-bunching behaviors.

In this paper, we consider a system with two coupling two-level single molecules as a photon pair source. The probabilities of emitting one, two-, and multi- photons: P₁(t), P₂(t), and P_multi(t), are respectively calculated at Δ _A = Δ ω /2 via generating function approach. In a short time region, the probability of emitting two photons is three times greater than the probabilities of emission one and more photons. That means that in a short time, the system is a good two-photon source. In a long time region, the correlation coefficient R_AB is introduced to describe the coincidence of the emission photons, and on the weak driving field condition, the correlation coefficient and the Mandel’ s Q parameter Q_tot show that the emitted photons are bunching. The system could emit the photon pair train driven by a weak field. As the driving field increases, the correlation between the emission photons is destroyed by the driving field.