Second-order coherence function of a plasmonic nanoantenna fed by a single- photon source

We study the second-order coherence function of a plasmonic nanoantenna fed by near-field of a single-photon source incoherently pumped in the continuous wave regime. We consider the case of a strong Purcell effect, when the single-photon source radiates almost entirely in the mode of a nanoantenna. We show that when the energy of thermal fluctuations, kT , of the nanoantenna is much smaller than the interaction energy between the electromagnetic field of the nanoantenna mode and the single-photon source, R Ω  , the statistics of the emission is close to that of thermal radiation. In the opposite limit, R kT Ω >>  , the nanoantenna radiates single photons. In the last case, we demonstrate the possibility of overcoming the radiation intensity of an individual single-photon source. This result opens the possibility of creating a high-intensity single-photon source. © 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

SPSs suitable for nanophotonic applications have an important drawback in that their radiation rate is low [13]. The characteristic radiation rate of SPSs based on solid-state quantum emitters does not exceed one radiation event per nanosecond. This radiation rate could be increased by placing an SPS inside an open resonator, i.e. using the Purcell effect [14]. This increase is proportional to the quality factor of the resonator and is inversely proportional to the volume of the resonator mode.
In nano-optics, a system consisting of an antenna and an SPS should be nanosized. Metallic plasmonic nanoantennas satisfy this requirement. In such a system, the role of the resonator mode is played by localized surface plasmons. Although the Q-factor of plasmonic structures is relatively low, due to the small volume of the modes, the Purcell factor reaches a value of 2 4 For many applications, it is important to know how the antenna-SPS system radiates. Note that when the Purcell factor is large, the SPS mainly radiates in resonator mode [17,18]. In other words, an excited SPS passes the main part of the energy to a nanoantenna, which then reradiates this energy. Since the characteristic radiation rate of plasmonic structures is several orders of magnitude greater than the radiation rate of SPS, we achieve a desirable increase in intensity.
On the other hand, since a nanoantenna without an SPS is in thermal equilibrium, it radiates as a black body with a second-order coherence function (2) (0) 2 g = [19,20]. Thus, the system may no longer radiate single photons, even though (2) (0) 0 g = for an SPS. This has been confirmed by recent experimental results involving measurements of the radiation statistics of plasmonic structures interacting with SPSs [21][22][23][24][25][26][27][28][29][30][31][32]. In the overwhelming majority of these experiments, (2) (0) g of the radiation of an antenna-SPS system has a value of a few tenths. Some of these experiments even demonstrate super-Poisson statistics, with (2) 25,29,31]. However, if an SPS passes only one photon into the antenna mode, we can expect this photon to be radiated by the antenna before thermalization. It has recently been theoretically shown that a plasmonic nanoantenna may produce single-photon radiation if it is excited by coherent pumping [33], or if coherent population trapping is used in a three-level system [34]. However, incoherent pumping is also widely used and more easily achieved in practical realizations of SPSs.
In this paper, we demonstrate that it is possible for an antenna-SPS system to emit single photons under incoherent pumping of the SPS. Using computer simulation, we show that for a plasmonic nanoantenna-SPS system, the values of the second-order coherence function, (2) , are in the range from 0 to 2, depending on the ratio of the energy of thermal fluctuations, , kT of the nanoantenna and the interaction energy, , R Ω  between the mode of the nanoantenna and the SPS. For , the Purcell factor is small, as is the part of the energy transferred from the emitter to the antenna; as a result, the nanoantenna radiates as a black body with (2) the Purcell factor and the radiation rate are large. In such a case, the rearrangement of the quantum states of the nanoantenna effectively gives single-photon emission. In this case, (2) g can reach zero. The obtained result can be used to create nanoscale ultrafast SPSs based on plasmonic nanoantennas.

The model
We consider a plasmonic nanoantenna, which size is much less than the radiating wavelength in a free space, fed by an SPS. We assume that the SPS is a two-level system (TLS) interacting with only one of the nanoantenna modes and transmits its energy to the nanoantenna through near-field interaction. The Hamiltonian of such a system has the form [20,35]: ω ω σ σ σ where TLS ω and M ω are frequencies of the TLS transition and the antenna mode, respectively.
The first term in Eq. (1) describes the nanoantenna mode; operators â + and â are the creation and annihilation operators of a plasmon in the mode, and satisfy the commutation relation ˆ,  (for more detailed information about the second quantization procedure for the near field in dissipative dispersive media, see Refs [36][37][38][39].).
In the absence of the interaction between the nanoantenna and the dipole emitter, the eigenstates of the system consist of the nanoantenna and dipole emitter eigenstates, ,  To describe the losses, we introduce three reservoirs interacting with the system. The Hamiltonian of these reservoirs has the form [19,40]: The first term describes the electromagnetic field of the free space, which is responsible for the radiative losses of the system. Operators , b λ The first term in Eq.
is the electric field after the second quantization procedure, and , , / The second term describes dephasing of the TLS, i.e. the process of emission and absorption of a quantum of the reservoir excitation in which the energy of the system does not change, but the average dipole moment (non-diagonal elements of its density matrix) decays [19,40]. The last term describes the interaction of phonons in the metal and the nanoantenna mode.
Using the Born-Markov approximation, and excluding reservoir variables, we obtain the master equation for the density matrix in the Lindblad form [19,41]: where the Lindblad superoperator, describes the relaxation processes in the system due to interaction with reservoirs with rates , where i T is the temperature of the i-th reservoir. Note that in Eqs. (6) and (7), we add incoherent pumping of the TLS by introducing the term pump , γ which corresponds to the transition between the eigenlevels with increasing energy [43]. We assume that the temperature of the free space reservoir is zero, rad 0 T = , and the temperature of the pumping reservoir is pump 0, T = − so that interaction with this results in an energy transfer only from the reservoir to the system. The temperature of the reservoir of Joule losses can change. We investigate the dependence of the system behavior on this temperature. We suppose that the radiative and nonradiative losses of the nanoantenna and the TLS dephasing remain the same as for a non-interacting antenna and SPS.
From the system of equations in (8), we can obtain the dynamics of the diagonal elements of the density matrix. We can then use these to calculate all the average values of the operators of interest at any moment in time, as ˆˆ( ( ) ) ( ) . In the following, we consider the behavior of the second-order coherence function (2) (0) g .

The plasmonic nanoantenna as a single-photon source
The second-order coherence function (2) Here, we assume that the nanoantenna makes the main contribution to the radiation (which is much greater than that of the SPS). This assumption is reasonable because, as it has been mentioned in [18] (2) (0) g crosses over from 2 to 0. In the limit , R kT Ω >>  (2) (0) g tends to 0, corresponding to the radiation of single photons. Thus, for a sufficiently strong interaction and a low pumping rate, the plasmonic nanoantenna emits single photons, in agreement with experiment [23,27,32].
It should be noted that an increase in the pumping rate, pump rad , γ γ causes (2) (0) g to tend to unity, and the light from the system becomes coherent (see the dashed and dot-dashed curves in Fig. 1). This behavior corresponds to the coherent generation of the near-field in the nanoantenna; in this case, the system turns to a nanolaser. However, when only one SPS is used, this regime cannot be achieved, since the corresponding pumping rate is very high and cannot be obtained in experiments (see also Ref [45].). Thus, in the case of nanoantenna fed by one SPS, the real pumping rate is much lower than the threshold value. The case of zero pumping corresponds to the situation in which only the reservoir with a temperature greater than zero provides energy to the system. Note that at room temperature, in the optical region, the black body radiation is negligible and the system essentially does not radiate. To create radiation which can be detected, pump γ should have a reasonable value that is greater than zero. Thus, it is possible to observe single-photon emission from a nanoantenna by setting the required temperature and the pumping power of the TLS. The effect described here was obtained using a numerical simulation of Eq. (8). To clarify the mechanism of this effect, we consider a simplified model of the original problem.

Low-quantum excitation limit
To understand the behavior described in the previous section, we consider a simplified model of the system consisting of a nanoantenna coupled to an SPS. Let us assume that the pumping power is zero and take into account only the excitations of the lower states, Eq. (2), which give first nonzero contributions to ( ) Suppose for a moment that we have only the interaction with the reservoir of Joule losses, with temperature T . In this case, the system comes to thermal equilibrium with the reservoir, and the diagonal elements of the density matrix are then distributed according to Gibbs distribution [19], i.e.: where m E is the energy of the m -th eigenstate. Using Eqs. (10) to (12), we can calculate (2) (0) g : ( ) ω ω = = , we have: Hence, in the limits / 1 R kT Ω >>  and / 1 R kT Ω <<  (Fig. 1, dashed and solid curves, respectively), at zero pumping rate, we obtain: (2) Expressions (15) and (16) are in qualitative agreement with the results of numerical simulation, as shown in Fig. 1. The obtained result can be qualitatively explained as follows. When the energy of thermal fluctuations, , kT is much higher than the interaction energy,

Radiation
As can be see pumping rate exceed the in power at whic useful in prac which the nan time a radiatio rate of ener    In both cases, the radiation intensity rad S H  rates are larger than that for a single TLS, and may reach one radiation process per picosecond (for the parameters used in Refs [27]. and [32], the values are larger by two and three orders of magnitude, respectively). The quantitative distinction between the experimental results obtained in Refs [27]. and [32] (for the value of (2) (0) g and the radiation rates) is due to the significant difference in the Rabi constants of the systems and, consequently, the Purcell factors. Moreover, according to Fig. 3 and Eq. (17), a small negative detuning, ,

M TLS
ω ω < can reduce the value of (2) (0) g even further, as observed in the experiment in Ref [27].

Conclusion
As discussed in the introduction, an attempt to increase the radiation rate of isolated SPSs by using plasmonic nanoantennas is expected to lead to the deterioration of single-photon radiation properties due to the contribution of nanoantenna (open-cavity) radiation to the total emission of the system. In the present paper, we demonstrate the possibility of a nanoantenna fed by an SPS radiating single photons at high rates. We show that the second-order coherence function of radiation, (2)