# Photon antibunching control in a quantum dot and metallic nanoparticle hybrid system with non-Markovian dynamics

###### Abstract

Photon-number statistics of the emitted photons from a quantum dot placed in the vicinity of a metallic nanoparticle (with either shell or solid-sphere geometry) in the non-Markovian regime is investigated theoretically. In the model scheme, the quantum dot is considered as a InAs three-level system in -type configuration with two transition channels. One of channels is driven by a polarized classical field while the two channels are coupled to the plasmon modes. Plasmon resonance modes of a nanoshell, in contrast of a nanosphere, are tunable at demand frequency by controlling the thickness and the materials of the core and the embedding media. The results reveal that the emitted photons from the hybrid system under consideration are antibunched. Moreover, the antibunching behavior of the emitted photons can be controlled by the geometrical parameters of the system, namely, the quantum dot-metal nano particle separation distance, as well as the system’s physical parameters including the detuning frequency of the quantum dot transitions with respect to the surface plasmon modes, and the Rabi frequency of the polarized driving field. Additionally, the studied system has the potential to be a highly controllable single-photon source.

###### pacs:

42.50.Nn, 42.50.Ar, 78.67.Hc, 73.20.Mf## I Introduction

Single-photon sources, as a crucial ingredient of quantum information technology, are amongst the most widely investigated quantum systems during the last four decades or so 1 ; 2 ; 3 ; 4 . Numerous applications for single-photon sources have been proposed in the fields of quantum cryptography 6 ; 7 ; 8 ; 9 , quantum repeaters 10 ; 11 , and quantum computation 12 . In addition, the low-noise nature of nonclassical photons makes them as ideal candidates for application in the fundamental measurement problems 13 ; 14 ; 15 . To date, various theoretical schemes and experimental demonstrations have been carried out on the generation of single-photon emitters. Some examples include atomic cascade transition in calcium atoms 16 , single ions in traps 17 , parametric down-conversion 19 ; 18 , single molecules coupled to a resonant cavity20 , semiconductor quantum dots (QDs) 4 ; 21 ; 22 ; 22a , defect centers in diamond 23 ; 24 , single-walled carbon nanotubes 25 , and photonic crystals 25-1 .

Another way of realizing an on-demand single photon source is to use the hybrid systems composed of a semiconductor QD and a metal nanoparticle (MNP) 26 . It is well known that the environment of an emitter changes its decay rate (Purcell effect) 26-1 . The interaction of an MNP with light leads to non-propagating excitations of the conduction electrons of the MNP; the quanta of these excitations are called localized surface plasmons (LSPs) 26-2 . The evanescent near-field associated to the LSPs increases the local density of states (LDOS) around the MNP dramatically 27 ; 28 ; 29 ; 30 ; 30-1 ; 30-2 . When a QD, as an emitter, is located in the evanescent field of an MNP, the decay rate of the QD is affected through the LDOS and increases significantly 27 ; 28 ; 29 ; 30-2 . The physical features of the QD-MNP system can be controlled by the geometry of the hybrid system, i.e., the shape of the MNP and the QD-MNP separation distance. Depending on the geometry, the system can operate either in the strong-coupling or the weak-coupling regime and exhibits the non-Markovian 27 ; 31 or Markovian behavior 30 .

In recent years, significant theoretical studies have been performed on the photon-number statistics of emitted light from a variety of hybrid systems in order to provide efficient methods for the controllable generation of antibunched photons. In Ref. 31-1 the conditions for the realization of photon antibunching of molecular fluorescence in a hybrid system of a single molecule and a plasmonic nanostructure composed of four nanostrips have been theoretically investigated by using the Green’s tensor technique 31-2 . The photon-number statistics from the resonance fluorescence of a two-level atom near a metallic nanosphere driven by a laser field with finite bandwidth has also been studied, and it has been shown that the statistics can be controlled by the location of the atom around the metal nanosphere, the intensity and the bandwidth of the driving laser, and detuning from the atomic resonance 32 . In addition, a theoretical framework based on the combination of the field-susceptibility/Green’s-tensor technique with the optical Bloch equations has been developed 33 to describe the photon statistics of a quantum system coupled with complex dielectric or metallic nanostructures. By using an approach based on the Green’s function method and a time-convolutionless master equation, the dynamics of the photon-photon correlation function in a hybrid system composed of a solid-state qubit placed near an infinite planar surface of a dissipative metal has been studied 27 under the Markov approximation. For a hybrid structure consisting of an optically driven two-level QD coupled to a metallic nanoparticle cluster it has been shown 34 that the single-photon emission can be efficiently controlled by the geometrical parameters of the system.

In this paper, we theoretically investigate the photon-number statistics of the light emitted from a single semiconductor QD with -type configuration in the vicinity of either a metal nanoshell or a solid nanosphere in the non-Markovian regime. The present study follows two main purposes. First, we aim to introduce the hybrid system as a nonclassical photon source. Second, we intend to investigate whether the photon-number statistics can be controlled by the geometrical and physical parameters of the hybrid system. Our theoretical description of the system involves the quantization of electromagnetic field in the presence of an MNP within the framework of the classical dyadic Green’s function approach including quantum noise sources which is appropriate for dispersive and absorbing media.

The ohmic nature of metals has a substantial impact on the reduction of the effective QD-MNP interaction, specially at high frequencies due to the interband transitions. One effective way to deal with this phenomenon is to use geometries in which the surface plasmons are induced in low energies. The plasmon resonance frequencies of a nanoshell are adjustable by the thickness of the nanoshell and the dielectric permittivities of the core and the shell materials. Therefore, the resonance frequency modes of a nanoshell can appear at much lower energies than those of a nanosphere 35 ; 48 ; 49 ; 50 ; 51 ; 52 .

The paper is structured as follows. In Sec.II, we first describe the theoretical model of the hybrid system composed of a single QD coupled to an MNP within the framework of the master equation approach. Then, we derive an expression for the normalized second-order autocorrelation function for the photons emitted by the QD. Numerical results and discussions are presented in Sec.III, where we explore the controllability of the photon-number statistics through the physical as well as geometrical parameters of the hybrid system. Finally, we present our conclusions in Sec. V.

## Ii Description of the Hybrid System

As shown in Fig.1, the physical system consists of a QD which is located at distance from the surface of a nanoshell of inner radius , outer radius , and frequency-dependent permittivity . The QD-nanoshell system is embedded in a homogeneous background medium with relative permittivity . As a realistic example like references 27-1 ; 27-2 ; 27-3 , we choose a -type three-level InAs QD with permittivity 55 ; 54 . The discrete energy states and have energies and , respectively, with respect to the state , such that .

It has been shown experimentally 54 that the dipole moments and in the InAs QD are aligned perpendicular to each other. Thus, the atomic dipole moment operator is taken as in which is real. Moreover, we consider the classical driving field as a -polarized field. Therefore, the transition is coupled to the classical driving field with Rabi frequency , and detuning . On the other hand, it couples to the x component of surface plasmon modes on the MNP with detuning .

The transition is coupled to the elementary z-excitations of the MNP with frequency and detuning . The classical driving field does not couple to the transition because they are aligned along the perpendicular directions.

### ii.1 System Hamiltonian

The total Hamiltonian of the whole system can be written as {linenomath*}

(1) |

where denotes the free Hamiltonian of the QD, describes the Hamiltonian of the medium-assisted quantized electromagnetic field, and refers to the QD-MNP interaction Hamiltonian which are given, respectively, by

(2a) | |||

(2b) | |||

(2c) |

Here, are the Pauli operators, and denote, respectively, the bosonic annihilation and creation operators for the elementary excitations of the lossy metal nanoparticle satisfying the commutation relation 37 , and is the transition dipole moment between and levels. The interaction Hamiltonian of Eq.(2c) has been written in the rotating-wave approximation. Moreover, is the electric field operator at the position of the QD and is defined in the following.

The part in the total Hamiltonian of Eq.(1) accounts for the coupling of the QD to the external driving field and is given by where the effective Rabi frequency is defined as with in which is the dielectric constant of the QD. In this definition, contains both the direct classical monochromatic field of frequency and amplitude , and the scattered field from the MNP which would be defined through the classical dyadic Green’s function as 29 {linenomath*}

(3) |

Here is the phase change associated with the scattered laser field.

Quantization of the electromagnetic fields in the presence of an absorbing and dispersive medium via dyadic Green’s function approach leads to an explicit expression for the electric field operator in the following form 37 {linenomath*}

(4) |

where is the imaginary part of the frequency-dependent dielectric permittivity of absorbing medium. Also, in this equation is the dyadic Green’s function of the system describing the system response at to a point source at . The Green’s function is obtained through two contributions, where is the direct contribution from the radiation sources in free-space solution and is the reflection contribution coming from the interaction of the dipole with the materials. In a frame rotating at the laser frequency the total Hamiltonian of Eq.(1) reads with

(5a) | |||

(5b) | |||

(5c) |

### ii.2 Dynamics of the system

The time evolution of the whole system which is composed of the QD, the MNP, and the coherent laser field, in the interaction picture is determined through the Liouville equation 38 ; 39 {linenomath*}

(6) |

Here, the Hamiltonian and the total density matrix in the interaction picture are defined as: and . The time evolution of the total density matrix in the Schrödinger picture is given by {linenomath*}

(7) |

By formally integrating Eq.(6), and inserting such a formal solution for on the right hand side of Eq.(6) we arrive at {linenomath*}

(8) |

We assume that the states of the system and the reservoir are initially uncorrelated, so that in which and stand for the initial density matrix of the system and the reservoir, respectively. We also assume that the reservoir is in thermal equilibrium. After partial tracing over the reservoir degrees of freedom and applying a second-order Born approximation, the master equation of the reduced density matrix of the QD in the interaction picture, , is obtained as {linenomath*}

(9) | |||

where we have used because the system-reservoir interaction has no diagonal elements in the representation in which the Hamiltonian of the thermal reservoir is diagonal. By substituting Eq.(9) into Eq.(7) and partial tracing over the reservoir variables one arrives at {linenomath*}

(10) |

Since the plasmonic modes are excited at optical frequencies whose energy scales are much higher than the thermal energy , it is reasonable to assume that the system is at zero temperature 29 ; 30 . Therefore, we can use the following correlation functions for the reservoir operators: and . On the other hand, taking a look at Eq.(II.2), one can recognize that this master equation is a non-Markovian master equation in the time convolution form.

After calculating the integrand on the right hand side of Eq.(II.2) explicitly we arrive at {linenomath*}

(11) |

where , . By this definition , for on-resonance driving (i.e., , , and in which the spectral density is defined as {linenomath*}

(12) |

The last two terms on the right hand side of Eq.(II.2) correspond to the pure dephasing and spin relaxation of lower energy levels of the QD, respectively. Here, in which the operators indicate the pure dephasing and and are the pure dephasing rates of the QD states and , respectively; while are the operators of spin relaxation and indicates the spin relaxation of lower energy states 54-1 . Furthermore, in deriving Eq.(II.2) we have made use of the relation 37 .

### ii.3 Photon-number statistics

One of the main purposes of the present contribution is to explore the photon-number statistics of the light emitted from the QD-MNP hybrid system. In particular, we intend to analyze the influence of the geometry of the MNP on the statistical properties of the emitted photons. The photon-number statistics can be determined by the normalized second-order photon autocorrelation function, i.e., the conditional probability of detecting the second photon at time when the first photon has already been detected at 38 . For a given quantum emitter system with an available excited state and a single or multiple channel(s) of relaxation, this autocorrelation function can be written as 33 ; 26-2 {linenomath*}

(13) |

where and indicates the steady-state population of the excited state.

In order to calculate the correlation function for the system under consideration, we apply the Laplace transform method to solve the master equation (II.2). Laplace transforming the both sides of Eq.(II.2) yields {linenomath*}

(14) | |||

(15) |

Here , , and . Considering the initial conditions , and the Laplace transform of the excited-state population is obtained as

(16) |

For convenience, we define the total spontaneous emission rate of the state as in which and with

(17a) | |||

(17b) | |||

(17c) | |||

(17d) |

Moreover, the Lamb shift is given by 41 . In Eq.(16) the functions , and are defined by

(18a) | |||

(18b) | |||

(18c) |

where and . In order to obtain the time evolution of the excited state population, , we should use numerical methods (see section III).

## Iii Results and Discussion

In this section, we present and discuss various numerical results and calculations to analyze the quantum statistics of the photons emitted from the hybrid QD-MNP system under consideration (Fig.1). Throughout the calculations,
we use atomic units ( , , , and ). For the numerical calculations, we consider a three-level self-assembled InAs QD as the emitter with z- and x-oriented dipole moments of nm and eV and eV 54 . The QD is placed at distance from the outer surface of a silver nanoshell composed of a spherical core of radius and permittivity surrounded by a concentric Ag shell of radius and frequency-dependent permittivity , embedded in a medium with permittivity .

An important ingredient for our numerical calculation is the dielectric permittivity of MNP. At low photon frequencies below the interband transitions region, the permittivity function of metals can be well described by the Drude model. For silver the interband effects already start to occur for energies in excess of 1eV and thus the validity of the Drude model breaks down at high frequencies 42 ; 43 . Therefore, in the numerical calculations we will use the measured dielectric data of silver reported in Ref.43 . The parameters used for silver are characterized by the plasma frequency of the bulk and collision rate of the free electrons 44 .

### iii.1 LDOS of the system

Density of states which change locally around the MNP provides us good insights about the interaction of the QD-MNP and the frequencies of surface plasmons which depend strictly on the orientation of the dipole moment of the QD. The scaled LDOS of a hybrid system is defined as in which being the free space density of states 30-1 . Here, {linenomath*}

(19) | |||

(20) |

where is the spherical Hankel function of the first kind and the coefficients and are given by 45

(21) | |||

with , , , , , and

(23) | |||

(24) |

In addition, denotes the spherical Bessel function of the first kind.

In Fig.2 we have plotted the scaled LDOS, , to examine the influence of the core and medium materials on the resonance frequencies of the plasmon modes. As can be seen in Fig.2(a), with increasing the core dielectric permittivity the resonance frequencies of plasmon modes shift toward lower frequencies. Similarly, Fig.2(b) shows that the resonance frequencies of the plasmon modes experience a slight redshift once we fix the core dielectric and change the embedding medium dielectric constant from 1 to 2. These results are in agreement with the experimental findings reported in Ref. 49 .

In general, the results show that increasing the dielectric permittivities of the core and the embedding medium lead to the redshift of the plasmon resonances. This behavior has a simple physical interpretation. As is well known, within the quasi-static approach, surface plasmons are collective electromagnetic oscillations at metallic surfaces over a fixed positively charged background, with induced surface charges providing the restoring force. Increasing the dielectric permittivity of the surrounding medium causes the strength of the surface charges to be effectively reduced, leading to a decreased restoring force and consequently, the plasmon energies are lowered. By increasing the core and the medium relative permittivities we adjust the surface plasmon resonance on demand frequency. By tuning the dielectric functions of background and core of the nanoshell to 55 the surface plasmons resonance frequencies for with is for both and surface plasmons which can couple effectively with InAs QD 54 .

In Figs.3(a) and 3(b)) we have plotted the scaled LDOS for different QD-MNP separation distances of a nm nanoshell composed of a spherical core of radius nm and permittivity 47 surrounded by a concentric Ag shell of radius nm and frequency-dependent permittivity , embedded in a medium with permittivity 47 .

The resonance frequencies of surface plasmons can be obtained by the poles of the dyadic Green’s function. The features of the plasmon resonances in a metallic nanoshell are described by the plasmon hybridization model 46 . In this model, the surface plasmon modes of internal and external surfaces (or core and shell surfaces) are combined in two symmetric and antisymmetric modes. Accordingly, these new branch modes possess low () and high () energies. The thickness of the nanoshell controls the strength of the interaction between the sphere-like and the cavity-like plasmon modes. With increasing the thickness of the nanoshell these modes behave more independently. The important parameters, which possess important influences on the plasmon resonance frequency, are the thickness of MNP , the core material dielectric constant (), and the embedding medium dielectric constant ().