Green’s functions technique for calculating the emission spectrum in a quantum dot-cavity system

 

Edgar A. Gómez,^1* J. D. Hernández-Rivero,² Herbert Vinck-Posada³

*E-mail: eagomez@uniquindio.edu.co
Universidad del Quindo, A.A. 2639, 630004 Armenia, Colombia.
Departamento de Fsica, Universidade Federal de Minas Gerais, A.A. 486, 31270-901 Pampulha, Belo Horizonte, Minas Gerais, Brazil.
Universidad Nacional de Colombia, A.A. 055051, 111321 Bogota, Colombia.

Received: 6 September 2016, Accepted: 5 November 2016
Edited by: J. P. Paz
Licence: Creative Commons Attribution 3.0
DOI: http://dx.doi.org/10.4279/PIP.080008

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Abstract

We introduce the Green’s functions technique as an alternative theory to the quantum regression theorem formalism for calculating the two-time correlation functions in open quantum systems at the steady state. In order to investigate the potential of this theoret-ical approach, we consider a dissipative system composed of a single quantum dot inside a semiconductor cavity and the emission spectrum is computed due to the quantum dot as well as the cavity. We propose an algorithm based on the Green’s functions technique for computing the emission spectrum that can easily be adapted to more complex open quantum systems. We found that the numerical results based on the Green’s functions technique are in perfect agreement with the quantum regression theorem formalism. More-over, it allows overcoming the inherent theoretical difficulties associated with the direct application of the quantum regression theorem in open quantum systems.

Keywords microcavity, master equation, open quantum system, QD-Cavity

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I. Introduction

The measurement and control of light produced by quantum systems have been the focus of interest of the cavity quantum electrodynamics [1, 2]. Spe-cially, the emission of light powered by solid-state devices coupled to nanocavities is an extensive area of research due to its promising technological appli-cations, such as infrared and low-threshold lasers [3,4], single and entangled photon sources [5,6], as well as various applications in quantum cryptogra-phy [7] and quantum information theory [8]. Experiments with semiconductor quantum dots (QDs) embedded in microcavities have revealed a plethora of quantum effects and offer desirable properties for harnessing coherent quantum phenomena at the single photon level. For example, the Purcell en-hancement [9], photon antibunching [10], vacuum Rabi splitting [11] and strong light matter coupling [12]. These and many other quantum phenom-ena are being confirmed experimentally by observ-ing the power spectral density of the light (PSD) emitted by the quantum-dot cavity systems (QD-cavity). Thus, the PSD, or the so-called emission spectrum of the system, becomes the only relevant information that allows to study the properties of the light via measurements of correlation functions, as it is stated by the Wiener-Khintchine theorem [13]. In order to compute the emission spectrum of the QD-cavity systems in the framework of open quantum systems, different approaches have been elaborated from the theoretical point of view. For example, the method of thermodynamic Green’s functions has been applied to the determination of the susceptibilities and absorption spectrum of atomic systems embedded in nanocavities [14], the time-resolved photo-luminescence approach whose application allows to determine the emission spec-trum when an additional subsystem is considered, the socalled the photon reservoir [15]. These theo-retical approaches are based on several approximations and therefore, they have their own limitations when they are considered in more general scenarios. In consequence, these methods are not used extensively.

Frequently, the emission spectrum of QD-cavity systems is computed through the Quantum Regres-sion Theorem (QRT) [16–18], since it relates the evolution of mean values of observables and the two-time correlation functions. It is worth men-tioning that the QRT approach can be difficult to implement in a computer program because com-putational complexity increases significantly as the number of QDs or modes inside the cavity are being considered; more precisely, the dimensionality asso-ciated with the Hilbert space is large. In general, the QRT approach is time-consuming because it is required to solve a large system of coupled differential equations and numerical instabilities that can arise. Moreover, theoretical complications related to dynamics of the operators involved can appear, as we will point out in the next section. In spite of this, the QRT approach is widely used in theoret-ical works, for example, in studies about photoluminescence spectra of coupled light-matter systems in microcavities in the presence of a continuous and incoherent pumping [19, 20]. Also, in studies considering the relation between dynamical regimes and entanglement in QD-cavity systems [21,22]. In the past, the Green’s functions technique (GFT) was successfully applied for calculating the emis-sion spectrum for a very simple quantum system, e.g., the micro-maser [23]. Nevertheless, this ap-proach has not been widely noticed in many significant situations in open quantum systems. The purpose of this work is to provide a simple and efficient numerical method based on the GFT in or-der to overcome the inherent difficulties associated with the direct application of the QRT approach by solving the dynamics of the system in the frequency domain directly. This paper is structured as follows: the theoretical background of the QRT as well as the GFT are presented in section II. A concrete application of our proposed methodology for calculating the emission spectrum of the QD-cavity system is considered in section III. Moreover, for comparison purposes with the GFT, we discuss in some detail the methodology of the QRT for calculating the emission spectrum of the cavity. The numerical results for the emission spectrum of the quantum dot, as well as of the cavity obtained from both the GFT and the QRT, are shown in section IV. A discussion about our findings is summarized in section V.

II. Theoretical background

i. Quantum regression theorem

One of the most important measurements when the light excites resonantly a QD-cavity system is the emission spectrum of the system. From a theoretical point of view, it is assumed that it corre-sponds to a stationary and ergodic process which can be calculated as a PSD of light by using the well-known Wiener-Khintchine theorem [13]. This theorem states that the emission spectrum is given by the Fourier Transform of the two-time correlation function of the operator field a; explicitly, it is

 

ii. Green’s functions technique

Let us consider a QD-cavity system and an operator A which does not opérate on the reservoirs, trien its single-time expectation valué in the Heisenberg representation is given by

 

The density operator system-reservoir can be evolved írom an initial state at time 0 to an arbi-trary time t via ps®R,(t) = Ui(t,0)ps®R,(0)U(t,0), with £/(í, 0) being a unitary time-evolution operator involving the Hamiltonian terms oí the system and reservoirs. Moreover, the operator p~s<g>R{t) = Ps(t) ^ Pñ(í) depicts the composite density operator oí the system and reservoir. It is worth point-ing out that tilde means that the operator has been transformed to the Heisenberg representation and that the dynamics oí the system depend directly on PS0R(t) f°r all times. The validity oí the Marko-vian approximation requires that the state oí the system is sufficiently well described when it is con-sidered that ps(t) = Trp,(p~s<g>R(t)). Therefore, it is sufficient to write p~s<g>R(t) = Ps(t) ® PB.{t) f°r all times. If we assume that at t = 0 the initial state of the system is the steady state, then /5s(g>fi(0) = Pg¡L_R- Here, the superscript ”(ss)” should be understood to be the steady state of the system-reservoir. After tracing over degrees of free-dom of the reservoirs, we have that the Eq. (2) takes the form

 

 

where we have taken into account that the super-operator C acts only on the system Hilbert space and not on the reservoir. It is straightforward to conclude that G{t) is an operator that obeys the same dynamical equations as ps{r). More precisely, dG{r)/dT = CG{t) with the boundary condition G(0) = B(0)/5s(0) at the steady state of the system.

We also conclude that the two-time correlation íunction in the long-time limit can be written as

where G(t) = Tr r [Ü(t)Bp'g¡L_RÜ' (t)] is defined as the Green’s functions operator and the operators A,

B and p_sJ_R are in the achrodmger representation.

A = A(0) and B = B(0) are operators considered at the steady state oí the system. In the remainder oí the paper, we assume that U(t) = £/(t, 0). Particularly, the Eq. (9) takes the form oí the Eq. (1) after performing the integral transformation. More precisely, by taking the real part oí the Laplace transform to the Eq. (9), we obtain an expression in terms of the Green’s functions operator in the frequency domain as follows

Notice that the operators A and B should be defined appropriately for describing the emission spectrum due to the cavity or the quantum dot. Moreover, the wide tilde is used to indicate that the Laplace transform was taken. The superscript ”(ss)” should be understood to be the steady state of the reduced density operator of the system. After taking the Laplace transform of the Eq. (9), we obtain an expression for the emission spectrum of the system in terms of the Green’s functions operator in the frequency domain as follows

Prior to leaving this section, we mention that this result will be the starting point for calculating the emission spectrum due to the cavity as well as the quantum dot by considering the photon and fermionic operators in a separated way.

III. Application to the QD-cavity system

i. Model

In order to illustrate the potential of the Green’s function technique for calculating the emission spectrum in a QD-cavity system, we will consider an open quantum system composed of a quantum dot interacting with a confined mode of the electro-magnetic field inside a semiconductor cavity. This quantum system is well described by the Jaynes-Cummings Hamiltonian [24]

where the quantum dot is described as a fermionic system with only two possible states, e.g., |G) and |X) are the ground and excited state. a = |G)(X| and a (<r' = |X)(G| and a') are the annihilation (creation) operators for the fermionic system and the cavity mode, respectively. The parameter g is the light-matter coupling constant. Moreover, note that we have set Ti = 1. We also define the detun-ing between frequencies of the quantum dot and the cavity mode as A = wj — w_a, where wj and ui_a are the energies associated to an exciton and the photons inside the cavity, respectively. This Hamiltonian system is far from describing any real physical situation since it is completely integrable [25] and no measurements could be done since the light remains always inside the cavity. In order to incorpórate the effects of the environ-ment on the dynamics of the system, we consider the usual approach to model an open quantum system by considering a whole system-reservoir Hamiltonian which is frequently split in three parts. Namely, H = Hs + Hsr + Hr, where Hs defines the Hamiltonian term of the QD-cavity system as it is defined in the Eq. (12). The Hamiltonian terms Hsr and Hr corresponding to a bilin-ear coupling between the system-reservoir and its respective reservoirs Hr have been discussed in de-tail by Perea et al. in [26]. The reader can find a detailed discussion of the Markovian master equa-tion in [27, 28]. In the framework of open quantum systems, different reservoirs have been pro-posed in order to describe the dissipation, deco-herence or decays. Particularly, for QD-cavity systems, a reservoir is considered for describing thephysical situation where the photons are absorbed in a semiconductor and electron-hole pairs (exci-tons) can be produced which can be associated to either electrical injection or the capture of excitons optically created at frequencies larger than the typical ones of our system. This process corresponds to the socalled continuous and incoherent pumping of the QD. Also, when the excitons are coupled to the leaky modes of the cavity with energy different than the cavity mode, there is a residual density of states inside the cavity and this process is responsible for the spontaneous emission (radiative recom-bination) to an independent reservoir of photons. Another physical process is known as the coherent emission and it is due to the direct dissipation of the cavity mode, more precisely, the cavity mode is coupled to the continuum of photonic modes out of the cavity. For obtaining the master equation for the QD-cavity system, it is convenient to consider the interaction picture with respect to Hs + Hsr and assume the validity of the Born-Markov approximation. After tracing out the degrees of freedom of all the reservoirs, one arrives to the Lindblad master equation for the reduced density matrix of the system equations for the reduced density matrix of the system in the bared basis. It is an extended Hilbert space formed by taking the tensor product of the state vectors for each of the system components, {|G}, \X)} (g> {|"-}}^Lo- In this basis, the reduced density matrix ps can be written in terms of its matrix elements as psan,¡3m = {(xn\ps(T)\/3m). Henee, the Eq. (13) explicitly reads

The parameter 7 is the decay rate due to the spontaneous emission, k is the decay rate of the cavity photons across the cavity mirrors, and P is the rate at which the excitons are being pumped. Figure 1 shows a scheme of the simplified model of the QD-cavity system showing the processes of continuous pumping P and cavity loses k. The physical process begins when the light from the pumping láser enters into the cavity and excites one of the quantum dots in the QD layer. Thus, light from this source couples to the cavity and a fraction of photons escapes through the partly transparent mirror from the cavity and goes to the spectrometer for measurements of the emission spectrum. A general approach for solving the dynamics of the system consists in writing the corresponding Bloch

Note that we use the convention that all Índices written in Greek alphabet are used for the fermionic states and take only two possible valúes \G), \X). The Índices written in Latin alphabet are used for the Fock states which take the possible valúes 0,1, 2, 3 ... Additionally, it is worth mentioning that our proposed method does not require to solve a system of coupled differential equations, instead of it, we solve a reduced set of algebraic equations that speed up the numerical solution. Prior to leaving this section, we point out that the number of excitations of the system is defined by the operator N = a/á + a'a. The closed system and the number of excitations of the system is con-served, Le., [Hs,N] = 0. It allows us to organize the states of the system through the number of excitations criterion such that the density matrix elements pGn,Gm Pxn—i,Xn—i, f>Gn,Xn-i and P~Xn-i,Gn are related by having the same number of quanta* sub-spaces of a fixed number of excitation evolve independently from each other. The Fig. 2 shows a schematic representation of the action of the dissipative processes involved in the dynamics of the system according to the excitation number (N_exc).

 

Figure 1: The picture represents a QD-cavity sys-tem showing the processes of continuous pumping P and cavity loses h¿.

Figure 2: Ladder of bared states for a two-level quantum dot coupled to a single cavity mode. The double headed green arrow depicts the matter cou-pling constant q dashed red lines the emission of the cavity mode k solid black lines the exciton pumping rate P and solid blue lines the sponta-neous emission rate 7.

 

ii. Emission spectrum of the cavity based on the GFT

In order to compute the emission spectrum of the cavity, we will consider the two-time correlation function accordingly with the Eq. (9) for the field operator as follows

As we pointed out in section II, this operator must obey the same master equation as the reduced den-sity operator of the system. In fact, the terms that only contribute in the Eq. (18) are given by the matrix elements Gfj_m^_n{r) = (/3m| G{t) \^n) of the Green’s functions operator. This is due to the fact that the projection operator \¡3m) {^n\ enters into G{t) in the same way as into the reduced density operator of the system.

In order to identify these matrix elements, it should be considered that for the QD-cavity system, the dynamics of all coherences asymptotically van-ish and there only remains the reduced density matrix elements which are ruled by the number of excitations criterion, i.e., pon,Gm PXn-i,Xn-i, PGn,Xn-i, PXn-i,Gn- Thus, the Eq. (17) can be written as follows

Note that from this expression, it is easy to identify the nonzero matrix elements of the Green’s functions operator that contribute to the emission spec-trum. Finally, after performing the Laplace trans-form to the Eq. (18), we have that the emission spectrum of the cavity is given by

It is worth mentioning that the initial conditions may be obtained by evaluating the Green’s func-tion operator at t = 0. Moreover, by using the fact that the time evolution operators become the iden-tity and TrR[p^^s ] = 1. We obtain a set of initial conditions given by

Note that this set of initial conditions corresponds to the steady state of the reduced density matrix of the system. A general algorithm based on the GFT for computing the emission spectrum is presented in the appendix. We mention that this approach can be adapted easily for calculating the emission spectrum due to the cavity as well as the quantum dot.

iii. Emission spectrum of the quantum dot based on the GFT

In order to compute the emission spectrum of the quantum dot, we will consider the two-time corre-lation function given by Eq. (9), but for the case of the matter operator

where we have considered that the matter operator is given by <r(0) = a at the steady state. It is straightforward to show, after performing the partial trace over the degrees of freedom of the system, that the two-time correlation function reads

IV. Results and discussion

In this section, we compare the numerical calcula-tions based on the GFT and the QRT approach for the emission spectrum of the cavity as well as the quantum dot. In particular, the QD-cavity system can display two different dynamical regimes by changing the parameters of the system and two regimes can be achieved when the loss and pump rates are modified. In fact, the relation g > \k — 71/4 holds for the strong coupling regime and the relation g < \k — 7I/4 remains valid for the weak coupling regime. Figure 3 shows the numeri-cal results for the emission spectrum associated to the cavity in the strong coupling regime, where the emission spectrum oí the cavity based on the GFT is shown clS el solid blue line and the emission spectrum based on the QRT approach clS el dashed red line. The parameters oí the system are g = 1 meV, 7 = 0.005 meV, n = 0.2 meV, P = 0.3 meV, A = 2 meV and uj_a = 1000 meV. Particularly for this set of parameters valúes, we can identify two different peaks which are associated to the energy of the cavity and the quantum dot, they are uj_a k, 998.3 meV and lux ~ 1000.3 meV. We have considered the rel-ative error as a quantitative measure of the discrep-ancy between the GFT and the QRT approaches. More precisely, by monitoring the numerical com-putations of the emission spectrum, we have esti-mated that the máximum relative error is on the order of 10-³ in all numerical calculations that we have performed. Similarly, the emission spectrum of the cavity based on the GFT (solid blue line) and the QRT approach (dashed red line) for the strong coupling regime are shown in Fig. 4. Here, the parameters valúes of the system are given by g = 1 meV, 7 = 0.005 meV, n = 2 meV, P = 0.005 meV, A = 0.0 meV and io_a = 1000 meV. Note that we have considered the resonant case, more precisely, the same energy valúes for the cavity and the quantum dot. Here, the emission spectrums do not match but repel each other, resulting in a struc-ture of two sepárate peaks for a distance of approxi-mately two times the coupling constant, Le., 'lg k, 2 meV. It is worth mentioning that this quantum ef-fect is well-known as Rabi splitting in QD-cavity systems. The emission spectrum of the quantum dot in the weak coupling regime is shown in Fig. 5. The numerical result for the emission spectrum of the quantum dot based on the GFT is shown as a solid blue line and the corresponding numerical result for QRT approach is shown clS el dashed red line. We set the weak coupling regime by consider-ing high valúes of the decay and pump rates k = 5 meV and P = 1 meV, respectively. The rest of the parameters valúes are g = 1 meV, 7 = 0.1 meV, A = 5 meV and uj_a = 1000 meV. We conclude that the method based on the GFT is in perfect agreement with the QRT approach and reproduces very well the emission spectrum of the QD-cavity system.

For comparison purposes with our GFT approach, we have also implemented the numerical method based on the QRT for the QD-cavity system (see details in section iv). In the con-ducted simulations, we have considered the same truncation level in the bare-state basis, e.g., Nexc = 10. Moreover, we have solved numerically the dynamical equations of the system given by the Eq. (34) until time t_max = 2¹⁷ ps for obtaining an acceptable resolution in frequency domain, it is Aw « 0.048 meV. In order to test the performance of the GFT approach in terms of efficiency, we have compared the computational time involved on the numerical calculation of the emission spectrum of the cavity at four different excitation numbers N_exc. Table 1 shows in first column the excitation number. Second and third columns show the elapsed time (CPU time) in seconds during the simulations for the GFT and the QRT approach, respectively. It is worth mentioning that we have considered, for this comparison, exactly the same resolution in the frequency domain and the numerical calculations were carried out with the same parameters valúes as in Fig. 4 for both methods. It is straightforward to observe that the QRT approach is time-consuming compared with the GFT approach when the excitation number is increased. From the computational point of view, it is due to the fact that the QRT approach requires solving a large number of coupled differential equations in contrast to the GFT approach which requires a relatively small system of algebraic equations.

V. Conclusions

We have presented the GFT as an alternative methodology to the QRT approach for calculating the two-time correlation functions in open quantum systems. We have applied the GFT and the QRT approach for calculating the emission spectrum in a QD-cavity system. In particular, the performance of the GFT in terms of accuracy and eñiciency by comparison of the emission spectrum of the cavity and the quantum dot is demonstrated, as well as by comparison of the computational times involved during the numerical simulations. In fact, we have shown that the GFT offers a computational advan-tage, namely, the speeding up numerical calculations. We conclude that the GFT allows to overeóme the inherent theoretical diñiculties presented

Figure 5: Emission spectrum of the quantum dot based on the GFT as a solid blue line and the corre-sponding numerical calculation based on the QRT approach clS el dashed red line. The parameters valúes are k = 5 meV, P = 1 meV, g = 1 meV, 7 = 0.1 meV, A = 5 meV and u>„ = 1000 meV

in the QRT approach, Le., to find a closure condi-tion on the set of operators involved in the dynam-ical equations. We mention that our methodology based on the GFT can be extended for calculating the emission spectrum in significant situations where the quantum dots are in biexcitonic regime or when the quantum dots are coupled to photonic cavities.

Acknowledgements - EAG acknowledges the finacial support from Vicerrectoría de Investigaciones at Universidad del Quindío through research grant No. 752. HVP acknowledges the financial support from Colciencias, within the project with code 11017249692, and HERMES code 31361.

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