TIME-DEPENDENT PHOTON CORRELATIONS FOR INCOHERENTLY PUMPED QUANTUM DOT STRONGLY COUPLED TO THE CAVITY MODE
A. V. Poshakinskiy* A. N. Poddubny
Ioffe Physical-Technical Institute, Russian Academy of Sciences 194021, St. Petersburg, Russia
Received September 25, 2013
The time dependence of correlations between the photons emitted from a microcavity with an embedded quantum dot under incoherent pumping is studied theoretically. Analytic expressions for the second-order correlation function </(J)(i) are presented in strong and weak coupling regimes. The qualitative difference between the incoherent and coherent pumping schemes in the strong coupling case is revealed: under incoherent pumping, the correlation function demonstrates pronounced Rabi oscillations, but in the resonant pumping case, these oscillations are suppressed. At high incoherent pumping, the correlations decay monoexponentially. The decay time nonmonotonically depends on the pumping value and has a maximum corresponding to the self-quenching transition.
DOI: 10.7868/S0044451014020059
1. INTRODUCTION
Semiconductor quantum dots form a promising platform for quantum optics devices, including singlephoton emitters and emitters of entangled photon pairs fl 4]. The quantum dot-based light sources can be characterized by means of photon photon correlation spectroscopy, i.e., by measuring the second-order correlation function g{'2\t) between two photons with a delay t [5, 6]. Multiple experimental observations of the antibunching f.g(2^(0) < 1] of the photons emitted from quantum dots are already available [7 11]. One of the possible routes to further enhancing the performance of these light sources is to resonantly couple the quantum dot exciton to the photon mode confined inside the microcavity in all three spatial directions [3]. The physics of such quantum microcavites becomes especially rich in the strong-coupling regime, where the new quasipar-ticles, exciton polaritons, are formed due to the interaction between excitons and cavity photons [1,3,12 15].
Here, we study the time dependence of the second-order correlations between the photons emitted from a quantum dot microcavity under stationary incoherent pumping. Experimentally, this regime can be re-
* E-mail: poshakinskiv'fl'mail.ioffe.ru
alized in quantum dot microcavities driven by electrical pumping [16] or continuous optical pumping [14]. The coexistence of (i) the strong-coupling regime and (ii) the stationary incoherent pumping regime makes the time dynamics of the correlations very specific.
The strong-coupling regime [17 19] qualitatively distinguishes the system from the conventional laser, described by the Scully Lamb theory [20]. Moreover, the incoherent pumping makes it different from the single-atom laser in the strong-coupling regime, which has been demonstrated experimentally and analyzed theoretically [21, 22]. Such systems are typically coherently pumped by resonant light [4,23 26]. As we show in Sec. 3, the photon photon correlations for a resonantly pumped atom and for an incoherently pumped quantum dot are very different. While both systems show antibunching, the time-dependent correlator g^(t) demonstrates oscillations at the vacuum Rabi splitting frequency in the incoherent pumping case, but not in the case of a resonantly pumped atom [21]. Recent experiments for incoherently pumped laser with a single quantum dot in the strong-coupling regime [14], as well as the comprehensive theoretical analysis in [27 29], were focused 011 the stationary correlator .g(2^(0) at zero time delay. Detailed analysis of time-dependent correlations was limited to the regime with a large exciton photon detun-
ing [30, 31] or weak coupling [10], where the polaritons are not formed.
Hence, there is still a need to develop a detailed theory accounting for the specifics of the fast-increasing field of quantum-dot-based cavity quantum electrodynamics. Here, we focus on the temporal dynamics of correlations in the strong-coupling regime and show that it provides additional information on the lifetime of polariton eigenstates and the energy splitting between them. Our main goal is to derive transparent analytic answers for the time-resolved correlator g^ (t) as a function of the incoherent pumping intensity in both strong and weak coupling regimes.
The rest of the paper is organized as follows. In Sec. 2, the model and the calculation approach are described. Section 3 is devoted to the role of the pumping mechanism and demonstrates the difference between incoherent and resonant pumping schemes. Sections 4 and 5 respectively present the theory developed in the strong and weak coupling regime. The results are summarized in Sec. 6. Auxiliary derivations are given in Appendices A and B.
2. MODEL
We consider a zero-dimensional microcavity where a single photon mode is coupled to a single exciton state of the quantum dot. Polarization degrees of freedom of both photons and excitons are disregarded for simplicity. Under these assumptions, the Hamiltonian of the system has the standard form fl]
H = huJo^c- + huJQlM) + hg{c)b + dA), (1)
where u>o is the resonance frequency of the cavity, tuned to the exciton resonance, c and (J are the boson annihilation and creation operators for the cavity mode ([<■*,fi] = 1), b = |G) {A'| and tf = |-Y) {<j| are the corresponding operators for the single-exciton mode, |A') and |G) are respective states with one exciton and no excitons, and g is the light exciton coupling constant. Equation (1) corresponds to a quantum dot smaller than the exciton Bohr radius. To consider the case of a large quantum dot, one should generalize the model following Refs. [32, 33].
To determine the intensity of emission from the cavity, we should also introduce the processes of particles generation and decay. We consider incoherent continuous pumping of excitons into the quantum dot with the rate W (see Fig. la). The "microscopic" discussion of the pumping mechanism can be found in Ref. [32], while the distinction between incoherent and coherent
pumping schemes is discussed in Sec. 3. The exciton mode is characterized by the nonradiative damping r_\-. Photons can escape the cavity through the mirrors with the rate Fc- Hence, the full system state is described by a density matrix p and its evolution is determined by the equation dp/dt = C[p] with the Liouvillian [1]
£\P] = p]+TcLc[p]+TxLh[p]+WLht [pi (2)
where La[p] = (2upu) — u)up — pu)u)/2 are the Lind-blad terms, accounting for damping and pumping. The stationary density matrix p0 satisfies the equation £[Po] = 0. We can calculate the number of photons in the cavity ATc = ((:>i(:) and the exciton occupation number Nx = (b^b) as
Nc = Tr(rV/?o). Nx=Tv{tfbpQ)< (3)
where Tr stands for the operator trace and angular brackets denote the quantum mechanical expectation value. The luminescence spectrum of the system is given by [5]
oo
IM .X Rc j dt eiul <f+(0)f(#)> . (4)
"o
A detailed study of the dependence of these first-order correlators on the pumping and on other parameters can be found in Refs. [28, 34]. The goal of this paper is to analyze the time dependence of the second-order correlator that characterizes fluctuations of the emission intensity from the cavity. They are described by the correlator g^(t) determining the probability to register two photons with the time delay t [5]:
gm(t)=^(cH0)cHt)c(t)c(0)). (5)
Equation (5) presents the simplest definition of the correlation function, suitable for the analytic treatment in what follows. A more general expression, taking the finite response rate and spectral window of the photon detector for two- and multiple-photon correlations into account is given in Ref. [6]. The calculation of g^(t) is based on the quantum regression theorem [5]
g^(t)=^Tv[^cX(t)], (6)
where the evolution of the operator \ (/) = e£l[cpo(J] is governed by the dynamic equation
4~ = £[\]< \(0) = Wt. (7)
Fig. 1. The sketch of (a) an incoherently pumped quantum dot in a microcavity system and (b) a resonantly pumped 3-level atomic cavity system. Panels с and d show the comparison of the luminescence spectra for these systems, while panels e and /present the </J)(i) dependence. The parameters chosen are у/Тс = 10, Г_\- = 0.1Гс, and И /IV = 0.1 for the quantum dot in the microcavity system (panels c,c) and у/Тс = Ю, Г_\- = Г/л- = Г/о = 0.1Гс and il/2y = 0.01 for the atomic
cavity system (panels d,f)
For zero time delay, Eq. (6) assumes the form
.9i2)(0) = -^Tr(rtrVpo). (8)
-vc
For large time delays, the correlator tends to unity, -¥■ oo) = 1, because the probabilities of detection of two photons become independent.
3. COMPARISON OF INCOHERENT AND COHERENT PUMPING
In this section, we compare the characteristics of emitted photons in the cases of coherent and incoherent pumping. We focus on the strong-coupling regime, when the light exciton coupling g is stronger than the decay rates of the exciton and photon. We demonstrate below that these two pumping schemes are qual-
itativoly different even at a small pumping rate. The incoherent pumping scheme is used for the quantum dot in a microcavity as sketched in Fig. la and was described in Sec. 2. The density matrix equations can be conveniently analyzed using the basis of eigenstates of Hamiltonian (1), which are well defined in the strong-coupling regime (g Fc,Fa-). The eigenstates are given by [35]
|0) = |0, G), |m,±) =
|m.G) ± |m — 1, A')
0)
m > 1.
where \m,G) and |m, A') are the respective states with m photons and no excitons or one exciton. The energy spectrum forms the Jaynes Cummings ladder
Eg — 0, Em j- — îllhulg i
(10)
Each rung of the ladder contains two states split by the Rabi frequency 2\fnig, increasing with the rung number m. In the limit of vanishing pumping W -C Fe, the luminescence spectrum is determined by transitions from the lowest occupied excited levels |1,±) to the ground state |0), and therefore contains two pea
Для дальнейшего прочтения статьи необходимо приобрести полный текст. Статьи высылаются в формате PDF на указанную при оплате почту. Время доставки составляет менее 10 минут. Стоимость одной статьи — 150 рублей.