## Abstract

In this paper, we introduce a hybrid three-dimensional photonic-crystal cavity with an embedded quantum dot, and investigate the dynamics of the cavity-quantum dot system. The general procedure of modelling such a practical structure is presented, where the master equation is solved on the basis of the parameters obtained from defect mode analyses. According to our study, this structure can be engineered to achieve a nearly deterministic single photon gun. The excitation power is found to have an optimal value in terms of photon emission efficiency. Large excitation pulse duration is believed to cause a spurious peak in the second-order coherence measurement.

©2004 Optical Society of America

## 1. Introduction

Since Yablonovitch first proposed to modify the spontaneous emission by way of a periodic dielectric structure [1], two-dimensional planar photonic-crystal cavities and micro-pillars with one-dimensional photonic band gaps have been widely used to enhance the light emission efficiency for many applications, such as defect mode lasers [2] and sources of single photons [3, 4]. The advantage of a photonic-crystal cavity, from which these applications have benefited, lies in its inherent flexibility, which allows the fine-tuning of defect mode properties. On the other hand, the embedded quantum dot (QD) will not suffer from random motion, which tends to occur in atomic systems [5]. In light of these intriguing features, the cavity-QD system has been proposed to achieve entangled photon pair emission [6], strong coupling [7], and quantum information processing [8].

Here, we propose a structure capable of further exploiting the above-mentioned benefits. A detailed theoretical analysis is then applied on this structure. The photon emission properties based on the proposed structure is presented, and several related issues are further discussed.

## 2. The photonic-crystal structure

The structure, shown in Fig. 1(a), can be considered as a hybrid three-dimensional photonic-crystal, which inherits both from the one-dimensional distributed-Bragg reflector (DBR) consisting of alternating GaAs and AlAs layers [9] and from the two-dimensional photonic-crystal slab [10]. A cavity can be introduced by removing one of the patterned holes and increasing the thickness of one of the GaAs layers, as shown in Fig. 1(b). Although this three-dimensional photonic-crystal does not possess a complete bandgap extending over the entire Brillouin zone, a well behaved defect mode still exits according to our numerical results presented later, and its advantages over its two prototypes, namely, more flexibility and more mechanical robustness, make this structure a promising candidate for a device in quantum information systems.

Obviously, the fabrication thereof will also be the combination of those of the two prototypes: first, the stacked DBR is formed by epitaxial growth with sparsely distributed QDs located approximately at the central *xy*-plane of the spacer region (the dark region in FIG. 1(b), of which the thickness is labeled as *s*); then, the patterned holes are formed by dry etching technique in a way such that the center of the defect is in alignment with one QD. Despite the unavoidable challenges in fabrication, current epitaxial and etching techniques allow such structure to be practical.

## 3. Model of analysis

In order to investigate the dynamic behavior of the system, several parameters should be determined beforehand, namely, the spontaneous decay rate of the QD γ_{0}, the cavity decay rate κ and the QD-cavity coupling strength *g*. Here, the QD is assumed to be a self-assembled InAs QD surrounded by GaAs matrix with a radiation wavelength around 876 nm [11], and the corresponding γ_{0} is typically 0.56 GHz [7, 12, 13]. The other two parameters (*g*, κ) are obtained by solving the Maxwell’s equations with the boundary conditions set by this structure [7, 14, 15]. Therefore, we start our study with the defect mode analysis.

For an electromagnetic problem in a practical geometry as complex as a photonic-crystal cavity, one of the most accurate ways of solving it is the finite-difference time-domain (FDTD) method [16]. When performing numerical simulations using FDTD, we set as a rule of thumb that the longitudinal period *p* is 8*a*/15 (shown in Fig. 1(a), where a is the inter-hole spacing.), the thickness of the spacer region is *s*=16*a*/15, and the thickness of other layers of GaAs or AlAs is 0.5*p*. The defect is surrounded transversally by holes arranged in a 5-layer hexagonal lattice, and longitudinally by 6 periods of DBR stacks. A uniform mesh is applied to discretize the structure, and the resolution of the discretization is 15 grids per inter-hole spacing. The whole computational region is truncated by perfectly matched layers to absorb the out-going waves. Without loss of generality, we choose to analyze the *x*-dipole mode by exciting the cavity with an *x*-polarized initial field [10]. The result is illustrated in Fig. 2.

The resonance frequency is found to be *a*/λ=0.263. If the QD is resonant with the cavity mode, i.e. λ=876 nm and *a*=230 nm, the quality factor *Q* of the *x*-dipole mode can be estimated to be 641; this corresponds to a cavity decay rate κ=1678 GHz according to κ=(π*c*)/(λ*Q*), where *c* is the speed of light in vacuum. The mode volume *V*, which is defined as

is found to be 6.2×10^{-21} m^{3}, i.e., approximately 0.42(λ/*n*)^{3}. The maximum QD-cavity coupling strength g is then estimated to be 441 GHz according to the relation

where *ε* is the dielectric constant at the location of the QD, μ is the dipole moment matrix element between the single-exciton and the ground states, and *V*
_{0} = (3*cλ*
^{2}
*ε*
_{0})/(2*πγ*
_{0}
*ε*). Clearly, the parameters (*g*, κ, γ_{0}) satisfies the criterion of the so called bad cavity limit [5]: *γ*
_{0} ≪*g*
^{2}/κ≪κ, which is proper for the application as an active photon emitter.

With the necessary parameters at hand, we proceed to analyze the dynamics of the photonic-crystal cavity-QD system. Considering only the single-exciton state, we treat the quantum dot as a two-level system. We shall however come back to the validity of this simplification later in this paper. Under the assumption that the exciton is resonantly coupled to the cavity mode, the coherent part (under the rotating wave approximation) of the Hamiltonian in the interaction picture can be expressed as

where *a* and *a*
^{†} are the annihilation and the creation operators for the quantized cavity field, respectively; *σ*= |*G*〉〈*X*| and its hermitian conjugate *σ* are the QD exciton projection
operators, where |*G*〉 and |*X*〉 denote the ground state and single-exciton state of the QD,
respectively; *r(t)* is the coupling between the QD exciton and the external classical field. Taking the dissipation due to both the QD spontaneous decay and the cavity decay into account, the master equation of our system becomes [17]

where *ρ* denotes the density operator of the system. In order to measure the characteristic of the system, we examine the *P(t)* as defined in Ref. [18]

which can be interpreted as the average photon number detected during the time interval from 0 to *t*, provided that an ideal photodetector is used. In our numerical simulation, we have used the classical pump field of the form

where the pulse duration ~ 2*T*
_{0} is at the order of several picoseconds for a typical experimental setup (see, for example, Ref. [11]).

## 4. Results and discussions

The numerical result is plotted in Fig. 3, where the parameters used are (*g*, κ, γ_{0}, *r*
_{0})=(441,1678,0.56,500) GHz and 2*T*
_{0}=3 ps, among which (*g*, κ) are from the preceding calculations. It is seen that we are expected to detect approximately one photon after a time interval longer than the system decay time scale (determined by *g*
^{2}/κ ~ 100 GHz) but still significantly shorter than the time scale determined by the pump repetition rate (typically 100 MHz [11]). This indicates that the system can emit almost deterministically one photon for each excitation pulse.

As shown in Fig. 3, *P(t)* approaches a saturation value after a sufficient decay of the system. Unlike the case reported in [18], however, we find the saturation value *P*(+∞) strongly dependent on the peak pump rate *r _{0}*: as

*r*increases,

_{0}*P*(+∞) first reaches a maximum value

*P*

_{max}and then decrease, as shown in Fig. 4.

To verify this numerical result and to gain more physical insight, we study the dynamics of the system using the non-Hermitian effective Hamiltonian [19]

The wavefunction can be written as

where |*μ*,*n*〉 (*μ* = *G*,*X* and *n* = 0,1) denotes the QD exciton state |*μ*〉 with *n* cavity photons. The Schrodinger equation yields

$$i{\dot{a}}_{2}=r\left(t\right){a}_{1}+g{a}_{3}-i\frac{{\gamma}_{0}}{2}{a}_{2}$$

$$i{\dot{a}}_{3}=g{a}_{2}-i\kappa {a}_{3},$$

with an initial condition *a*
_{1}(0) = 1, *a*
_{2}(0) = *a*
_{3}(0) = 0 . For simplicity, the pump is taken to be a rectangular pulse with excitation rate *r*
_{0} and pulse duration *T*. By assuming the excitation dominates the dynamics when *t*<*T*, we can derive an adiabatic solution of Eq. (9)

$${a}_{2}\approx \{\begin{array}{c}\mathrm{sin}\left({r}_{0}t\right)\phantom{\rule{1em}{0ex}}t<T\phantom{\rule{9em}{0ex}}\\ \mathrm{sin}\left({r}_{0}T\right)\mathrm{exp}\left\{-\left(\frac{{g}^{2}}{\kappa}+\frac{{\gamma}_{0}}{2}\right)t\right\}\phantom{\rule{.5em}{0ex}}t\ge T\end{array}\phantom{\rule{.2em}{0ex}}$$

$${a}_{1}\approx -i{\int}_{0}^{t}r\left(t\prime \right){a}_{2}\left(t\prime \right)dt\prime .$$

If the pulse duration *T* is short enough, *P(t)* can be rewritten as

where *F* = 2*g*
^{2}/(κ*γ*
_{0}) is the Purcell factor [20–22]. It is seen clearly from Eq. (11) that the QD decays at an enhanced rate of 2*g*
^{2}/κ+*γ*
_{0} and the fraction of the QD decaying into the cavity mode is *F*/(*F*+*1*). With present parameters, the Purcell factor *F* is much greater than one. Therefore,

This shows that the saturation value of *P(t)* strongly depends on the pulse area 2*r*
_{0}
*T*. Excessive pulse tends to lower the inversion level, thus degrading the photon emission efficiency. An actual QD, however, is somewhat different from a pure two-level system, especially after a full excitation pumped by a π pulse: aside from interacting with a single-exciton described by the present model, excessive pulse will create multiple-excitons, typically biexcitons, as well. Therefore, the prediction of this model is accurate to the extent where no multiple-exciton effect comes into play. But the fact that over-area pulse tends to lower the inversion level, and hence the photon emission efficiency, still holds true even beyond the validity of this model. So *P*(+∞) is bound to exhibit a maximum value *P*
_{max} with respect to *r*
_{0}, which can be verified experimentally. We believe presumably that this effect may account for the slight decrease in luminance intensity and external quantum efficiency in earlier experiments [3, 11].

Unlike the λ-type atom in Law’s proposal [18], a QD has some probability of being re-excited after decaying to the ground state by emitting a cavity photon, provided the pump pulse is sufficiently wide. Therefore, successive emission of more than one photon triggered by a single pump pulse is possible, which would lead to a *P*
_{max} greater than one. Due to the cavity-enhanced QD decay, which is characterized by 2*g*
^{2}/κ+*γ*
_{0}~100 GHz, the temporal
separation between these successive photons will be at the order of 0.01 ns. However, a typical Hanbury Brown and Twiss (HBT) experiment configuration has a temporal resolution of 0.5 ns [11]. Therefore, these successively emitted photons, though antibunched in nature, cannot be discriminated experimentally, thus giving rise to a spurious non-zero value of the second order function *g*
^{(2)}(0). Such effect is not prominent in earlier experiments since the enhanced decay rate is not significant [3]. But for a cavity as studied in this paper, whose Purcell factor is much greater than one, it is expected to cause some observable influence.

By decreasing the pulse duration, the probability of more than one photon triggered by a single pump pulse can be significantly reduced. Fig. 5 illustrates the pulse duration dependence of the maximum emitted photon number *P*
_{max} . Clearly, to avoid this kind of photon emission while still maintaining a high decay rate, i.e. high efficiency, the excitation pulse should be sufficiently short relative to the QD decay time.

This work is merely a preliminary look on the possible CQED effects in the proposed structure. Several improvements can be made by taking advantages of the flexibility of the photonic-crystal. For example, by increasing the layers of the air holes around the defect, the probability of the in-plane leak of a cavity photon can be reduced exponentially [10], thus further facilitating the photon collection. By tuning the geometries of the holes around the defect, the quality factor of the cavity can be further improved without sacrificing the mode volume [23, 24]. Furthermore, the vacuum Rabi splitting, or the strong-coupling, is likely to occur [7], making this structure a possible quantum computing element with potential scaling ability [25].

## 5. Conclusions

In summary, we have introduced a hybrid photonic-crystal cavity, which, together with an embedded quantum dot, can be used as a nonclassical light source in quantum information systems. The study of cavity quantum electrodynamics in this structure presents a general procedure of modeling practical structures. Our analysis shows that the system can emit one photon almost deterministically under the pulsed excitation. The pump power is found to have an optimum value in terms of photon emission efficiency, beyond which the excessive pump pulse tends to lower the inversion level of the quantum dot and hence the photon emission efficiency. This maximum efficiency is dependent on the pump pulse duration. If the pulse duration is long enough to allow the re-excitation of the quantum dot, successive photon emission generated by a single pump pulse will occur. These antibunched photons will cause a spurious non-zero value of *g*
^{(2)}(0).

## Acknowledgments

We thank Mr. L. You, B. Bai and N. Ma for helpful discussions. This work was supported by National Natural Science Foundation of China (No. 60244001 and 60290084) and the Foundation of Key Projects of Basic Research (TG 200003601).

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