A numerical investigation of pulse propagation in a quantum dot structure in the regime of electromagnetically induced transparency is reported. The quantum dot is described as a cone on top of a wetting layer and the calculated energy levels and dipole moments are used in an effective three-level model. Pulse propagation characteristics such as degree of slowdown, absorption, and pulse distortion are investigated with respect to their dependence on the dephasing rates and pulse width. It is seen how Rabi oscillations can seriously distort the pulse when the spectral width of the pulse becomes too large compared to the width of the EIT window.
©2007 Optical Society of America
The experimental demonstration of ultra-slow light in cold atoms  using the effect of electromagnetically induced transparency (EIT) has spurred a lot of research activities due both to fundamental interests as well as possible applications. While the fundamental aspects of EIT can be investigated in cold atoms and gaseous media, practical applications, e.g. for telecommunications, need to be based on solid-state material and should also preferably allow integration with other functionalities. Semiconductor quantum dot (QD) material incorporated in a waveguide geometry has been suggested as an obvious candidate [2, 3] since the discrete level structure admits the realization of the level configuration for electromagnetically induced transparency. However, in contrast to cold atomic media, semiconductor structures have short dephasing times; room-temperature values are typically in the range of sub-hundred to a few hundreds of femtoseconds due to efficient electron-electron scattering mechanisms. It is important to investigate under which conditions EIT can be experimentally observed in such structures.
We here consider the case of self-assembled quantum dots formed on top of a wetting layer. We follow  and investigate the case where the coupling signal couples two subband levels in the conduction band and by the induced level Rabi-shift makes the medium transparent to the interband probe signal as well as increases its group refractive index. We extend the investigations of  by considering the case of a pulsed signal and making a systematic investigation of parameter dependencies, in particular emphasizing the roles of dephasing rates and pulse width. Approximate analytical and semi-analytical investigations of pulse propagation and distortion have recently been reported [4, 5, 6, 7] based on frequency domain calculations and estimations. In this paper we focus on the time domain behavior and show the role of Rabi and dipole oscillations in leading to pulse break-up when the pulse spectral width approaches the spectral width over which EIT is effective. Employing dipole moments calculated for realistic self-assembled dots, we quantify the variation of the effective group index, pulse distortion, and pulse absorption with dephasing times and pulse width. In particular the dependence on dephasing is important, since reported dephasing times vary over many orders of magnitude, among other things depending on the temperature [2, 8].
It should be mentioned that there exists an alternative method for achieving light slow-down in semiconductors based on the excitation of coherent population oscillations by beating of the probe signal with a nearly-degenerate pump signal . In this case the degree of light slowdown is not limited by the dephasing time but rather by the lifetime, which is typically significantly longer in semiconductors. Using this technique light slow down has been achieved in short semiconductor waveguides operated at room temperature . However, since the physics of the two effects differ significantly we shall not go into detail with this scheme.
The paper is organized as follows: In section 2.1 we introduce the theoretical model which is based on the Maxwell-Bloch equations for a three-level system. In section 2.2 we calculate the transition matrix element (TME) for the QD model considered; this is done for different QD and wetting layer sizes. Section 2.3 summarizes some analytical results from the continuous wave (CW) field analysis of the model. In section 3.1 we show an example of how Rabi oscillations are induced for spectrally wide pulses, which cause pulse distortions. In section 3.2 we perform a parameter study of the pulse propagation characteristics and their dependence on dephasing rates and pulse width, which is our main result. Finally, the conclusions are summarized in section 4.
The physical model used for the simulation of EIT in this article is a 1D model consisting of a waveguide with a high density of InAs QDs embedded in GaAs. The waveguide is surrounded by a lower index material which we leave unspecified. A schematic of the QD situated on top of a wetting layer (WL) is shown in Fig. 1 along with the simplified level structure. We employ the so-called ladder scheme , where a strong coupling beam excites the intersubband transition 2-3 in the conduction band, thereby modifying the susceptibility of a probe signal tuned to the interband transition 1-2. Below we first describe the general three-level density matrix equations used to model the interaction of the quantum dot medium with the coupling and probe beams and after that we present calculations of the dipole matrix elements appropriate to the transitions in Fig. 1(b).
The combined interacting system of the probe pulse and QD material is modelled by a semi-classical approach. The three coupled levels in the QDs are described using the density matrix formalism and the probe field by Maxwell’s equations in the form of a reduced wave equation.
2.1. Light-matter coupling
The equation of motion for the density matrix is expanded in the eigenstates of the Hamiltonian, Ĥ0, for the stationary QD system, which are considered in Sec. 2.2. The time evolution of the elements of the density matrix is governed by
Here Ĥ I describes the interaction between the material and the field. Using the dipole operator, dˆ = - e r, Ĥ I can be written explicitly as
where Ẽ p,c(x,t) is the slowly-varying complex envelope. The subscripts p and c refer to probe and coupling fields and it has been used that both fields are polarized in the y-direction and propagate in the x-direction. The last term in Eq. (1) is a phenomenological decay/dephasing term with γnm being the decay/dephasing rate, see .
The operator Ĥ 0 satisfies the eigenvalue equation Ĥ 0|n〉 =En|n〉. It should be noted that in-homogeneous broadening, due to fluctuations in the QD size, is not taken into account in this treatment. This is a major approximation, which is fulfilled if the inhomogeneous broadening is smaller than the homogeneous broadening implied by the value of the dephasing rate. This is not yet the case for quantum dot structures grown using the self-organizing Stranski-Krastanov method, although several groups are working to achieve it. Nonetheless, our results demonstrate important limitations in reaching EIT due to dephasing effects even for perfect samples. Furthermore, the role of inhomogeneous broadening was investigated for the CW case in  and a scheme of multiple pumps was suggested in order to mitigate the effect.
The matrix elements of the interaction Hamiltonian are evaluated using the dipole approximation. As mentioned we consider a 1D model, with the field evolving along the x-axis and with a field profile in the transverse directions that is assumed constant by virtue of a confining waveguide. The waveguide contains a high density of quantum dots with an average distance much less than the optical wavelength and the medium is thus modelled as an effective medium with a continuous x-dependent excitation and population. Utilizing the symmetry of the QD and the properties of the lattice Bloch functions, see Sec. 2.2, we find that the only non-zero matrix elements are 〈1[y∣2∣〉 and 〈2∫y∣3〉. To simplify the equations we introduce the slowly varying envelopes σij, by the following definitions
Applying the rotating wave approximation we obtain the optical Bloch equations,
We define the detuning of the fields as ∆p = ω21 - ωp and ∆c = ω32 - ωc, where ωnm = (En -Em)/h̅, the transition matrix elements as μnm = e 〈n∣y∣m∣, and the Rabi frequencies as Ωp = μ12 Ẽ * p/2h̅ and Ωc = μ23 E̅ * c/2h̅. It should be noted that another commonly used definition of the Rabi frequency is: Ω = μE/hμ, which differs by a factor of two from the definition used here.
The electric field is governed by Maxwell’s wave equation, with a polarization source term induced by the active material. In this treatment we consider the coupling field to be a CW field much stronger than the probe field and under the assumption that the excited state ∣2〉 remains almost unpopulated we shall neglect the real excitation of carriers from 2 to 3, thereby implying a constant coupling field along the propagation direction of the waveguide. The polarization seen by the probe is written in the form,
where N is the x-dependent QD density. Within the slowly varying envelope approximation we obtain the following reduced wave equation for the probe envelope
where n b is the effective index of the waveguide, approximated as the index of GaAs, and Γ is the confinement factor of the waveguide. For simplicity we assume Γ = 1.
2.2. QD model and transition matrix elements
The basis states needed to expand the density operator are obtained by solving the Schrödinger equation for the QD system in the effective mass approximation, which is accurate when the QD dimensions are much larger than the lattice constant. The full wave function is then given by ψ(r) = u(r)F(r), where u(r) is a lattice periodic Bloch function. The function F(r) is determined from the following eigenvalue equation
where m *(r) is the effective mass of the respective band and V(r) is the confining potential. This equation is subject to the same standard boundary conditions as the Schrödinger equation.
As discussed above we consider an InAs cone shaped QD on top of a wetting layer, embedded in bulk GaAs [11, 12]. A schematic of the QD system along with relevant dimensions is shown in Fig. 1(a). The system is assumed rotationally symmetric with respect to the z-axis and the rotational symmetry is exploited by the use of cylindrical coordinates. The dimensions Lz and R 0 are numerical boundaries employed in a finite element method. The light gray areas are GaAs and dark gray InAs, hence the different band gaps will create a confining potential for the electrons in the conduction band and the holes in the valence band. The problem is now a standard particle-in-a-box, indicated in Fig. 1(b), where the potential depth, ∆Ej (j = c,v), is given in Tab. 1 for both the conduction (c) and valence (v) band.
The states will be labelled ∣jnm〉 where j refers to the band (c or v), the quantum number n belongs to the r and z directions and acts as a main quantum number, and m is the magnetic quantum number. The envelope “wave functions” can now be written
where the functions fjnm (r, z) are determined numerically using a finite element method. The states ∣jnm〉 and ∣jn(-m)〉 are energy degenerate for m≠0, and we can choose fjnm = fj)n(-m).
As mentioned above the only non-zero TME are 〈1 ∣y∣2〉 and 〈2∣y∣3〉. We choose the states ∣n〉 to be the following: ∣1〉 = ∣uv〉 ∣v10〉, ∣2〉 = ∣uc〉 ∣c10〉, and ∣3〉 = ∣uc〉 (∣c1, 1〉 - ∣c1, -1〉)/(i√2). The state ∣3〉 has the same energy as the orthogonal state ∣3+〉 = ∣uc〉 (∣c1, 1〉 + ∣c1, -1〉)/√2 but since 〈2∣y∣3+〉 = 0 the latter does not couple to ∣2〉 via electric dipole interactions.
We first consider the interband TME 〈1 ∣y∣ 2〉, which can be approximated by the following
where 〈uv∣y∣uc〉 = (0.699 × 1015 nm s)ω21 is taken from . The remaining integral is simply the overlap between the two envelope states after integration over the angular coordinate φ.
The intersubband TME 〈2∣y∣3〉 can be approximated by the following
where y = r sin φ has been inserted and again the angular part has been evaluated in the remaining integral.
The TME calculated from Eq. (5) and (6) are shown in Fig. (2) for varying wetting layer thickness d and quantum dot height h. The remaining parameters used in this calculation are listed in Table 1. It is of great interest to maximize both TMEs, since a large μ12 will enhance the group index and a large μ23 will lower the power requirements for the coupling field. In the figures we see a clear tendency for larger TME as we increase the thickness of the WL and the height of the QD. In both cases the spatial extent of the envelope wave functions increases and due to better confinement in both the conduction and the valence band the matrix element increases. The cut-off in the graphs for d > 1.25 nm has been made where the state ∣3〉 is no longer bound to the QD.
Calculations of the relevant TME for a GaAs/InAs QD system have been performed previously [2, 3] but by adopting a simple description of the QD as a disc structure and without the presence of a WL. The effect of the wetting layer was included in the calculations presented in [11, 12], but results for the interband and intersubband matrix elements were not presented. In all of these calculations, excitonic as well as strain effects were neglected. For detailed comparisons to experiments such effects need to be included, but this will be at the prize of a considerable increase in the complexity of the model. From our results we conclude that the inclusion of a WL, apart from being more realistic, seems to give a larger TME according to Fig. 2.
2.3. Continuous wave analysis
In the case where both the coupling and probe beams are CW signals and when the probe Rabi frequency is small compared to the Rabi frequency of the coupling beam it is possible to find an analytical approximation for the effective complex susceptibility for the probe field at a specific point in space [2, 3]. Even though we in this article mainly deal with the time domain behavior of pulse propagation, it is useful to consider the main properties of this susceptibility. In our notation, the expression for the susceptibility (with ∆c = 0) seen by the probe is
It is interesting to note that the expression only depends on the dephasing rates γ12 and γ13, the other dephasing and relaxation rates are decoupled by the small probe power assumption.
Figure 3(a) shows the real, χ′, and imaginary, χ″, part of the susceptibility as a function of the normalized detuning - ∆p /∣Ωc∣, where we have assumed ∆c = 0 for maximum EIT effect. This illustrates a typical EIT situation. The splitting of the second level due to the strong coupling signal is clearly seen by the two peaks in χ″, and the large slope in χ′ indicating a high group index is also apparent. In the figure we have indicated the magnitude of the level splitting equal to 2Ωc and the role of γ12 as linewidth for the absorption peaks. For this illustration we have used values of the dephasing rates so that γ12,γ13 ≪ Ωc, which is desirable for efficient EIT. Usually it is assumed that γ13 ≪ γ12 in order to achieve a regime of “pure” EIT , but this is usually not fulfilled in semiconductors, where the dephasing rates in systems realised so far appear to be comparable. We therefore assume γ = γ12 = γ13.
Figure 3(b) shows the calculated group index and imaginary part of the susceptibility versus Rabi frequency of the coupling beam for zero detuning, χ″(∆p = 0). The group index is a measure of the degree of slow-down compared to vacuum and is calculated as n g =Re[n] + ωp∂ωp, Re[n], where n is the complex refractive index. This is connected to the susceptibility through n 2 = n 2 b + χ. The imaginary part χ″ is proportional to the absorption and needs to be low in order to avoid excessive absorption which is detrimental for any application. The different series are for different values of the dephasing rates; in practise one observes a large variation with temperature of the dephasing rate , although the detailed dependence of individual rates is not known in general.
The figure shows that a high group index also implies a large absorption, at least when one does not have the freedom to lower the dephasing rate γ13 independently of γ12. We also see, for fixed Rabi frequency, the large influence of the absolute value of the dephasing on the group index as well as the absorption. The large effect of the dephasing rates which can be concluded from this analysis is one of the main motivations for investigating these dependencies further in Sec. 3.
The frequency dependence of the susceptibility of course implies dispersion effects leading to distortion in the case of short pulses propagating through the medium. Such effects, e.g, pulse broadening, were estimated in [4, 6] based on the frequency dependence of the susceptibility. Furthermore, in the paper  analytical approximations for the temporal pulse shape were derived based on the frequency domain description. However, these approximations do not allow to consider the case of pulse spectral widths comparable to the EIT window, for the case of interest here, where γ12 ≃ γ13. Our numerical simulations of the temporal properties of the pulses take full account of the dynamics of the medium as well as propagation effects, and offers additional insight into the physics of EIT in the transient regime.
We numerically analyze the distortion of a pulse being transmitted through the quantum dot medium exposed to a CW coupling beam using the model developed above. We consider a Gaussian probe pulse
where 𝓔p is the peak amplitude of the probe field and Wt is the intensity full width at half maximum (FWHM), which is related to the spectral FWHM W ω through W ω Wt = 4ln2. As mentioned the coupling field is a strong CW field with amplitude, 𝓔c. EIT is most effective when the coupling field is much stronger than the probe, hence we initiate all simulations so that 𝓔c/𝓔p = 50 corresponding to μ23𝓔c/ μ12𝓔p = 219 forthe given dipole moments. The detunings, ∆p and ∆c, are set to zero to achieve the maximum EIT effect. This implies that frequencies of the two fields are ωp = ω21 and ωc = ω32, i.e. resonant. We emphasize, however, that the model is able to consider the practically relevant case of stronger probe pulses.
3.1. Rabi oscillations
As a specific example of the pulse evolution Fig. 4 shows two simulations of pulses with different spectral width. The active QD medium is situated in-between the black lines whereas the remaining medium is “cold” with a constant refractive index given by the background index. The probe pulse enters the “cold” medium at the left boundary. The left plots show the contour lines of the probe field envelope with the corresponding envelope of the 1-2 dipole shown in the right plot. The two upper plots are for a relatively long pulse, whose spectrum is narrow compared to the EIT window, W ω = 2Ωc/10. The propagation through the QD medium is in this case well described by the group index and absorption inferred from the CW analysis and the pulse experiences no significant dispersion.
For shorter pulses which are spectrally wide compared to the EIT window, the CW analysis can no longer predict the dynamic behavior of the probe pulse. A simulation for an extreme case (W ω, = 2Ωc/0.7) is shown in Fig. 4(c) and 4(d). Different frequency components of the pulse now experience different group indices and different levels of absorption which leads to strong dispersion and the main pulse is broken up. This is the same result predicted by frequency domain calculations when the probe pulse is assumed to undergo linear propagation governed by a susceptibility induced by the couling field. From a dynamical point of view the pulse contains frequencies which are resonant with the split ∣2〉 state. This induces Rabi oscillations in the dipoles which in turn drive the electric field and give rise to the oscillating tails seen in the field envelope, as also observed in references [5, 6].
3.2. Discussion of parameter dependencies
For a more quantitative evaluation of the parameter dependence we introduce three quantities that measure important properties of the transmitted pulse. As a measure of distortion resulting from temporal broadening, we use the standard RMS (root mean square) width σ defined in the following way
Here, N is proportional to the pulse energy and serves as an x-dependent normalization constant. The RMS width after transmission normalized by the initial RMS width, σrel = σout/σin, indicates the relative broadening of the pulse. Another commonly used measure, the FWHM width, is not suited for describing the distortion experienced by spectrally wide pulses, due to the appearance of oscillating tails. For applications the absorption of the pulse is most important and we quantify this through the transmission coefficient, T = 𝓝out/𝓝in. Finally, we calculate the group index of the pulse by comparing 〈t〉out and 〈t〉in, with and without the slap of active material. For symmetric pulses, 〈t〉 gives the position of the peak amplitude. However, the pulses do not always remain symmetric during propagation and in this case 〈t〉 measures the position of the intensity center of mass.
For semiconductors the rate of dephasing is known to range from 1 × 109 s-1 at very low temperatures to 10 × 1012 s-1 at room temperature  and the detailed study of the dependence of the EIT effect on the dephasing rate is of major importance. In our simulations we have chosen to fix γ13 and W ω at certain values and vary γ12 over a wide range. We are not aware of measurements for semiconductors, where γ12 and γ13 are measured independently. Since both rates are influenced by scattering with phonons and carrier-carrier scattering one could expect the rates to be similar for semiconductors, but this may not be the case for strongly confined quantum dot structures where the phase space for scattering is strongly reduced. In order to explore the phenomena occuring in different parameter regions, in particular covering the case of atomic EIT, where one usually assumes γ13 ≪ γ12, we shall therefore allow a large variation in the dephasing rates.
Results are presented in Fig. 5 for γ13 = 10 × 109 s-1 and W ω/2Ωc varying in the range from 0.1 to 0.4. This value of γ13 is realistic for a semiconductor material at low temperatures. We will mainly focus on the case where W ω = 2Ωc/2.5, as this spectral width is comparable to the EIT window and behavior which deviates from the ideal situation can be expected.
For low dephasing rate (#1,γ12 ≈ 1 × 109 s-1), Fig. 5(b) shows a large pulse broadening. The Rabi oscillations induced by the pulse decay very slowly and the transmitted pulse develops a very long tail as seen in Fig. 5(d). The relative broadening in this case deviates strongly from the case of spectrally narrow pulses, whereas the group index and transmission are not that different. The slightly higher value of n g compared to n g,CW , is due to the tail acquired by the pulse, which displaces 〈t〉out from the main peak. The main pulse basically experiences the group index predicted by the CW analysis.
For higher dephasing rate (#2, γ12 ≈ 100 × 109 s-1), we observe a large reduction in the degree of pulse broadening, but only a small change in the group index and transmission. The larger dephasing rate dampens the Rabi oscillations in the active region and as a consequence the tail after transmission is much less pronounced, implying a smaller broadening. In the region between #2 and #3 the value of γ12 becomes larger than 2Ωc, and the large decoherence rate washes out the splitting of state ∣2〉 and the active material again acquires the characteristics associated with an absorption resonance. The transition between #2 and #4 is illustrated in Fig. 6(a) and (b) through the CW susceptibility.
Although not suitable for applications, the regime where γ12 > 2Ωc displays some peculiar behavior which is relevant due to the large dephasing rates encountered in semiconductors. At point #3, γ12 ≈ 1012 s-1, in Fig. 5(b) the pulse width attains a maximum while the transmission is minimized. The pulse in this case experiences two very wide absorption peaks and a dip at the center frequency, due to the relatively low γ13. Due to the finite spectral width of the pulse a relatively large damping occurs. The linear part of the refractive index still covers a significant part of the pulse and the pulse has a rather large group index, although significantly reduced from the value predicted by the CW analysis.
Increasing γ12 to ≈ 10 × 1012 s-1, #4, a large reduction in pulse width and a significant increase in transmission is observed. This is due to the absolute decrease of the susceptibility as γ12 is increased and the corresponding reduction in the perturbation of the pulse induced by the medium. This results in the pulse seen in Fig. 5(d), which has a shape similar to the input pulse but a reduced peak height.
To illustrate the pulse changes for varying pulse width and fixed dephasing rate, the transmitted pulses #5 to #7, are plotted in Fig. 5(e). It is clearly seen how the spectrally narrowest pulse preserves its initial Gaussian shape and how the spectrally wide pulse is distorted the most.
Calculations have also been performed for a smaller and a larger value of the 1-3 dephasing rate, i.e., γ13 = 109 s-1 and γ13 = 100 × 109 s-1, see Fig. 7. For the case of the low γ13 we see the same qualitative features as for the case analyzed above. The quantitative differences are mainly due to the lower absorption obtained when γ13 attains this small value. For the larger dephasing rate, γ13 = 100 × 109 s-1, however, the transmission drops to a very small value of 10-5 for the first part of the graph, while the relative broadening and group index are approximately constant until γ12 > 2Ωc, beyond which point there is hardly any influence of the coupling beam. Figure 6(c) shows the pulses corresponding to points #8 and #9, in which cases the absorption is very large.
We have presented a theoretical investigation of slow light based on the effect of electromag-netically induced transparency in semiconductor quantum dots. We consider the ladder scheme suggested in  and describe the dynamics of the medium by Bloch equations with transition matrix elements calculated for conical semiconductor quantum dots situated on a wetting layer. The dynamical equations including propagation effects are numerically analyzed for a range of different parameter values. In particular, since dephasing rates in semiconductors are known to vary widely depending, e.g., on temperature  we emphasize the influence of dephasing rates and pulse widths on the properties of the transmitted pulses. In comparison to earlier analysis of quantum dot systems [2, 3] we thus extend the results from a CW calculation of the effective susceptibility to a dynamical analysis including propagation effects, as well as an improved calculation of the dipole elements involved in the ladder scheme for self-organized quantum dots.
Under ideal EIT conditions, when the pulse spectral width is narrow compared to the EIT window, and the dephasing rate γ13 characterizing the EIT resonance is small compared to the dephasing rate of the two-level transition γ12), we observe slow down and reduced absorption as predicted by a CW analysis . When this condition is not satisfied and the spectrum of the pulse overlaps with the absorption peaks we observe pulse distortion effects. The pulse properties are quantified by detailed calculations of the mean slow-down effect as well as the degree of pulse-broadening. In some cases the pulse develops a significant oscillating tail, and from the dynamical analysis this is seen to result from damped Rabi oscillations excited in the medium, thus limiting the practical applications of such pulses. We show the detailed variation of pulse properties with dephasing rates and pulse widths and explain the qualitatively different regimes of operation.
Numerical examples of pulse propagation in the regime of EIT were presented in , but in a more general setting where the different roles of the two important dephasing rates, γ12 and γ13, were ignored. Approximate calculations of distortion effects based on linear frequency domain analysis were presented in [4, 5, 6] and our results support and extend these findings. The numerical approach allows to consider cases where the pulse spectral width is comparable to or larger than the EIT window and we furthermore obtain the temporal profile of the output pulse and the dynamics of the medium, which provides further insight into the physics of EIT in quantum dots.
The authors acknowledge support from EU through the IST project “QPHOTON” as well as support from the Danish Research Councils through the project “QUEST”.
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