Analytical description of gain depletion and recovery in quantum dot optical amplifiers

We present an analytical description of ultrashort femtosecond pump–probe experiments and investigate the gain response of quantum dot (QD) semiconductor optical amplifiers. The calculation provides a full analytical solution of numerical studies to recent experiments in such structures (Gomis-Bresco et al 2008 Phys. Rev. Lett. 101 256803). Our approach is based on QD Bloch equations, which are analytically evaluated within the third-order temporal perturbation theory (χ(3) level). In particular, we study the influence of the coherence and the population dynamics of two confined QD levels on the gain as a function of the delay time between the pump and the probe pulse. We discuss how to engineer optimal conditions for high-performance QD amplifiers, which are characterized by an ultrafast gain recovery and a pronounced gain depletion.


Introduction
Self-assembled quantum dots (QDs) are considered excellent candidates for application as semiconductor optical amplifiers (QD-SOAs) in ultrafast ethernet networks [1][2][3][4][5]. They show a picosecond gain recovery time, which is much faster than conventional or quantum well SOAs [1,6,7]. This can be ascribed to the presence of the two-dimensional wetting layer (WL), which acts as a charge carrier reservoir enabling an efficient refilling of QD states [8][9][10][11][12][13][14][15]. However, the amplification of ultrafast optical pulse trains with pulse repetition rates of up to 1 THz is still a great challenge for current research [4,7]. Recent experiments show that there is a limiting delay time, below which even at high injection currents QDs cannot be completely refilled [7]. A comprehensive understanding of the gain dynamics is the key to designing high-speed QD-SOAs. The theoretical methods applied to address this problem reach from simple rate equation models [5,16,17] to microscopic approaches, including (quantum) kinetic calculations of scattering processes [7-9, 12, 18]. The latter are more sophisticated, but their evaluation involves demanding numerical calculations, in particular, if the temporal dynamics from femtoseconds to picoseconds need to be resolved. The aim of this work is an analytical description of the gain response of QD-SOAs, which allows an efficient investigation of optimal conditions for an ultrafast gain recovery and a pronounced gain depletion. Most of the experimental work is focused on pump-probe experiments, where gain depletion and recovery are measured as a response to a probe pulse.
We model femtosecond pump-probe experiments within the formalism of the density matrix theory, where all input parameters can be calculated on a microscopic footing [8,9,15]. Since we are interested in modeling experiments with ultrashort pump and probe pulses, coherence effects are of major importance. Applying a third-order temporal perturbation theory (χ (3) level), we obtain analytical solutions, which are valid for small pulse areas. A similar approach has been applied by Lindberg and Koch [19,20]. Their focus, however, was on the investigation of the dynamic Stark effect in semiconductor absorbers. 3 corresponding population probabilities f b (t, τ ) with b = e, h [7,10,[21][22][23]: with τ as the delay time between the pump and the probe pulses. Because carrier relaxation processes within the QD states are much faster than capture processes from the WL states, the energetically lowest lying states are considered to be crucial for the gain dynamics. The neglected inhomogeneous broadening is expected to overestimate the gain, but it is assumed to have no influence on the characteristic features in the gain spectrum.
In equations (1) and (2), the light-matter interaction is taken into account within the semiclassical approach, i.e. the charge carriers are treated quantum mechanically, while the field is considered to be classical. Because we focus on the ultrafast dynamics of gain switched devices, the influence of spontaneous emission is negligible. The strength of the light-matter interaction is given by the Rabi frequency pp (t, τ ) = d vc E pp (t, τ )/h with the interband dipole moment [7] d vc = 0.6 e 0 nm and the electrical field E pp (t, τ ) = E p e −|t|/α p e −iν p t 0 + E 0 e −|t−τ |/α 0 e −iν 0 t 0 containing both the probe (E 0 ) and the pump pulse (E p ) contributions. For mathematical reasons, we apply (as often used in the literature) [21] an exponential shape for the pulse to obtain a complete analytical solution, which gives insights into the complex gain dynamics. The pulse shape does not qualitatively change the gain response [7]. The conclusions drawn in this paper remain unchanged in the case of a Gaussian pulse. In agreement with typical experiments [7], the pump pulse is assumed to be c p = E p /E 0 = 100 times larger than the time-delayed probe pulse. The pump/probe pulse amplitude E p/0 =h p/0 /(2d vc α p/0 ) contains the pulse area p/0 and the pulse width α p/0 . Phase shifts on a macroscopic time scale t 0 , typical for heterodyne experiments, are described by the factors e −iν p t 0 and e −iν 0 t 0 , respectively, with a fixed time t 0 , which is varied during many repetitions of the experiment. The phase contributions written explicitly with the frequencies ν p and ν 0 provide the possibility of distinguishing between pump and probe contributions using heterodyne detection schemes [24]. The Coulomb in-scattering rate S in b , the corresponding scattering time T 1,b and the dephasing time T 2 describe the Auger transitions between localized QD and continuous WL states [8,9,[11][12][13][14]. They are calculated microscopically beyond the Hartree Fock level by considering the screened Coulomb potential up to the second-order Born-Markov approximation.
S in b , T 1,b and T 2 show a nonlinear dependence on the WL charge carrier density n wl and on the temperature T of the WL charge carriers. Details of the evaluation can be found in [7,12,15]. Here, we focus on providing corresponding analytical solutions for equations (1) and (2). Because a QD-SOA runs within the gain regime, where n wl is in the range of 10 12 cm −2 , it is assumed that the Coulomb interaction dominates over the electron-phonon coupling [9,12,13]. However, more complex theories [9] could also be used to provide microscopic input, including electron-phonon interaction. Furthermore, since we treat the WL states as a reservoir, n wl and T can be varied as external parameters. The decay of coherence, expressed by the microscopic polarization p(t, τ ), is determined by the T 2 time, which can be separated into two different contributions, T −1 2 = T −1 2,scat + T −1 2,pure . The first part accounts for scattering processes between the QD and WL states. The dephasing of optical transitions, however, can also occur via pure dephasing processes which do not lead to a change of population. This contribution is found to play a crucial role in QDs due to their large separation between energy levels [9,25]. Because the goal of this work is an analytical solution for the gain dynamics in QD-SOAs, we describe a QD-WL structure, where QDs are considered as two-level systems. This is a good approximation for small QDs with strong confinement and large energy separation between different levels [26]. However, equations (1) and (2) can be generalized in a straightforward way at the price of a demanding numerical evaluation [7]. Here, we focus on providing an analytical solution, which allows the reader to perform systematic investigations of the gain dynamics.
The dynamics of density matrix elements f b (t, τ ), p(t, τ ) allow the calculation of the pump-probe gain coefficient g(ω, τ ), which is a function of the probe frequency, ω, and the delay time, τ , between the pump and probe pulses. It corresponds to the negative absorption coefficient and therefore can be expressed in the frequency space as the ratio between the probe pulse-induced macroscopic polarization,P 0 (ω, τ ), and the electrical field, E 0 (ω), of the probe pulse [22]: In a pump-probe experiment, only the probe signal is measured, which probes the changes arising from the pump pulse. In a heterodyne experiment [24], the probe pulse-induced macroscopic polarization,P 0 (ω, τ ), is determined by all contributions proportional to the probe frequency, e −iν 0 t 0 . It can be separated from the full polarization, P = d vc p(t, τ ), by a repetition of the experiment with different times, t 0 , leading to a separation of the probe pulse part of the emitted field. This ensures that only the response coming from the probe pulse (after the influence of the pump pulse) is obtained [27]. In this work, the gain dynamics are calculated for the case in which the pulses are in resonance to the QD band gap energy, ω = ω QD (i.e. detuning effects are not considered). The first step is the calculation of the microscopic polarization by evaluating the Bloch equations. See equations (1) and (2). In general, this set of differential equations can only be solved numerically. However, an analytic solution can be achieved using an iterative integration procedure [19,20,28]. We solve the equations on the χ (3) level considering contributions of up to the third order in the field amplitude, E(t, τ ). This perturbative approach is shown to be a good approximation for small pump pulse areas, [28]. For the following results, the pump pulse area has been fixed at p = 0.01π . Because we focus on the stationary gain regime before the arrival of the pump pulse, the starting point of the iteration is the stationary value for the population probability, f 0,b = S in b T 1,b , which can easily be calculated from equation (2) assuming that the pulse has not yet arrived. Since the scattering time, T 1,b , is determined by the in-and out-scattering rates with (1) yields a first-order differential equation for the microscopic polarization. Its solution can be found analytically: Inserting p (1) (t ) into equation (2) yields the second-order population probability, f (2) (t ): Finally, equation (5) plugged again into equation (1) gives the third-order microscopic polarization, p (3) (t):  (1) and (2)) requires a case differentiation: (a) positive delay times, τ 0, i.e. the probe pulse enters the sample after the pump pulse, and (b) negative delay times, τ < 0. An analytical solution for the gain dynamics can be obtained by decomposing the problem in different cases depicted in (a) and (b).
For small pulse areas, the iterative integration can be truncated at this level. Our aim is to calculate the gain dynamics for all times, t, and for all delay times, τ , between the pump and the probe pulse. Because the pulses are assumed to have an exponential shape, the integrations in equations (4)-(6) can be performed analytically but require a distinction of different cases. Assuming the pump pulse to be ahead of the probe pulse, i.e. τ 0, we distinguish (I) t 0, t τ , (II) t < 0, t < τ and (III) t > 0, t < τ (see figure 1(a)). The situation with the pump pulse behind the probe pulse, i.e. τ < 0, can be treated in an similar way, as shown in figure 1(b). As a result, the general solution is composed of six different parts, III,i (t, τ ), with i = (1, 2) distinguishing the case with positive and negative delay times, respectively: with 0 = +1 and p = −1. Furthermore, l ∈ (0, p) denotes the probe (0) and pump (p) pulse contribution, and b ∈ (e, h) stands for electrons and holes. Here, the following abbreviations have been introduced: The analytical solution from equations (7)-(9) describes the dynamics for all times, t and τ . The points T 1,b = T 2 , T 1,b = jα, T 2 = jα j ∈ (1, 2, 3) need to be excluded (see equation (10)), but the corresponding solutions for these points can be found in a straightforward way. The solution of p (3) II,i (t, τ ) from equation (8), valid for negative times t and for t < τ , is identical for positive and negative delay times τ since in this case e −| t−τ | = e (t−τ ) independently of τ . Its coefficients, B 1,l , B 2,l,l , are given by the following relations: with˜ l = l (e −τ/α 0 δ l,0 + 1) and l = (0, p) denoting the probe and the pump pulse contribution, respectively. As expected, the microscopic polarization shows a dependency on the dephasing time, T 2 , the decay time, T 1,b and the pump pulse width, α p . In section 3, we will discuss the influence of these three parameters on the gain recovery time and the gain depletion in detail.
To be able to describe the dynamics for all possible parameters, the part solutions p (3) I,i (t, τ ) and p (3) III,i (t, τ ) also need to be taken into account. Their coefficients are rather complex, showing a complicated dependence on T 2 , T 1,b and α p . (See the supplementary material, available from stacks.iop.org/NJP/12/063012/mmedia.) However, in section 3.2 we will discuss a relatively simple limiting case, which already gives a good description of the gain dynamics in the limit of small dephasing times, T 2 . Figure 2 shows an example of the solution for the microscopic polarization p (3) (t, τ ), the electron, and the hole population probabilities, f (2) b (t, τ ), as a function of time for an exemplary delay time, here τ = 1 ps. The parameters, such as the decay time, T 1,b , and the scattering part of the dephasing time, T 2,scat , are calculated microscopically [8,15] 9)). For comparison, the pump and probes (×10) pulse are also shown. room temperature) [8], assuring that the system is within the gain regime. We note that the dephasing time, T 2 , also includes the pure dephasing contribution, T 2,pure . The overall T 2 time is found to be approximately 25 fs, which is in good agreement with experimental values [7,29]. The width of the pump and the probe pulse is assumed to be α p/0 = 300 fs. While p (3) (t, τ ) directly reflects the pump pulse since T 2 α p (polarization is slaved by the field), f (2) p (t, τ ) shows two time scales. In the first hundreds of femtoseconds, the incident pump pulse depletes the ground state, resulting in a decrease in f (2) b (t, τ ). Reaching the picosecond time scale, the charge carriers are captured from the neighboring WL states via Coulomb-driven carrier-carrier scattering processes, leading to an increase in f (2) b (t, τ ) towards the stationary value, f 0,b . The dynamics of electrons and holes are similar. The holes reach their stationary value faster, since their scattering channel is more efficient at the considered WL carrier density [8].

Gain dynamics
In this section, we investigate systematically the dependence of the gain recovery time and the gain depletion on the decay time, T 1 b , the dephasing time, T 2 and the probe pulse width, α p . After deriving the density matrix elements, f (2) b (t, τ ) and p (3) (t, τ ), the determination of the gain coefficient, g(ω, τ ), is straightforward. The probe pulse-induced macroscopic polarization, P 0 (t, τ ), and the electrical field, E probe (t), just need to be Fourier transformed. The imaginary part of their ratio gives the gain coefficient, g(t, τ ) (see equation (3)). Plugging the solution from equations (7) to (9) into equation (3) gives the gain coefficient, g(ω = ω QD , τ ), as a function of the delay time, τ . Within the modeled femtosecond pump-probe experiment, a comparatively strong pump pulse is applied (with a 100 times larger area than the probe pulse). As a result of the ultrafast pumping, the QD ground state is depleted, leading to a reduction in the gain coefficient, g(ω, τ ), which is tested by a weak probe pulse. Figure 3 shows a characteristic gain curve with a typical dip at τ = 0, where the overlap between the pump and the probe pulse is The overlap between the pump and the probe pulse leads to a characteristic quadratic-like minimum in the gain curve (coherent artifact). After a while, the gain recovers again, accounting for capture of charge carriers from the WL states. For high injection currents and repetition rates smaller than 1 THz, the gain recovers completely on a picosecond time scale.
maximal. The origin of the dip can be understood in analogy to pump-probe experiments, where different propagation directions (and not frequencies) are used to independently measure pump and probe transmission. Here, the resulting interference pattern of the two pulses produces a spatial modulation of the polarization. The generated optical grating scatters part of the pump beam in the probe direction, resulting in a quadratic-like shape of the gain peak [24], i.e. the signal recovers already on a femtosecond time scale. This scattering effect, which does not describe the intrinsic gain dynamics, is often called a coherent artifact. As soon as the Coulombdriven scattering processes start to be important, the QD ground state is refilled with charge carriers from the WL states: the gain recovers towards its initial value. Its recovery time is ruled by the strength of Coulomb scattering rate, S in b , and the corresponding decay time, T 1,b . As shown in figure 3, the gain basically follows the population dynamics after the influence of coherence effects has diminished.

Optimal gain recovery and depletion
A prerequisite for high-performance QD-SOAs is a fast gain recovery and a strong gain depletion. These two characteristics turn out to be difficult to fulfill at the same time. Accelerating the gain recovery time leads to a considerable reduction in the absolute gain, and vice versa. The analytical solution of the gain dynamics can be exploited to find an optimal region to engineer ideal QD-SOAs. Figures 4 and 5 illustrate the dependence of the gain recovery and the gain depletion, respectively, on the dephasing time, T 2 , the decay time, T 1 , and the pump pulse width, α p . Because the electron and hole dynamics are found to be similar (cf figure 2), we focus on the influence of the electron decay time, T 1,e . The variation in T 2  figure 3) expressed in % as a function of the dephasing time, T 2 , the electron decay time, T 1,e and the pump pulse width, α p . The probe pulse is kept constant with α 0 = 300 fs. While one of the three quantities is varied, the other two are fixed to the values α p = 300 fs, T 2 = 25 fs and T 1 = T 1,e = 3.4 ps (microscopically calculated). The figure shows that the population decay time, T 1 , is the crucial channel determining the efficiency of gain recovery. There is also an optimal pump pulse width, α p ≈ 100 fs, at which the gain recovery is maximal. The inset illustrates the dependence on the decay time, T 1 , for a larger range. and T 1 time corresponds to the variation in the injection current density, the temperature, the size of QDs, etc [7,8,12]. The decoupling of T 2 and T 1 time can be ascribed to pure dephasing contributions to the dephasing time, which have no influence on the population dynamics and its decay time. In particular, pure dephasing via phonon contribution clearly leads to a decoupling of T 1 and T 2 time. Figure 4 confirms the expectation that the T 1 time is the crucial parameter to reach an efficient gain recovery. For T 1 < 0.8 ps, the gain is completely recovered after 10 ps. Increasing the T 1 time, i.e. reducing the strength of capture processes, leads to a less efficient gain recovery. At the same time, the gain depletion is slightly enhanced, but this effect is negligibly small (note the factor 500 in figure 5). In contrast, the variation in the dephasing time, T 2 , has a strong influence on gain depletion, as can be seen in figure 5. The longer the dephasing time, the weaker is the decay of the coherence (see equation (1)) and the stronger the QD ground state is depleted. As a result, however, the gain needs more time to recover to its initial value. The gain recovery efficiency decreases with increasing T 2 (see figure 4). Finally, variation of the pump pulse width α p shows an interesting feature: there is an optimal value, α p ≈ 100 fs, at which the gain recovery is maximal. Note that this value is sensitive to the dephasing time, T 2 . Enhancement of α p reduces the gain depletion considerably, since a temporally broad pulse has a spectrally narrow overlap with the QD ground state. The appearance of an optimal pump pulse width is a consequence of competing processes responsible for filling (inscattering processes, S in b in equation (2)) and pulse-induced depleting of the QD ground state. Our goal is to find an optimal regime at which the QD-SOAs show an ultrafast gain recovery and a pronounced gain depletion at the same time. Figures 4 and 5 suggest the following compromise solution: (i) minimize the population decay time, T 1 , by optimizing the scattering channels (e.g. increasing the current, enhancing the temperature or using smaller QDs with stronger confinement), (ii) maximize the dephasing time, T 2 , by controlling the channels for pure dephasing processes and (iii) choose the pump pulse width, α p , with the optimal gain recovery,

Discussion of the analytical result
The full analytical solution for the gain dynamics is rather complex, containing numerous terms. However, in many situations the solution can be simplified considerably. In this section, we assume (i) similar pump and probe pulse widths (α p ≈ α 0 ), (ii) similar scattering times for electrons and holes (T 1,e ≈ T 1,h ) and (iii) dephasing times, T 2 , shorter than the pump/probe pulse widths and shorter than the decay time, T 1 (T 2 < α 0,p < T 1 ). Furthermore, assuming the pump pulse to be much stronger than the probe pulse, i.e. E 0 /E p 1, contributions proportional to 3 0 and 2 0 p can be neglected. These assumptions are realistic for modeled pump-probe experiments [7]. In the considered limit, we obtain a rather simple three-term (two-term) solution for positive (negative) delay times: g I (ω, τ ) = g 1 e −τ/α p + g 2 e −τ/T 1 + g 3 e −τ/T 2 (τ 0), (13) g II (ω, τ ) = g 4 e τ/α p + g 5 e τ/T 2 (τ < 0).
The equations contain the contributions stemming from the dephasing time, T 2 , the decay time, T 1 , and the pulse width, α. Mixed terms are negligible within the applied assumptions. The solution reproduces well the characteristic gain behavior, including all considered time scales (see the red line in figure 6). The corresponding coefficients follow from equation (3)  It is compared with the red line exhibiting the simple three-term (two-term) solution discussed below. Furthermore, the green line exhibits the contribution stemming only from the T 1 term (see equation (15)).

Conclusions
We have presented an analytical solution for the gain dynamics of QD-SOAs. The approach is based on microscopic QD-Bloch equations, which are evaluated within the third-order temporal perturbation theory. We have investigated the influence of the decay time, T 1 , the dephasing time, T 2 , and the pump pulse width, α, on the gain as a function of the delay time between the pump and the probe pulse. The results can be used to obtain optimal conditions to achieve an ultrafast gain recovery and a pronounced gain depletion at the same time.