We report direct observations of Rabi oscillations and self-induced transparency in a quantum dot optical amplifier operating at room temperature. The experiments make use of pulses whose durations are shorter than the coherence time which are characterized using Cross-Frequency-Resolved Optical Gating. A numerical model which solves the Maxwell and Schrödinger equations and accounts for the inhomogeneously broadened nature of the quantum dot gain medium confirms the experimental results. The model is also used to explain the relationship between the observability of Rabi oscillations, the pulse duration and the homogeneous and inhomogeneous spectral widths of the semiconductor.
© 2013 Optical Society of America
Direct observations of Rabi oscillations and self-induced transparency in a room temperature electrically driven Quantum Dash (QDash) laser-amplifier were recently demonstrated in a series of experiments [1,2], which were confirmed by a detailed Maxwell-Schrödinger model [1,3]. Two key elements enable those observations. The first is operation with pulses whose duration (~150 fs) is shorter than the room temperature coherence time . The second is the ability to fully characterize the pulse complex electric field, following propagation along the amplifier waveguide, using a high resolution Cross-Frequency-Resolved Optical Gating (X-FROG) system . The measured X-FROG traces exhibit clear oscillations in amplitude and phase when the laser-amplifier is biased in the gain regime (where the oscillations represent Rabi oscillations), while in absorption, the combination of pulse narrowing and the phase profile signify self-induced transparency.
In , Rabi oscillations were demonstrated in a quantum cascade laser operating at 30K. Many other experiments identified coherent effects such as Rabi oscillations [7–12] and self-induced transparency [13,14] in low temperature semiconductors. At room temperature, clear signatures of coherent biexcitons were observed in in ZnO/ZnMgO quantum wells and reported in  while off-resonance interactions in a quantum well optical amplifier, which were explained by the process of adiabatic following are described in .
This paper reports on direct observation of Rabi oscillations and self-induced transparency in a Quantum Dot (QD) optical amplifier. The device is based on a newly developed high gain InAs/InP QD medium [17–19]. We demonstrate a systematic evolution of Rabi oscillations with increasing input pulse energy and bias. The results resemble those for the QDash amplifier  proving that in both types of gain material, the device behaves as an effective two-level system, an observation made also in  for the quantum cascade laser. The ability to induce and experience quantum-coherent effects, in a room temperature QD optical amplifier, designed for conventional telecomm systems, opens the way to exploit such practical devices in a variety of applications such as quantum communication, data processing, storage and sensing.
This paper also describes an extension of the model presented in [1,3] to include the inhomogeneously broadened nature of the self-assembled QDs. The new model confirms the experimental results. In addition, it enables to resolve a peculiarity stemming from the fact that the spectral width of the 150 fs pulse (about 25 nm) is much wider than the room temperature homogeneous linewidth of the semiconductor (7-8 nm). The pulse therefore excites resonantly several transitions, from different QD ensembles within the inhomogeneous linewidth, while off-resonantly exciting several others, which are expected to exhibit somewhat different Rabi frequencies. This should in principle distort the overall response to a degree where no oscillations are observable. Clear oscillations are experimentally observed however and the new inhomogeneous model resolves this ambiguity.
2. Experimental observations
Important developments in MBE growth of InP QDs  emitting in the telecom wavelength range (1550 nm) enable control over the dots morphology  yielding symmetric QDs with high density (approximately , obtained from atomic force analysis and atom probe tomography ) which offer high optical gain . The experiments we report here used a 1.5 mm-long optical amplifier based on such QDs. The ridge width was 2.5 μm and the facet reflectivity was reduced to 0.01% using a multi-layer AR coating. The active region comprised four InAs dot layers formed by a nominal deposition of five monolayers of InAs separated by 20 nm In0.528Al0.238Ga0.234As barrier layers. The high dot density plays a crucial role in the ability to observe Rabi oscillations which are imprinted on an intense pulse since the criteria for their observability has been shown  to be strongly related to number of emitters in the material.
The propagation of 150 fs wide pulses was analyzed by characterizing the complex envelope of the electric field (amplitude and phase) at the amplifier output. High-resolution X-FROG measurements [5,20] were used. A schematic description of the X-FROG set-up is shown in Fig. 1. A 150 fs long pulse from an optical parametric oscillator is injected to the waveguide of the QD amplifier. A replica of the pulse (separately characterized by standard FROG) is used to gate the pulse at the amplifier output using sum-frequency generation in a non-linear crystal. The product is measured by a spectrometer. Controlling the delay between the gating pulse and the output pulse allows generating a map of spectra versus delay, from which the complex envelope of the output pulse is retrieved by a computerized algorithm.
Measurements of input energy dependent amplitude and instantaneous frequency (chirp) profiles for a bias level of 250 mA are presented in Fig. 2. The profiles evolve from a single peaked pulse whose instantaneous frequency exhibits one valley to a double peaked pulse with an oscillatory chirp. Such profiles have been shown  to unequivocally represent the evolution from classical saturation (under moderate input pulse energy) to Rabi oscillations when the pulses are sufficiently energetic. The original observations [1,2] were for a QDash gain medium and the present results show that the same physical processes occur in a QD optical amplifier.
The evolution of the instantaneous frequency profiles originates from the fact that the electronic state of a semiconductor is imprinted on the instantaneous frequency profile by the plasma effect as detailed in . The refractive index of the semiconductor is inversely proportional to the excited carrier population. When carriers are depleted, due to stimulated emission induced by an amplified pulse, the index of refraction increases, thereby decreasing the instantaneous frequency. The pulse leading edge is therefore red shifted and conversely the trailing edge experiences a blue shift. The overall effect appears as a valley in the instantaneous frequency profile.
The lowest energy pulse in Fig. 2 represents a simple single gain event; it shows no amplitude distortion and a corresponding single instantaneous frequency valley. Such a pulse is said to have an area  smaller than π so it does not cause a complete Rabi oscillation. As the input pulse energy (area) increases, the amplitude deforms gradually. A pulse with an area of approximately 4π yields two complete Rabi flops causing the pulse to break up and obtain an amplitude profile with two peaks. Consistent with the results of , the instantaneous frequency also evolves into two distinct valleys signifying the occurrence of two distinct gain events. Furthermore, the first gain event (valley) occurs earlier for larger pulse energies. This is due to the higher Rabi frequency, , of the more intense pulses.
Figure 3 shows similar results but for a lower bias – 150 mA. Once more, the systematic input pulse energy dependent measurements show clearly that the system evolves from simple saturation to one that is governed by a complete flopping of the populations. Comparing the intensity profiles of Fig. 3 to those of Fig. 2 reveals that the second peak exhibits a lower intensity than the leading peak. Recalling that the pulse breakup is caused by the alternating sign of the gain due to the population flopping, the appearance of the second peak at lower relative intensity is due to the fact that the pulse develops a smaller area upon propagation and hence the second gain event occurs at a later time, and the Rabi cycle is incomplete. The instantaneous frequency traces of Fig. 3 exhibit once more the expected profiles with clear signatures of almost two complete Rabi oscillations.
The coherent light-matter interactions were also examined in the absorption regime with the results shown in Fig. 4. Here the initial occupation probability of the electrons is higher in the valence band state than in the conduction band state. This means that a complete flopping of the populations transforms the medium into gain conditions. An interaction of this kind is called self-induced transparency (SIT) and was observed for the first time in a room temperature semiconductor optical amplifier in . Propagation in the SIT regime causes pulse compression since its central portion experiences gain while the leading and trailing edges are absorbed. The amplitude traces of Fig. 4 reveal that the pulses are somewhat wider than the input pulse. This is due to a competing effect: two-photon absorption (TPA). Since TPA is a highly nonlinear process, it affects more the peak of the pulse and less its wings, causing some pulse broadening which opposes the compression due to SIT. Nevertheless, comparing the pulse widths of the three traces in Fig. 4 we note that the pulses do compress as the input power increases. Moreover, they are clearly narrower than the ones under gain conditions (Figs. 2 and 3) proving that significant pulse shortening is caused by the state of the material.
Extracting information from the instantaneous frequency under the absorption conditions is more difficult since the lack of excited carriers makes their effect on the refractive index less apparent. Additional processes such as TPA play a significant role in determining the index. This was studied thoroughly by Zilkie  where it was shown that under absorbing conditions, changes in the refractive index due to changes in carrier density (known as the alpha parameter) have the opposite sign compared to the gain regime. The measurements of the instantaneous frequency profiles in Fig. 4 show a valley, similar to the gain regime, are consistent with the principles laid out in .
3. Numerical investigation
The observed signatures of coherent light matter interactions were confirmed by a numerical model, which solves simultaneously Maxwell and Schrödinger equations. The experiments in  were confirmed using a homogeneous model [1,3], which assumed the laser-amplifier to comprise a series of two-level systems having a single effective transition energy. The two-level systems were fed through incoherent relaxations from a high-energy carrier reservoir and the assumed coherence time was uniform. While that model [1,3] contained several simplifying assumptions, it successfully reconstructed the measured complex electric field proving the existence of Rabi oscillations and self-induced transparency . The model also converged to the classical limit, describing, for example, the build-up  of Beer's law for a step-function excitation of an optical amplifier, or reconstructing four-wave mixing in a QD laser .
The homogeneous model [1,3] was extended now to account for the inhomogeneous nature of the gain broadening that stems from size distribution of the self-assembled QDs. This extension offers a more realistic model of the medium and leads also to better insights into the interaction between the broadband pulse and many, relatively narrow-banded, components across the gain spectrum of the device. The inhomogeneous gain spectrum comprises a distribution of resonant transitions having the same homogeneous linewidths as depicted schematically in Fig. 5(a). These two-level systems do not interact directly, but only via capture and escape processes (whose rates are dictated by the principle of detailed balance ) to and from a common carrier reservoir that feeds them, as shown schematically in Fig. 5(b). The valance band of the QDs comprises many closely separated energy levels which are strongly coupled to each other and are therefore considered as a single state.
The density of dots with a particular transition frequency is distributed according to
The excitation level under the electrical drive used in all the experiments we report was such that no emission was detected from any level other than the ground state. This is consistent with recent photoluminescence measurements  which show excited state emission in similar QDs only under very high excitation levels. Therefore, the present model assumes that the QDs comprise only one bound state.
The incoherent relaxations in the model are introduced by a set of rate equations for the electron and hole populations of the reservoir (and) and for the occupation probabilities in the conduction and valence bands of each QD type, noted by or, respectively, as in the density-matrix formalism:
The stimulated emission term, originates form the Schrödinger equation of the two-level systems under the dipole approximation for the interaction with the electromagnetic field. is the electric field, the dipole moment of the QDs and Planck's factor. and are the coherence terms (off-diagonal) in the density-matrix formalism. The part of the Schrödinger equation describing the evolution of the coherence terms is:
With the inhomogeneously broadened model at hand, it is possible to explain why the homogeneous model [1,3] (with single resonant frequency and Rabi oscillation period) can reconstruct the interaction of a short pulse (having a broad spectrum, represented schematically in Fig. 5(a)) with an inhomogeneously broadened medium.
We recall that a 150 fs input pulse has a spectral width of about 25 nm (13 meV) and hence it interacts resonantly with several separate resonances whose room temperature homogeneous linewidth is about 8 nm (4 meV) . The inhomogeneous model calculates the occupation probability of each level in each sub-system. Figure 6(a) describes the corresponding population inversion during the interaction with the pulse and across the gain spectrum. The interaction acts as if in resonance with that part of the gain, which overlaps the pulse spectrum. Only a few Rabi flops take place, and their periods are rather uniform across the interacting spectrum of sub-levels. The resultant overall calculated signature (on the pulse amplitude and instantaneous frequency profiles), depicted by the blue curves in Fig. 7, exhibits a double peaked amplitude profile and correspondingly two minima in the chirp profile. These resemble well the signatures obtained with the homogeneous model [1,3] (under the condition that its homogeneous linewidth is taken to be roughly equal the spectral width of the pulse). This plot also reproduces qualitatively the measurements described in Figs. 2 and 3.
In contrast, when simulating a longer, 500 fs input pulse (whose spectral width is only 8 nm, 4 meV) having the same peak power, such that the Rabi period at resonance is the same, the occupation probabilities undergo more oscillation cycles during the pulse and a change in oscillation periods is clearly observed for off-resonant sub-levels, as depicted in Fig. 6(b). The latter observation is consistent with the expectations of the fundamental theory . The overall response is generated in this case by a superposition of various contributions whose frequencies are sufficiently different so that their interference during the long, 500 fs, pulse duration smears the oscillation patterns imprinted on both the amplitude and instantaneous frequency profiles as seen in the green traces of Fig. 7.
Stating this differently, pulses whose durations approach the inverse of the inhomogeneously broadened spectral width (usually termed T2*), overlap with most of the gain spectrum, thus interacting resonantly with an effectively two-level-like system. For long pulses however, the Rabi oscillations decay after T2* due to the destructive interference of the different polarizations which originate from different transitions, and the overall oscillatory nature of the response is destroyed. This de-phasing is another manifestation of convergence from the semi-classical description (where coherent effects dominate) to the classical saturation regime.
This work presents a direct observation of room temperature coherent light matter interactions in an InAs/InP QD amplifier operating near 1550 nm. This may have a major impact on the practical implementation of many proposed quantum effects in communication, processing, storage and sensing. The interactions are expressed as Rabi oscillations and self-induced transparency imprinted on a 150 fs long pulse after propagating along the amplifier. The observations resemble earlier results in QDash amplifiers. We also present a numerical Maxwell-Schrödinger model, which extends the one in , to account for the inhomogeneously broadened nature of the resonant medium. This new model is used to confirm the experimental signature of Rabi oscillations and also to clarify the relationship between the observability of coherent interactions in an inhomogeneously broadened gain spectrum and the pulse width. It asserts that for a pulse whose spectral width is close to the inhomogeneous linewidth, the signature of Rabi oscillations may be regarded as that obtained from an effectively homogeneously broadened gain medium. In contrast, for long pulses, the overall response comprises different contributions from various spectral regions, each having a somewhat different Rabi frequency. These last for a sufficiently long time so that the overall response is smeared. Its exact nature can only be determined by considering the inhomogeneity of the gain broadening.
This work was partially supported by the Israel Science Foundation. Ouri Karni acknowledges financial support of the Gutwirth Foundation. Amir Capua thanks the Wolf and Clore foundations for their financial support. The material growth and device fabrication was partially supported by the EU project “DeLight” and the Marie-Curie Project “Mitepho”. The technical assistance by Anna Rippien and Florian Schnabel is acknowledged.
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