1. Introduction

Coherent interaction of radiation with matter is at the very heart of quantum optics. The most prominent example is the periodic exchange of energy between a two-level system and the photon field, so-called Rabi oscillations [1]. Exploiting the effects of coherence is an important step in photonic devices and might enable optical data storage and solid state quantum computing [2]. However, the ideal two-level system, in general, represents a too simplified model of real world solid state systems.

To study light-matter interactions beyond the textbook model, two-dimensional semiconductor heterostructures in combination with terahertz time-domain spectroscopy (THz TDS) may serve as an ideal experimental test bed. The number of quantized energy levels, their spacing, and the transition dipole moments can be engineered at the structure growth stage, giving full control over the atomic system [3]. In addition, the coherent generation and detection of single-cycle pulses of electromagnetic radiation enables the simultaneous measurement of changes in amplitude and phase induced by the sample under study [4]. Thereby, the photons have an energy in the range of a few tens of meV corresponding to typical subband spacings in semiconductor quantum wells, making THz TDS the ideal tool for investigating intersubband carrier dynamics [5]. Unlike optical pump-probe techniques, THz pulses do not generate additional electron-hole pairs and, thus, allow the undistorted characterization of the electronic quantum state. In addition, the coherent polarization, which is connected to the off-diagonal elements of the density matrix, can be probed directly without relying on population transfer. For instance, THz TDS has been successfully applied to the phase-resolved study of gain dynamics in THz quantum cascade lasers and the observation of gain clamping [6].

So far, the spectroscopy of intersubband excitations in quantum wells subject to intense THz irradiation has been mainly limited to narrow-band pulses from free-electron lasers [7, 8], or has been based on optical probing of interband transitions [9–12]. The direct observation of intersubband dynamics using few-cycle pulses has been either restricted to the linear regime [5, 6, 13, 14] or has been carried out at higher frequencies in the mid-infrared region [15–18]. Thereby, the THz spectral range offers unique advantages. For instance, the vacuum Rabi splitting can become a significant fraction of the intersubband transition frequency, giving rise to intersubband polaritons and the so-called ultrastrong coupling regime [18, 19].

In this Letter, we demonstrate the direct observation of non-equilibrium intersubband dynamics in a modulation-doped multiple quantum well (MQW) sample induced by intense, broadband terahertz pulses. The THz field dependent transmission of the intersubband system is probed using a depletion-modulation technique, which allows to selectively measure the nonlinear interaction of the THz pulses and the electrons in the quantum well. Of special interest is thereby the multiple octave spanning bandwidth of the used single-cycle THz pulses, which allows the phase-locked coupling of adjacent intersubband transitions. In more detail, we discuss the efficient, coherent population transfer from the ground state to higher lying levels of the quantum well, the THz induced undressing of intersubband plasmons, and the THz Stark effect.

2. Experiment

The experimental setup is based on an Er-doped fiber laser (Menlo Systems) that provides both the seed pulses for a regenerative Ti:Sapphire amplifier (Spectra Physics), operating at a center wavelength of 780 nm with a repetition rate of 1 kHz, and probe pulses for electro-optic detection of the THz transients at 1560 nm. For detection of the THz transients, a (110)-cut 300 μm thick GaAs crystal is used, which offers a detection bandwidth up to 6 THz. By using a fast digitizer (National Instruments), the probe pulses are directly sampled at the repetition rate of the fiber laser (80 MHz), which makes the use of an optical chopper unnecessary. This scheme thereby improves the noise characteristics (a SNR of 1:2000 is routinely achieved) and the versatility of the THz TDS spectrometer as compared to the case where the same amplified pulse is used for generation and detection of the THz transients. The setup is similar to the setup presented by Sell et al. in Ref. [20].

The single-cycle THz pulses are generated by optical rectification in a 400 μm thick (110)-cut GaP crystal [21]. At the GaP crystal, the pump pulses have a duration of 130 fs with a maximum pulse energy of 3.3 mJ. The generated THz pulses have a bandwidth of 5.5 THz with the maximum peak electric field incident on the sample being estimated to 20 kV/cm. The electric field strength is determined from the electro-optic modulation signal without sample, thereby taking into account the Fresnel transmission coefficient of the silicon windows of the optical cryostat.

For the experiments, the single-cycle THz pulses are tightly focused on the cleaved facet of the sample (see Fig. 1(a)), which serves also as THz waveguide. The THz electric field is oriented parallel to the growth direction to comply with the polarization selection rules for inter-subband transitions [3]. To block any bypassing THz light, the sample is mounted on a metallic holder with a narrow aperture. The multiple quantum well structure consists of ten periods of symmetrically modulation doped, 52 nm wide GaAs wells separated by 160 nm Al_0.3Ga_0.7As barriers grown on a semi-insulating GaAs substrate. For electrically contacting the wells, the sample has an aluminum Schottky gate on the surface and Au / Ge alloy Ohmic contacts to the quantum wells. From capacitance-voltage measurements, the areal density of carriers per well is estimated to n_2d = 2.55 × 10¹⁰ cm⁻². The sample is mounted in an optical cryostat and is kept at 5 K throughout the entire experiment. By applying a negative bias voltage of −10 V between the Schottky gate and the Ohmic contacts, the wells can be entirely depleted (see Fig. 1(b)). This allows us to use an electrical modulation technique to selectively measure the effect of the electronic polarization on the transmitted THz pulses [5]. To this end, we apply a 500 Hz square signal phase locked to the 1 kHz THz pulses. Thus, every second THz pulse probes the intersubband excitation. The black curve in Fig. 1(c) shows the reference electric field E_ref of the transmitted THz pulse after passing through the depleted sample. Approximately 10% of the incident peak electric field are detected after the sample. The THz pulse has undergone significant dispersion due to the frequency dependent refractive index of the 500 μm thick GaAs waveguide. The measured modulation signal ΔE = E_{zero bias} − E_ref makes almost 12% of the reference electric field (red dashed curve). The achievable signal to noise ratio is still above 100:1.

 

Fig. 1 Overview of experiment. (a) The MQW sample is mounted on a copper holder with aperture. (b) By applying a negative bias to the Schottky contact, the quantum wells are depleted. (c) Transmitted (black, solid) and modulation signal (red, dashed) for weak THz fields. The two transients are to scale. (d) Sketch of the lowest four levels of the QW. The spectrum of the incident pulse spans the ground and first two excited levels. Black arrows indicate allowed transitions, red dashed arrows show a 2-photon transition.

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Figure 1(d) shows the level scheme of the quantum wells. The transition frequencies have been calculated self-consistently including the depolarization shift [3]. At 5 K, all electrons are assumed to be in the ground state. Thus, with the bandwidth of the driving pulse, indicated by the blue bar, only the first three states can be accessed. The pump spectrum peaks around 1.4 THz, which is close to the first transition |1〉 → |2〉. For sufficiently high electric field strengths, Rabi oscillations between the ground and first excited state will lead to a level splitting, which can be probed by the next higher transition |2〉 → |3〉 (Autler-Townes effect). Furthermore, the direct transition |1〉 → |3〉 is dipole forbidden, but state |3〉 can be populated via two interfering pathways, either through the intermediate state |2〉 or directly by a 2-photon resonant transition (indicated by red arrows). This quantum interference might lead to an enhanced transmission similar to electromagnetically induced transparency [15, 22]. In addition, the absorption strength of each transition is dependent on the relative occupation numbers of the involved levels. Thus, for strong driving, we expect to observe saturation of the intersubband transitions.

3. Results

The expected non-linear optical effects in the multiple quantum well structure are strongly dependent on the electric field amplitude of the THz driving pulse. Figure 2 shows the measured modulation signal, ΔE, for different values of the pump power incident on the GaP crystal. At the lowest pump energy of 125 μJ, the incident THz peak field is estimated to be 0.8 kV/cm. The modulation signal shown in Fig. 2(a) corresponds to the expected free-induction decay of a weakly driven two-level system (black curve) [6], corresponding to the |1〉 → |2〉 transition. The oscillations are monochromatic with a center frequency of 1.5 THz and a dephasing time of 2.3 ps. When the THz amplitude is increased, a beating pattern gradually emerges, indicated by the black arrows in Fig. 2(b). This beating is a clear indication for a second frequency component in the spectrum. We attribute this to increased occupation of the first excited level, which activates the next transition, |2〉 → |3〉. In addition, the modulation signal decays faster (≈ 1.5ps), which indicates an additional dephasing mechanism compared to the simple two-level model. Around 6.8 kV/cm (Fig. 2(c)), the beating pattern gets more pronounced and resembles the photon echoes produced by the free-induction decay in water molecules, for example [23]. This pulse-like structure is an indication for a broadened modulation spectrum which contains many frequency components. The character of the modulation signal changes again for very high pump powers (Fig. 2(d)). Both the oscillation period (≈ 2.4THz) and the decay time (≈ 1.7ps) are much faster than in Fig. 2(a).

 

Fig. 2 Modulation signals for different pump powers. The insets show the normalized modulation spectra with frequency in THz. (a) For the lowest THz field amplitude, the free induction decay is almost monochromatic (red). The black dashed curve shows the results of FDTD simulations for a two-level system. (b) For 4.5 kV/cm, a beating becomes obvious. The temporal position of the beat nodes are indicated by the arrows. (c) For higher peak fields, the total pulse becomes shorter and shows echo-like features. (d) Above 12 kV/cm, the free induction decay contains higher frequencies.

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Further insight into the field dependence of the intersubband excitations can be gained by calculating the (single-pass) absorption spectra according to α(ω) ∝ Im(−iΔE(ω)/E_ref), which is valid under the assumption, that the depleted sample is non-absorbing and ΔE(ω) ≪ E_ref [24]. Figure 3(a) shows the absorption spectra for increasing values of the THz peak electric field from 0.8 kV/cm to 20 kV/cm (from bottom to top). Apart from a small peak at 2.4 THz, there is only one significant feature around 1.5 THz for the lowest driving fields, which corresponds to the fundamental transition |1〉 → |2〉. Thus, essentially all electrons can be assumed to be initially in the ground state. For increasing pump field strengths, electrons are efficiently transferred to higher lying states and the higher transitions start to appear in the spectra: the |2〉 → |3〉 transition around 2.4 THz and the |3〉 → |4〉 transition at 3.3 THz. The observed behavior is consistent with the scenario of sequential pumping of electrons from the ground state to higher excited states by the broadband THz pulse. We note at this point, that the absence of signatures of Rabi oscillations both in the THz transients, as well as in the absorption spectra, can be attributed most probably to the broadband nature of the driving pulse, which acts as an additional dephasing mechanism.

 

Fig. 3 (a) Absorption spectra for increasing THz peak electric fields (from bottom to top). The lowest three transitions, |1〉 → |2〉 (A), |2〉 → |3〉 (B), and |3〉 → |4〉 (C) are visible. The curves are vertically offset for clarity. (b) Integrated absorbance of the three transitions A, B, and C as function of the THz pulse energy. (c) Fractional occupations of the lowest three quantum well levels as function of the incident THz pulse energy. (d) Comparison of measured and calculated frequency shift of the fundamental transition as function of the THz pulse energy.

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Figure 3(b) shows the integrated absorbance of the three lowest transitions as function of the normalized THz pulse energy. The pulse energy has been determined by temporally integrating the squared reference transient and has been normalized to the maximum pulse energy used in the experiment (corresponding to a peak field of 20 kV/cm). The intersubband absorption amplitudes show a clear dependence on the pump power. The decrease of the absorption peak associated to the |1〉 → |2〉 transition is a sign for THz induced saturation of the intersubband transition. The absorption strength of the |2〉 → |3〉 transition appears only for higher pulse energies and shows a clear maximum. Finally, the third transition gradually appears. This is a clear indication for an efficient transfer of population from the ground state to higher lying levels within the THz pulse duration. Thereby, the pulse duration is shorter or on the order of the dephasing times of the subband transitions. These have been determined to τ₁₂ = 2ps, τ₂₃ = 1ps, and τ₃₄ = 1.3ps, by comparison of our data (Fig. 2) to results of one-dimensional finite-difference time-domain simulations [25]. Thus, the population transfer is at least partially coherent. Figure 3(c) shows the energy dependent level occupations extracted from the relative absorption strengths of the three lowest transitions. Thereby, we made use of α_ij ∝ n_2d(N_i − N_j) f_ijτ_ij with f_ij being the oscillator strength of the transition [26], and the assumption that the electrons are only efficiently transferred up to the third level (N₁ + N₂ + N₃ = 1). The ground state population is efficiently transferred to the second excited level and shows the typical two-level saturation behavior. For higher pulse energies, also the third level becomes populated with a maximum fractional occupation of over 8%.

In addition to the efficient population transfer, the frequency of the fundamental transition is shifted to lower values for increasing amplitude of the driving field. This effect has been predicted theoretically to be related to an undressing of the collective intersubband excitation which is causing the depolarization shift of the resonance frequency [27]. Till now it has only been observed experimentally using intense narrow-band THz radiation from a free electron laser with peak powers on the order of 1 kW [8]. The dark-blue triangles in Fig. 3(d) show the measured absorption frequency associated to the |1〉 → |2〉 transition. For increasing THz pulse energy, the transition is red shifted and shows a clear minimum. However, for higher pulse energies, the frequency is increasing linearly. To the best of our knowledge, this behavior has not been reported so far.

According to Ref. [26], the depolarization shifted transition frequency is given by ${\tilde{\omega }}_{12}={\omega }_{12}\sqrt{1+\alpha }$, where α ∝ (N₁ − N₂). Due to the efficient population transfer, N₁ − N₂ is reduced, which leads to the undressing of the collective excitations [7]. The green circles in Fig. 3(d) represent the calculated frequency shift using the level occupations from Fig. 3(c). For low THz fields, the transition frequency is red shifted, which corresponds to the experimental data (dark blue triangles) and to the results reported in Refs. [7,8]. However, this simple model fails to explain the additional linear blue shift that is observed experimentally for higher THz fields.

In the following, we argue that this behavior can be explained by taking into account the Stark shift induced by the THz electric field propagating in the waveguide, which leads to ${\tilde{\omega }}_{12}=\mathrm{\Delta }\omega \left(E_2\right)+{\omega }_{12}\sqrt{1+\alpha }$. In a single infinite quantum well, the transition frequency from the ground to the first excited subband scales to lowest order as $\mathrm{\Delta }\omega \left(E_z\right)\propto E_z^2$, where E_z is the electric field applied perpendicular to the quantum well plane [28]. We found an excellent agreement between the theoretical and experimental curves assuming that for the highest THz pulse energy, an effective dc electric field of E_z = 1.7kV/cm is acting on the quantum well (shown as light-blue triangles in Fig. 3(d)). This value corresponds to the electric field associated with the low-frequency part of the THz pulses with frequencies below the transition frequency (< 1.5THz). As the low frequency part of the spectrum is both phase-locked and co-propagating with the 1.5 THz photons, which are probing the intersubband transition, the THz pulse is effectively probing the quantum well subject to an effective dc electric field. Thus, the appearance of the THz Stark shift is a consequence of the large bandwidth of the single-cycle THz pulses and cannot be observed in experiments using narrow-band excitation from a free-electron laser, for example. Further insight into this phenomenon could be gained by FDTD simulations that take into account the modification of the electronic wave functions in the quantum wells subject to the local electric field of the THz pulses, and by additional experiments with frequency shaped THz pulses. These will be subject to future work.

4. Conclusion

In conclusion, we have demonstrated the direct observation of non-equilibrium dynamics of intersubband transitions subject to intense few-cycle terahertz pulses. Using a table-top experiment, we could demonstrate a plurality of effects which have been accessed so far only with single frequency excitation using free electron lasers. Thereby, we observed the coherent transfer of electrons within the THz pulse duration up to the third excited level. This fast population transfer could also be used to explain the observed red shift of the fundamental transition, an effect known as the undressing of the collective excitation of the intersubband plasmon resonance. In addition, we have been able to observe an additional blue shift, which could be associated to a Stark shift of the first two quantum well levels induced by the THz electric field itself.