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Abstract
We report the observation of the intersubband AC-Stark effect in a single wide GaAs/AlGaAs quantum well. In a three-level configuration, the n = 2 to n = 3 intersubband transition is resonantly pumped at 3.5 THz using a free-electron laser. The induced spectral changes are probed using THz time-domain spectroscopy with a broadband pulse extending up to 4 THz. We observe an Autler-Townes splitting at the 1 − 2 intersubband transition as well as an indication of a Mollow triplet at the 2 − 3 transition, both evidencing the dressed states. For longer delay times, a relaxation of the hot-electron system with a time constant of around 420 ps is measured.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
A quantum mechanical two-level system resonantly driven by a strong periodic electric field gives rise to new coupled electron-photon or dressed states. In this non-perturbative regime of light-matter interaction both upper and lower state are split by the Rabi frequency $\Omega _R$, a measure of the coupling strength in the system (see Fig. 1(a)).
Here $\vec {\mu }_{ij}$ is the electric dipole moment of the transition between levels $i$ and $j$, and $\vec {E}(t)$ the coupling electric field. If the Rabi frequency is larger than the line broadening, one can observe this splitting, then called Rabi splitting or AC-Stark splitting [1,2]. We note that this has to be distinguished from the vacuum Rabi splitting [3], which occurs in the interaction of an electronic transition with a high-quality optical cavity without driving field. One route to an experimental observation is to use a second, weaker probe beam monitoring a transition to a third level. Now, depending on the exact way of pumping and probing the possible transitions in such a three-level system and on the relative size of the Rabi frequency and relaxation rates, one can observe the Autler-Townes (AT) splitting or quantum interference effects such as electromagnetically induced transparency (EIT) [4,5] or gain without inversion [6]. All these effects were demonstrated decades ago in atomic systems [7–12]. In solids, and in particular in semiconductors, investigating the AC-Stark effect is much more challenging due to the short (usually sub-ps) scattering rates, yet observations of such phenomena have been reported in quantum wells mainly in relation to excitonic (optical AC-Stark effect) [13–16] and intraexcitonic transitions (intraexcitonic AC-Stark effect) [17–20]. Quantum dots, due to their atomic-like energy levels, have been widely studied as well [21–23]. But especially interesting are intersubband transitions (ISBT) in engineered quantum well (QW) systems, which are also the basis of quantum cascade lasers, due to their atomic-like ($\delta$-function) joint density of states. Using such systems, there exist few reports related to the above effects. These experiments have been performed either by probing a THz transition in the near infrared (across the band gap) [24] or in the mid infrared around $(100\ldots 200)\,\textrm {meV}$[6,25,26]. Our investigation of the AC-Stark effect in solids may lead to new approaches in a similar way as developed for atoms, e.g. a deeper understanding of coherent control [27], new imaging techniques [28] or electric field measurement techniques [29,30].
Fig. 1. AC-Stark effect. Panel (a) depicts a two-level system interacting with a light field. The light-matter coupling lifts the degeneracy of the coupling between photon and two-level system and reveals the split dressed states with an energy difference proportional to the Rabi frequency $\Omega _R$. For a three-level ladder system including an additional weak electric probe field, one finds the (b) EIT-ladder and the (c) AT-ladder configuration. Here, the dressed-state splitting has been omitted in (c) for the sake of simplicity. A weak probe electric field can probe the Autler-Townes doublet between a pair of split states and the unperturbed state, while probing the dressed states themselves gives rise to a Mollow triplet.
As a particular challenge, we explore the possibility to observe these effects exclusively in the THz range. The motivation for this is threefold: (i) In the THz range the typical transition energies are below the longitudinal optical (LO) phonon energy in most III-V semiconductors, so we can expect slower relaxation rates [31–33]. (ii) Time-domain THz spectroscopy provides the possibility to probe the induced spectral changes by a spectrally broadband beam. In combination with a THz free-electron laser as a pump source, this allows for an archetypical narrowband-pump–broadband-probe configuration. Here the THz free-electron laser has in particular a much narrower bandwidth (relative bandwidth $\Delta \lambda /\lambda \approx (1\ldots 2)\,\%$) than typical high-field THz sources based on table-top lasers above $1\,\textrm {TH}_{\textrm {z}}$ [34,35]. (iii) With strong enough driving fields one can, in principle, reach a highly non-perturbative regime where the Rabi energy approaches the transition energy and thus the rotating wave approximation (RWA) is not valid anymore.
In our experiment, we use a single, modulation-doped wide GaAs/AlGaAs quantum well with three subbands lying below the LO phonon energy. The use of a single well eliminates thickness fluctuations usually present in multi-quantum well systems [36,37] and reduces related scattering effects, yet the magnitude of the intersubband absorption is accordingly smaller. We pump the 2-3 transition resonantly and probe the resulting changes in the absorption spectrum. This is the so-called EIT-ladder configuration [5,38] (see Fig. 1(b)), which, in principle, allows for the observation of EIT, if the characteristic time constants fulfill certain conditions. We note that pumping the 1-2 transition and probing the 2-3 transition would correspond to the so-called AT-ladder configuration [5,38] (see Fig. 1(c)), where EIT cannot occur even theoretically [5,39,40].
This work reports about the intersubband AC-Stark effect in an EIT-ladder configuration, where we pump the 2-3 transition at the frequency $\omega _{23}$ in the non-perturbative, but still weak-to-intermediate coupling regime ($\Omega _R \ll \omega _{23}$). Probing the 1-2 transition, we are able to observe the Autler-Townes splitting of the $n=2$ state, while probing near the 2-3 transition, we even see indications of a Mollow triplet in absorption of the driven transition (see Fig. 1(b)). In solids such a triplet has been reported only a few times up to now, whether in resonance fluorescence [41,42] or absorption [23,43] and it is still an interesting feature of the AC-Stark splitting to study. As a side-effect of the present experiment we determine an energy relaxation time of the heated electron distribution around $400\,\textrm {ps}$, limited by acoustic phonon scattering.
2. Experimental details and methods
2.1 Sample: a modulation-doped, wide single GaAs quantum well
Our GaAs/Al$_x$Ga$_{1-x}$ As sample was grown by molecular beam epitaxy on a semi-insulating GaAs substrate with a composition of $x=0.2$ confirmed by energy-dispersive X-ray spectroscopy (not shown here). The $42\,\textrm {nm}$ wide single GaAs QW was modulation-doped by means of two Si $\delta$-doping layers within the AlGaAs barriers. Schottky and ohmic contacts were fabricated to provide a full control of the electron density in the QW by externally applying a gate voltage. A design optimized for modifying electrically the electron density resulted in asymmetric doping layers with respect to the QW center; for the layer structure and conduction band diagram see Fig. 2. From an extra sample with electrical contacts in a Van-der-Pauw geometry, we determined an electron density of about $2 \cdot 10^{11}\,\textrm {cm}^{-2}$ in the QW using magneto-transport measurements. Here, we applied a gate voltage such that the QW shape is expected to be close to the flat-band condition. For a comparison with calculated electron densities we refer to Appendix A.
Fig. 2. Single GaAs/AlGaAs quantum well sample structure. Panel (a) shows the cross-sectional sample structure imaged with a transmission electron microscope. The different contrasts show the GaAs and the AlGaAs layers, which are labeled with their thicknesses for the sake of clarity. With a very weak contrast difference, the positions of the $\delta$-doping layers are visible, which result from different growth temperatures in this region (Si below the detection limit of energy-dispersive X-ray spectroscopy). Panel (b) shows bandstructure simulations performed with the nextnano3 software. In this simulation, a gate voltage of $-0.4\,\textrm {V}$ is applied to the gate-contact, while the ohmic contact is set to ground. Hence, under this flat-band condition, the quantum well is filled with electrons and has a symmetric shape. Within these simulations, we assume a simple model for the chemical potential (Fermi-level $\mathcal{E} _F$), which is constant between the ohmic contact and the doping layer close to the gate and then changing linearly towards the gate contact.
Bandstructure simulations (nextnano3 [44,45]) of our sample show 7 confined subbands in the QW potential, where at most up to 2 subbands are occupied considering the carrier density at $10\,\textrm {K}$ in our experiments. For improvements of the THz transmission at the sample interfaces, we additionally polished the side facets of the sample in a $45^{\circ }$-waveguide geometry such that THz pulses undergo two total internal reflections on the front surface, corresponding to four passes through the QW (see Fig. 3(a)).
Fig. 3. Experimental details. Since our measurement is performed in transmission, we guide pump and probe pulses through our sample polished in a $45^{\circ }$-waveguide geometry (a). In our pump-probe experiment (b), strong pump pulses resonant to the second ISBT induce a splitting of states with a separation proportional to the Rabi frequency $\Omega _R$. Our broadband probe pulses cover a frequency range between $0.5\,\textrm {TH}_{\textrm {z}}$ and $4\,\textrm {TH}_{\textrm {z}}$ such that we probe both split states simultaneously. Panel (c) depicts the 2D absorption spectrum of our GaAs/AlGaAs quantum well modulated between a gate voltage of $-2\,\textrm {V}$ (depletion) and $-0.5\,\textrm {V}$ (population). We observe absorption lines related to the first ISBT at $2.25\,\textrm {TH}_{\textrm {z}}$ ($9\,\textrm {meV}$) and the second ISBT at $3.5\,\textrm {TH}_{\textrm {z}}$ ($14.5\,\textrm {meV}$). The gray shaded area in the background represents the probe transmission spectrum through the depleted QW sample and a slit aperture. The absorption measurement has been acquired at $10\,\textrm {K}$.
2.2 THz time-domain spectroscopy
Weak THz-probe pulses are generated in a $400\,\mathrm{\mu}\textrm {m}$ thick GaP crystal by optical rectification utilizing strong $50\,\textrm {fs}$-short near-infrared (NIR) pulses from a Ti:Sapphire laser-amplifier system at a central wavelength of $800\,\textrm {nm}$ with a repetition rate of 250 kHz and pulse energies of up to $6\,\mathrm{\mu}\textrm {J}$. The probe pulses are detected using electro-optic sampling in a $300\,\mathrm{\mu}\textrm {m}$ thick GaP crystal and a balanced photodetector scheme [46]. There, the THz field is sampled by changing the probe delay time between a sampling NIR beam and the THz field (not shown in experimental scheme Fig. 3(b)). With the calculated electric field amplitude at the detector position [47], we estimate the peak THz-field at the sample position to be $\leq 150\,\textrm {V}\cdot \textrm {cm}^{-1}$ with a spot size of about $500\,\mathrm{\mu}\textrm {m}$ full width at half maximum (FWHM). Our experimental setup allows for studying our sample with broadband THz pulses, whereby after the sample transmission, we observe frequencies ranging from $0.5\,\textrm {TH}_{\textrm {z}}$ to $4\,\textrm {TH}_{\textrm {z}}$. In order to determine the absorption spectrum of our QW, we measure two different signals. The first signal is the electrically gated transmission $\Delta E_{\textrm {THz}}(t) = \left [E_{\textrm {T}}(t) - E_0(t)\right ]$ through the sample. Here, we modulate electrically the electron density inside the QW by applying an AC-bias (from $U_{\textrm {depl}}$ to $U_{\textrm {high}}$) of $971\,\textrm {H}_{\textrm {z}}$ to the sample (see Fig. 3(b)). Capacitance-voltage measurements reveal that the charge carriers equilibrate within the chosen AC modulation frequency. Hence, $\Delta E_{\textrm {THz}}(t)$ is then the difference between the THz transmission of the QW in the depleted state ($E_0(t)$) and the filled state ($E_{\textrm {T}}(t)$). The second signal is the average THz transmission through the sample between depleted and filled state $\left [ E_0(t) + E_{\textrm {T}}(t) \right ]/2$. We measure here an average signal, because the THz transmission is mechanically modulated with a chopper at a frequency of 771 Hz, while the electrical modulation is still applied. Both THz traces are then acquired simultaneously using two lock-in amplifiers such that we synchronize the chopper’s modulation frequency and the sample’s AC-bias modulation frequency to avoid any phase shift, which would be typical in sequential measurements. Hence, we are able to calculate the 2D electrical conductivity $\sigma ^{\textrm {2D}}$[48,49], where the real part is directly proportional to the 2D absorption $\alpha _{2D}$.
Without strong pump pulses and applying only the broadband pulses of the THz time-domain setup, we are able to measure the linear absorption of our wide QW. Figure 3(c) shows the absorption spectrum, where transitions between the first and second subband at $\nu _{12} = 2.25\,\textrm {TH}_{\textrm {z}}$ and the second and third subband at about $\nu _{23} = 3.5\,\textrm {TH}_{\textrm {z}}$ are evidenced. The absorption spectrum is acquired slightly off the flat-band condition in order to have a stronger absorption amplitude but still avoiding a high population of the second subband. The electric dipole matrix elements of both ISBTs are $\vert \vec {\mu }_{12}\vert =$ 89 eÅ and $\vert \vec {\mu }_{23}\vert =$ 85 eÅ, respectively, which we determine with the help of numeric simulations utilizing the software nextnano3. Furthermore, these simulations confirm the absorption frequency of the second ISBT, while for the first ISBT, a resonance at $1.9\,\textrm {TH}_{\textrm {z}}$ is predicted. The clear mismatch arises from the fact that the simulation neglects collective effects caused by the relatively high electron density. In our experiment the important collective effect is the depolarization shift [50,51], which is particularly strong for low transition energies. Here, the single-electron interaction with the THz radiation is screened by an ensemble of electrons causing a collective excitation, which requires more energy for the ISBT. A detailed characterization of this effect is essential to understand the pump-probe measurements and a further description as well as simulations are discussed in Appendix A. All measurements are performed at approximately $10\,\textrm {K}$, where our sample is cooled inside a liquid He-flow cryostat.
2.3 All-THz pump-probe spectroscopy setup
Our all-THz pump-probe setup combines a table-top THz time-domain setup as described above with the free-electron laser (FEL) FELBE (Free-electron laser at the Electron Linear accelerator with high Brilliance and low Emittance) at the Helmholtz-Zentrum Dresden-Rossendorf (HZDR) (see Fig. 3(b)). The synchronization between the FEL pump and table-top probe pulses is provided by a master clock oscillating at 13 MHz. For further details about the synchronization as well as for both laser systems we refer to Ref. [52]. Strong pump pulses with a duration of $16\,\textrm {ps}$ are generated by FELBE operating at a repetition rate of 13 MHz. The central frequency is tuned to $\nu _{\textrm {pump}} = 3.5\,\textrm {TH}_{\textrm {z}}$ with a narrowband spectrum of about 50 GHz FWHM. Here, the pump frequency $\nu _{\textrm {pump}}$ is set in resonance to the second ISBT $\nu _{23}$ ($\mathcal{E} _{23}$) to induce the AC-Stark splitting (see Fig. 1(b)). The pump beam itself is guided through a delay stage (pump-probe delay time), coupled into the setup by a Si-beamsplitter and focused to and transmitted through the sample, after which it is finally blocked. To control the temporal overlap between pump and probe pulses at the QW position, the pump-probe delay is changed. Hence, we observe an AC-Stark splitting near zero delay, while both pulses are present; a signal observed at much longer delays is a measure of the electron relaxation dynamics (see section 3.3). For a pulse energy of $0.6\,\textrm {nJ}$ and a spot size of about $500\,\mathrm{\mu}\textrm {m}$ FWHM, we estimate a peak electric field amplitude of $\vert \vec {E} \vert _{\textrm {peak}} = 2.2\,\textrm {kV}\cdot \textrm {cm}^{-1}$ at the GaAs QW. Both the Autler-Townes doublet and the Mollow triplet are then probed simultaneously with the broadband THz pulses of our setup.
3. All-THz pump-probe spectroscopy of the AC-Stark effect
With a pump-frequency resonant to the second ISBT, the second and third subbands get dressed by the pump light field and split into two doublets, where the degeneracy is lifted by the coupling energy $\hbar \Omega _R$. Hence, the AC-Stark splitting depends strongly on the electric field amplitude proportional to the Rabi frequency (see Eq. (1)), which is a measure of the coupling strength. In our experiment, we change the pump-probe delay and thus probe an absorption splitting at various temporal positions during the pump pulse. In Fig. 4(a), we depict the 2D absorption spectrum for several pump-probe time delays. We observe spectral dynamics of both ISBTs simultaneously starting with the unperturbed ISBT at about $t_0 = -10\,\textrm {ps}$ relative to the pump-probe time overlap. As we probe the maximum splitting at the peak intensity of the pump pulses (time overlap), we observe a decreased absorption amplitude and a broadening of both absorption lines. Here, the maximum absorption line splitting of the first and the second ISBT is slightly shifted in time. The reason is a strong positive chirp of the broadband probe pulses due to a strong dispersion close to the GaAs Reststrahlenband [53] acquired while passing through the sample. In other words, the phase velocity of higher frequencies is slower than the phase velocity of lower frequencies and the pulse get stretched. Consequently, when moving the probe pulse from negative towards positive pump-probe time delays across the pump pulse, the “late” part of the probe pulse (i.e. the trailing edge containing the higher frequencies) is overlapping with pump pulses at first. When the pump-probe delay time becomes strongly positive and the temporal overlap terminates, then the “early” part of the probe (i.e. the leading edge of the probe containing the lower frequencies) overlaps with the pump finally. For even later pump-probe delays, the absorption strength recovers (also because of population relaxation back to the ground state) and the linewidth of both ISBT gets narrower again.
Fig. 4. All-THz pump-probe measurements with a pump electric field amplitude of $2.2\,\textrm {kV}\cdot \textrm {cm}^{-1}$ in GaAs at the QW and a central frequency at $3.5\,\textrm {TH}_{\textrm {z}}$. Panel (a) depicts probe spectra at different pump-probe delays. For the sake of clarity, we zoom at pump-probe delays (b) $t_2 = 23\,\textrm {ps}$ and (c) $t_1 = -1\,\textrm {ps}$, where a maximum splitting of the Autler-Townes doublet and the Mollow triplet, respectively, is visible with the unperturbed spectrum in the background (grey, before time overlap $t_0 = -10\,\textrm {ps}$). The applied AC-bias for the modulation ranges from $U_{\textrm {depl}} = -2\,\textrm {V}$ (depletion) and $U_{\textrm {high}} = -0.5\,\textrm {V}$ (population). All spectra have been normalized to the maximum absorption of the unperturbed system and have been acquired at $10\,\textrm {K}$.
3.1 Intersubband AC-Stark effect: Autler-Townes doublet
In our experiment, we couple pump and probe pulses according to an EIT-ladder configuration, where we probe the absorption between the first and the second split subband to observe an Autler-Townes doublet. Figure 4(b) depicts a zoom on frequencies around the first ISBT at the pump-probe time overlap $t_2$. The absorption line at $\nu _{12} = 2.25\,\textrm {TH}_{\textrm {z}}$ is split into a doublet at $1.99\,\textrm {TH}_{\textrm {z}}$ and $2.24\,\textrm {TH}_{\textrm {z}}$, while the absorption amplitude is almost symmetrically reduced by about $80\,\%$. The slight asymmetry in amplitude can be caused by an off-resonant light coupling or maybe is even an effect beyond strong coupling [19,54]. However, we determine a Rabi frequency of about $0.25\,\textrm {TH}_{\textrm {z}}$, which is only roughly half the calculated value $\Omega _R / (2\pi ) = 0.47\,\textrm {TH}_{\textrm {z}}$ (Eq. (1) of an ideal system. This theoretical value is an upper limit due to an intrinsic timing jitter of FEL pulses and spatial effects in the experiment. The magnitude of the timing jitter is about $1\,\textrm {ps}$, which adds up in a smeared-out instantaneous peak electric field amplitude at the time overlap. This effective electric field amplitude then causes a smaller spectral splitting. Even more, the spatial pump-probe overlap also limits our experimental resolution, since both beams have approximately the same spot size. As a consequence, we record a spatially averaged splitting of slightly different electric field amplitudes. A desirable much smaller probe focus is difficult to reach in this frequency range, but also a small pump focus would be needed to reach high peak electric fields for inducing the AC-Stark effect. Another spatial and temporal resolution limit of the splitting is due to the multipass waveguide geometry, where pump and probe pulses overlap twice at the QW.
Although we expect a symmetric splitting around $\nu _{12}$ for resonant pumping, we observe a red shift of the doublet. Typically, such a red shift is expected for non-resonant pumping and depends on the pump frequency detuning, where an anti-crossing behaviour of the absorption peak position is observed [17,54]. In such a case, the split absorption lines merge at later pump-probe delays back to the initial absorption frequency of the unperturbed system. In contrast, we observe a persistent red shift for even later pump-probe delay times ($t_4$ in Fig. 4(a)). The most likely explanation is that the red shift is a consequence of an elevated electron temperature induced by the FEL pump pulses. Linear absorption experiments of thermally excited subbands, in which we increased the lattice temperature and consequently the electron temperature of the sample, indicate a similar red shift (not shown here). Indeed, a population redistribution of electrons towards the second subband causes an unscreening of the depolarization shift [51] leading to a red shift (see discussion in Appendix A). The complete absorption spectrum at $t_3$ in Fig. 4(a) supports this interpretation, where we observe a larger absorption amplitude of the second ISBT $\nu _{23}$ compared to the first ISBT $\nu _{12}$. Another indicator is that the absorption amplitude of the second ISBT is even higher at $t_4$ than before the pump-probe overlap at $t_0$.
3.2 Intersubband AC-Stark effect: Mollow triplet
In our experiment, we benefit from our broad probe spectrum such that we resolve both ISBTs simultaneously without the need for a frequency sweep. In fact, we do not only observe an Autler-Townes doublet, but we also observe an indication of a Mollow triplet in absorption as well (depicted in Fig. 4(c)). Here, a symmetric splitting of the second ISBT absorption line into three peaks appears at the pump-probe time overlap. Considering theory, we expect a symmetric splitting by pumping the second ISBT resonantly, but a higher central absorption amplitude as well [55]. However, as a result, we determine a Rabi frequency of $0.3\,\textrm {TH}_{\textrm {z}}$. Here we find a small difference between the Rabi frequency extracted from the Autler-Townes doublet and the Mollow triplet. A plausible explanation is a slight difference in the pump probe delay. It seems that for the Mollow triplet absorption spectrum we reach a better timing between pump pulses and the trailing edge of the chirped probe pulses. As a remark, the shoulders and peaks at the low- and high-frequency edge of the absorption spectrum shown in Fig. 4 are most likely spectral-noise artifacts. The high amplitude may appear due to the division necessary for calculating the absorption spectrum. Several measurements did not show a systematic behavior in amplitude and frequency position of these artifacts.
At last, we want to classify the light-matter coupling regime in our experiment. Zaks et al. [18] determined a ratio of $\Omega _R / ( 2\pi \cdot \nu _{\textrm {pump}}) \approx 50\%$ for their experiments indicating the ultra-strong coupling regime beyond the rotating wave approximation. Here, they observed a red shift as well and assigned it to an effect within this regime. In our case, we are able to explain the additional red shift consistently with an elevated electron temperature and a consequent unscreening of the depolarization shift. Furthermore, we determine a ratio of about $\Omega _R / ( 2\pi \cdot \nu _{23}) = 7\%$, which is much lower as compared to the experiments of Zaks et al.. Hence, we conclude that all measurements are within the rotating wave approximation in the strong coupling regime. Reaching higher fields and thus approaching the ultra-strong coupling regime has proven to be difficult in our experiment; possible reasons are related to the waveguide sample geometry, sample heating and a low signal-to-noise ratio.
3.3 All-THz pump-probe dynamics
In addition to measuring THz absorption spectra of split states in the QW, we can change the pump-probe delay (see Fig. 3(a)) to determine electron relaxation dynamics. In our experiment, the second subband is partially occupied, and hence strong pump pulses induce a heating of the electronic system thermalizing and cooling back to equilibrium.
From the pump-probe signal in the right part of Fig. 5, we can extract two time scales. The short time constant ($16\,\textrm {ps}$) of the initial signal drop is related to the FEL pump pulse width. This is the time window we use for the spectral probing of the coherent AC-Stark effect. This bleaching signal recovers on a much slower time scale with a time constant of $420\,\textrm {ps}$, which must be related to the relaxation dynamics of the electrons. We can learn in fact more from the already mentioned (actually unwanted) chirp, namely that the measured relaxation dynamics correspond to the first ISBT. The induced positive chirp from the Reststrahlenband of GaAs separates the probe frequencies in time such that we observe a $\approx 3\,\textrm {ps}$ time shift between the ISBT absorption frequencies $\nu _{12}$ and $\nu _{23}$. The left part of Fig. 5 shows the electrically gated transmission, which is proportional to the 2D absorption Eq. (2), and the corresponding spectrogram of a short-time Fourier transform. There, a clear separation in time of the ISBT absorption frequencies is visible. This results from the fact that the THz-pulse itself is chirped or temporally stretched. Considering our chirped pulses and the fact that we measure pump-probe dynamics at the amplitude maximum (dotted line) of the electrically gated transmission related to frequencies around $2.25\,\textrm {TH}_{\textrm {z}}$, we assign the observed dynamics to the first ISBT between the subbands $n=1$ and $n=2$. Hence, we conclude to observe a population redistribution of thermally excited carriers followed by a cooling of electrons towards the equilibrium state. This is consistent with the unscreening of the depolarization shift of the first ISBT $\nu _{12}$ and the higher absorption amplitude of the second ISBT $\nu _{23}$ long after the pump-probe time overlap (remember Fig. 4(a) at $t_4$). Compared to literature, the extracted time constant appears to be limited by acoustic-phonon scattering [32,56–58] typical for ISBT below the Reststrahlenband and in accordance to values reported previously.
Fig. 5. Connection between the electrically gated transmission and pump-probe dynamics. The left part of the figure depicts the electrically gated transmission through the sample and the corresponding spectrogram of a short-time Fourier transform. A dotted line from the maximum amplitude in the THz time trace connects the right part of the figure that shows the pump-probe dynamics at $2.2\,\textrm {kV}\cdot \textrm {cm}^{-1}$ in GaAs. This line indicates the fixed probe delay of the electro-optic sampling measurement at which the pump-probe delay is varied and a pump-probe curve measured. The pump-probe measurement is fitted (red line) with a function, which is mainly a convolution between a Gaussian pulse (black dashed line) and a single exponential decay (material function). From the fit, we extract a pulse width of $\tau _{\textrm {FWHM}} = 16\,\textrm {ps}$ and a decay time of $\tau _{\textrm {decay}} = 420\,\textrm {ps}$. The measurement has been acquired at $10\,\textrm {K}$.
4. Conclusion
In summary, we have investigated the intersubband AC-Stark effect with an archetypical all-THz pump-probe setup combining strong pump pulses being particularly narrowband ($\Delta \lambda /\lambda \approx (1\ldots 2)\,\%$) from the free-electron laser FELBE and weak broadband-probe pulses. Probing the dressed state $n=2$ from the ground state $n=1$, we have measured an Autler-Townes doublet with a Rabi frequency of about $0.25\,\textrm {TH}_{\textrm {z}}$. Moreover, we could observe an indication of a Mollow triplet (probing both dressed states $n=2$ and $n=3$). Several experimental issues such as timing jitter or spatial averaging of the probed electric field of pump pulses limited these investigations to the weak-to-intermediate coupling regime within the rotating wave approximation indicated by the ratio $\Omega _R / ( 2\pi \cdot \nu _{23}) = 7\%$. Furthermore, our results reveal important effects, namely the depolarization shift and a positive chirp, influencing the Autler-Townes absorption signature. As a side effect of the experimental setup, we were able to determine a slow relaxation time of about $420\,\textrm {ps}$ corresponding to the intersubband relaxation $2\rightarrow 1$. Despite the slower relaxation processes below the optical phonon energy, it is still challenging to observe the Autler-Townes effect for intersubband transitions clearly, even using a relatively sophisticated all-THz pump-probe approach.
Appendix A: depolarization shift
In order to calculate the ISBT absorption spectrum, we utilize a simulation software for semiconductor nanodevices called nextnano3 [44,45] (nextnano GmBH). The software solves mainly the Poisson and the stationary Schrödinger equation self-consistently. As a result, we determine the bandstructure, subband energies, wavefunctions, ISBT energies, dipole matrix elements, oscillator strengths and other parameters. However, the software does not include the calculation of the depolarization shift. We include the effect following the model of Załużny et al.[50] such that we can write $\mathcal{E} _{12} \rightarrow \mathcal{E} _{12}^{dep}$ with:
Fig. 6. Measured absorption spectra and simulated peak positions of the ISBTs. Panel (a) depicts the normalized 2D absorption for different modulation voltages from depletion $U_{\textrm {depl}} = -2\,\textrm {V}$ to a higher voltage $U_{\textrm {high}}$ (up to $0\,\textrm {V}$). All spectra have been acquired and calculated at low temperatures of $10\,\textrm {K}$. Simulations (b) show the frequency position of the first $\mathcal{E} _{12}$ (gray/black) and second $\mathcal{E} _{23}$ (red) ISBT for different calculated electron densities in a modulation-doped QW. The gray crosses represent the frequency position of the first ISBT neglecting the depolarization shift, while the black dots show a strong blue shift of $\mathcal{E} _{12}$ including this effect. The yellow horizontal bar marks the simulated flat-band condition of the QW predicted by nextnano3 simulations. For both panels (a) and (b) the green horizontal bar marks the voltage and the simulated electron density, respectively, which we assume to be comparable with our measurements. Note the different frequency scale.
Funding
Bundesministerium für Bildung und Forschung (05K14CRA).
Acknowledgments
The authors thank Alexej Pashkin for discussions, Artur Erbe, Bernd Scheumann and the NanoFaRo for fabrication support of the samples and P. Michel and the FELBE team for running the free-electron laser FELBE. Portions of this work were published in “THz pump-probe spectroscopy of the intersubband AC-Stark effect in a GaAs quantum well”, PhD Thesis Dissertation, TU Dresden, 2019.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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