## Abstract

We investigate nonlinear mid-infrared detection via two-photon transitions involving two bound subbands and one continuum resonance in an n-type multiple quantum well. By varying the excitation energy, we have tuned the two-photon transition from resonant, yielding optimum resonant enhancement with a real intermediate state, to nearly-resonant, with a virtual but resonantly enhanced intermediate state. For autocorrelation purposes, the latter configuration improves time resolution whilst partially retaining a resonant enhancement of the two-photon transition strength.

©2008 Optical Society of America

## 1. Introduction

Intersubband transitions in quantum wells (QW) have found important applications in quantum well infrared photodetectors (QWIPs) [1] and quantum cascade lasers [2]. In addition, QW-based device structures also provide opportunities to create model systems for basic concepts in solid state physics, like quantum interference [3], coherent transport [4], electromagnetically induced transparency [5], and gain without inversion [6]. Recently, optical nonlinearities involving intersubband transitions in QWs have attracted increasing attention for wavelength conversion in quantum cascade lasers [7].

We have previously demonstrated quadratic photodetection by two-photon transitions involving three equidistant energy levels |1〉, |2〉, and |3〉 at energies *E*
_{1}, *E*
_{2}, and *E*
_{3}, respectively, namely two bound states and one continuum resonance [8, 9]. In this three-level configuration, two photons are required to excite an electron into the continuum (see inset of Fig. 1(a)), such that the photocurrent depends quadratically on incident power. This is in contrast to QWIPs, where transitions from one single bound state to the continuum result in a linear dependence.

Interferometric photocurrent autocorrelation measurements under femtosecond excitation then allow us to probe the dynamics associated with the intermediate resonant state. Since the intersubband relaxation time *T*
_{1} and dephasing time *T*
_{2} of the |1〉→|2〉 transition are much longer than the scattering time associated with the continuum resonance |3〉, these experiments allow us to extract *T*
_{1} and *T*
_{2} [8]. The approach has proven to be useful for systematic studies of the dynamical behavior of intersubband transitions [10], in particular to discriminate between different scattering processes. The method thus represents an alternative to four-wave mixing [11], both experiments being based on the third-order nonlinear susceptibility *χ*
^{(3)}.

In this paper, we report on quadratic photodetection involving two-photon transitions in the nearly-resonant regime. While a resonant intermediate state has the principal advantage of enhancing drastically the two-photon transition efficiency, non-resonant two-photon transitions allow for better temporal resolution, not limited any more by population or phase relaxation, and spectrally wider wavelength regime. By changing the excitation energy, we have influenced the resonant character of the two-photon transition. In this way, the intermediate state of the two-photon transition can be changed from resonant to virtual, which partially maintains resonant enhancement while improving temporal resolution. We also treat the two-photon photocurrent dynamics theoretically by solving the Liouville equation of a three-level system.

## 2. Sample characteristics

The sample for this study is an In_{0.10}Ga_{0.90}As/Al_{0.31}Ga_{0.69}As multipleQWstructure containing 20 periods of 7.3 nm wide QWs and 46 nm wide barriers. Nominally the central 5 nm of each QW are doped with Si to a sheet concentration of 8×10^{11}cm^{-2}, corresponding to a Fermi energy *E _{F}* of about 15 meV. The multiple QW is embedded between n-type contact layers. The sample was grown by molecular beam epitaxy (MBE) on a [100]-oriented semi-insulating GaAs substrate and processed into mesa detectors of 120×120

*µ*m

^{2}and 240×240

*µ*m

^{2}in area with ohmic contact metallization covering the top of the mesas. The absorption of the unprocessed sample was measured at a temperature of 77 K in Brewster-angle geometry [1] using a Fourier-transform infrared (FTIR) spectrometer. For photocurrent measurements, the infrared radiation is coupled into the structures via standard 45-degree facets [1, 12]. Both Brewster and facet geometries give rise to an electric field component along the growth direction to satisfy the polarization rules for intersubband absorption.

The spectral dependence is summarized in Fig. 1(a). The |1〉→|2〉 transition gives rise to intersubband absorption at a peak wavelength of *λ _{peak}*=7.6

*µ*m (

*E*

_{2}-

*E*

_{1}=163.2 meV) with 11 meV full width at half-maximum. Figure 1(a) also shows the spectral dependence of the photocurrent measured with a FTIR spectrometer under glowbar illumination, with the sample held at an elevated temperature of 160 K. As illustrated in Fig. 1(b), thermal population at the energy range of the second subband, well above the Fermi energy

*E*, gives rise to a (linear) photocurrent originating from the |1〉→|2〉 and |2〉→|3〉 transitions, such that one photon at the associated energy is capable to excite these electrons above the barrier energy

_{F}*E*[8].

_{B}A clear photocurrent signal is also caused by the |1〉→|3〉 transition. Even though this transition is parity forbidden in a symmetric QW, the symmetry is sufficiently broken by residual asymmetry induced by the asymmetric dopant distribution and by the externally applied electric field. In addition, the parity selection rule does not strictly apply to continuum states. The step-like increase at around 280 meV indicates the photoconductive energy threshold; the broad absorption line is characteristic for bound-to-continuum transitions and relaxes the conditions for resonant two-photon transitions to be observed. The associated photocurrent signal is about three times larger than the peak at 163 meV since only about 0.1% of the carriers are thermally excited into the second subband at this temperature.

Close inspection of the photocurrent in the spectral vicinity of the |1〉→|2〉 transition reveals a narrow photocurrent peak with the same spectral position and linewidth as the absorption, and a spectrally broad component with a shoulder at 140 meV. We therefore associate the narrow peak with the |1〉→|2〉 bound-to-bound transition and the broad component with the |2〉→|3〉 bound-to-continuum transition (see Fig. 1(b)). The energy width of the latter is comparable to the width of the |1〉→|3〉 transition, which further supports our assignment. Concerning the excitation mechanisms, the former contribution involves the |1〉→|2〉 transition of thermally excited electrons in level |1〉 with high enough kinetic energy, such that the total energy of the final state in the second subband is close to or above the barrier, while the latter stems from |2〉→|3〉 bound-to-continuum excitation of electrons in the thermally populated second subband (see the inset of Fig. 2(a)). Since both mechanisms involve the same thermal activation energy and the lowest two subbands have essentially the same density of states, it is not unexpected that the two excitation paths lead to comparable signal strengths.

Concerning two-photon transitions in this structure, the |1〉→|2〉 transition energy is some-what larger than the onset of the |2〉→|3〉 transition. According to Fig. 1(a), the width of the continuum resonance is wide enough to ensure resonantly enhanced two-photon absorption to be always present as long as the photon energy is resonant with or close to the |1〉→|2〉 transition. The level configuration thus yields the opportunity for resonant two-photon excitation at the |1〉→|2〉 transition energy of *hν*=*E*
_{2}-*E*
_{1}=163 meV (full double arrows). A second opportunity is *detuned, close-to-resonant* excitation, where the excitation energy is detuned in a way that the final state remains inside the energy regime of the third level, but with the intermediate state moved out of resonance. For the latter we chose an excitation energy of *hν*=146 meV<*E*
_{2}-*E*
_{1} (dashed double arrows).

## 3. Quadratic photocurrent autocorrelation

By applying a voltage across the active region, electrons being photoexcited into the continuum resonance are efficiently ejected from the QW and drift towards the collector contact (inset of Fig. 2(a)). Since each electron has to absorb two photons in order to reach the continuum resonance, the resulting photocurrent scales quadratically with the incident power density [8].

To investigate two-photon transitions, we operate the device at low temperature (77 K) in order to suppress the thermally activated [8], linear photocurrent contribution studied in Fig. 1, and at moderate bias of 1–2 V to prevent electrons from tunneling out of the intermediate state [13]. For quadratic autocorrelation measurements, it is crucial that these linear contributions do not cause any measurable signal. The chosen experimental conditions ensure complete suppression of linear contributions of the photocurrent below the noise level of the measurement.

To study the dynamics associated with intersubband excitation, we use mid-infrared pulses of 165 fs duration with a center wavelength tunable between 6 and 18 *µ*m. The radiation is generated by difference frequency mixing of the signal and idler beams of an optical parametric oscillator pumped by a mode-locked Ti:Sapphire laser [14]. Using a GaSe crystal, pulses of 50 W peak power (0.7 mW average power) are obtained, which are split in a Michelson interferometer and superimposed collinearly on the sample with variable delay time. The pulses obtained using this setup were nearly transform-limited, as was inferred from the observed Gaussian shapes both of the envelope of the linear interferogram and of the spectrum [14].

In Fig. 2, measured autocorrelation traces are therefore compared to a calculated “ideal” autocorrelation trace of a Gaussian pulse. The most striking feature in interferometric autocorrelation is an 8:1 ratio between the signals occurring at zero delay and at large delay, which is induced by quadratic power dependence and constructive interference [8, 15].

In spite of the extremely high optical nonlinearity of two-photon transitions involving a resonant intermediate state, quadratic detection always leads to some population of the intermediate level, such that the achievable time resolution is limited by the associated population and phase relaxation times. It is therefore desirable to exploit this nonlinearity by close-to-resonant operation involving a virtual intermediate state. In this case, the two-photon absorption cross section should already exhibit some resonant enhancement without sacrificing dynamical resolution due to populations in the intermediate subband. By varying the excitation energy, we have therefore tuned the two-photon transition from resonant, with a real intermediate state for optimum resonant enhancement, to near-resonant, with a virtual intermediate state.

Since the intermediate state in the present samples is energetically closer to the barrier than to the ground subband, two-photon photoemission can be induced at a photon energy below *E*
_{2}-*E*
_{1}. Figure 2 shows the signal obtained for (a) resonant excitation at *λ*=7.6 *µ*m and (b) detuned excitation at *λ*=8.5 *µ*m. In the latter case, only a small fraction of the signal arises from excitation via the intermediate state. The signal shape thus approaches that of an ideal autocorrelation (Fig. 2(c)). Therefore, the intrinsic response is faster as compared to resonant excitation where it is determined by intermediate state electron phase and energy relaxation.

Fourier-transforms of these autocorrelation traces are shown in Fig. 2(d). For detuned excitation at *λ*=8.5 *µ*m, the data exhibit almost perfect agreement with the Fourier-transform of the ideal autocorrelation at the same excitation wavelength. In addition to the excitation frequency, a second peak appears at the second harmonic, which is characteristic for quadratic autocorrelation; it is caused by the non-sinusoidal shape of the fringes. For resonant excitation (7.6 *µ*m), we observe a narrowing of the spectral width at the fundamental frequency due to the longer dephasing time; the intersubband transition acts like a narrow-band oscillator. Thus, the spectral detection width in fact increases for non-resonant excitation.

The larger spectral detection width and faster response are at the expense of a signal decrease by more than two orders of magnitude, even though we estimate that the near-resonant intermediate state increases the signal by at least one order of magnitude as compared to the completely non-resonant case. This nearly resonant configuration is thus interesting for strong mid-infrared ps or fs sources, e. g., free-electron lasers [15], whereas a resonant two-photon detector is better suited for weaker sources like mode-locked quantum cascade lasers [16].

## 4. Numerical simulation of two-photon photocurrents

By solving the Liouville equation of a three-level system, we have numerically calculated entire photocurrent autocorrelation traces. The approach has been described in detail by Hattori *et al.* [17], who applied the model previously to the two-photon response of photomultiplier tubes.

As a typical example, Fig. 3(a) shows a signal trace assuming intersubband relaxation times of *T*
_{1}=530 fs and *T*
_{2}=120 fs, calculated for Gaussian pulses of σ=165 fs duration. Here the signal is normalized to the baseline level generated by the two pulses at large delay time, i. e., *τ*≫*T*
_{1},*T*
_{2},σ. The interferometric second-order autocorrelation of the pulses is shown for comparison in Fig. 3(b). In both cases, the signal at *τ*=0 shows a 8 : 1 peak-to-background-ratio resulting from the quadratic power dependence. For *τ*≠0, two features related to the finite relaxation times are found. First, the interference fringes show a slower decay, which is attributed to the dephasing of electrons photoexcited into the intermediate state. Second, for larger delay times the finite lifetime of intermediate state electrons results in an exponential decrease of the signal towards the asymptotic value. This calculation also confirms the validity of the model function used in our previous work [8, 9, 10].

Experimentally, detuning from resonance as in Fig. 2(b) still results in residual excitation of the intermediate level, such that the shape of the associated autocorrelation trace is in between those of Figs. 3(a) and (b).

## 5. Summary

We have investigated the dynamics of two-photon intersubband photocurrents in In_{0.10}Ga_{0.90}As/Al_{0.31}Ga_{0.69}As QWs involving two bound subbands and one continuum resonance. Varying the excitation energy, we have compared quadratic photocurrent autocorrelations in this artificial three level system under resonant and close-to-resonant excitation of the intermediate level. Here the latter configuration improves temporal resolution of the autocorrelation while partially retaining a resonant enhancement of the two-photon transition strength. The approach has been justified by numerical simulations based on the Liouville equation of a three-level system.

## Acknowledgments

- The authors are grateful to P. Koidl (Freiburg) and M. Helm (Dresden) for helpful discussions. HCL thanks the Alexander von Humboldt foundation for the Bessel Award and the renewed research stay in Dresden.

## References and links

**1. **H. Schneider and H. C. Liu, *Quantum Well Infrared Photodetectors: Physics and Applications* (Springer, 2006).

**2. **C. Sirtori and R. Teissier, “Quantum Cascade Lasers: Overview of Basic Principles and State of the Art,” in *Intersubband Transitions in Quantum Structures*, R. Paiella, ed. (McGraw-Hill, 2006), pp. 1–44.

**3. **J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, and A. Y. Cho, “Quantum Interference in Intersubband Transitions,” in *Intersubband Transitions in Quantum Wells: Physics and Device Applications II*, Semicond. Semimet.62, H. C. Liu and F. Capasso, eds. (Academic Press, 2000), pp. 101–128. [CrossRef]

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**6. **M. D. Frogley, J. F. Dynes, M. Beck, J. Faist, and C. C. Phillips, “Gain without inversion in semiconductor nanostructures,” Nature Mat. **5**, 175–178 (2006). [CrossRef]

**7. **C. Gmachl, O. Malis, and A. Belyanin, “Optical Nonlinearities in Intersubband transitions and Quantum Cascade Lasers,” in *Intersubband Transitions in Quantum Structures*, R. Paiella, ed. (McGraw-Hill, 2006), pp. 181–235.

**8. **H. Schneider, T. Maier, H. C. Liu, M. Walther, and P. Koidl, “Ultra-sensitive femtosecond two-photon detector with resonantly enhanced nonlinear absorption,” Opt. Lett. **30**, 287–289 (2005). [CrossRef] [PubMed]

**9. **T. Maier, H. Schneider, H. C. Liu, M. Walther, and P. Koidl, “Two-photon QWIPs for quadratic detection of weak mid-infrared pulsed lasers,” Infrared Phys. Technol. **47**, 182–187 (2005). [CrossRef]

**10. **H. Schneider, T. Maier, M. Walther, and H. C. Liu, “Two-photon photocurrent spectroscopy of electron intersubband relaxation and dephasing in quantum wells,” Appl. Phys. Lett. **91**, 191116 (2007). [CrossRef]

**11. **T. Elsaesser, “Ultrafast Dynamics of Intersubband Excitations in Quantum Wells and Quantum Cascade Structures,” in *Intersubband Transitions in Quantum Structures*, R. Paiella, ed. (McGraw-Hill, 2006), pp. 181–235.

**12. **H. Schneider, C. Schnbein, P. Koidl, and G. Weimann, “Influence of optical interference in quantum well infrared photodetectors with 45*°* facet geometry,” Appl. Phys. Lett. **74**, 16 (1999). [CrossRef]

**13. **T. Maier, H. Schneider, M. Walther, P. Koidl, and H. C. Liu, “Resonant two-photon photoemission in quantum well infrared photodetectors,” Appl. Phys. Lett **84**, 5162–5164 (2004). [CrossRef]

**14. **S. Ehret and H. Schneider, “Generation of subpicosecond infrared pulses tunable between 5.2 *µ*m and 18 *µ*m at a repetition rate of 76 MHz,” Appl. Phys. B **66**, 27–30 (1998). [CrossRef]

**15. **H. Schneider, O. Drachenko, S. Winnerl, M. Helm, and M. Walther, “Quadratic autocorrelation of free-electron laser radiation and photocurrent saturation in two-photon quantum-well infrared photodetectors,” Appl. Phys. Lett. **89**, 133508 (2006). [CrossRef]

**16. **R. Paiella, F. Capasso, C. Gmachl, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, A. Y. Cho, and H. C. Liu, “Self-Mode-Locking of Quantum Cascade Lasers with Ultrafast Optical Nonlinearities,” Science **290**, 1739–1742 (2000). [CrossRef] [PubMed]

**17. **T. Hattori, Y. Kawashima, M. Daikoku, H. Inouye, and H. Nakatsuka, “Femtosecond Two-Photon Response Dynamics of Photomultiplier Tubes,” Jpn. J. Appl. Phys. **39**, 4793–4798 (2000). [CrossRef]