## THz-driven nonlinear intersubband dynamics in quantum wells |

Optics Express, Vol. 20, Issue 21, pp. 23053-23060 (2012)

http://dx.doi.org/10.1364/OE.20.023053

Acrobat PDF (1065 KB)

### Abstract

In this work, we demonstrate the direct observation of non-equilibrium intersubband dynamics in a modulation-doped multiple quantum well sample induced by intense terahertz pulses. The transmission spectra show a clear dependence on the incident THz field strength, which gives rise to a multitude of nonlinear optical effects that go beyond the standard textbook two-level description of light-matter interaction. 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. Examples of this interaction include the efficient, coherent population transfer, the THz induced undressing of collective excitations, and the THz Stark effect.

© 2012 OSA

## 1. Introduction

1. I. I. Rabi, “Space quantization in a gyrating magnetic field,” Phys. Rev. **51**, 652–654 (1937). [CrossRef]

2. W. Tittel, M. Afzelius, T. Chanelière, R. L. Cone, S. Kröll, S. A. Moiseev, and M. Sellars, “Photon-echo quantum memory in solid state systems,” Laser & Photon. Rev. **4**, 244–267 (2010). [CrossRef] [PubMed]

4. P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging - modern techniques and applications,” Laser & Photon. Rev. **5**, 124–166 (2011). [CrossRef] [PubMed]

5. J. N. Heyman, R. Kersting, and K. Unterrainer, “Time-domain measurement of intersubband oscillations in a quantum well,” Appl. Phys. Lett. **72**, 644–646 (1998). [CrossRef]

6. J. Kröll, J. Darmo, S. Dhillon, X. Marcadet, M. Calligaro, C. Sirtori, and K. Unterrainer, “Phase-resolved measurements of stimulated emission in a laser,” Nature **449**, 698–701 (2007). [CrossRef] [PubMed]

7. M. S. Sherwin, K. Craig, B. Galdrikian, J. Heyman, A. Markelz, K. Campman, S. Fafard, P. F. Hopkins, and A. Gossard, “Nonlinear quantum dynamics in semiconductor quantum wells,” Physica D **83**, 229–242 (1995). [CrossRef]

8. K. Craig, B. Galdrikian, J. N. Heyman, A. G. Markelz, J. B. Williams, M. S. Sherwin, K. Campman, P. F. Hopkins, and A. C. Gossard, “Undressing a collective intersubband excitation in a quantum well,” Phys. Rev. Lett. **76**, 2382–2385 (1996). [CrossRef] [PubMed]

9. S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science **310**, 651–653 (2005). [CrossRef] [PubMed]

12. B. Zaks, D. Stehr, T.-A. Truong, P. M. Petroff, S. Hughes, and M. S. Sherwin, “THz-driven quantum wells: Coulomb interactions and Stark shifts in the ultrastrong coupling regime,” New J. Phys. **13**, 083009 (2011). [CrossRef]

5. J. N. Heyman, R. Kersting, and K. Unterrainer, “Time-domain measurement of intersubband oscillations in a quantum well,” Appl. Phys. Lett. **72**, 644–646 (1998). [CrossRef]

6. J. Kröll, J. Darmo, S. Dhillon, X. Marcadet, M. Calligaro, C. Sirtori, and K. Unterrainer, “Phase-resolved measurements of stimulated emission in a laser,” Nature **449**, 698–701 (2007). [CrossRef] [PubMed]

13. R. Kersting, R. Bratschitsch, G. Strasser, K. Unterrainer, and J. N. Heyman, “Sampling a terahertz dipole transition with subcycle time resolution,” Opt. Lett. **25**, 272–274 (2000). [CrossRef]

14. M. Wagner, M. Helm, M. S. Sherwin, and D. Stehr, “Coherent control of THz intersubband polarization in a voltage controlled single quantum well,” Appl. Phys. Lett. **99**, 131109 (2011). [CrossRef]

15. G. B. Serapiglia, E. Paspalakis, C. Sirtori, K. L. Vodopyanov, and C. C. Phillips, “Laser-induced quantum coherence in a semiconductor quantum well,” Phys. Rev. Lett. **84**, 1019–1022 (2000). [CrossRef] [PubMed]

18. G. Günter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol, L. Sorba, S. D. Liberato, C. Ciuti, A. Tredicucci, A. Leitenstorfer, and R. Huber, “Sub-cycle switch-on of ultrastrong light-matter interaction,” Nature **458**, 178–181 (2009). [CrossRef] [PubMed]

18. G. Günter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol, L. Sorba, S. D. Liberato, C. Ciuti, A. Tredicucci, A. Leitenstorfer, and R. Huber, “Sub-cycle switch-on of ultrastrong light-matter interaction,” Nature **458**, 178–181 (2009). [CrossRef] [PubMed]

19. C. Ciuti, G. Bastard, and I. Carusotto, “Quantum vacuum properties of the intersubband cavity polariton field,” Phys. Rev. B **72**, 115303 (2005). [CrossRef]

## 2. Experiment

*et al.*in Ref. [20

20. A. Sell, A. Leitenstorfer, and R. Huber, “Phase-locked generation and field-resolved detection of widely tunable terahertz pulses with amplitudes exceeding 100MV/cm,” Opt. Lett. **33**, 2767–2769 (2008). [CrossRef] [PubMed]

21. D. Dietze, K. Unterrainer, and J. Darmo, “Dynamically phase-matched terahertz generation,” Opt. Lett. **37**, 1047–1049 (2012). [CrossRef] [PubMed]

_{0.3}Ga

_{0.7}As 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*

_{2}

*= 2.55 × 10*

_{d}^{10}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

5. J. N. Heyman, R. Kersting, and K. Unterrainer, “Time-domain measurement of intersubband oscillations in a quantum well,” Appl. Phys. Lett. **72**, 644–646 (1998). [CrossRef]

*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 Δ

*E*=

*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.

15. G. B. Serapiglia, E. Paspalakis, C. Sirtori, K. L. Vodopyanov, and C. C. Phillips, “Laser-induced quantum coherence in a semiconductor quantum well,” Phys. Rev. Lett. **84**, 1019–1022 (2000). [CrossRef] [PubMed]

## 3. Results

*E*, 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

6. J. Kröll, J. Darmo, S. Dhillon, X. Marcadet, M. Calligaro, C. Sirtori, and K. Unterrainer, “Phase-resolved measurements of stimulated emission in a laser,” Nature **449**, 698–701 (2007). [CrossRef] [PubMed]

23. H. Harde, S. Keiding, and D. Grischkowsky, “THz commensurate echoes: Periodic rephasing of molecular transitions in free-induction decay,” Phys. Rev. Lett. **66**, 1834–1837 (1991). [CrossRef] [PubMed]

*α*(

*ω*) ∝ Im(−

*i*Δ

*E*(

*ω*)/

*E*

_{ref}), which is valid under the assumption, that the depleted sample is non-absorbing and Δ

*E*(

*ω*) ≪

*E*

_{ref}[24

24. D. Golde, M. Wagner, D. Stehr, H. Schneider, M. Helm, A. M. Andrews, T. Roch, G. Strasser, M. Kira, and S. W. Koch, “Fano signatures in the intersubband terahertz response of optically excited semiconductor quantum wells,” Phys. Rev. Lett. **102**, 127403 (2009). [CrossRef] [PubMed]

*τ*

_{12}= 2ps,

*τ*

_{23}= 1ps, and

*τ*

_{34}= 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*

_{2}

*(*

_{d}*N*−

_{i}*N*)

_{j}*f*

_{ij}*τ*with

_{ij}*f*being the oscillator strength of the transition [26], and the assumption that the electrons are only efficiently transferred up to the third level (

_{ij}*N*

_{1}+

*N*

_{2}+

*N*

_{3}= 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%.

*α*∝ (

*N*

_{1}−

*N*

_{2}). Due to the efficient population transfer,

*N*

_{1}−

*N*

_{2}is reduced, which leads to the undressing of the collective excitations [7

7. M. S. Sherwin, K. Craig, B. Galdrikian, J. Heyman, A. Markelz, K. Campman, S. Fafard, P. F. Hopkins, and A. Gossard, “Nonlinear quantum dynamics in semiconductor quantum wells,” Physica D **83**, 229–242 (1995). [CrossRef]

7. M. S. Sherwin, K. Craig, B. Galdrikian, J. Heyman, A. Markelz, K. Campman, S. Fafard, P. F. Hopkins, and A. Gossard, “Nonlinear quantum dynamics in semiconductor quantum wells,” Physica D **83**, 229–242 (1995). [CrossRef]

8. K. Craig, B. Galdrikian, J. N. Heyman, A. G. Markelz, J. B. Williams, M. S. Sherwin, K. Campman, P. F. Hopkins, and A. C. Gossard, “Undressing a collective intersubband excitation in a quantum well,” Phys. Rev. Lett. **76**, 2382–2385 (1996). [CrossRef] [PubMed]

*E*is the electric field applied perpendicular to the quantum well plane [28

_{z}28. M. Matsuura and T. Kamizato, “Subbands and excitons in a quantum well in an electric field,” Phys. Rev. B **33**, 8385–8389 (1986). [CrossRef]

*dc*electric field of

*E*= 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

_{z}*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

## Acknowledgments

## References and links

1. | I. I. Rabi, “Space quantization in a gyrating magnetic field,” Phys. Rev. |

2. | W. Tittel, M. Afzelius, T. Chanelière, R. L. Cone, S. Kröll, S. A. Moiseev, and M. Sellars, “Photon-echo quantum memory in solid state systems,” Laser & Photon. Rev. |

3. | H. C. Liu and F c (Eds.), |

4. | P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging - modern techniques and applications,” Laser & Photon. Rev. |

5. | J. N. Heyman, R. Kersting, and K. Unterrainer, “Time-domain measurement of intersubband oscillations in a quantum well,” Appl. Phys. Lett. |

6. | J. Kröll, J. Darmo, S. Dhillon, X. Marcadet, M. Calligaro, C. Sirtori, and K. Unterrainer, “Phase-resolved measurements of stimulated emission in a laser,” Nature |

7. | M. S. Sherwin, K. Craig, B. Galdrikian, J. Heyman, A. Markelz, K. Campman, S. Fafard, P. F. Hopkins, and A. Gossard, “Nonlinear quantum dynamics in semiconductor quantum wells,” Physica D |

8. | K. Craig, B. Galdrikian, J. N. Heyman, A. G. Markelz, J. B. Williams, M. S. Sherwin, K. Campman, P. F. Hopkins, and A. C. Gossard, “Undressing a collective intersubband excitation in a quantum well,” Phys. Rev. Lett. |

9. | S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science |

10. | J. R. Danielson, Y.-S. Lee, J. P. Prineas, J. T. Steiner, M. Kira, and S. W. Koch, “Interaction of strong single-cycle terahertz pulses with semiconductor quantum wells,” Phys. Rev. Lett. |

11. | M. Wagner, H. Schneider, D. Stehr, S. Winnerl, A. M. Andrews, S. Schartner, G. Strasser, and M. Helm, “Observation of the intraexcitonic Autler-Townes effect in GaAs/AlGaAs semiconductor quantum wells,” Phys. Rev. Lett. |

12. | B. Zaks, D. Stehr, T.-A. Truong, P. M. Petroff, S. Hughes, and M. S. Sherwin, “THz-driven quantum wells: Coulomb interactions and Stark shifts in the ultrastrong coupling regime,” New J. Phys. |

13. | R. Kersting, R. Bratschitsch, G. Strasser, K. Unterrainer, and J. N. Heyman, “Sampling a terahertz dipole transition with subcycle time resolution,” Opt. Lett. |

14. | M. Wagner, M. Helm, M. S. Sherwin, and D. Stehr, “Coherent control of THz intersubband polarization in a voltage controlled single quantum well,” Appl. Phys. Lett. |

15. | G. B. Serapiglia, E. Paspalakis, C. Sirtori, K. L. Vodopyanov, and C. C. Phillips, “Laser-induced quantum coherence in a semiconductor quantum well,” Phys. Rev. Lett. |

16. | C. W. Luo, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Phase-resolved nonlinear response of a two-dimensional electron gas under femtosecond intersubband excitation,” Phys. Rev. Lett. |

17. | J. F. Dynes, M. D. Frogley, M. Beck, J. Faist, and C. C. Phillips, “A Stark splitting and quantum interference with intersubband transitions in quantum wells,” Phys. Rev. Lett. |

18. | G. Günter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol, L. Sorba, S. D. Liberato, C. Ciuti, A. Tredicucci, A. Leitenstorfer, and R. Huber, “Sub-cycle switch-on of ultrastrong light-matter interaction,” Nature |

19. | C. Ciuti, G. Bastard, and I. Carusotto, “Quantum vacuum properties of the intersubband cavity polariton field,” Phys. Rev. B |

20. | A. Sell, A. Leitenstorfer, and R. Huber, “Phase-locked generation and field-resolved detection of widely tunable terahertz pulses with amplitudes exceeding 100MV/cm,” Opt. Lett. |

21. | D. Dietze, K. Unterrainer, and J. Darmo, “Dynamically phase-matched terahertz generation,” Opt. Lett. |

22. | H. Schmidt, D. E. Nikonov, K. L. Campman, K. D. Maranowski, A. C. Gossard, and A. Imamoğlu, “Quantum interference in semiconductor quantum wells,” Laser Phys. |

23. | H. Harde, S. Keiding, and D. Grischkowsky, “THz commensurate echoes: Periodic rephasing of molecular transitions in free-induction decay,” Phys. Rev. Lett. |

24. | D. Golde, M. Wagner, D. Stehr, H. Schneider, M. Helm, A. M. Andrews, T. Roch, G. Strasser, M. Kira, and S. W. Koch, “Fano signatures in the intersubband terahertz response of optically excited semiconductor quantum wells,” Phys. Rev. Lett. |

25. | D. Dietze, “Nonlinear terahertz spectroscopy of semiconductor heterostructures,” Ph.D. thesis, Vienna University of Technology (2012). |

26. | M. Helm, |

27. | M. Zalużny, “Influence of the depolarization effect on the nonlinear intersubband absorption spectra of quantum wells,” Phys. Rev. B |

28. | M. Matsuura and T. Kamizato, “Subbands and excitons in a quantum well in an electric field,” Phys. Rev. B |

**OCIS Codes**

(190.5970) Nonlinear optics : Semiconductor nonlinear optics including MQW

(320.7110) Ultrafast optics : Ultrafast nonlinear optics

(300.6495) Spectroscopy : Spectroscopy, teraherz

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: July 20, 2012

Revised Manuscript: August 20, 2012

Manuscript Accepted: August 20, 2012

Published: September 24, 2012

**Citation**

D. Dietze, J. Darmo, and K. Unterrainer, "THz-driven nonlinear intersubband dynamics in quantum wells," Opt. Express **20**, 23053-23060 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-21-23053

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### References

- I. I. Rabi, “Space quantization in a gyrating magnetic field,” Phys. Rev.51, 652–654 (1937). [CrossRef]
- W. Tittel, M. Afzelius, T. Chanelière, R. L. Cone, S. Kröll, S. A. Moiseev, and M. Sellars, “Photon-echo quantum memory in solid state systems,” Laser & Photon. Rev.4, 244–267 (2010). [CrossRef] [PubMed]
- H. C. Liu and F c (Eds.), Intersubband Transitions in Quantum Wells: Physics and Device Applications I, vol. 62 of Semiconductors and Semimetals (Academic Press, 2000).
- P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging - modern techniques and applications,” Laser & Photon. Rev.5, 124–166 (2011). [CrossRef] [PubMed]
- J. N. Heyman, R. Kersting, and K. Unterrainer, “Time-domain measurement of intersubband oscillations in a quantum well,” Appl. Phys. Lett.72, 644–646 (1998). [CrossRef]
- J. Kröll, J. Darmo, S. Dhillon, X. Marcadet, M. Calligaro, C. Sirtori, and K. Unterrainer, “Phase-resolved measurements of stimulated emission in a laser,” Nature449, 698–701 (2007). [CrossRef] [PubMed]
- M. S. Sherwin, K. Craig, B. Galdrikian, J. Heyman, A. Markelz, K. Campman, S. Fafard, P. F. Hopkins, and A. Gossard, “Nonlinear quantum dynamics in semiconductor quantum wells,” Physica D83, 229–242 (1995). [CrossRef]
- K. Craig, B. Galdrikian, J. N. Heyman, A. G. Markelz, J. B. Williams, M. S. Sherwin, K. Campman, P. F. Hopkins, and A. C. Gossard, “Undressing a collective intersubband excitation in a quantum well,” Phys. Rev. Lett.76, 2382–2385 (1996). [CrossRef] [PubMed]
- S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science310, 651–653 (2005). [CrossRef] [PubMed]
- J. R. Danielson, Y.-S. Lee, J. P. Prineas, J. T. Steiner, M. Kira, and S. W. Koch, “Interaction of strong single-cycle terahertz pulses with semiconductor quantum wells,” Phys. Rev. Lett.99, 237401 (2007). [CrossRef]
- M. Wagner, H. Schneider, D. Stehr, S. Winnerl, A. M. Andrews, S. Schartner, G. Strasser, and M. Helm, “Observation of the intraexcitonic Autler-Townes effect in GaAs/AlGaAs semiconductor quantum wells,” Phys. Rev. Lett.105, 167401 (2010). [CrossRef]
- B. Zaks, D. Stehr, T.-A. Truong, P. M. Petroff, S. Hughes, and M. S. Sherwin, “THz-driven quantum wells: Coulomb interactions and Stark shifts in the ultrastrong coupling regime,” New J. Phys.13, 083009 (2011). [CrossRef]
- R. Kersting, R. Bratschitsch, G. Strasser, K. Unterrainer, and J. N. Heyman, “Sampling a terahertz dipole transition with subcycle time resolution,” Opt. Lett.25, 272–274 (2000). [CrossRef]
- M. Wagner, M. Helm, M. S. Sherwin, and D. Stehr, “Coherent control of THz intersubband polarization in a voltage controlled single quantum well,” Appl. Phys. Lett.99, 131109 (2011). [CrossRef]
- G. B. Serapiglia, E. Paspalakis, C. Sirtori, K. L. Vodopyanov, and C. C. Phillips, “Laser-induced quantum coherence in a semiconductor quantum well,” Phys. Rev. Lett.84, 1019–1022 (2000). [CrossRef] [PubMed]
- C. W. Luo, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Phase-resolved nonlinear response of a two-dimensional electron gas under femtosecond intersubband excitation,” Phys. Rev. Lett.92, 047402 (2004). [CrossRef] [PubMed]
- J. F. Dynes, M. D. Frogley, M. Beck, J. Faist, and C. C. Phillips, “A Stark splitting and quantum interference with intersubband transitions in quantum wells,” Phys. Rev. Lett.94, 157403 (2005). [CrossRef] [PubMed]
- G. Günter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol, L. Sorba, S. D. Liberato, C. Ciuti, A. Tredicucci, A. Leitenstorfer, and R. Huber, “Sub-cycle switch-on of ultrastrong light-matter interaction,” Nature458, 178–181 (2009). [CrossRef] [PubMed]
- C. Ciuti, G. Bastard, and I. Carusotto, “Quantum vacuum properties of the intersubband cavity polariton field,” Phys. Rev. B72, 115303 (2005). [CrossRef]
- A. Sell, A. Leitenstorfer, and R. Huber, “Phase-locked generation and field-resolved detection of widely tunable terahertz pulses with amplitudes exceeding 100MV/cm,” Opt. Lett.33, 2767–2769 (2008). [CrossRef] [PubMed]
- D. Dietze, K. Unterrainer, and J. Darmo, “Dynamically phase-matched terahertz generation,” Opt. Lett.37, 1047–1049 (2012). [CrossRef] [PubMed]
- H. Schmidt, D. E. Nikonov, K. L. Campman, K. D. Maranowski, A. C. Gossard, and A. Imamoğlu, “Quantum interference in semiconductor quantum wells,” Laser Phys.9, 797–812 (1999).
- H. Harde, S. Keiding, and D. Grischkowsky, “THz commensurate echoes: Periodic rephasing of molecular transitions in free-induction decay,” Phys. Rev. Lett.66, 1834–1837 (1991). [CrossRef] [PubMed]
- D. Golde, M. Wagner, D. Stehr, H. Schneider, M. Helm, A. M. Andrews, T. Roch, G. Strasser, M. Kira, and S. W. Koch, “Fano signatures in the intersubband terahertz response of optically excited semiconductor quantum wells,” Phys. Rev. Lett.102, 127403 (2009). [CrossRef] [PubMed]
- D. Dietze, “Nonlinear terahertz spectroscopy of semiconductor heterostructures,” Ph.D. thesis, Vienna University of Technology (2012).
- M. Helm, The Basic Physics of Intersubband Transitions, vol. 62 of Semiconductors and Semimetals (Academic Press, 2000).
- M. Zalużny, “Influence of the depolarization effect on the nonlinear intersubband absorption spectra of quantum wells,” Phys. Rev. B47, 3995–3998 (1993). [CrossRef]
- M. Matsuura and T. Kamizato, “Subbands and excitons in a quantum well in an electric field,” Phys. Rev. B33, 8385–8389 (1986). [CrossRef]

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