Abstract

In this paper, we review our recent experimental studies on electromagnetically induced transparency (EIT) from electron spin coherences in semiconductor quantum wells. Coherent Raman resonances, manifestations of EIT from electron spin coherences at relatively low pump intensities, were demonstrated in both V-type and Λ-type three-level systems via heavy-hole exciton and trion transitions in undoped and doped quantum wells, respectively. Coherent Raman resonances from electron spin coherences via light-hole transitions were also demonstrated in a waveguide geometry that enables a long optical interaction length as well as a large absorption coefficient. Experimental approaches that can avoid or reduce detrimental many-body effects in quantum wells are suggested for the realization of nearly ideal EIT processes.

© 2012 Optical Society of America

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2011 (2)

H. Bluhm, S. Foletti, I. Neder, M. Rudner, D. Mahalu, V. Umansky, and A. Yacoby, “Dephasing time of GaAs electron-spin qubits coupled to a nuclear bath exceeding 200 μs,” Nat. Phys. 7, 109–113 (2011).
[CrossRef]

C. Phelps, J. Prineas, and H. Wang, “Excitonic nonlinear optical response from correlation-enhanced tunneling in mixed-type GaAs quantum wells,” Phys. Rev. B 83, 153302 (2011).
[CrossRef]

2010 (1)

M. Sladkov, A. U. Chaubal, M. P. Bakker, A. R. Onur, D. Reuter, A. D. Wieck, and C. H. van der Wal, “Electromagnetically induced transparency with an ensemble of donor-bound electron spins in a semiconductor,” Phys. Rev. B 82, 121308 (2010).
[CrossRef]

2009 (6)

N. H. Kwong, S. Schumacher, and R. Binder, “Electron-spin beat susceptibility of excitons in semiconductor quantum wells,” Phys. Rev. Lett. 103, 056405 (2009).
[CrossRef]

C. Phelps, T. Sweeney, R. T. Cox, and H. Wang, “Ultrafast coherent electron spin flip in a modulation-doped CdTe quantum well,” Phys. Rev. Lett. 102, 237402 (2009).
[CrossRef]

D. Barettin, J. Houmark, B. Lassen, M. Willatzen, T. R. Nielsen, J. Mork, and A. P. Jauho, “Optical properties and optimization of electromagnetically induced transparency in strained InAs/GaAs quantum dot structures,” Phys. Rev. B 80, 235304 (2009).
[CrossRef]

T. R. Nielsen, A. Lavrinenko, and J. Mork, “Slow light in quantum dot photonic crystal waveguides,” Appl. Phys. Lett. 94, 113111 (2009).
[CrossRef]

J. Houmark, T. R. Nielsen, J. Mork, and A. P. Jauho, “Comparison of electromagnetically induced transparency schemes in semiconductor quantum dot structures: Impact of many-body interactions,” Phys. Rev. B 79, 115420 (2009).
[CrossRef]

P. Lunnemann and J. Mork, “Reducing the impact of inhomogeneous broadening on quantum dot based electromagnetically induced transparency,” Appl. Phys. Lett. 94, 071108(2009).
[CrossRef]

2008 (4)

S. Marcinkevicius, A. Gushterov, and J. P. Reithmaier, “Transient electromagnetically induced transparency in self-assembled quantum dots,” Appl. Phys. Lett. 92, 041113 (2008).
[CrossRef]

X. Xu, B. Sun, P. R. Berman, D. G. Steel, A. S. Bracker, D. Gammon, and L. J. Sham, “Coherent population trapping of an electron spin in a single negatively charged quantum dot,” Nat. Phys. 4, 692–695 (2008).
[CrossRef]

K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature 452, 67–71 (2008).
[CrossRef]

S. O’Leary and H. Wang, “Manipulating nonlinear optical responses from spin-polarized electrons in a two-dimensional electron gas via exciton injection,” Phys. Rev. B 77, 165309 (2008).
[CrossRef]

2007 (2)

T. Wang, R. Rajapakse, and S. F. Yelin, “Electromagnetically induced transparency and slow light with n-doped GaAs,” Opt. Commun. 272, 154–160 (2007).
[CrossRef]

S. O’Leary, H. Wang, and J. P. Prineas, “Coherent Zeeman resonance from electron spin coherence in a mixed-type GaAs/AlAs quantum well,” Opt. Lett. 32, 569–571 (2007).
[CrossRef]

2006 (3)

S. Michael, W. W. Chow, and H. C. Schneider, “Coulomb corrections to the slowdown factor in quantum-dot quantum coherence,” Appl. Phys. Lett. 89, 181114 (2006).
[CrossRef]

C. Santori, P. Tamarat, P. Neumann, J. Wrachtrup, D. Fattal, R. G. Beausoleil, J. Rabeau, P. Olivero, A. D. Greentree, S. Prawer, F. Jelezko, and P. Hemmer, “Coherent population trapping of single spins in diamond under optical excitation,” Phys. Rev. Lett. 97, 247401 (2006).
[CrossRef]

P. Kolchin, S. Du, C. Belthangady, G. Y. Yin, and S. E. Harris, “Generation of narrow-bandwidth paired photons: Use of a single driving laser,” Physi. Rev. Lett. 97, 113602 (2006).
[CrossRef]

2005 (10)

T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, S. Y. Lan, T. A. B. Kennedy, and A. Kuzmich, “Storage and retrieval of single photons transmitted between remote quantum memories,” Nature 438, 833–836 (2005).
[CrossRef]

M. D. Eisaman, A. Andre, F. Massou, M. Fleischhauer, A. S. Zibrov, and M. D. Lukin, “Electromagnetically induced transparency with tunable single-photon pulses,” Nature 438, 837–841 (2005).
[CrossRef]

A. M. C. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” Science 308, 672–674 (2005).
[CrossRef]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[CrossRef]

J. F. Dynes, M. D. Frogley, J. Rodger, and C. C. Phillips, “Optically mediated coherent population trapping in asymmetric semiconductor quantum wells,” Phys. Rev. B 72, 085323 (2005).
[CrossRef]

J. F. Dynes, M. D. Frogley, M. Beck, J. Faist, and C. C. Phillips, “Ac stark splitting and quantum interference with intersubband transitions in quantum wells,” Phys. Rev. Lett. 94, 157403 (2005).
[CrossRef]

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]

I. Bar-Joseph, “Trions in GaAs quantum wells,” Semicond. Sci. Technol. 20, R29–R39 (2005).
[CrossRef]

S. Sarkar, P. Palinginis, P.-C. Ku, C. J. Chang-Hasnain, N. H. Kwong, R. Binder, and H. Wang, “Inducing electron spin coherence in GaAs quantum well waveguides: spin coherence without spin precession,” Phys. Rev. B 72, 035343 (2005).
[CrossRef]

K.-M. C. Fu, C. Santori, C. Stanley, M. C. Holland, and Y. Yamamoto, “Coherent population trapping of electron spins in a high-purity n-type GaAs semiconductor,” Phys. Rev. Lett. 95, 187405 (2005).
[CrossRef]

2004 (4)

P. Palinginis and H. Wang, “Coherent Raman resonance from electron spin coherence in GaAs quantum wells,” Phys. Rev. B 70, 153307 (2004).
[CrossRef]

P. Palinginis and H. Wang, “Vanishing and emerging of absorption quantum beats from electron spin coherence in GaAs quantum wells,” Phys. Rev. Lett. 92, 037402 (2004).
[CrossRef]

M. C. Phillips and H. Wang, “Exciton spin coherence and electromagnetically induced transparency in the transient optical response of GaAs quantum wells,” Phys. Rev. B 69, 115337 (2004).
[CrossRef]

V. M. Axt and T. Kuhn, “Femtosecond spectroscopy in semiconductors: a key to coherences, correlations and quantum kinetics,” Rep. Prog. Phys. 67, 433–512 (2004).
[CrossRef]

2003 (8)

C. J. Chang-Hasnain, P. C. Ku, J. Kim, and S. L. Chuang, “Variable optical buffer using slow light in semiconductor nanostructures,” Proc. IEEE 91, 1884–1897 (2003).
[CrossRef]

W. W. Chow, H. C. Schneider, and M. C. Phillips, “Theory of quantum-coherence phenomena in semiconductor quantum dots,” Phys. Rev. A 68, 053802 (2003).
[CrossRef]

M. C. Phillips, H. Wang, I. Rumyantsev, N. H. Kwong, R. Takayama, and R. Binder, “Electromagnetically induced transparency in semiconductors via biexciton coherence,” Phys. Rev. Lett. 91, 183602 (2003).
[CrossRef]

M. Phillips and H. Wang, “Electromagnetically induced transparency due to intervalence band coherence in a GaAs quantum well,” Opt. Lett. 28, 831–833 (2003).
[CrossRef]

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. M. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423, 731–734 (2003).
[CrossRef]

C. H. van der Wal, M. D. Eisaman, A. André, R. L. Walsworth, D. F. Phillips, A. S. Zibrov, and M. D. Lukin, “Atomic memory for correlated photon states,” Science 301, 196–200 (2003).
[CrossRef]

M. D. Lukin, “Colloquium: trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. 75, 457–472 (2003).
[CrossRef]

T. Li, H. Wang, N. Kwong, and R. Binder, “Electromagnetically induced transparency via electron spin coherence in a quantum well waveguide,” Opt. Express 11, 3298–3303 (2003).
[CrossRef]

2002 (2)

J. M. Shacklette and S. T. Cundiff, “Role of excitation-induced shift in the coherent optical response of semiconductors,” Phys. Rev. B 66, 045309 (2002).
[CrossRef]

M. Phillips and H. Wang, “Spin coherence and electromagnetically induced transparency via exciton correlations,” Phys. Rev. Lett. 89, 186401 (2002).
[CrossRef]

2001 (6)

D. S. Chemla and J. Shah, “Many-body and correlation effects in semiconductors,” Nature 411, 549–557 (2001).
[CrossRef]

L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001).
[CrossRef]

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409, 490–493 (2001).
[CrossRef]

D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86, 783–786 (2001).
[CrossRef]

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2001).
[CrossRef]

P. R. Hemmer, A. V. Turukhin, M. S. Shahriar, and J. A. Musser, “Raman-excited spin coherences in nitrogen-vacancy color centers in diamond,” Opt. Lett. 26, 361–363 (2001).
[CrossRef]

2000 (2)

A. Imamoglu, “Electromagnetically induced transparency with two dimensional electron spins,” Opt. Commun. 179, 179–182 (2000).
[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]

1999 (3)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

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D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation,” Phys. Rev. Lett. 83, 1767–1770 (1999).
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1998 (1)

J. M. Kikkawa and D. D. Awschalom, “Resonant spin amplification in n-type GaAs,” Phys. Rev. Lett. 80, 4313–4316 (1998).
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1997 (3)

J. Kikkawa, I. Smorchkova, N. Samarth, and D. Awschalom, “Room-temperature spin memory in two-dimensional electron gases,” Science 277, 1284–1287 (1997).
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S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36–42 (1997).
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S. Crooker, D. Awschalom, J. J. Baumberg, F. Flack, and N. Samarth, “Optical spin resonance and transverse spin relaxation in magnetic semiconductor quantum wells,” Phys. Rev. B 56, 7574–7588 (1997).
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1996 (1)

E. Arimondo, “Coherent population trapping in laser spectroscopy,” Prog. Opt. 35, 257–354 (1996).
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1995 (2)

M. Lindberg and R. Binder, “Dark states in coherent semiconductor spectroscopy,” Phys. Rev. Lett. 75, 1403–1406 (1995).
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1993 (2)

K. Kheng, R. T. Cox, M. Y. d’ Aubigné, F. Bassani, K. Saminadayar, and S. Tatarenko, “Observation of negatively charged excitons X− in semiconductor quantum wells,” Phys. Rev. Lett. 71, 1752–1755 (1993).
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H. Wang, K. Ferrio, D. G. Steel, Y. Z. Hu, R. Binder, and S. W. Koch, “Transient nonlinear optical response from excitation induced dephasing in GaAs,” Phys. Rev. Lett. 71, 1261–1264 (1993).
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1992 (2)

I. Galbraith, P. Dawson, and C. T. Foxon, “Optical nonlinearities in mixed type I-type II GaAs/AlAs multiple quantum wells,” Phys. Rev. B 45, 13499 (1992).
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V. M. Axt and T. Kuhn, “Femtosecond spectroscopy in semiconductors: a key to coherences, correlations and quantum kinetics,” Rep. Prog. Phys. 67, 433–512 (2004).
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J. F. Dynes, M. D. Frogley, M. Beck, J. Faist, and C. C. Phillips, “Ac stark splitting and quantum interference with intersubband transitions in quantum wells,” Phys. Rev. Lett. 94, 157403 (2005).
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C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409, 490–493 (2001).
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P. Kolchin, S. Du, C. Belthangady, G. Y. Yin, and S. E. Harris, “Generation of narrow-bandwidth paired photons: Use of a single driving laser,” Physi. Rev. Lett. 97, 113602 (2006).
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D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation,” Phys. Rev. Lett. 83, 1767–1770 (1999).
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K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature 452, 67–71 (2008).
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A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. M. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423, 731–734 (2003).
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S. Michael, W. W. Chow, and H. C. Schneider, “Coulomb corrections to the slowdown factor in quantum-dot quantum coherence,” Appl. Phys. Lett. 89, 181114 (2006).
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W. W. Chow, H. C. Schneider, and M. C. Phillips, “Theory of quantum-coherence phenomena in semiconductor quantum dots,” Phys. Rev. A 68, 053802 (2003).
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C. J. Chang-Hasnain, P. C. Ku, J. Kim, and S. L. Chuang, “Variable optical buffer using slow light in semiconductor nanostructures,” Proc. IEEE 91, 1884–1897 (2003).
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C. Phelps, T. Sweeney, R. T. Cox, and H. Wang, “Ultrafast coherent electron spin flip in a modulation-doped CdTe quantum well,” Phys. Rev. Lett. 102, 237402 (2009).
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K. Kheng, R. T. Cox, M. Y. d’ Aubigné, F. Bassani, K. Saminadayar, and S. Tatarenko, “Observation of negatively charged excitons X− in semiconductor quantum wells,” Phys. Rev. Lett. 71, 1752–1755 (1993).
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S. Crooker, D. Awschalom, J. J. Baumberg, F. Flack, and N. Samarth, “Optical spin resonance and transverse spin relaxation in magnetic semiconductor quantum wells,” Phys. Rev. B 56, 7574–7588 (1997).
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K. Kheng, R. T. Cox, M. Y. d’ Aubigné, F. Bassani, K. Saminadayar, and S. Tatarenko, “Observation of negatively charged excitons X− in semiconductor quantum wells,” Phys. Rev. Lett. 71, 1752–1755 (1993).
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A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. M. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423, 731–734 (2003).
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C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409, 490–493 (2001).
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J. F. Dynes, M. D. Frogley, J. Rodger, and C. C. Phillips, “Optically mediated coherent population trapping in asymmetric semiconductor quantum wells,” Phys. Rev. B 72, 085323 (2005).
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J. F. Dynes, M. D. Frogley, M. Beck, J. Faist, and C. C. Phillips, “Ac stark splitting and quantum interference with intersubband transitions in quantum wells,” Phys. Rev. Lett. 94, 157403 (2005).
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J. F. Dynes, M. D. Frogley, M. Beck, J. Faist, and C. C. Phillips, “Ac stark splitting and quantum interference with intersubband transitions in quantum wells,” Phys. Rev. Lett. 94, 157403 (2005).
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C. Santori, P. Tamarat, P. Neumann, J. Wrachtrup, D. Fattal, R. G. Beausoleil, J. Rabeau, P. Olivero, A. D. Greentree, S. Prawer, F. Jelezko, and P. Hemmer, “Coherent population trapping of single spins in diamond under optical excitation,” Phys. Rev. Lett. 97, 247401 (2006).
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S. E. Harris, J. E. Field, and A. Kasapi, “Dispersive properties of electromagnetically induced transparency,” Phys. Rev. A 46, R29–R32 (1992).
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S. Crooker, D. Awschalom, J. J. Baumberg, F. Flack, and N. Samarth, “Optical spin resonance and transverse spin relaxation in magnetic semiconductor quantum wells,” Phys. Rev. B 56, 7574–7588 (1997).
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J. F. Dynes, M. D. Frogley, M. Beck, J. Faist, and C. C. Phillips, “Ac stark splitting and quantum interference with intersubband transitions in quantum wells,” Phys. Rev. Lett. 94, 157403 (2005).
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J. S. Weiner, D. S. Chemla, D. A. B. Miller, H. A. Haus, A. C. Gossard, W. Wiegmann, and C. A. Burrus, “Highly anisotropic optical properties of single quantum well waveguides,” Appl. Phys. Lett. 47, 664–667 (1985).
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P. Kolchin, S. Du, C. Belthangady, G. Y. Yin, and S. E. Harris, “Generation of narrow-bandwidth paired photons: Use of a single driving laser,” Physi. Rev. Lett. 97, 113602 (2006).
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Figures (13)

Fig. 1.
Fig. 1.

Schematic of a Λ-type three-level system with two external optical fields coupling to two respective dipole transitions. εa and εb are the probe and pump fields, respectively.

Fig. 2.
Fig. 2.

Theoretically calculated probe absorption spectra for a Λ-type three-level system with the pump Rabi frequency indicated in the figure. Other parameters used are δb=0 and γab=0.01γa.

Fig. 3.
Fig. 3.

Polarization selection rules for the dipole transitions between the conduction and the HH and LH valence bands in a GaAs QW.

Fig. 4.
Fig. 4.

Polarization selection rules for the HH transitions in a GaAs QW subject to an external magnetic field along the x axis (Voigt geometry), with the z axis being the QW growth axis. (a) Circularly polarized fields. (b) Linearly polarized fields.

Fig. 5.
Fig. 5.

(a) DT response obtained at 10K from a 13nm GaAs QW subject to an external magnetic field along the x axis (Voigt geometry), with B=0.5T, 2 T, and 4 T from top to bottom. Both the pump and probe have the same circular polarization. (b) Calculated DT response using the V-type system discussed in the text. Parameters taken from the experiment: γ/2π=2.5GHz, γs/2π=200MHz, |ge|=0.26, and exciton radiative recombination rate of Γ/2π=200MHz. The two dipole transitions in the V-system are assumed identical except for the transition energies.

Fig. 6.
Fig. 6.

(a)–(d) Polarization dependence of the DT responses obtained at 10K from a 13nm GaAs QW subject to an external magnetic field along the x axis (Voigt geometry) with B=2T. The polarization configuration of the pump and probe for each DT response is indicated in the figure.

Fig. 7.
Fig. 7.

(a) Schematic of the polarization configuration and experimental geometry for experiments in a GaAs QW slab waveguide. (b) Linear transmission spectrum of the GaAs QW slab waveguide obtained at 10K for a TE-polarized field propagating along the waveguide. (c) The same as (b) except that the optical field is TM-polarized.

Fig. 8.
Fig. 8.

DT responses from a GaAs QW slab waveguide obtained at 50K. The pump and probe, propagating along the same direction in the waveguide, are TE and TM polarized, respectively. (a) B=0T. The solid line shows a Lorentzian fit. (b) B=0.25T. The external magnetic field is along the z axis.

Fig. 9.
Fig. 9.

Polarization dependence of the DT responses from a GaAs QW slab waveguide obtained at 20K and B=0.25T. (a) The pump and probe, propagating along the same direction in the waveguide, are TE and TM polarized, respectively. (b) Both the pump and probe are TE polarized. (c) Theoretically calculated DT response under the experimental conditions of (a). (d) Theoretically calculated DT response under the experimental conditions of (b). Parameters are taken from the experiment with γ/2π=0.16THz and γs/2π=Γ/2π=1GHz.

Fig. 10.
Fig. 10.

Energy level diagram and polarization selection rules for the HH trion transitions in an external magnetic field along the x axis (Voigt geometry), with the z axis being the QW growth axis. The trion state is labeled with the jz of the valance band state (the two electrons in the trion have the opposite spins). (a) Circularly polarized fields. (b) Linearly polarized fields.

Fig. 11.
Fig. 11.

(a) Linear absorption spectrum of a modulation-doped CdTe QW obtained at 10K. (b) DT response obtained near the trion absorption resonance at 10K. An external magnetic field with B=0.3T is applied along the x axis (Voigt geometry).

Fig. 12.
Fig. 12.

(a) Schematic of a GaAs/AlAs mixed-type QW structure. (b) Photoluminescence spectra of the wide well obtained at 12K in the presence of a He–Ne laser, with the intensity of the He–Ne laser indicated in the figure. (c) Absorption spectra of the wide well obtained at 10K in the absence (red dashed line) and presence (black dotted line) of a 0.2mW/cm2 He–Ne laser.

Fig. 13.
Fig. 13.

(a) DT response obtained at 10K from a GaAs/AlAs mixed-type QW, with the pump laser tuned to the HH trion absorption resonance. An external magnetic field with B=0.3T is applied along the x axis (Voigt geometry). (b) The DT response with increasing excess electron population, injected by a laser with λ=535nm. From the top to bottom, the injection laser intensity is 0, 0.05, and 0.1mW/cm2.

Equations (8)

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p˜˙a(t)=(iδaγa)p˜a(t)iΩa(t)2[ne(t)na(t)]+iΩb(t)2p˜ba(t),
p˜˙b(t)=(iδbγb)p˜b(t)iΩb(t)2[ne(t)nb(t)]+iΩa(t)2p˜ba(t)*,
p˜˙ba(t)=[i(δaδb)γab]p˜ba(t)iΩa(t)2p˜b(t)*+iΩb(t)2p˜a(t),
p˜˙a(1)(t)=(iδaγa)p˜a(1)(t)+iΩa(t)2+iΩb(t)2p˜ba(1)(t),
p˜˙ba(1)(t)=[i(δaδb)γab]p˜ba(1)(t)+iΩb(t)2p˜a(1)(t).
p˜a(1)=iΩa2i(δaδb)γab(iδaγa)[i(δaδb)γab]+Ωb2/4,
|jz=1/2=13|Y11|+23|Y10|,
|jz=1/2=13|Y11|+23|Y10|,

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