## Coherent spectroscopy of semiconductors

Optics Express, Vol. 16, Issue 7, pp. 4639-4664 (2008)

http://dx.doi.org/10.1364/OE.16.004639

Acrobat PDF (2136 KB)

### Abstract

The coherent optical response of semiconductors has been the subject of substantial research over the last couple of decades. The interest has been motivated by unique aspects of the interaction between light and semiconductors that are revealed by coherent techniques. The ability to probe the dynamics of charge carriers has been a significant driver. This paper presents a review of selected results in coherent optical spectroscopy of semiconductors.

© 2008 Optical Society of America

## 1. Introduction

1. R. J. Elliott, “Intensity of Optical Absorption by Excitons,” Phys. Rev. **108**, 1384–1389 (1957). [CrossRef]

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

4. W. Chow, S. Koch, and M. Sargent, *Semiconductor-Laser Physics* (Springer-Verlag, Berlin, 1994). [CrossRef]

9. T. Meier, P. Thomas, and S. Koch, *Coherent Semiconductor Optics* (Springer-Verlag, Berlin, 2007). [CrossRef]

## 2. Materials

*m*=0.0662

_{c}*m*and two valence bands, known as the heavy-hole band, with effective mass

_{e}*m*=0.34

_{hh}*m*

_{0}, and as the light-hole band, with

*m*=0.094

_{lh}*m*

_{0}(see Fig. 1). The conduction band (cb) has a spin of

*k*=0, however mechanical strain due to mounting and differentials in thermal expansion can easily impose sufficient strain on thin samples so that the exciton transition is often split.

*Ga*

_{x}_{1-x}As, typically grown by molecular beam epitaxy (MBE). This material choice is due to the fact that AlGaAs is nearly lattice matched to GaAs for all Al fractions. The direct band gap of Al

*Ga*

_{x}_{1-x}As is

*E*(

_{g}*x*)=1.424+1.247

*x*eV at room temperature temperature for

*x*<0.45 [11

11. S. Adachi, “GaAs, AlAs and Al_{x}Ga_{1}-_{x}As: Material Parameters for use in research and device applications,” J. Appl. Phys. **58**, R1–R29 (1985). [CrossRef]

*x*>0.45 the lowest band gap of Al

*Ga*

_{x}_{1-x}As becomes indirect.

_{0.3}Ga

_{0.7}As barriers are used, resulting in a 245 meV deep well in the conduction band and a 130 meV deep well in the hh valence band.

12. R. L. Greene, K. K. Bajaj, and D. E. Phelps, “Energy levels of Wannier excitons in GaAs-Ga_{1-x}Al* _{x}*As quantumwell structures,” Phys. Rev. B

**29**, 1807–1812 (1984). [CrossRef]

_{0.3}Ga

_{0.7}As. There are 10 wells, which increases the absorption but also can increase the inhomogeneity due to well-to-well fluctuations. The heavy-hole and light-hole exciton resonances are clearly evident.

## 3. Methods

### 3.1. Decoherence

### 3.2. Linear spectroscopy

### 3.3. Nonlinear spectroscopy

*P*is the induced polarization,

*E*is the electric field of the incident light and

*χ*

^{(n)}is the

*n*order susceptibility. (More generally to include polarization,

^{th}*E*and

*P*are vectors and

*χ*

^{(n)}is an

*n*+1 order tensor). The

*χ*

^{(1)}term is responsible for the linear response, i.e., the absorption and index of refraction of the material. It can be shown that all even orders of

*χ*are zero for a material with inversion symmetry. Thus the lowest order nonlinearity that occurs in all materials is due to

*χ*

^{(3)}. To produce a signal that is at approximately the same frequency as the incident light, this means that three incident fields interact to produce the signal. However, it is often the case that there are only two incident laser beams, in which case one of them acts twice.

13. D. S. Kim, J. Shah, D. A. B. Miller, T. C. Damen, A. Vinattieri, W. Schafer, and L. N. Pfeiffer, “Femtosecond pulse distortion in GaAs quantum-wells and its effect on pump-probe or 4-wave-mixing experiments,” Phys. Rev. B **50**, 18,240–18,249 (1994). [CrossRef]

*αl*<1, where

*α*is the Beers law absorption coefficient and

*l*is the effective thickness, i.e., the transmitted intensity is

*I*=

_{t}*I*

_{i}e*for an incident intensity of*

^{-αl}*I*. In high quality semiconductor quantum wells at low temperature, which have an exciton absorption width of 1 meV or less, this rule typically means an upper limit of 10 quantum wells in the sample. Disorder can also have significant effects on the pulse distortion [14

_{i}14. T. Stroucken, A. Knorr, C. Anthony, A. Schulze, P. Thomas, S. W. Koch, M. Koch, S. T. Cundiff, J. Feldmann, and E. O. Göbel, “Light-propagation and disorder effects in semiconductor multiple-quantum wells,” Phys. Rev. Lett. **74**, 2391–2394 (1995). [CrossRef] [PubMed]

### 3.3.1. Two-pulse transient four-wave-mixing

**k**

_{a}and

**k**

_{b}are incident on the sample. For τ>0 pulse

**k**

_{a}arrives first. Typically the angle between

**k**

_{a}and

**k**

_{b}is small. Their nonlinear interaction gives rise to signal in the direction

^{k}

_{s}=2

**k**

_{b}-

**k**

_{a}under the right conditions. The basic geometry for TFWM is sketched in Fig. 3. Two-pulse TFWM has three main variants, time-integrated (TI-TFWM), time-resolved (TR-TFWM) and spectrally resolved (SRTFWM). The three variants are distinguished by how the signal is detected. In TI-TFWM, a slow detector is used, which essential integrates the signal over time. In TR-TFWM, the signal is time-resolved, typically using a reference pulse and upconversion in a non-linear crystal [15]. Scanning the reference pulse maps out the signal as a function of “real” time, which is typically designated as

*t*. In SR-TFWM, a spectrometer is used to record the spectrum of the signal. Both TR- and SR-TFWM are intrinsically two-dimensional experiments as data is usually taken as function of two variables.

**k**

_{b}-

**k**

_{a}. The spatial modulation occurs because of the angle between the two beams. The initial phase of the coherence is set by

**k**

_{a}, and in general will vary across the sample because

**k**

_{a}is at an angle. This phase will then evolve until the second pulse arrives. At locations where the second pulse is in phase with the coherence, it will constructively interfere to create excited state population, while at locations where the second pulse is out of phase with respect to the coherence, it will return the sample to the ground state. In turn, the absorption is modulated by the spatially varying excited state population, and thus there is an effective diffraction grating, which can in turn scatters a portion of the second pulse in the

**k**

_{s}direction.

16. T. Yajima and Y. Taira, “Spatial optical parametric coupling of picosecond light-pulses and transverse relaxation effect in resonant media,” J. Phys. Soc. Japan **47**, 1620–1626 (1979). [CrossRef]

*γ*is the dephasing rate and

_{ph}*θ*(

*x*) is the Heavyside step function. In an inhomogeneously broadened system, the TFWM signal is a photon echo centered at

*t*=τ, where

*t*=0 corresponds to the arrival of the second pulse at the sample. In the limit of strong inhomogeneous broadening, the TI-TFWM signal is

*γ*in the presence of inhomogeneous broadening is one of the strengths of TFWM. However, one of its drawbacks is the need to know whether or not inhomogeneous broadening is present in order to properly interpret the results. In certain circumstances, comparison to the linear absorption linewidth can reveal which formula to use. The linear absorption linewidth will give the inhomogeneous width if there is inhomogeneous broadening and the homogeneous width otherwise. However, there can be ambiguity if the inhomogeneous and homogeneous widths are comparable. Additionally, if the homogeneous width depends on excitation conditions, as it often does in semiconductors, comparison to an absorption linewidth can be misleading. Temporally or spectrally resolving the signal can help remove the ambiguity.

_{ph}17. A. Honold, L. Schultheis, J. Kuhl, and C. W. Tu, “Reflected degenerate 4-wave mixing on GaAs single quantum wells,” Appl. Phys. Lett. **52**, 2105–2107 (1988). [CrossRef]

### 3.3.2. Three-pulse transient four-wave-mixing

**k**

*,*

_{a}**k**

*and*

_{b}**k**

*. The nonlinear interaction of the pulses in the sample gives rise to a signal in the direction*

_{c}**k**

*=-*

_{s}**k**

*+*

_{a}**k**

*+*

_{b}**k**

*. The delay between pulse*

_{c}**k**

*and*

_{a}**k**

*is usually designated as τ with τ>0 for*

_{b}**k**

*arriving first. Typically the delay between*

_{a}**k**

*and*

_{b}**k**

*is designate as*

_{c}*T*with

*T*>0 for

**k**

*arriving after*

_{c}**k**

*[18]. There are many different geometries for 3P-TFWM, a few of most of common are shown in Fig. 4.*

_{b}*γ*by scanning

_{gr}*T*[19

19. A. M Weiner, S. D. Silvestri, and E. P. Ippen, “3-pulse scattering for femtosecond dephasing studies - theory and experiment,” J. Opt. Soc. Am. B **2**, 654–662 (1985). [CrossRef]

*γ*becomes a 4, just as in the two-pulse case. However, 3P-TFWM can distinguish between homogeneously and inhomogeneously broadened systems by measuring the signal as a function of τ around 0 for a fixed

_{ph}*T*>0. If the signal is symmetric about zero, the system is homogeneously broadened, whereas if it only occurs for τ>0 it is inhomogeneously broadened. With the introduction of inhomogeneous broadening, spectral diffusion also needs to be considered as 3P-TFWM is sensitive to it. Spectral diffusion is a process by which an excitation initially at one frequency can shift to another frequency. Varying both

*T*and τ can provide clear signatures of spectral diffusion. Similarly, in situations where the Markovian approximation is not valid, the correlation function of the frequency fluctuations that give rise to dephasing can be extracted using a technique known as three pulse echo peak shift (3PEPS) spectroscopy [19

19. A. M Weiner, S. D. Silvestri, and E. P. Ippen, “3-pulse scattering for femtosecond dephasing studies - theory and experiment,” J. Opt. Soc. Am. B **2**, 654–662 (1985). [CrossRef]

20. W. P. de Boeij, M. S. Pshenichnikov, and D. A. Wiersma, “On the relation between the echo-peak shift and Brownian-oscillator correlation function,” Chem. Phys. Lett. **253**, 53–60 (1996). [CrossRef]

21. T. Joo, Y. Jia, J.-Y. Yu, M. J. Lang, and G. R. Fleming, “Third-order nonlinear time domain probes of solvation dynamics,” J. Chem. Phys. **104**, 6089–6108 (1996). [CrossRef]

*T*. By varying the angle between the first two excitation pulses, the effects of spatial diffusion and population decay can be separated. The grating relaxation rate is

*γ*=2

_{gr}*γ*+8

_{pop}*π*

^{2}

*D*Λ

^{-2}where

*γ*is the population relaxation time,

_{pop}*D*is the spatial diffusion coefficient and Λ=

*n*λ/2

*sinθ*is the grating spacing for an angle

*θ*between the beams

**k**

*and*

_{a}**k**

*. If the first two pulses are coincident in time, this measurement is often known as a “transient grating” experiment.*

_{b}### 3.3.3. Spectrally resolved transient absorption

22. M. Joffre, D. Hulin, A. Migus, A. Antonetti, C. B. A. Laguillaume, N. Peyghambarian, M. Lindberg, and S. W. Koch, “Coherent effects in pump probe spectroscopy of excitons,” Opt. Lett. **13**, 276–278 (1988). [CrossRef] [PubMed]

### 3.3.4. Two dimensional Fourier transform spectroscopy

24. D. Jonas, “Two-dimensional femtosecond spectroscopy,” Annu. Rev. Phys. Chem. **54**, 425–463 (2003). [CrossRef] [PubMed]

25. T. H. Zhang, C. N. Borca, X. Q. Li, and S. T. Cundiff, “Optical two-dimensional Fourier transform spectroscopy with active interferometric stabilization,” Opt. Express **13**, 7432–7441 (2005). [CrossRef] [PubMed]

## 4. Decoherence in semiconductors

### 4.1. Excitons

26. J. Hegarty, M. D. Sturge, A. C. Gossard, and W. Wiegmann, “Resonant degenerate 4-wave mixing in GaAs multiquantum well structures,” Appl. Phys. Lett. **40**, 132–134 (1982). [CrossRef]

27. L. Schultheis, M. D. Sturge, and J. Hegarty, “Photon-echoes from two-dimensional excitons in GaAs-AlGaAs quantum wells,” Appl. Phys. Lett. **47**, 995–997 (1985). [CrossRef]

28. L. Schultheis, J. Kuhl, A. Honold, and C. W. Tu, “Picosecond phase coherence and orientational relaxation of excitons in GaAs,” Phys. Rev. Lett. **57**, 1797–1800 (1986). [CrossRef] [PubMed]

29. L. Schultheis, A. Honold, J. Kuhl, K. Köhler, and C. W. Tu, “Optical dephasing of homogeneously broadened two-dimensional exciton-transitions in GaAs quantum-wells,” Phys. Rev. B **34**, 9027–9030 (1986). [CrossRef]

30. A. Honold, L. Schultheis, J. Kuhl, and C. W. Tu, “Collision broadening of two-dimensional excitons in a GaAs single quantum well,” Phys. Rev. B **40**, 6442–6445 (1989). [CrossRef]

31. M. Koch, R. Hellmann, G. Bastian, J. Feldmann, E. O. Göbel, and P. Dawson, “Enhanced energy and phase relaxation of excitons in the presence of bare electrons,” Phys. Rev. B **51**, 13,887–13,890 (1995). [CrossRef]

32. S. Schmitt-Rink, D. S. Chemla, and D. A. B. Miller, “Theory of transient excitonic optical nonlinearities in semiconductor quantum-well structures,” Phys. Rev. B **32**, 6601–6609 (1985). [CrossRef]

33. M. Lindberg and S. W. Koch, “Effective Bloch equations for semiconductors,” Phys. Rev. B **38**, 3342–3350 (1988). [CrossRef]

### 4.2. Free electron-hole pairs

34. P. C. Becker, H. L. Fragnito, C. H. B. Cruz, R. L. Fork, J. E. Cunningham, J. E. Henry, and C. V. Shank, “Femtosecond photon-echoes from band-to-band transitions in GaAs,” Phys. Rev. Lett. **61**, 1647–1649 (1988). [CrossRef] [PubMed]

35. A. Lohner, K. Rick, P. Leisching, A. Leitenstorfer, T. Elsaesser, T. Kuhn, F. Rossi, and W. Stolz, “Coherent optical polarization of bulk GaAs studied by femtosecond photon-echo spectroscopy,” Phys. Rev. Lett. **71**, 77–80 (1993). [CrossRef] [PubMed]

36. A. Leitenstorfer, A. Lohner, K. Rick, P. Leisching, T. Elsaesser, T. Kuhn, F. Rossi, W. Stolz, and K. Ploog, “Excitonic and free-carrier polarizations of bulk GaAs studied by femtosecond coherent spectroscopy,” Phys. Rev. B **49**, 16,372–16,380 (1994). [CrossRef]

37. J.-Y. Bigot, M. T. Portella, R. W. Schoenlein, J. E. Cunningham, and C. V. Shank, “Two-dimensional carrier-carrier screening in a quantum well,” Phys. Rev. Lett. **67**, 636–639 (1991). [CrossRef] [PubMed]

### 4.3. Quantum beats and interference effects

38. E. O. Göbel, K. Leo, T. C. Damen, J. Shah, S. Schmitt-Rink, W. Schäfer, J. F. Muller, and K. Köhler, “Quantum beats of excitons in quantum-wells,” Phys. Rev. Lett. **64**, 1801–1804 (1990). [CrossRef] [PubMed]

39. M. Koch, J. Feldmann, G. von Plessen, E. O. Göbel, P. Thomas, and K. Kohler, “Quantum beats versus polarization interference - an experimental distinction,” Phys. Rev. Lett. **69**, 3631–3634 (1992). [CrossRef] [PubMed]

40. V. G. Lyssenko, J. Erland, I. Balslev, K. H. Pantke, B. S. Razbirin, and J. M. Hvam, “Nature of nonlinear 4-wave-mixing beats in semiconductors,” Phys. Rev. B **48**, 5720–5723 (1993). [CrossRef]

41. S. T. Cundiff, “Effects of correlation between inhomogeneously broadened transitions on quantum beats in transient 4-wave-mixing,” Phys. Rev. A **49**, 3114–3118 (1994). [CrossRef] [PubMed]

42. X. Zhu, M. S. Hybertsen, and P. B. Littlewood, “Quantum beats in photon echo from four-waving mixing,” Phys. Rev. Lett. **73**, 209 (1994). [CrossRef] [PubMed]

43. M. Koch, J. Feldmann, G. von Plessen, S. T. Cundiff, E. O. Göbel, P. Thomas, and K. Köhler, “Koch et al. reply,” Phys. Rev. Lett. **73**, 210 (1994). [CrossRef] [PubMed]

25. T. H. Zhang, C. N. Borca, X. Q. Li, and S. T. Cundiff, “Optical two-dimensional Fourier transform spectroscopy with active interferometric stabilization,” Opt. Express **13**, 7432–7441 (2005). [CrossRef] [PubMed]

44. K. Leo, T. C. Damen, J. Shah, E. O. Göbel, and K. Kohler, “Quantum beats of light hole and heavy hole excitons in quantum wells,” Appl. Phys. Lett. **57**, 19–21 (1990). [CrossRef]

45. B. Feuerbacher, J. Kuhl, R. Eccleston, and K. Ploog, “Quantum Beats between the Light and Heavy Hole Excitons in a Quantum Well,” Solid State Commun. **74**, 1279–1283 (1990). [CrossRef]

46. X. Chen, W. J. Walecki, O. Buccafusca, D. N. Fittinghoff, and A. L. Smirl, “Temporally and spectrally resolved amplitude and phase of coherent four-wave-mixing emission from GaAs quantum wells,” Phys. Rev. B **56**, 9738–9743 (1997). [CrossRef]

47. M. Phillips and H. Wang, “Coherent oscillation in four-wave mixing of interacting excitons,” Solid State Commun. **111**, 317–321 (1999). [CrossRef]

48. A. L. Smirl, M. J. Stevens, X. Chen, and O. Buccafusca, “Heavy-hole and light-hole oscillations in the coherent emission from quantum wells: Evidence for exciton-exciton correlations,” Phys. Rev. B **60**, 8267–8275 (1999). [CrossRef]

50. T. Aoki, G. Mohs, M. Kuwata-Gonokami, and A. A. Yamaguchi, “Influence of Exciton-Exciton Interaction on Quantum Beats,” Phys. Rev. Lett. **82**, 3108–3111 (1999). [CrossRef]

51. L. Banyai, D. B. T. Thoai, E. Reitsamer, H. Haug, D. Steinbach, M. U. Wehner, M. Wegener, T. Marschner, and W. Stolz, “Exciton-LO-phonon quantum kinetics - evidence of memory effects in bulk GaAs,” Phys. Rev. Lett. **75**, 2188–2191 (1995). [CrossRef] [PubMed]

52. J. Feldmann, T. Meier, G. von Plessen, M. Koch, E. O. Göbel, P. Thomas, G. Bacher, C. Hartmann, H. Schweizer, W. Schäfer, and H. Nickel, “Coherent dynamics of excitonic wave-packets,” Phys. Rev. Lett. **70**, 3027–3030 (1993). [CrossRef] [PubMed]

54. T. Meier, A. Schulze, P. Thomas, H. Vaupel, and K. Maschke, “Signatures of Fano resonances in four-wave-mixing experiments,” Phys. Rev. B **51**, 13,977–13,986 (1995). [CrossRef]

55. U. Siegner, M. A. Mycek, S. Glutsch, and D. S. Chemla, “Ultrafast coherent dynamics of Fano resonances in semiconductors,” Phys. Rev. Lett. **74**, 470–473 (1995). [CrossRef] [PubMed]

56. S. BarAd, M. V. Marquezini, S. Mukamel, and D. S. Chemla, “Quantum confined Fano interference,” Phys. Rev. Lett. **78**, 1363–1366 (1997). [CrossRef]

57. Y. H. Ahn, S. B. Choe, J. C. Woo, D. S. Kim, S. T. Cundiff, J. M. Shacklette, and Y. S. Lim, “Quantum Interference of Virtual and Real Amplitudes in a Semiconductor Exciton System,” Phys. Rev. Lett. **89**, 237403 (2002). [CrossRef] [PubMed]

## 5. Many-body signals

**k**

*-*

_{b}**k**

*if*

_{a}**k**

*arrived second. By the standard definition of the delay between the pulses, this signal corresponded to the delay being positive and no signal was expected for a negative delay signal, as expressed by the step functions in equations 2 and 3. The negative delay signal in semiconductors is due to many-body interactions. However, there are multiple phenomena that can give rise to such a signal.*

_{b}### 5.1. Local fields

58. K. Leo, M. Wegener, J. Shah, D. S. Chemla, E. O. Göbel, T. C. Damen, S. Schmitt-Rink, and W. Schäfer, “Effects of coherent polarization interactions on time-resolved degenerate 4-wave-mixing,” Phys. Rev. Lett. **65**, 1340–1343 (1990). [CrossRef] [PubMed]

59. M. Wegener, D. S. Chemla, S. Schmitt-Rink, and W. Schäfer, “Line-shape of time-resolved 4-wave-mixing,” Phys. Rev. A **42**, 5675–5683 (1990). [CrossRef] [PubMed]

*E*, by

_{A}*E*+

_{A}*LP*where

*P*is the polarization induced in the sample and

*L*is a constant that connects polarization to field. Inclusion of the local field produces a correction to the oscillation frequency, known as the Lorenz-Lorentz shift. At third order, it also produces a TFWM signal for negative delays. Specifically, it predicts that for a homogeneously broadened system, the signal for positive delay will decay as

**k**

*pulse arrives first, the free decay can persist until after the*

_{b}**k**

*pulse arrives, thereby producing a signal in the direction 2*

_{a}**k**

*-*

_{b}**k**

*.*

_{a}60. M. Lindberg, R. Binder, and S. W. Koch, “Theory of the semiconductor photon-echo,” Phys. Rev. A **45**, 1865–1875 (1992). [CrossRef] [PubMed]

61. W. Schäfer, F. Jahnke, and S. Schmitt-Rink, “Many-particle effects on transient 4-wave-mixing signals in semiconductors,” Phys. Rev. B **47**, 1217–1220 (1993). [CrossRef]

62. S. Weiss, M. A. Mycek, J. Y. Bigot, S. Schmitt-Rink, and D. S. Chemla, “Collective effects in excitonic free induction decay - do semiconductors and atoms emit coherent-light in different ways,” Phys. Rev. Lett. **69**, 2685–2688 (1992). [CrossRef] [PubMed]

63. D. S. Kim, J. Shah, T. C. Damen, W. Schäfer, F. Jahnke, S. Schmitt-Rink, and K. Köhler, “Unusually slow temporal evolution of femtosecond 4-wave-mixing signals in intrinsic GaAs quantum-wells - direct evidence for the dominance of interaction effects,” Phys. Rev. Lett. **69**, 2725–2728 (1992). [CrossRef] [PubMed]

### 5.2. Excitation induced effects

64. H. L. 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). [CrossRef] [PubMed]

65. Y. Z. Hu, R. Binder, S. W. Koch, S. T. Cundiff, H. Wang, and D. G. Steel, “Excitation and polarization effects in semiconductor 4-wave-mixing spectroscopy,” Phys. Rev. B **49**, 14,382–14,386 (1994). [CrossRef]

66. H Wang, K. B. Ferrio, D. G. Steel, P. R. Berman, Y. Z. Hu, R. Binder, and S.W. Koch, “Transient 4-wave-mixing line-shapes - effects of excitation-induced dephasing,” Phys. Rev. A **49**, R1551–R1554 (1994). [CrossRef] [PubMed]

67. 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]

**k**

*, creates a coherence. This coherence produces a free decay that radiates in the*

_{b}**k**

*direction. The second pulse,*

_{b}**k**

*, interacts with this polarization to produce a spatially modulated excited state population that acts on the coherence induced by the first pulse. In regions of large population, the original coherence in direction*

_{a}**k**

*begins to decay faster. After some time, there will be a spatial modulation of the coherence, which will change its radiation pattern so that it emits in the signal direction, 2*

_{b}**k**

*-*

_{b}**k**

*. The*

_{a}**k**

*pulse acts twice by both establishing the initial coherence and participating in the formation of the population grating. The*

_{b}**k**

*pulse only participates in forming the grating. This process can be thought of scattering of the initial coherence into the signal direction by the population.*

_{a}*γ*=

_{ph}*γ*

^{0}

_{ph}+

*γ*′

*N*, as is the resonance frequency,

_{ex}*ω*=

*ω*

^{0}+

*ω*′

*N*

_{ex}, where

*N*is the excited state population. The intensity of the TI-TFWM signal is

_{ex}*µ*is the dipole moment of the transition and

*N*is the density of oscillators. From this result it is easy to see that the effects of the local field, EID and EIS are difficult, even impossible, to distinguish in TI-TFWM. If the emitted field, not intensity, were measured, the situation is slightly better as there is a

*π*/2 phase shift between the signal due to EID and those due to EIS and the local field. However, it is not clear how to determine the phase of the signal. It is interesting to note that this expression does not give an intensity dependent decay rate, this is reflection of the fact that the signals due to EID and EIS are intrinsically non-perturbative [68

68. J. M. Shacklette and S. T. Cundiff, “Nonperturbative transient four-wave-mixing line shapes due to excitationinduced shift and excitation-induced dephasing,” J. Opt. Soc. Am. B **20**, 764–769 (2003). [CrossRef]

66. H Wang, K. B. Ferrio, D. G. Steel, P. R. Berman, Y. Z. Hu, R. Binder, and S.W. Koch, “Transient 4-wave-mixing line-shapes - effects of excitation-induced dephasing,” Phys. Rev. A **49**, R1551–R1554 (1994). [CrossRef] [PubMed]

69. V. M. Axt and A. Stahl, “A dynamics-controlled truncation scheme for the hierarchy of density-matrices in semiconductor optics,” Z. für Physik B **93**, 195–204 (1994). [CrossRef]

70. M. Lindberg, Y. Z. Hu, R. Binder, and S. W. Koch, “*χ*^{(3)} formalism in optically-excited semiconductors and its applications in 4-wave-mixing spectroscopy,” Phys. Rev. B **50**, 18,060–18,072 (1994). [CrossRef]

71. M. Z. Maialle and L. J. Sham, “Interacting electron theory of coherent nonlinear response,” Phys. Rev. Lett. **73**, 3310–3313 (1994). [CrossRef] [PubMed]

72. T. Östreich, K. Schönhammer, and L. J. Sham, “Theory of exciton-exciton correlation in nonlinear optical response,” Phys. Rev. B **58**, 12,920–12,936 (1998). [CrossRef]

73. S. R. Bolton, U. Neukirch, L. J. Sham, D. S. Chemla, and V. M. Axt, “Demonstration of sixth-order coulomb correlations in a semiconductor single quantum well,” Phys. Rev. Lett. **85**, 2002–2005 (2000). [CrossRef] [PubMed]

74. V. M. Axt, S. R. Bolton, U. Neukirch, L. J. Sham, and D. S. Chemla, “Evidence of six-particle Coulomb correlations in six-wave-mixing signals from a semiconductor quantum well,” Phys. Rev. B **63**, 115,303 (2001). [CrossRef]

### 5.3. Biexcitons

_{2}molecule. In a GaAs quantum well, the biexciton binding energy is typically around 1 meV [75

75. D. A. Kleinman, “Binding energy of biexcitons and bound excitons in quantum wells,” Phys. Rev. B **28**, 871–879 (1983). [CrossRef]

76. B. F. Feuerbacher, J. Kuhl, and K. Ploog, “Biexcitonic contribution to the degenerate-4-wave-mixing signal from a GaAs/Al_{x}Ga_{1-x}As quantum-well,” Phys. Rev. B **43**, 2439–2441 (1991). [CrossRef]

77. D. J. Lovering, R. T. Phillips, G. J. Denton, and G. W. Smith, “Resonant generation of biexcitons in a GaAs quantum-well,” Phys. Rev. Lett. **68**. [PubMed]

78. S. Bar-Ad and I. Bar-Joseph, “Exciton spin dynamics in GaAs heterostructures,” Phys. Rev. Lett. **68**, 349–352 (1992). [CrossRef] [PubMed]

79. K. Bott, O. Heller, D. Bennhardt, S. T. Cundiff, P. Thomas, E. J. Mayer, G. O. Smith, R. Eccleston, J. Kuhl, and K. Ploog, “Influence of exciton-exciton interactions on the coherent optical-response in GaAs quantum-wells,” Phys. Rev. B **48**, 17,418–17,426 (1993). [CrossRef]

80. E. J. Mayer, G. O. Smith, V. Heuckeroth, J. Kuhl, K. Bott, A. Schulze, T. Meier, D. Bennhardt, S. W. Koch, P. Thomas, R. Hey, and K. Ploog, “Evidence of biexcitonic contributions to 4-wave-mixing in GaAs quantum-wells,” Phys. Rev. B **50**, 14,730–14,733 (1994). [CrossRef]

81. K. B. Ferrio and D. G. Steel, “Observation of the ultrafast two-photon coherent biexciton oscillation in a GaAs/Al_{x}Ga_{1-x}As multiple quantum well,” Phys. Rev. B **54**, R5231–R5234 (1996). [CrossRef]

79. K. Bott, O. Heller, D. Bennhardt, S. T. Cundiff, P. Thomas, E. J. Mayer, G. O. Smith, R. Eccleston, J. Kuhl, and K. Ploog, “Influence of exciton-exciton interactions on the coherent optical-response in GaAs quantum-wells,” Phys. Rev. B **48**, 17,418–17,426 (1993). [CrossRef]

82. T. F. Albrecht, K. Bott, T. Meier, A. Schulze, M. Koch, S. T. Cundiff, J. Feldmann, W. Stolz, P. Thomas, S. W. Koch, and E. O. Göbel, “Disorder mediated biexcitonic beats in semiconductor quantum wells,” Phys. Rev. B **54**, 4436–4439 (1996). [CrossRef]

83. V. M. Axt and A. Stahl, “The role of the biexciton in a dynamic density-matrix theory of the semiconductor band-edge,” Z. Für Physik B **93**, 205–211 (1994). [CrossRef]

### 5.4. Coupling between exciton and continuum

84. D. S. Kim, J. Shah, J. E. Cunningham, T. C. Damen, W. Schäfer, M. Hartmann, and S. Schmitt-Rink, “Giant excitonic resonance in time-resolved 4-wave-mixing in quantum-wells,” Phys. Rev. Lett. **68**, 1006–1009 (1992). [CrossRef] [PubMed]

85. S. T. Cundiff, M. Koch, W. H. Knox, J. Shah, and W. Stolz, “Optical coherence in semiconductors: Strong emission mediated by nondegenerate interactions,” Phys. Rev. Lett. **77**, 1107–1110 (1996). [CrossRef] [PubMed]

86. M. U. Wehner, D. Steinbach, and M. Wegener, “Ultrafast coherent transients due to exciton-continuum scattering in bulk GaAs,” Phys. Rev. B **54**, R5211–R5214 (1996). [CrossRef]

87. D. Birkedal, V. G. Lyssenko, J. M. Hvam, and K. E. Sayed, “Continuum contribution to excitonic four-wave mixing due to interaction-induced nonlinearities,” Phys. Rev. B **54**, 14,250–14,253 (1996). [CrossRef]

88. V. M. Axt, B. Haase, and U. Neukirch, “Influence of Two-Pair Continuum Correlations Following Resonant Excitation of Excitons,” Phys. Rev. Lett. **86**, 4620–4623 (2001). [CrossRef] [PubMed]

### 5.5. Magnetic field effects

89. P. Kner, S. BarAd, M. V. Marquezini, D. S. Chemla, and W. Schafer, “Magnetically enhanced exciton-exciton correlations in semiconductors,” Phys. Rev. Lett. **78**, 1319–1322 (1997). [CrossRef]

90. N. A. Fromer, C. Schüller, D. S. Chemla, T. V. Shahbazyan, I. E. Perakis, K. Maranowski, and A. C. Gossard, “Electronic Dephasing in the Quantum Hall Regime,” Phys. Rev. Lett. **83**, 4646–4649 (1999). [CrossRef]

91. K. M. Dani, J. Tignon, M. Breit, D. S. Chemla, E. G. Kavousanaki, and I. E. Perakis, “Ultrafast Dynamics of Coherences in a Quantum Hall System,” Phys. Rev. Lett. **97**, 057401 (2006). [CrossRef] [PubMed]

### 5.6. Summary

39. M. Koch, J. Feldmann, G. von Plessen, E. O. Göbel, P. Thomas, and K. Kohler, “Quantum beats versus polarization interference - an experimental distinction,” Phys. Rev. Lett. **69**, 3631–3634 (1992). [CrossRef] [PubMed]

40. V. G. Lyssenko, J. Erland, I. Balslev, K. H. Pantke, B. S. Razbirin, and J. M. Hvam, “Nature of nonlinear 4-wave-mixing beats in semiconductors,” Phys. Rev. B **48**, 5720–5723 (1993). [CrossRef]

92. A. Euteneuer, E. Finger, M. Hofmann, W. Stolz, T. Meier, P. Thomas, S. W. Koch, W. W. Rühle, R. Hey, and K. Ploog, “Coherent excitation spectroscopy on inhomogeneous exciton ensembles,” Phys. Rev. Lett. **83**, 2073–2076 (1999). [CrossRef]

48. A. L. Smirl, M. J. Stevens, X. Chen, and O. Buccafusca, “Heavy-hole and light-hole oscillations in the coherent emission from quantum wells: Evidence for exciton-exciton correlations,” Phys. Rev. B **60**, 8267–8275 (1999). [CrossRef]

74. V. M. Axt, S. R. Bolton, U. Neukirch, L. J. Sham, and D. S. Chemla, “Evidence of six-particle Coulomb correlations in six-wave-mixing signals from a semiconductor quantum well,” Phys. Rev. B **63**, 115,303 (2001). [CrossRef]

88. V. M. Axt, B. Haase, and U. Neukirch, “Influence of Two-Pair Continuum Correlations Following Resonant Excitation of Excitons,” Phys. Rev. Lett. **86**, 4620–4623 (2001). [CrossRef] [PubMed]

93. G. Bartels, A. Stahl, V. M. Axt, B. Haase, U. Neukirch, and J. Gutowski, “Identification of Higher-Order Electronic Coherences in Semiconductors by their Signature in Four-Wave-Mixing Signals,” Phys. Rev. Lett. **81**, 5880–5883 (1998). [CrossRef]

94. C. N. Borca, T. H. Zhang, X. Q. Li, and S. T. Cundiff, “Optical two-dimensional Fourier transform spectroscopy of semiconductors,” Chem. Phys. Lett. **416**, 311–315 (2005). [CrossRef]

95. X. Q. Li, T. H. Zhang, C. N. Borca, and S. T. Cundiff, “Many-body interactions in semiconductors probed by optical two-dimensional Fourier transform spectroscopy,” Phys. Rev. Lett. **96**, 057406 (2006). [CrossRef] [PubMed]

96. I. Kuznetsova, P. Thomas, T. Meier, T. Zhang, X. Li, R. P. Mirin, and S. T. Cundiff, “Signatures of many-particle correlations in two-dimensional Fourier-transform spectra of semiconductor nanostructures,” Solid State Comm. **142**, 154–158 (2007). [CrossRef]

97. L. J. Yang, I. V. Schweigert, S. T. Cundiff, and S. Mukamel, “Two-dimensional optical spectroscopy of excitons in semiconductor quantum wells: Liouville-space pathway analysis,” Phys. Rev. B **75**, 125302 (2007). [CrossRef]

98. M. Erementchouk, M. N. Leuenberger, and L. J. Sham, “Many-body interaction in semiconductors probed with two-dimensional Fourier spectroscopy,” Phys. Rev. B **76**, 115307 (2007). [CrossRef]

## 6. Disorder

99. P. W. Anderson, “Absence of Diffusion in Certain Random Lattices,” Phys. Rev. **109**, 1492–1505 (1958). [CrossRef]

100. N. Mott, “The basis of the electron theory of metals, with special reference to the transition metals,” Proc. Phys. Soc. London **62**, 416 (1949). [CrossRef]

101. C. Weisbuch, R. Dingle, A. C. Gossard, and W. Wiegmann, “Optical characterization of interface disorder in GaAs-Ga_{1-x}Al_{x}As multi-quantum well structures,” Solid State Commun. **38**, 709–712 (1981). [CrossRef]

102. C. Lonsky, P. Thomas, and A. Weller, “Optical dephasing in disordered semiconductors,” Phys. Rev. Lett. **63**, 652–655 (1989). [CrossRef] [PubMed]

103. J. Hegarty, L. Goldner, and M. D. Sturge, “Localized and delocalized two-dimensional excitons in GaAs-AlGaAs multiple-quantum-well structures,” Phys. Rev. B **30**, 7346–7348 (1984). [CrossRef]

104. J. Hegarty and M. D. Sturge, “Studies of exciton localization in quantum-well structures by nonlinear-optical techniques,” J. Opt. Soc. Am. B **2**, 1143–1154 (1985). [CrossRef]

105. R. Hellmann, M. Koch, J. Feldmann, S. T. Cundiff, E. O. Göbel, D. R. Yakovlev, A. Waag, and G. Landwehr, “Homogeneous linewidth of excitons in semimagnetic CdTe/Cd_{1-x}Mn_{x}Te multiple-quantum wells,” Phys. Rev. B **48**, 2847–2850 (1993). [CrossRef]

106. T. Takagahara, “Localization and homogeneous dephasing relaxation of quasi-2-dimensional excitons in quantum-well heterostructures,” Phys. Rev. B **32**, 7013–7015 (1985). [CrossRef]

107. T. Takagahara, “Localization and energy-transfer of quasi-2-dimensional excitons in GaAs-AlAs quantum-well heterostructures,” Phys. Rev. B **31**, 6552–6573 (1985). [CrossRef]

108. T. Takagahara, “Excitonic relaxation processes in quantum well structures,” J. Luminescence **44**, 347–366 (1989). [CrossRef]

109. M. D. Webb, S. T. Cundiff, and D. G. Steel, “Stimulated-picosecond-photon-echo studies of localized exciton relaxation and dephasing in GaAs/Al_{x}Ga_{1-x}As multiple quantum-wells,” Phys. Rev. B **43**, 12,658–12,661 (1991). [CrossRef]

110. H. Wang, M. Jiang, and D. G. Steel, “Measurement of phonon-assisted migration of localized excitons in GaAs/AlGaAs multiple-quantum-well structures,” Phys. Rev. Lett. **65**, 1255–1258 (1990). [CrossRef] [PubMed]

111. S. G. Carter, Z. Chen, and S. T. Cundiff, “Echo peak-shift spectroscopy of non-Markovian exciton dynamics in quantum wells,” Phys. Rev. B **76**, 121303 (2007). [CrossRef]

27. L. Schultheis, M. D. Sturge, and J. Hegarty, “Photon-echoes from two-dimensional excitons in GaAs-AlGaAs quantum wells,” Appl. Phys. Lett. **47**, 995–997 (1985). [CrossRef]

112. G. Noll, U. Siegner, S. G. Shevel, and E. O. Göbel, “Picosecond stimulated photon-echo due to intrinsic excitations in semiconductor mixed-crystals,” Phys. Rev. Lett. **64**, 792–795 (1990). [CrossRef] [PubMed]

113. M. D. Webb, S. T. Cundiff, and D. G. Steel, “Observation of time-resolved picosecond stimulated photon-echoes and free polarization decay in GaAs/AlGaAs multiple quantum-wells,” Phys. Rev. Lett. **66**, 934–937 (1991). [CrossRef] [PubMed]

113. M. D. Webb, S. T. Cundiff, and D. G. Steel, “Observation of time-resolved picosecond stimulated photon-echoes and free polarization decay in GaAs/AlGaAs multiple quantum-wells,” Phys. Rev. Lett. **66**, 934–937 (1991). [CrossRef] [PubMed]

114. S. T. Cundiff, H. Wang, and D. G. Steel, “Polarization-dependent picosecond excitonic nonlinearities and the complexities of disorder,” Phys. Rev. B **46**, 7248–7251 (1992). [CrossRef]

115. S. T. Cundiff and D. G. Steel, “Coherent transient spectroscopy of excitons in GaAs-AlGaAs quantum-wells,” IEEE J. Quantum Electron. **28**, 2423–2433 (1992). [CrossRef]

116. H. H. Yaffe, Y. Prior, J. P. Harbison, and L. T. Florez, “Polarization dependence and selection rules of transient four-wave mixing in GaAs quantum-well excitons,” J. Opt. Soc. Am. B **10**(4), 578–583 (1993). [CrossRef]

117. D. Bennhardt, P. Thomas, R. Eccleston, E. J. Mayer, and J. Kuhl, “Polarization dependence of four-wave-mixing signals in quantum wells,” Phys. Rev. B **47**, 13,485–13,490 (1993). [CrossRef]

114. S. T. Cundiff, H. Wang, and D. G. Steel, “Polarization-dependent picosecond excitonic nonlinearities and the complexities of disorder,” Phys. Rev. B **46**, 7248–7251 (1992). [CrossRef]

118. S. Weiser, T. Meier, J. Mobius, A. Euteneuer, E. J. Mayer, W. Stolz, M. Hofmann, W. W. Ruhle, P. Thomas, and S. W. Koch, “Disorder-induced dephasing in semiconductors,” Phys. Rev. B **61**, 13,088–13,098 (2000). [CrossRef]

119. I. Kuznetsova, T. Meier, S. T. Cundiff, and P. Thomas, “Determination of homogeneous and inhomogeneous broadening in semiconductor nanostructures by two-dimensional Fourier-transform optical spectroscopy,” Phys. Rev. B **76**, 153301 (2007). [CrossRef]

104. J. Hegarty and M. D. Sturge, “Studies of exciton localization in quantum-well structures by nonlinear-optical techniques,” J. Opt. Soc. Am. B **2**, 1143–1154 (1985). [CrossRef]

115. S. T. Cundiff and D. G. Steel, “Coherent transient spectroscopy of excitons in GaAs-AlGaAs quantum-wells,” IEEE J. Quantum Electron. **28**, 2423–2433 (1992). [CrossRef]

120. A. G. V. Spivey and S. T. Cundiff, “Inhomogeneous dephasing of heavy-hole and light-hole exciton coherences in GaAs quantum wells,” J. Opt. Soc. Am. B **24**, 664–670 (2007). [CrossRef]

121. F. Jahnke, M. Koch, T. Meier, J. Feldmann, W. Schäfer, H. Nickel, P. Thomas, S. W. Koch, and E. O. Göbel, “Simultaneous influence of disorder and coulomb interaction on photon-echoes in semiconductors,” Phys. Rev. B **50**, 8114–8117 (1994). [CrossRef]

39. M. Koch, J. Feldmann, G. von Plessen, E. O. Göbel, P. Thomas, and K. Kohler, “Quantum beats versus polarization interference - an experimental distinction,” Phys. Rev. Lett. **69**, 3631–3634 (1992). [CrossRef] [PubMed]

## 7. Quantum dots

123. D. Birkedal, J. Bloch, J. Shah, L. N. Pfeiffer, and K. West, “Femtosecond dynamics and absorbance of selforganized InAs quantum dots emitting near 1.3 mu m at room temperature,” Appl. Phys. Lett. **77**, 2201–2203 (2000). [CrossRef]

124. K. L. Silverman, R. P. Mirin, S. T. Cundiff, and A. G. Norman, “Direct measurement of polarization resolved transition dipole moment in InGaAs/GaAs quantum dots,” Appl. Phys. Lett. **82**, 4552–4554 (2003). [CrossRef]

125. A. Mecozzi, J. Mørk, and M. Hofmann, “Transient four-wave mixing with a collinear pump and probe,” Opt. Lett. **21**, 1017 (1996). [CrossRef] [PubMed]

126. P. Borri, W. Langbein, J. Mørk, J. M. Hvam, F. Heinrichsdorff, M.-H. Mao, and D. Bimberg, “Dephasing in InAs/GaAs quantum dots,” Phys. Rev. B **60**, 7784–7787 (1999). [CrossRef]

127. P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong Dephasing Time in InGaAs Quantum Dots,” Phys. Rev. Lett. **87**, 157401 (2001). [CrossRef] [PubMed]

128. W. Langbein and B. Patton, “Heterodyne spectral interferometry for multidimensional nonlinear spectroscopy of individual quantum systems,” Opt. Lett. **31**, 1151–1153 (2006). [CrossRef] [PubMed]

129. D. Birkedal, K. Leosson, and J. M. Hvam, “Long Lived Coherence in Self-Assembled Quantum Dots,” Phys. Rev. Lett. **87**, 227401 (2001). [CrossRef] [PubMed]

130. P. Borri, W. Langbein, U. Woggon, M. Schwab, M. Bayer, S. Fafard, Z. Wasilewski, and P. Hawrylak, “Exciton Dephasing in Quantum Dot Molecules,” Phys. Rev. Lett. **91**, 267401 (2003). [CrossRef]

## 8. Other phenomena

### 8.1. Rabi flopping

131. I. I. Rabi, “Space Quantization in a Gyrating Magnetic Field,” Phys. Rev. **51**, 652–654 (1937). [CrossRef]

132. R. Binder, S. W. Koch, M. Lindberg, N. Peyghambarian, and W. Schäfer, “Ultrafast adiabatic following in semiconductors,” Phys. Rev. Lett. **65**, 899–902 (1990). [CrossRef] [PubMed]

133. S. T. Cundiff, A. Knorr, J. Feldmann, S.W. Koch, E. O. Göbel, and H. Nickel, “Rabi flopping in semiconductors,” Phys. Rev. Lett. **73**, 1178–1181 (1994). [CrossRef] [PubMed]

134. A. Schulzgen, R. Binder, M. E. Donovan, T. Lindberg, K. Wundke, H. M. Gibbs, G. Khitrova, and N. Peyghambarian, “Direct observation of excitonic Rabi oscillations in semiconductors,” Phys. Rev. Lett. **82**, 2346–2349 (1999). [CrossRef]

135. P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Rabi oscillations in the excitonic ground-state transition of InGaAs quantum dots,” Phys. Rev. B **66**, 081306 (2002). [CrossRef]

136. T. H. Stievater, X. Li, D. G. Steel, D. Gammon, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “Rabi Oscillations of Excitons in Single Quantum Dots,” Phys. Rev. Lett. **87**, 133603 (2001). [CrossRef] [PubMed]

137. H. Kamada, H. Gotoh, J. Temmyo, T. Takagahara, and H. Ando, “Exciton Rabi Oscillation in a Single Quantum Dot,” Phys. Rev. Lett. **87**, 246401 (2001). [CrossRef] [PubMed]

138. X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An All-Optical Quantum Gate in a Semiconductor Quantum Dot,” Science **301**, 809–811. [PubMed]

### 8.2. Non-radiative “Raman” coherence

140. M. Lindberg and R. Binder, “Dark States in Coherent Semiconductor Spectroscopy,” Phys. Rev. Lett. **75**, 1403–1406 (1995). [CrossRef] [PubMed]

141. K. B. Ferrio and D. G. Steel, “Raman quantum beats of interacting excitons,” Phys. Rev. Lett. **80**, 786–789 (1998). [CrossRef]

142. S. A. Hawkins, E. J. Gansen, M. J. Stevens, A. L. Smirl, I. Rumyantsev, R. Takayama, N. H. Kwong, R. Binder, and D. G. Steel, “Differential measurements of Raman coherence and two-exciton correlations in quantum wells,” Phys. Rev. B **68**, 035313 (2003). [CrossRef]

143. M. E. Donovan, A. Schülzgen, J. Lee, P.-A. Blanche, N. Peyghambarian, G. Khitrova, H. M. Gibbs, I. Rumyantsev, N. H. Kwong, R. Takayama, Z. S. Yang, and R. Binder, “Evidence for Intervalence Band Coherences in Semiconductor QuantumWells via Coherently Coupled Optical Stark Shifts,” Phys. Rev. Lett. **87**, 237402 (2001). [CrossRef] [PubMed]

144. A. G. V. Spivey, C. N. Borca, and S. T. Cundiff, “Correlation coefficient for dephasing of light-hole excitons and heavy-hole excitons in GaAs quantum wells,” Solid State Commun. **145**, 303–307 (2008). [CrossRef]

## 9. Summary and outlook

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43. | M. Koch, J. Feldmann, G. von Plessen, S. T. Cundiff, E. O. Göbel, P. Thomas, and K. Köhler, “Koch et al. reply,” Phys. Rev. Lett. |

44. | K. Leo, T. C. Damen, J. Shah, E. O. Göbel, and K. Kohler, “Quantum beats of light hole and heavy hole excitons in quantum wells,” Appl. Phys. Lett. |

45. | B. Feuerbacher, J. Kuhl, R. Eccleston, and K. Ploog, “Quantum Beats between the Light and Heavy Hole Excitons in a Quantum Well,” Solid State Commun. |

46. | X. Chen, W. J. Walecki, O. Buccafusca, D. N. Fittinghoff, and A. L. Smirl, “Temporally and spectrally resolved amplitude and phase of coherent four-wave-mixing emission from GaAs quantum wells,” Phys. Rev. B |

47. | M. Phillips and H. Wang, “Coherent oscillation in four-wave mixing of interacting excitons,” Solid State Commun. |

48. | A. L. Smirl, M. J. Stevens, X. Chen, and O. Buccafusca, “Heavy-hole and light-hole oscillations in the coherent emission from quantum wells: Evidence for exciton-exciton correlations,” Phys. Rev. B |

49. | T. Zhang, I. Kuznetsova, T. Meier, X. Li, R. Mirin, P. Thomas, and S. Cundiff, “Polarization-dependent optical 2D Fourier transform spectroscopy of semiconductors,” Proc. Nat. Acad. Sci. |

50. | T. Aoki, G. Mohs, M. Kuwata-Gonokami, and A. A. Yamaguchi, “Influence of Exciton-Exciton Interaction on Quantum Beats,” Phys. Rev. Lett. |

51. | L. Banyai, D. B. T. Thoai, E. Reitsamer, H. Haug, D. Steinbach, M. U. Wehner, M. Wegener, T. Marschner, and W. Stolz, “Exciton-LO-phonon quantum kinetics - evidence of memory effects in bulk GaAs,” Phys. Rev. Lett. |

52. | J. Feldmann, T. Meier, G. von Plessen, M. Koch, E. O. Göbel, P. Thomas, G. Bacher, C. Hartmann, H. Schweizer, W. Schäfer, and H. Nickel, “Coherent dynamics of excitonic wave-packets,” Phys. Rev. Lett. |

53. | M. Koch, S. Cundiff, W. Knox, J. Shah, and W. Stolz, “Magnetoexciton quantum beats: influence of Coulomb correlations,” Solid State Commun. |

54. | T. Meier, A. Schulze, P. Thomas, H. Vaupel, and K. Maschke, “Signatures of Fano resonances in four-wave-mixing experiments,” Phys. Rev. B |

55. | U. Siegner, M. A. Mycek, S. Glutsch, and D. S. Chemla, “Ultrafast coherent dynamics of Fano resonances in semiconductors,” Phys. Rev. Lett. |

56. | S. BarAd, M. V. Marquezini, S. Mukamel, and D. S. Chemla, “Quantum confined Fano interference,” Phys. Rev. Lett. |

57. | Y. H. Ahn, S. B. Choe, J. C. Woo, D. S. Kim, S. T. Cundiff, J. M. Shacklette, and Y. S. Lim, “Quantum Interference of Virtual and Real Amplitudes in a Semiconductor Exciton System,” Phys. Rev. Lett. |

58. | K. Leo, M. Wegener, J. Shah, D. S. Chemla, E. O. Göbel, T. C. Damen, S. Schmitt-Rink, and W. Schäfer, “Effects of coherent polarization interactions on time-resolved degenerate 4-wave-mixing,” Phys. Rev. Lett. |

59. | M. Wegener, D. S. Chemla, S. Schmitt-Rink, and W. Schäfer, “Line-shape of time-resolved 4-wave-mixing,” Phys. Rev. A |

60. | M. Lindberg, R. Binder, and S. W. Koch, “Theory of the semiconductor photon-echo,” Phys. Rev. A |

61. | W. Schäfer, F. Jahnke, and S. Schmitt-Rink, “Many-particle effects on transient 4-wave-mixing signals in semiconductors,” Phys. Rev. B |

62. | S. Weiss, M. A. Mycek, J. Y. Bigot, S. Schmitt-Rink, and D. S. Chemla, “Collective effects in excitonic free induction decay - do semiconductors and atoms emit coherent-light in different ways,” Phys. Rev. Lett. |

63. | D. S. Kim, J. Shah, T. C. Damen, W. Schäfer, F. Jahnke, S. Schmitt-Rink, and K. Köhler, “Unusually slow temporal evolution of femtosecond 4-wave-mixing signals in intrinsic GaAs quantum-wells - direct evidence for the dominance of interaction effects,” Phys. Rev. Lett. |

64. | H. L. 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. |

65. | Y. Z. Hu, R. Binder, S. W. Koch, S. T. Cundiff, H. Wang, and D. G. Steel, “Excitation and polarization effects in semiconductor 4-wave-mixing spectroscopy,” Phys. Rev. B |

66. | H Wang, K. B. Ferrio, D. G. Steel, P. R. Berman, Y. Z. Hu, R. Binder, and S.W. Koch, “Transient 4-wave-mixing line-shapes - effects of excitation-induced dephasing,” Phys. Rev. A |

67. | J. M. Shacklette and S. T. Cundiff, “Role of excitation-induced shift in the coherent optical response of semiconductors,” Phys. Rev. B |

68. | J. M. Shacklette and S. T. Cundiff, “Nonperturbative transient four-wave-mixing line shapes due to excitationinduced shift and excitation-induced dephasing,” J. Opt. Soc. Am. B |

69. | V. M. Axt and A. Stahl, “A dynamics-controlled truncation scheme for the hierarchy of density-matrices in semiconductor optics,” Z. für Physik B |

70. | M. Lindberg, Y. Z. Hu, R. Binder, and S. W. Koch, “ |

71. | M. Z. Maialle and L. J. Sham, “Interacting electron theory of coherent nonlinear response,” Phys. Rev. Lett. |

72. | T. Östreich, K. Schönhammer, and L. J. Sham, “Theory of exciton-exciton correlation in nonlinear optical response,” Phys. Rev. B |

73. | S. R. Bolton, U. Neukirch, L. J. Sham, D. S. Chemla, and V. M. Axt, “Demonstration of sixth-order coulomb correlations in a semiconductor single quantum well,” Phys. Rev. Lett. |

74. | V. M. Axt, S. R. Bolton, U. Neukirch, L. J. Sham, and D. S. Chemla, “Evidence of six-particle Coulomb correlations in six-wave-mixing signals from a semiconductor quantum well,” Phys. Rev. B |

75. | D. A. Kleinman, “Binding energy of biexcitons and bound excitons in quantum wells,” Phys. Rev. B |

76. | B. F. Feuerbacher, J. Kuhl, and K. Ploog, “Biexcitonic contribution to the degenerate-4-wave-mixing signal from a GaAs/Al |

77. | D. J. Lovering, R. T. Phillips, G. J. Denton, and G. W. Smith, “Resonant generation of biexcitons in a GaAs quantum-well,” Phys. Rev. Lett. |

78. | S. Bar-Ad and I. Bar-Joseph, “Exciton spin dynamics in GaAs heterostructures,” Phys. Rev. Lett. |

79. | K. Bott, O. Heller, D. Bennhardt, S. T. Cundiff, P. Thomas, E. J. Mayer, G. O. Smith, R. Eccleston, J. Kuhl, and K. Ploog, “Influence of exciton-exciton interactions on the coherent optical-response in GaAs quantum-wells,” Phys. Rev. B |

80. | E. J. Mayer, G. O. Smith, V. Heuckeroth, J. Kuhl, K. Bott, A. Schulze, T. Meier, D. Bennhardt, S. W. Koch, P. Thomas, R. Hey, and K. Ploog, “Evidence of biexcitonic contributions to 4-wave-mixing in GaAs quantum-wells,” Phys. Rev. B |

81. | K. B. Ferrio and D. G. Steel, “Observation of the ultrafast two-photon coherent biexciton oscillation in a GaAs/Al |

82. | T. F. Albrecht, K. Bott, T. Meier, A. Schulze, M. Koch, S. T. Cundiff, J. Feldmann, W. Stolz, P. Thomas, S. W. Koch, and E. O. Göbel, “Disorder mediated biexcitonic beats in semiconductor quantum wells,” Phys. Rev. B |

83. | V. M. Axt and A. Stahl, “The role of the biexciton in a dynamic density-matrix theory of the semiconductor band-edge,” Z. Für Physik B |

84. | D. S. Kim, J. Shah, J. E. Cunningham, T. C. Damen, W. Schäfer, M. Hartmann, and S. Schmitt-Rink, “Giant excitonic resonance in time-resolved 4-wave-mixing in quantum-wells,” Phys. Rev. Lett. |

85. | S. T. Cundiff, M. Koch, W. H. Knox, J. Shah, and W. Stolz, “Optical coherence in semiconductors: Strong emission mediated by nondegenerate interactions,” Phys. Rev. Lett. |

86. | M. U. Wehner, D. Steinbach, and M. Wegener, “Ultrafast coherent transients due to exciton-continuum scattering in bulk GaAs,” Phys. Rev. B |

87. | D. Birkedal, V. G. Lyssenko, J. M. Hvam, and K. E. Sayed, “Continuum contribution to excitonic four-wave mixing due to interaction-induced nonlinearities,” Phys. Rev. B |

88. | V. M. Axt, B. Haase, and U. Neukirch, “Influence of Two-Pair Continuum Correlations Following Resonant Excitation of Excitons,” Phys. Rev. Lett. |

89. | P. Kner, S. BarAd, M. V. Marquezini, D. S. Chemla, and W. Schafer, “Magnetically enhanced exciton-exciton correlations in semiconductors,” Phys. Rev. Lett. |

90. | N. A. Fromer, C. Schüller, D. S. Chemla, T. V. Shahbazyan, I. E. Perakis, K. Maranowski, and A. C. Gossard, “Electronic Dephasing in the Quantum Hall Regime,” Phys. Rev. Lett. |

91. | K. M. Dani, J. Tignon, M. Breit, D. S. Chemla, E. G. Kavousanaki, and I. E. Perakis, “Ultrafast Dynamics of Coherences in a Quantum Hall System,” Phys. Rev. Lett. |

92. | A. Euteneuer, E. Finger, M. Hofmann, W. Stolz, T. Meier, P. Thomas, S. W. Koch, W. W. Rühle, R. Hey, and K. Ploog, “Coherent excitation spectroscopy on inhomogeneous exciton ensembles,” Phys. Rev. Lett. |

93. | G. Bartels, A. Stahl, V. M. Axt, B. Haase, U. Neukirch, and J. Gutowski, “Identification of Higher-Order Electronic Coherences in Semiconductors by their Signature in Four-Wave-Mixing Signals,” Phys. Rev. Lett. |

94. | C. N. Borca, T. H. Zhang, X. Q. Li, and S. T. Cundiff, “Optical two-dimensional Fourier transform spectroscopy of semiconductors,” Chem. Phys. Lett. |

95. | X. Q. Li, T. H. Zhang, C. N. Borca, and S. T. Cundiff, “Many-body interactions in semiconductors probed by optical two-dimensional Fourier transform spectroscopy,” Phys. Rev. Lett. |

96. | I. Kuznetsova, P. Thomas, T. Meier, T. Zhang, X. Li, R. P. Mirin, and S. T. Cundiff, “Signatures of many-particle correlations in two-dimensional Fourier-transform spectra of semiconductor nanostructures,” Solid State Comm. |

97. | L. J. Yang, I. V. Schweigert, S. T. Cundiff, and S. Mukamel, “Two-dimensional optical spectroscopy of excitons in semiconductor quantum wells: Liouville-space pathway analysis,” Phys. Rev. B |

98. | M. Erementchouk, M. N. Leuenberger, and L. J. Sham, “Many-body interaction in semiconductors probed with two-dimensional Fourier spectroscopy,” Phys. Rev. B |

99. | P. W. Anderson, “Absence of Diffusion in Certain Random Lattices,” Phys. Rev. |

100. | N. Mott, “The basis of the electron theory of metals, with special reference to the transition metals,” Proc. Phys. Soc. London |

101. | C. Weisbuch, R. Dingle, A. C. Gossard, and W. Wiegmann, “Optical characterization of interface disorder in GaAs-Ga |

102. | C. Lonsky, P. Thomas, and A. Weller, “Optical dephasing in disordered semiconductors,” Phys. Rev. Lett. |

103. | J. Hegarty, L. Goldner, and M. D. Sturge, “Localized and delocalized two-dimensional excitons in GaAs-AlGaAs multiple-quantum-well structures,” Phys. Rev. B |

104. | J. Hegarty and M. D. Sturge, “Studies of exciton localization in quantum-well structures by nonlinear-optical techniques,” J. Opt. Soc. Am. B |

105. | R. Hellmann, M. Koch, J. Feldmann, S. T. Cundiff, E. O. Göbel, D. R. Yakovlev, A. Waag, and G. Landwehr, “Homogeneous linewidth of excitons in semimagnetic CdTe/Cd |

106. | T. Takagahara, “Localization and homogeneous dephasing relaxation of quasi-2-dimensional excitons in quantum-well heterostructures,” Phys. Rev. B |

107. | T. Takagahara, “Localization and energy-transfer of quasi-2-dimensional excitons in GaAs-AlAs quantum-well heterostructures,” Phys. Rev. B |

108. | T. Takagahara, “Excitonic relaxation processes in quantum well structures,” J. Luminescence |

109. | M. D. Webb, S. T. Cundiff, and D. G. Steel, “Stimulated-picosecond-photon-echo studies of localized exciton relaxation and dephasing in GaAs/Al |

110. | H. Wang, M. Jiang, and D. G. Steel, “Measurement of phonon-assisted migration of localized excitons in GaAs/AlGaAs multiple-quantum-well structures,” Phys. Rev. Lett. |

111. | S. G. Carter, Z. Chen, and S. T. Cundiff, “Echo peak-shift spectroscopy of non-Markovian exciton dynamics in quantum wells,” Phys. Rev. B |

112. | G. Noll, U. Siegner, S. G. Shevel, and E. O. Göbel, “Picosecond stimulated photon-echo due to intrinsic excitations in semiconductor mixed-crystals,” Phys. Rev. Lett. |

113. | M. D. Webb, S. T. Cundiff, and D. G. Steel, “Observation of time-resolved picosecond stimulated photon-echoes and free polarization decay in GaAs/AlGaAs multiple quantum-wells,” Phys. Rev. Lett. |

114. | S. T. Cundiff, H. Wang, and D. G. Steel, “Polarization-dependent picosecond excitonic nonlinearities and the complexities of disorder,” Phys. Rev. B |

115. | S. T. Cundiff and D. G. Steel, “Coherent transient spectroscopy of excitons in GaAs-AlGaAs quantum-wells,” IEEE J. Quantum Electron. |

116. | H. H. Yaffe, Y. Prior, J. P. Harbison, and L. T. Florez, “Polarization dependence and selection rules of transient four-wave mixing in GaAs quantum-well excitons,” J. Opt. Soc. Am. B |

117. | D. Bennhardt, P. Thomas, R. Eccleston, E. J. Mayer, and J. Kuhl, “Polarization dependence of four-wave-mixing signals in quantum wells,” Phys. Rev. B |

118. | S. Weiser, T. Meier, J. Mobius, A. Euteneuer, E. J. Mayer, W. Stolz, M. Hofmann, W. W. Ruhle, P. Thomas, and S. W. Koch, “Disorder-induced dephasing in semiconductors,” Phys. Rev. B |

119. | I. Kuznetsova, T. Meier, S. T. Cundiff, and P. Thomas, “Determination of homogeneous and inhomogeneous broadening in semiconductor nanostructures by two-dimensional Fourier-transform optical spectroscopy,” Phys. Rev. B |

120. | A. G. V. Spivey and S. T. Cundiff, “Inhomogeneous dephasing of heavy-hole and light-hole exciton coherences in GaAs quantum wells,” J. Opt. Soc. Am. B |

121. | F. Jahnke, M. Koch, T. Meier, J. Feldmann, W. Schäfer, H. Nickel, P. Thomas, S. W. Koch, and E. O. Göbel, “Simultaneous influence of disorder and coulomb interaction on photon-echoes in semiconductors,” Phys. Rev. B |

122. | D. Bimberg, M. Grundmann, and N. N. Ledentsov, |

123. | D. Birkedal, J. Bloch, J. Shah, L. N. Pfeiffer, and K. West, “Femtosecond dynamics and absorbance of selforganized InAs quantum dots emitting near 1.3 mu m at room temperature,” Appl. Phys. Lett. |

124. | K. L. Silverman, R. P. Mirin, S. T. Cundiff, and A. G. Norman, “Direct measurement of polarization resolved transition dipole moment in InGaAs/GaAs quantum dots,” Appl. Phys. Lett. |

125. | A. Mecozzi, J. Mørk, and M. Hofmann, “Transient four-wave mixing with a collinear pump and probe,” Opt. Lett. |

126. | P. Borri, W. Langbein, J. Mørk, J. M. Hvam, F. Heinrichsdorff, M.-H. Mao, and D. Bimberg, “Dephasing in InAs/GaAs quantum dots,” Phys. Rev. B |

127. | P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong Dephasing Time in InGaAs Quantum Dots,” Phys. Rev. Lett. |

128. | W. Langbein and B. Patton, “Heterodyne spectral interferometry for multidimensional nonlinear spectroscopy of individual quantum systems,” Opt. Lett. |

129. | D. Birkedal, K. Leosson, and J. M. Hvam, “Long Lived Coherence in Self-Assembled Quantum Dots,” Phys. Rev. Lett. |

130. | P. Borri, W. Langbein, U. Woggon, M. Schwab, M. Bayer, S. Fafard, Z. Wasilewski, and P. Hawrylak, “Exciton Dephasing in Quantum Dot Molecules,” Phys. Rev. Lett. |

131. | I. I. Rabi, “Space Quantization in a Gyrating Magnetic Field,” Phys. Rev. |

132. | R. Binder, S. W. Koch, M. Lindberg, N. Peyghambarian, and W. Schäfer, “Ultrafast adiabatic following in semiconductors,” Phys. Rev. Lett. |

133. | S. T. Cundiff, A. Knorr, J. Feldmann, S.W. Koch, E. O. Göbel, and H. Nickel, “Rabi flopping in semiconductors,” Phys. Rev. Lett. |

134. | A. Schulzgen, R. Binder, M. E. Donovan, T. Lindberg, K. Wundke, H. M. Gibbs, G. Khitrova, and N. Peyghambarian, “Direct observation of excitonic Rabi oscillations in semiconductors,” Phys. Rev. Lett. |

135. | P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Rabi oscillations in the excitonic ground-state transition of InGaAs quantum dots,” Phys. Rev. B |

136. | T. H. Stievater, X. Li, D. G. Steel, D. Gammon, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “Rabi Oscillations of Excitons in Single Quantum Dots,” Phys. Rev. Lett. |

137. | H. Kamada, H. Gotoh, J. Temmyo, T. Takagahara, and H. Ando, “Exciton Rabi Oscillation in a Single Quantum Dot,” Phys. Rev. Lett. |

138. | X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An All-Optical Quantum Gate in a Semiconductor Quantum Dot,” Science |

139. | H. Giessen, A. Knorr, S. Haas, S. W. Koch, S. Linden, J. Kuhl, M. Hetterich, M. Grun, and C. Klingshirn, “Selfinduced transmission on a free exciton resonance in a semiconductor,” Phys. Rev. Lett. |

140. | M. Lindberg and R. Binder, “Dark States in Coherent Semiconductor Spectroscopy,” Phys. Rev. Lett. |

141. | K. B. Ferrio and D. G. Steel, “Raman quantum beats of interacting excitons,” Phys. Rev. Lett. |

142. | S. A. Hawkins, E. J. Gansen, M. J. Stevens, A. L. Smirl, I. Rumyantsev, R. Takayama, N. H. Kwong, R. Binder, and D. G. Steel, “Differential measurements of Raman coherence and two-exciton correlations in quantum wells,” Phys. Rev. B |

143. | M. E. Donovan, A. Schülzgen, J. Lee, P.-A. Blanche, N. Peyghambarian, G. Khitrova, H. M. Gibbs, I. Rumyantsev, N. H. Kwong, R. Takayama, Z. S. Yang, and R. Binder, “Evidence for Intervalence Band Coherences in Semiconductor QuantumWells via Coherently Coupled Optical Stark Shifts,” Phys. Rev. Lett. |

144. | A. G. V. Spivey, C. N. Borca, and S. T. Cundiff, “Correlation coefficient for dephasing of light-hole excitons and heavy-hole excitons in GaAs quantum wells,” Solid State Commun. |

**OCIS Codes**

(160.6000) Materials : Semiconductor materials

(300.6470) Spectroscopy : Spectroscopy, semiconductors

(320.7150) Ultrafast optics : Ultrafast spectroscopy

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: January 29, 2008

Revised Manuscript: March 15, 2008

Manuscript Accepted: March 15, 2008

Published: March 20, 2008

**Virtual Issues**

Focus Serial: Frontiers of Nonlinear Optics (2007) *Optics Express*

**Citation**

Steven T. Cundiff, "Coherent spectroscopy of semiconductors," Opt. Express **16**, 4639-4664 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-7-4639

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