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
Coherent optical spectroscopy encompasses a powerful set of techniques for studying scattering processes that effect optically created excitations in a variety of systems. The power of the techniques comes from their sensitivity to the phase changes that occur when the optical excitations interact with their environment, including interactions with other optical excitations. Although linear absorption spectroscopy, perhaps the simplest optical measurement, can be considered a coherent experiment, and it can be sensitive to interactions under the right conditions, the effects of scattering are usually masked by a distribution of oscillator frequencies, which is known as inhomogeneous broadening. This difficulty can be circumvented by using nonlinear techniques, which can reveal the effect of scattering even when it is masked in the linear absorption spectrum.
Nonlinear optical spectroscopic techniques are typically classified as working either in the time-domain (also known as transient techniques), where information is obtained by varying the timing of incident optical pulses, or in the frequency-domain, where information is obtained by varying the frequency of incident continuous wave laser beams. However, a strict time versus frequency distinction can be blurred by, for example, spectrally resolving a signal produced using pulses or by Fourier transform techniques. The most important distinction is whether the incident laser beams are pulsed or continuous wave. Transient techniques are typically most sensitive to the fastest dynamics in the sample and are convenient for time-scales of a nanosecond and below, where the relevant delays are easily achieved by time-of-flight in free space.
Direct gap semiconductors are excellent candidates for transient coherent spectroscopy. Optical excitation creates electron-hole pairs, which are expected to experience scattering, from other carriers, phonons or disorder, on picosecond to sub-picosecond timescales. Furthermore, optical excitation can create non-equilibrium carrier distributions, which are expected to relax to equilibrium on sub-picosecond timescales. At low temperature, where phonon scattering is sufficiently suppressed to reveal other processes, the electron-hole pairs can form an exciton, a bound state with a hydrogenic wavefunction for the relative coordinate . Excitons result in strong resonant absorption features just below the gap, and consequently tend to dominate the nonlinear optical response.
Coherent spectroscopy was originally applied to semiconductors with the goal of understanding carrier dynamics, however it was quickly discovered that the coherent optical response was completely dominated by “many-body” effects. Understanding these many-body effects themselves then became the focus of much of the subsequent experimental and theoretical work. This interest was motivated by both fundamental interest in understanding many-body physics [2, 3] and applied interest in improving the ability to model optoelectronic devices such as semiconductor diode lasers .
This paper will give an overview of the work on coherent nonlinear optical spectroscopy of semiconductors that has taken place over the last 20+ years. The discussion will be mainly descriptive, trying to communicate the concepts without getting mired in details, either theoretical or experimental. To compensate for the lack of detail, extensive references will be given to help the interested reader delve deeper. Nevertheless, not every paper in the field can be mentioned and/or referenced. For more extensive discussions, the reader can also see the books by Haug and Koch , by Shah , by Wegener and Schäfer , edited by Takagahara  and by Meier, Thomas and Koch .
To start, typical semiconductor materials and structures will be briefly described. To a large extent an attempt will be made to focus on results that do not depend on the specific properties of individual samples, but rather to elucidate general outcomes. Following the description of materials, the primary methods will be discussed. Again the goal is to present the generic techniques. If alternate or variant methods provided specific insight, they will be discussed later. The discussion of results will start with exciton decoherence, as these were some of the first experiments that stimulated further work. Following these first experiments, a series of results showed the coherent response was very sensitive to many-body interactions. These interactions can be phenomenologically classified, which will guide the presentation. As mentioned above, a key motivation for using coherent spectroscopy is its ability to remove the effects of having an inhomogeneous distribution of resonant frequencies. In semiconductors, such inhomogeneity usually arises from structural disorder. The effects of disorder and its interplay with many-body effects will be discussed. One effect of disorder is to localize the center-of-mass wavefunction, thus a discussion of disorder leads naturally to quantum dots, where structures are purposely grown or fabricated to produce strong localization. Finally, a section for other phenomena will be presented.
Semiconductors can be classified as having a direct or indirect band gap. In a direct gap material, the maximum of the valence band is aligned in k-space with the minimum of the conduction band. Although both absorb light efficiently, typically indirect gap materials emit light very poorly. Thus direct gap materials are usually the focus coherent optical experiments. Gallium arsenide (GaAs) is the canonical example of a direct gap semiconductor and silicon is the most common example of an indirect semiconductor. Most samples discussed here are based on GaAs, including simple bulk GaAs crystals, ternary alloys of GaAs including AlGaAs and InGaAs, and most importantly, heterostructures based on these materials. GaAs and related compounds are known as III–V compounds as they are made up of materials from columns III and V of the periodic table.
At low temperature (<10K), GaAs has a direct band gap of 1.519 eV . Close to the fundamental gap, there is a single conduction band with an effective mass mc=0.0662me and two valence bands, known as the heavy-hole band, with effective mass mhh=0.34m 0, and as the light-hole band, with mlh=0.094m 0 (see Fig. 1). The conduction band (cb) has a spin of , the heavy-hole (hh) valence band has a spin of and the light-hole (lh) valence band has a spin of . The exciton binding energy is 4.1 meV, thus a strong excitonic feature is typically observed in bulk GaAs at 1.515 eV at low temperature. Typically, other bands do not strongly affect the coherent optical response. In unstrained bulk GaAs, the two valence bands are degenerate at 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.
The most common heterostructures are composed of GaAs and AlxGa1-xAs, 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 AlxGa1-xAs is Eg(x)=1.424+1.247x eV at room temperature temperature for x<0.45 . For x>0.45 the lowest band gap of AlxGa1-xAs becomes indirect.
The most common heterostructure is probably a simple quantum well consisting of a thin layer of GaAs sandwiched between two layers of AlGaAs. If the layer is sufficiently thin, quantum confinement will occur in the direction perpendicular to the layer. For a 10 nm thick quantum well, quantum confinement will increase the band gap by about 25 meV for hh-cb transitions and 32 meV for lh-cb transitions. The band offset is 35% in the valence band. Typically Al0.3Ga0.7As 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.
Quantum confinement also increases the exciton binding energy. In two dimensions, the binding energy is 4 times that of three dimensions. However, the finite barrier height means that this limit is not reached, but rather a maximum exciton binding energy of around 10 meV occurs for a 10 nm wide quantum well .
A linear absorption spectrum for a typical quantum well structure at 10 K is shown in Fig. 2. This sample has 10 nm wide wells and 10 nm wide barriers consisting of Al0.3Ga0.7As. 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.
After the III–V materials, the II–VI materials are probably the next most common for study with optical techniques. Several II–VI materials, such as CdTe and ZnTe also have direct gaps and suitable alloys exist for growing heterostructures, including quantum wells. Originally, interest in II–VI materials was mainly motivated by their possible use in green or blue laser diodes, however wide-gap III–V materials, such as GaN, have proven to be better. Interest continues in II–VI materials because it is possible to incorporate Manganese, which has interesting magnetic properties, while maintaining good optical properties.
Finally, it is worth noting that muchwork has been done on semiconductor nanocrytals grown in solution using colloidal chemistry. These nanocrystals are also called quantum dots. This material system will not be discussed here.
Coherent spectroscopy measures decoherence, the loss of optical coherence created by an incident light field. Decoherence is typically due to scattering events. To understand the spectroscopic techniques used to measure the decoherence, we first need to discuss its origins.
Optical excitation of a transition results in a coherent superposition between the ground and excited states. Fluctuations in the energy spacing between the ground and excited states, which correspond to fluctuations in the frequency of the transition, cause decoherence. Decoherence can be considered in terms of an ensemble, where different members in the ensemble get out of phase with one another, or for a single emitter, where a phase fluctuations set finite duration to the emitted wave. The fluctuations in the transition energy result from interactions with the environment, for example when atoms in a vapor collide or when a phonon in a crystal causes the lattice spacing to fluctuate. An elastic scattering event leaves the energy of the system unchanged, whereas inelastic scattering causes energy relaxation. Either case causes decoherence.
3.2. Linear spectroscopy
In certain limited circumstances, simple linear spectra, such as the absorption spectrum, are sensitive to the decoherence rate. Absorption can be treated as destructive interference between the incident light and the light reradiated by the polarization induced in the sample. The polarization is an ensemble sum, thus as individual oscillators undergo scattering processes that change their phase, the polarization decays. Although the decay of the polarization could be measured in the time-domain, typically frequency domain measurements are easier.
The weakness of linear spectroscopy is that it cannot distinguish between decoherence due to scattering and dephasing due to the oscillators having a distribution of frequencies, known as inhomogeneous broadening. In semiconductors, inhomogeneous broadening is usually due to structural disorder, such as fluctuations in the width of quantum wells or inhomogeneous strain. Nonlinear techniques, as described in the following sections, can obtain the decoherence rate due to scattering, even in the presence of inhomogeneous broadening. This ability is the primary motivation for using nonlinear techniques.
3.3. Nonlinear spectroscopy
Nonlinear spectroscopic techniques are based on using sufficiently intense light so that the induced polarization is no longer linearly proportional to the electric field strength of the incident light, but rather higher powers must be considered, i.e.,
where P is the induced polarization, E is the electric field of the incident light and χ (n) is the nth order susceptibility. (More generally to include polarization, 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.
Nonlinear spectroscopy can be performed in either the time domain or frequency domain. For the reasons mentioned above, time domain, also known as transient, techniques are most commonly used to study semiconductors. Thus only transient techniques are described in this section.
In transient spectroscopy techniques, care must be taken that the results are not affected by pulse distortion, which can occur when there is a strong resonant absorption feature, such as the excitonic resonances in semiconductors . As a “rule of thumb”, it is best to keep the absorption depth, αl<1, where α is the Beers law absorption coefficient and l is the effective thickness, i.e., the transmitted intensity is It=Iie -αl for an incident intensity of Ii. 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 .
3.3.1. Two-pulse transient four-wave-mixing
A common coherent technique is two-pulse transient four wave mixing (TFWM). In this method, pulses separated by a delay τ and with wavevectors 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=2k 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 . 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.
Two-pulse TFWM can be physically described as follows: The first pulse creates a coherence (also called a polarization) in the sample. The second converts the coherence to an excited state population that has a spatial sinusoidal modulation with a wavevector 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.
The decay of the TFWM signal as a function of τ is determined by the decoherence rate in the sample, however it depends on whether the sample is homogeneously or inhomogeneously broadened . For a medium of homogeneously broadened two-level systems in the approximation that the incident pulses are infinitely short (i.e., they are delta functions in time) and within the Markovian approximation, the intensity of the TI-TFWM signal as a function of τ is
where γph is the dephasing rate and 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
The ability to measure γph 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.
It is also possible to detect a reflected TFWM signal . In a thin sample, the strength of the reflected signal is comparable to that of the transmitted signal. This signal is emitted in a direction that is the reflection about the sample plane of the ordinary signal direction. Detecting the reflected signal can be very useful in samples that are grown on an opaque substrate that cannot be removed.
3.3.2. Three-pulse transient four-wave-mixing
In three-pulse transient four-wave-mixing (3P-TFWM), three pulses are incident on the sample in directions k a, k b and k c. The nonlinear interaction of the pulses in the sample gives rise to a signal in the direction k s=-k a+k b+k c. The delay between pulse k a and k b is usually designated as τ with τ>0 for k a arriving first. Typically the delay between k b and k c is designate as T with T>0 for k c arriving after k b . There are many different geometries for 3P-TFWM, a few of most of common are shown in Fig. 4.
The physical description of 3P-TFWM is very similar to that of two-pulse TFWM, with the role of the second pulse in two-pulse TFWM being split between the second and third pulses in 3P-TFWM. Specifically, the second pulse interacts with the coherence created by the first pulse to form a population grating and then the third pulse scatters off this grating.
There are several advantages to 3P-TFWM. One advantage is the ability to measure both the dephasing rate, by scanning τ and the relaxation rate of the grating, γgr by scanning T . For a medium of homogeneously broadened two level systems, the time-integrated 3P-TFWM signal is
For an inhomogeneously broadened system, the factor of 2 multiplying γph 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 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, 20, 21].
Another advantage to 3P-TFWM is its ability to measure spatial diffusion. Spatial diffusion will washout the excited state population and thus cause an exponential decay of the signal as a function of 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 γgr=2γpop+8π 2 DΛ-2 where γpop is the population relaxation time, D is the spatial diffusion coefficient and Λ=nλ/2sinθ is the grating spacing for an angle θ between the beams k a and k b. If the first two pulses are coincident in time, this measurement is often known as a “transient grating” experiment.
3.3.3. Spectrally resolved transient absorption
In a transient absorption experiment, a pump pulse injects a population that saturates a transition, thereby decreasing its absorption. The decrease in absorption is sensed by a time delayed probe pulse. The decay of the population is mapped out by scanning the delay of the probe pulse. As the population dynamics are measured, not the coherent dynamics, transient absorption is not considered to be coherent spectroscopy.
However, by spectrally resolving the transmitted pulse, information about the coherent response can be obtained. Specifically, for a resonance the phenomena responsible for the signal can be classified as being saturation, a change in the line width or a change in the line center. However, this is strictly true only in the absence of inhomogeneous broadening.
Surprisingly, information can also be obtained from spectrally resolved transient absorption (SRTA) even when the “probe” pulse precedes the “pump” pulse . In this situation, the free decay induced by the probe pulse is perturbed by the pump pulse, giving rise to a characteristic oscillatory probe spectrum.
It is worth noting that SRTA can be treated theoretically in a manner very similar to TFWM. In both cases, the incident pulses induce a nonlinear polarization in the sample that radiates a signal field. In TFWM, the intensity of this signal field alone is detected. In SRTA, the signal field and the transmitted probe pulse are both detected, with the heterodyne between them typically being isolated by modulating the pump. However, care must taken to consider that there is also a field radiated in the probe direction due to the linear response of the sample, which will also may give a significant heterodyne signal.
3.3.4. Two dimensional Fourier transform spectroscopy
In the last few years, a powerful technique has been developed that combines the best features of TFWM and SRTA. This technique, known as two-dimensional Fourier transform spectroscopy (2DFTS), was inspired by work in nuclear magnetic resonance  and more recently in ultrafast chemistry .
2DFTS is based on 3P-TFWM, but with the enhancement that the phase of the emitted signal is measured and correlated with the phase between the excitation pulses. These phases are shown schematically in Fig. 5. This enhancement is powerful for several reasons: 1) Measuring the signal phase alone is important as it reveals information about phenomenological origin of the coherent response. 2) Correlating the phases allows coupling between transitions to be determined. 3) The various terms that contribute to the nonlinear optical response can separated, and in some cases isolated. 4) The homogeneous and inhomogeneous linewidths can be determined simultaneously and unambiguously. 5) The evolution of non-radiative coherences, such as Raman or two-quantum coherences, can be observed.
The implementation of 2DFTS in the optical regime is challenging because it requires interferometric stability .
4. Decoherence in semiconductors
Coherent spectroscopic techniques were initially applied to semiconductors to determine the dephasing time. Much of this work focused on the exciton resonances as they have dephasing times in picosecond range and large oscillator strength. However, some work was also done on unbound electron-hole pairs. Virtually all of this work was performed with the sample at low temperature (<10K) to suppress dephasing due to phonons. By studying the dependence of the dephasing rate on temperature and carrier density, the contributions from phonon scattering and carrier-carrier scattering could be determined.
One of the earliest demonstrations of a TFWM signal from the exciton resonance in a GaAs quantum well by Hegarty, et al., showed that a strong signal could be obtained, however the dephasing was not determined due to insufficient time resolution . In subsequent work, an upper limit could be determined from a comparison to calculations . The sample displayed significant broadening due to disorder, TR-TFWM experiments showed that the signal was a photon echo. The strong signal obtained in these experiments motivated further studies.
In a subsequent pair of papers, the dephasing time of excitons in bulk GaAs  and in a single quantum well  was measured (see Fig. 6). In both cases, times in the range of a few picoseconds were obtained. In the single quantum well, the dephasing time matched the absorption linewidth, showing that this sample was homogeneously broadened. By varying the temperature, it was also possible to separate the contribution due to phonons, which was determined to be dominated by acoustic phonon scattering. The linear temperature dependence also supported the idea that the excitons were delocalized.
To characterize the excitonic dephasing due to carrier-carrier scattering, the dephasing time was measured in the presence of excitons injected by a pre-pulse or free electrons-hole pairs injected by a CW beam with higher photon energy . Not surprisingly, free electron-hole pairs were much more efficient at dephasing excitons. The presence of dephasing due to excitonexciton scattering means that there are new terms in the nonlinear response, although it was not recognized at the time (see section 5.2). Measurements in mixed type-I/type-II structures showed that bare electrons (without holes) dephased excitons even more strongly .
In parallel with these experimental developments, the theory of the coherent response of semiconductors was being developed [32, 33]. These efforts resulted in the semiconductor Bloch equations (SBE) that have a form similar to the optical Bloch equations. However, these early results were derived within a mean field (Hartree-Fock) type approximation, which was later found to be insufficient to describe many of the experimentally observed effects.
4.2. Free electron-hole pairs
Experiments were also done on band-to-band transitions in GaAs  where the signal was presumed to be a photon echo, although that was not verified. Dephasing times of 20–60 fs were obtained by using 6–10 fs incident pulses. The broad bandwidth of these pulses necessitated them being tuned fairly far up into the band. In later work using 100 fs pulses, signals due to band-to-band transitions, but closer to the bottom of the band, were isolated by increasing the excitation density such that the excitonic response was saturated [35, 36]. In these later studies, the signal was time resolved, demonstrating that a photon echo was present. However the time resolution was insufficient to extract a dephasing time. Measurements where also made on quantum wells using 10 fs pulses .
4.3. Quantum beats and interference effects
When multiple resonances are excited by the incident pulses, beating will occur in the TFWM signal. The beats will occur at the frequency corresponding to difference between the frequencies of the resonances. Beating can be classified as being due simple electromagnetic interference in the light emitted by two uncoupled resonances, often called polarization interference, and quantum beats that are due to coupled resonances where a quantum mechanical oscillation in the wavefunction will occur. The first observation of beating in the TFWM signal from a semiconductor quantum well was attributed to the presence of distinct regions of the quantum well that differed in thickness by one monolayer . In that paper, the beats were described as “quantum beats” although it was actually unclear whether or not the excitons localized in spatially separated areas really constituted a quantum mechanically coupled system. We will return to this point later in the discussion of disorder.
It was found that both TR-TFWM  and SR-TFWM  could distinguish between beats due to polarization interference and true quantum beats. In the presence of inhomogeneous broadening, the correlation between the broadening of the two transitions needs to be taken into account [41, 42, 43]. 2DFTS can also readily distinguish between these two cases as well .
Beats were also observed between the heavy-hole and light-hole excitons in a GaAs quantum well [44, 45]. For colinear excitation, it was clear that these would be quantum beats, and the results were in good agreement with theory. For cocircular excitation, the magnetic substates appear to result in a pair of uncoupled transitions. However, experiments showed that they were indeed coupled, which was attributed to many-body interactions [46, 47, 48]. 2DFTS also showed that they are coupled for co-circular excitation, in agreement with theory predictions . The effect of many-body effects on quantum beats was also observed in GaN .
Oscillatory behavior in the TFWM signal was also observed due to non-Markovian behavior induced by LO-phonons . In order to observe these, pulses with a bandwidth greater than the phonon frequency had to be used. In analogy with molecules, the beats can be described as beats between the lowest two vibrational levels. In a non-Markovian picture, they arise because the two-time correlation function for the frequency fluctuations due to phonons is not monotonic, but rather has a peak at the time delay corresponding to the inverse of the phonon frequency.
An interesting interference effect also occurs when both the exciton and continuum states are excited simultaneously by a broad band pulse . In these experiments, a sample containing InGaAs quantum wells surrounded by GaAs barriers was used. The built-in strain due to the lattice mismatch shifted the light hole exciton to higher energies so that only the heavy hole exciton and its continuum was excited by the incident pulses, which created excitonic noneigenstates. The observed signal had a characteristic temporal modulation, which resembled the coherent dynamics of an electron wave packet in a hydrogen atom in which all bound and the lowest continuum states had been excited. Magnetoexciton wavepackets were also observed using TFWM .
TFWM was also found to be an excellent tool for studying Fano interference in semicondcutors . Fano interference occurs when there is quantum mechanical coupling between a discrete state and an energetically degenerate continuum. It produces a characteristic asymmetric lineshape in the frequency domain. It was observed in bulk GaAs under a high magnetic field . The interference occurred between higher-order magnetoexcitons and the continuum of lower lying Landau levels. The results showed that the decay of TFWM signal was entirely due to quantum interference, not dephasing. Subsequent studies examined how quantum confinement broke up the continuum, causing the Fano interference to disappear .
Interference was observed between the real excitons created by on resonance excitation and virtual excitons created by off resonant excitation . In these experiments, the first pulse was spectrally shaped to have two distinct frequency components with variable spacing. As the frequency difference between these componentswas changed, the phase of the resulting beating was observed to shift, showing the expected phase shift between real and virtual excitations.
5. Many-body signals
The studies presented in the previous section were motivated by the ability of coherent spectroscopy, mainly TFWM, to measure dephasing times. However, anomalies were observed in the signals, the most dramatic of which was the signal for “negative delay” in two-pulse TFWM experiments. Calculations for an ensemble of two-level systems showed that a signal was generated in a two-pulse TFWM experiment only when the pulse that acted twice arrived second , i.e., a signal was only generated in the direction 2k b-k a if k b 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.
5.1. Local fields
The initial observations negative delay signals (see Fig. 7) attributed them to local field effects [58, 59]. Local field effects can easily be incorporated into the optical Bloch equations by replacing the applied electric field, EA, by EA+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 and for negative delay, it will rise as , which was in agreement with the experiments (see Fig. 7).
The generation of a signal for negative delays due to local fields has a straightforward physical explanation. The incident pulses produce a first order polarization that radiates in the same direction as the pulse, i.e., the well known free decay (also known as the “free induction decay” from NMR). The reradiated field can then drive the system on its own. However, this field decays with the dephasing rate, and so can persist long after the pulse is gone. Thus when the k b pulse arrives first, the free decay can persist until after the k a pulse arrives, thereby producing a signal in the direction 2k b -k a.
More sophisticated calculations based on the SBE also showed signals for negative delay [60, 61]. Following the previously developed language, these were also called “local field corrections”, however their origin was not purely due to optical local fields, but also due to manybody effects. Unfortunately, this has engendered some confusion. The calculations predicted that the TR-TFWM signal would show interesting dynamics, as did the optical Bloch equations with a local field correction. These effects were observed experimentally [62, 63].
5.2. Excitation induced effects
The presence of strong excitation induced effects was observed in early measurements of exciton dephasing (see section 4.1). However, it was not realized at the time that these constituted many-body phenomena that resulted in new coherent signals. The fact that a dephasing rate that depended on the excitation level, known as “excitation induced dephasing” (EID), would produce a signal was first discussed in the context of the polarization dependence of the signal from bulk GaAs . Similar effects were observed in quantum wells . Calculations were made based on the optical Bloch equations with a correction to include EID .
While EID was fairly easy to observe, it was readily apparent in the fact that the decay of the TFWM signal depended on the intensity of the incident beams, an excitation induced shift (EIS) in the exciton resonance frequency can also produce a signal. However an EIS is not directly apparent in the TFWM signal in the same way as EID. SR-TFWM experiments provided evidence for contributions from EIS . In this work, SRTA measurements were performed to verify the EIS was present and determine its strength relative to EID.
Physically, the signal produced by EID can be understood in terms of a linewidth grating. Just as for the ordinary TFWM signal (which is due to saturation), the interference between the polarization produced by the first pulse and the second pulse results in a grating in the excited state population. This in turn results in a modulation of the width of the exciton resonance. Increasing the width of a resonance decreases the absorption at line center, while increasing the absorption in the line wings. Thus a spatial modulation in the absorption coefficient still occurs, which diffracts the second pulse into the signal direction. A similar explanation for EIS can be given, except the grating is in the center frequency.
However, the picture of a linewidth grating does not explain why a signal occurs for negative delay, or does it intuitively explain the slow rise of the signal. These effects can understood as follows. The first pulse in time, k b, creates a coherence. This coherence produces a free decay that radiates in the k b direction. The second pulse, k a, 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 k b 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, 2k b-k a. The k b pulse acts twice by both establishing the initial coherence and participating in the formation of the population grating. The k a 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.
The combined effects of local fields, EID and EIS can be treated using modified optical Bloch equations. To include the excitation induced effects, the dephasing rate is assumed to depend linearly on the excited state population, γph=γ 0 ph+γ′Nex, as is the resonance frequency, ω=ω 0+ω′N ex, where Nex is the excited state population. The intensity of the TI-TFWM signal is
where 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 , whereas this expression is obtained from perturbation theory. This problem can be fixed by adjusting the expressions .
Excitation induced effects presented a challenge for the theory as they correspond to manybody correlations beyond a Hartree-Fock approximation. In a Hartree-Fock approximation, the Coulomb interaction is treated as the perturbation. An alternate approach, known as “dynamics controlled truncation” (DCT) treats the light field as the perturbation and keeps all terms to third order in the light field, regardless of the order in terms of the Coulomb interaction [69, 70]. This approach is able to successfully reproduce the experimental observations. An alternate approach used a diagrammatic approach to provide both a quantitative theory and a physical description for a wide range of phenomena observed in TFWM experiments [71, 72].
Biexcitons are bound pairs of excitons, much like an H2 molecule. In a GaAs quantum well, the biexciton binding energy is typically around 1 meV . This shift is small enough so that it is difficult make a definitive assignment in SR-TFWM , although it was pointed out that biexcitons also give rise to signals at negative delay . Evidence for biexcitons was also found in pump-probe experiments using oppositely polarized pump and probe .
The polarization dependence of the TI-TFWM signal also gave evidence for contributions from biexcitons. Detailed calculations were performed using an exciton basis  that agreed well with experiments . The signal at negative delay arises because a non-radiative twoquantum coherence can be excited. The evolution of this non-radiative coherence was monitored using heterodyne detection .
As the biexciton results in a new resonance, it might be expected that it would result in beating in the TFWM signal. However, calculations show that this is not the case for the TITFWM signal from a homogenously broadened sample , although beats corresponding to the biexciton binding energy do show up in the TR-TFWM signal. Inhomogeneous broadening, which is typically due to disorder, can cause beating to show up in the TI-TFWM signal . Disorder can also have the effect of increasing the biexciton binding energy quite substantially. As a consequence, biexcitonic effects are often more apparent in strongly disordered samples.
Biexcitonic effects are also beyond a Hartree-Fock approximation, and thus were not present in the original development of the semiconductor Bloch equations. The development of DCT allowed them to be properly described .
5.4. Coupling between exciton and continuum
Many-body interactions between continuum states, i.e., free electron-hole pairs, and excitons also produce significant effects that become important when broadband pulses are used because they will simultaneously excite both excitons and continuum states. The first evidence for coupling was the fact that emission from the exciton dominates the signal even when spectral overlap with it is minimal . These results were initially explained as being due to the substantial difference in dephasing rates between continuum states and the exciton. While the difference in dephasing rates clearly plays a role, later results showed that coupling was also important.
The importance of coupling between the exciton and continuum states was pointed out in three papers [85, 86, 87] using slightly different variations on TFWM. The first of these papers examined the SR-TFWM signal from a InGaAs/GaAsP sample where the heavy-hole exciton and its continuum were well separated from the light-hole exciton, greatly simplifying the spectra. Using broadband pulses that predominately excited the continuum states produced a signal that was dominated by the exciton resonance [see Fig. 8(a)]. However, the decay of the TI-TFWM was anomalously fast when compared the width the TFWM spectrum. Furthermore, even when the spectrum of the first pulse was filtered so it did not overlap with the exciton but only excited the continuum, the exciton resonance still dominated the spectrum of the TFWM signal [see Fig. 8(b)].
These apparently anomalous results can be understood as being due to many-body coupling between the continuum and the exciton states. Since free-electron hole pairs strongly scatter excitons, resulting in non-degenerate EID and EIS, a spatial grating written into the continuum would be transferred to the exciton, even though it was not directly excited. The resulting line width and line center modulation of the exciton resonance can scatter light from the second pulse, which does overlap it slightly. Later studies showed that even for selective excitation near the 1s exciton resonance, a strong mixing of discrete and continuum degrees of freedom occurs .
5.5. Magnetic field effects
The application of a magnetic field to a semiconductor causes carriers to undergo cyclotron orbits, effectively confining them in the plane perpendicular to the field. The confinement can dramatically influence their interactions. It also results in the formation of Landau levels and quantum Hall physics.
TFWM experiments on exciton resonances in bulk GaAs showed dramatic changes as the strength of an applied magnetic field was increased to 10 T (see Fig. 9) . The magnetic field results in a strong signal for negative delays, a signature of many-body contributions.
The dephasing of electron-hole pairs in the presence of a two-dimensional electron gas was studied using TFWM in the quantum Hall effect regime . As the magnetic field was increased, a cross-over from Markovian to non-Markovian dynamics was observed. The dephasing time underwent jumps at even Landau level filling factors. The observations were qualitatively reproduced by a model based on scattering by the collective excitations of the two-dimensional electron gas. Further experiments were done using 3P-TFWM, which showed a strong oscillatory off-resonant signal . By comparison to a microscopic theory, quantitative information about dephasing and interference of many-particle coherences was extracted.
The results presented in this section have clearly shown that TFWM and all of its variants are sensitive to many-body effects including local fields, excitation induced dephasing, excitation induced shift and biexcitons. Indeed in many cases, the theoretical results show that the signals are completely dominated by many-body terms, and the contributions due to ordinary Pauli-blocking terms are negligible. The negative-delay signals in TI-TFWM support this conclusion as they are absent if many-body terms are neglected and they are often just as strong as the positive-delay signals. However, the various TFWM are not very good at distinguishing amongst the various many-body contributions and determining their relative strengths.
Determining the relative strengths of the various many-body contributions requires more powerful techniques than TI-TFWM. It was recognized early on that adding a second dimension, such as often done in TR-TFWM, where time-resolved signals are measured as function of delay between excitation pulses, or SR-TFWM, where the spectrum is measured as a function of delay, was powerful approach [39, 40]. An alternate approach, known as coherent excitation spectroscopy, performed SR-TFWM using a narrow band first pulse, which was scanned in frequency . Careful analysis of the dependence on the polarization of in incident pulses as well as polarization and temporal resolution  or polarization and spectral resolution [74, 88, 93] of the emitted field can provide separation of the different many-body contributions.
Recently, 2DFTS (see section 3.3.4) has been shown to provide very significant new information. Early magnitude spectra confirmed the importance of many-body coupling between the heavy-hole and light-hole exciton resonances as well as providing clear evidence for the coupling between continuum states and the exciton resonances . By determining the overall phase of the signal, such that the signal could be unambiguously divided into real and imaginary parts, the lineshape of the exciton resonance could determined. The relative strengths of EID and EIS could then be determined because each results in a distinctive lineshape, as confirmed by a phenomenological model . The experimental results triggered the development of many-body theory for 2DFTS [96, 97, 98]. Comparison of the theoretical results clearly showed good agreement, but only when correlation terms beyond Hartree-Fock were included (see Fig. 10) . Most dramatic is the cross-peak between the heavy-hole and light-hole excitons (lower left corner of the spectra in Fig. 10), which is clearly present in the experiment, but does not appear in the theory unless the full calculation is performed. Again, the power of determining the overall phase in 2DFTS is apparent in comparisons of lineshape. The lineshapes of the diagonal features for the heavy-hole and light-hole excitons are only reproduced by the full theory. Magnitude only spectra cannot show dispersive lineshapes, as occurs for the heavy-hole excitons. Phase resolved spectra, but with an arbitrary unknown phase factor, are also ambiguous as the lineshape can be adjusted from dispersive to absorptive by changes in the overall phase. The large amount of information provided by 2DFTS promises further insight into the coherent response of semiconductors.
Disorder can dramatically change the properties of materials, for example transforming a conductor into an insulator. If the disorder results in local regions of lower potential energy, it is not surprising that transport properties will be degraded as charge carriers are trapped. However, even when the disorder results in local regions of increased potential energy, the carriers can also become spatially localized due to interference between forward and backscattered wavefunctions . The localization of the wave-function is important as it means that the Bloch wave-functions of a perfect lattice do not apply and that the density of states will be modified. Often theoretical approaches assume Bloch wave-functions.
The study of disorder and its effects on excitons in semiconductor heterostructures has a two-fold motivation. The first is that disorder is a fact of life, it is present in all real samples. Of course, in some cases it can be ignored if its effects are weak, however it is often dangerous to start from that assumption. Indeed, many experiments have been shown to have essential features due to disorder. The second motivation is that since excitons are neutral particles, certain effects can be clearer than in studies of the conductivity of disordered systems. For charged particles, the Coulomb interaction alone can be sufficient to produce an insulating state .
An important, and usually dominant, form of disorder in semiconductor quantum wells is fluctuations in the width of the well. Even in highest quality samples, fluctuations of one monolayer are inevitable in the transition between the barrier and well materials. In wider regions of the well, the confinement energy will be lower, and the exciton center-of-mass wavefunction will tend to be spatially localized in the potential minimum. Figure 11 shows this schematically. In a multiple quantum well sample, well-to-well fluctuations can also give rise to inhomogeneous broadening, however early measurement showed that intra well fluctuations were important . It is also predicted that disorder could affect the homogeneous linewidth . Studies of the exciton diffusion coefficient using transient gratings and homogeneous linewidth using resonant Rayleigh scattering gave evidence for a mobility edge, separating localized and delocalized states, near the center of the exciton line [103, 104]. Experiments on CdTe quantum wells provided evidence for an upper mobility edge as well .
For localized excitons, migration amongst localization sites is an important relaxation mechanism, leading to dephasing and spectral diffusion. At the lowest temperature, migration occurs due to phonon assisted hopping [106, 107, 108], while at higher temperatures thermal activation to the delocalized states. 3P-TFWM experiments observed the temperature dependence predicted by these mechanisms . These measurements also displayed the signatures of spectral diffusion. Earlier frequency domain four-wave-mixing experiments had also observed the phonon-assisted hopping regime .
Using three pulse echo peak shift measurements, the correlation function for the frequency fluctuations in a disordered quantum well was measured . These measurement showed that the correlation function had an exponential decay at long times and agreed well with a spectral diffusion measurement. At short times, the correlation function actually showed an increase, which could not be reproduced by a spectral diffusion model. Using an underdamped oscillator model, could reproduce the short time behavior, but not the long time. Further theoretical work is needed to provide a comprehensive description.
Since disorder produces inhomongeneous broadening, the TFWM signal should be a photon echo, which was confirmed by TR-TFWM [27, 112, 113]. However, studies of the intensity dependence showed a transition from an echo to a free decay . Most likely this occurs due to changing many-body contributions, however it has not been fully explained.
Disorder was also shown to significantly affect the polarization dependence of the TFWM signal. It was observed that the signal was much weaker, and decayed faster when the excitation pulses were cross-linear as compared to co-linear polarized [114, 115, 116]. A connection between the polarization dependence and the inhomogeneous linewidth was observed, suggesting the disorder played a role . Furthermore, time resolving the signal showed that the signal changed from a photon echo for co-polarized excitation to a free decay for cross-polarized excitation . Eventually these observations where reproduced by calculations that included disorder and correlations beyond Hartree-Fock . 2DFTS promises further insight into these effects .
The presence of a mobility edge close to line center means that the dephasing rate varies significantly across the inhomogeneous distribution [104, 115]. For excitation pulses that cover the entire inhomogeneous distribution, non-exponential decays would be expected as some components decay faster than others . This effect was most evident as a changing in the beat period for HH-LH beats.
The importance of including both disorder and many-body effects was evident when experiments were done to resolve the issue of whether the beats observed from QW regions of different thickness were quantum beats or simply polarization interference . Using TR-TFWM , mixed behavior was observed.
7. Quantum dots
Most of the discussion thus far has focused on quantum wells where the carriers are quantum confined in only one dimension. It is possible to fabricate lower dimensional structures such as quantum wires and quantum dots, with confinement in two and three dimension, respectively. One approach was to start with a quantum well and then use electron beam lithography to produce structures small enough to exhibit quantum confinement in plane of the quantum well. However, such structures usually displayed poor optical quality due to carrier capture by defects induced the by etching. Utilizing the principles of self-organization during epitaxial growth have proven much more successful, particularly for producing quantum dots . As mentioned above, colloidal nanocrystals, which display quantum confinement in three dimensions, will not be discussed here.
Self organized quantum dots form during epitaxial growth when there is a mismatch in the lattice spacing across a heterojunction. Initially, a thin “wetting layer” grows, however as the growth continues, islands form because the resulting decrease in strain energy overcomes the surface energy. These islands have the appropriate dimensions to display quantum confinement in all three dimensions if the island material has a lower bandgap than the surrounding material. While known as “quantum dots”, which implies three dimensional symmetry, they are actually much thinner in the growth dimension than the transverse dimensions, thus they are more like quantum disks. The most common material system is InAs quantum dots grown on GaAs, although other materials display similar behavior. The size and density of the dots can be controlled to some extent by growth conditions. However, there is always a substantial size distribution, which results in strong inhomogeneous broadening of the optical resonance.
Performing optical measurement on self-organized quantum dots was challenging due to their relatively weak absorption being masked by non-resonant contributions from the surrounding matrix. The absorption coefficient, and hence dipole moment, was first measured indirectly in a pump-probe experiment . Subsequently absorption measurements, including the polarization dependence were performed in a waveguide .
Measurements of the TFWM signal from self-organized InAs quantum dots were also challenging. One approach was to use quantum dots embedded in a waveguide, which allowed for an increased interaction distance but required heterodyne detection to separate the signal from the excitation pulses . The first measurements on quantum dots using this technique were done at room temperature . Subsequent low temperature measurements showed that the dephasing time could approach 1 ns . This approach has also been extended to perform 2DFTS on single dots . This long dephasing time made it possible to isolate the quantum dot TFWM signal based on delay alone . By stacking quantum dots on top of each other during growth, it has also been possible to create quantum dot “molecules” .
8. Other phenomena
In this section, several interesting phenomena that do not fit neatly into the previous sections will be presented.
8.1. Rabi flopping
The oscillation of a two level system between the ground and excited states in the presence of a strong resonant driving field, often called transient nutation or Rabi flopping, is a basic quantum mechanical effect and a text book topic today . Rabi flopping is intrinsically a coherent phenomena, it must occur faster than dephasing. The Rabi flopping rate is proportional to the dipole moment and the strength of the driving electric field. Rabi flopping was theoretically predicted to occur in semiconductors, although with a doubling of the flopping rate due to many-body effects .
The experimental observation of Rabi flops in semiconductors was hindered not only by rapid dephasing, but also by the fact that the dephasing times are even shorter at elevated densities. And, to make matters worse in semiconductors, employing shorter pulses increases the hot carrier density, also increasing the dephasing. The interaction of two copropagating pulses provided the first evidence for Rabi flopping in semiconductors . Later work confirmed these results using narrow band pulses .
Rabi flopping has also been observed in self-organized quantum dots , and even in single quantum dots [136, 137]. Quantum dots have received much attention because of their possible use in quantum information processing .
Self-induced transparency is a phenomenon closely related to Rabi flopping. Self-induced transparency occurs when the leading edge of a pulse acts as a π/2 pulse and inverts the medium, while the second half of the pulse also acts as a π/2 pulse, driving the medium back to the ground state. As the medium is left unexcited, no absorption occurs. This process transfers energy from the leading to the trailing edge of the pulse, resulting an effective slowing of the propagation speed. The observation of self-induced transparency on the exciton resonances faced the same difficulties as Rabi-flopping .
8.2. Non-radiative “Raman” coherence
In a three-level system, one level will have a dipole allowed transition to the other two. In an inversion symmetric system, there will not be a dipole allowed transition between the other two levels. However, a coherence can still be established between those two levels, although only to second order in the applied field. As this will be a non-radiative coherence because it does not have a dipole. In a “Λ” system, where there are two lower states and one upper state with both lower states having a dipole allowed transition to the upper state, a coherence between the two lower levels is known as a “Raman” coherence. In a “V” system, where there is a single lower state and two upper states, the coherence between the two upper states is also known as a Raman coherence, although in a strict sense it is not. In a ladder system, where there are three equally spaced states with dipole allowed transitions from the lower state to the middle state and from the middle state to the upper state, a non-radiative two-quantum coherence can be formed between the lower and upper states. Non-radiative coherences are critical to such coherent phenomena such as dark-states and electromagnetically induced transparency.
For excitons in GaAs quantum wells, Raman coherence can occur between the heavy-hole and light-hole excitons. It has been predicted that these states can be used to form a dark-state and generate electromagnetically induced transparency . Two-quantum coherence can form between the ground state and biexciton state, as was discussed in section 5.3.
The first attempt to measure the heavy-hole-light-hole excitonic Raman coherence was made using a transient absorption measurement . It was pointed out that many-body interactions were critical to the observations. Later work separated interaction effects from the Raman coherence . Evidence for inter valence band Raman coherence was also seen the optical Stark effect where pumping the heavy-hole exciton resulted in a shift of the light-hole exciton . By using 3P-TFWM, it was also possible to compare the dephasing rates of the optical and Raman coherences, there by determining the correlation between heavy-hole and light-hole bands during a scattering event .
9. Summary and outlook
Over the last 20 years, the field of coherent spectroscopy of semiconductors has seen a substantial amount of work, which has solved many of the puzzles presented by early results and produced a good understanding of the coherent response of semiconductors near the fundamental bandgap. In particular it is now understood that many body effects dominate the signal. Furthermore, these many body effect require that correlation terms beyond Hartree-Fock are included in the theoretical description. Indeed, optically excited semiconductors are an ideal system for studying many-body effects. The effects of disorder are also better understood, especially in the regime where disorder dominates.
The recent development of optical two-dimensional Fourier transform spectroscopy provides the one of the most exciting new directions for the future. 2DFTS will provide a much more rigorous test of the theory as it can separate the various contributions to the signal and quantitatively compare them. The interplay between many-body effects and disorder, especially in the regime where these effects are of comparable strength, is still not well understood. 2DFTS should be particularly good at providing insight into how many-body interactions compete with localization due to disorder.
Self-organized quantum dots have also attracted significant attention recently, with particular emphasis on the isolated response of single dots. Understanding and controlling the interactions and couplings between dots will be an important next step. Much of the work on quantum dots is motivated by applications in quantum information, where they are often presented as “artificial atoms”. In this context, many-body effects are often seen as undesirable. One has to wonder if it might not be better to develop quantum information schemes that leverage many-body interactions rather than trying to suppress them.
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