Selective Enhancements in 2D Fourier Transform Optical

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Selective Enhancements in 2D Fourier Transform Optical Spectroscopy with Tailored Pulse Shapes Patrick Wen and Keith A. Nelson* Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States S Supporting Information *

ABSTRACT: Spectral features in two-dimensional Fourier transform optical spectroscopy were selectively enhanced using pulse shapes and sequences designed to amplify specific excited-state resonances. The enhancement was achieved by tailoring a small set of input parameters that control the amplitude and phase profiles of the excitation fields, coherently driving or suppressing selected resonances. The tailored pulse shapes were applied to enhance exciton and biexciton coherences in a semiconductor quantum well. Enhancement of selected resonances was demonstrated even in cases of spectrally overlapping features and complex many-body interactions. Modifications in the 2D spectral line shapes due to the tailored waveforms were calculated using the optical Bloch equations.

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amplification of peaks using tailored phase profiles in the first two excitation pulses.7 Atomic rubidium gas provided a good test system because its excited states are energetically well separated relative to their line widths. However, application of coherent control techniques in the 2D FT OPT of many systems of interest, such as molecular aggregates and nanostructured solid state systems, requires selective amplification of peaks that are overlapping. In addition, because the line shapes in 2DS can be distorted in undesirable ways by frequency-dependent phase profiles in the excitation pulses,8 modifications in the line shapes of 2DS due to applied phase profiles must be carefully analyzed. Finally, control techniques to selectively amplify multiplequantum coherences, and not just coherences in single electronic states, have yet to be developed for 2D FT OPT. Here, using only energy levels of the system as input parameters, tailored pulse sequences are designed to optimize specific peaks in the 2DS of gallium arsenide quantum wells (GaAs QWs). Optimal pulse shapes are designed using simple calculations based on perturbative coherent control techniques that have been previously developed in one-dimensional spectroscopy: double-pulse (DP) shapes that selectively enhance different resonance frequencies and pulses with a frequencydependent phase window (PW) that selectively enhance different two-quantum resonances.9,10 The pulse shapes are incorporated in the laboratory using pulse-shaping-based 2D FT OPT, as already demonstrated.5,11−14 As has been extensively studied recently,2,15−20 2DS of GaAs QWs are dominated by

INTRODUCTION Two-dimensional Fourier transform optical spectroscopy (2D FT OPT), as in 2D FT NMR, alleviates congestion in spectra by spreading the information content over two dimensions. Coupled states are directly resolved as cross-peaks and, depending on the excitation pulse sequence, inhomogeneous broadening can be removed and two-quantum coherences can be isolated from emissive states that normally dominate spectra.1 Even with these advantages, broadening of peaks relative to the energy separation between them may still produce twodimensional spectra (2DS) that are difficult to interpret. A state with a weak dipole moment relative to that of a neighboring state may be completely obscured by the line shape of the neighboring peak. These problems are often worse when multiple-quantum coherences are measured because there are often a greater number of multiple-quantum peaks, because correlations often exist among many combinations of singlequantum states, and the peaks are often broader due to faster homogeneous dephasing times.2 The isolation of weak or overlapping peaks may be further improved by tailoring waveform amplitude and/or phase profiles to excite or probe specific resonances, as has been well developed in the strong-field regime for 2D NMR.3 Toward this goal, evolutionary algorithms in 2D FT OPT have been theoretically and experimentally explored.4−6 In these works, the shape of excitation fields are based on adaptive feedback loops that optimize target spectral signatures of a particular sample. Generally, the optimal pulse shapes are not easily adaptable to enhancing spectral signatures in other systems. Tailored pulse shapes that are based on calculations considering a limited number of system parameters may be adaptable to many systems. Such an approach has recently been tested in the 2D FT OPT of atomic rubidium gas by the selective © XXXX American Chemical Society

Special Issue: Prof. John C. Wright Festschrift Received: January 31, 2013 Revised: April 27, 2013

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Figure 1. (a) Diagram showing the relative energy orderings of the ground (g), heavy-hole or light-hole exciton (H or L), and biexciton states (HH, LH, or LL). (b) and (c) Feynman pathways showing coherent excitation of the various exciton and biexciton states in rephasing (b) and two-quantum (c) scans. The first two excitation fields in two-quantum scans are kept time coincident for all 2DS in this work.

surface of a 2D spatial light modulator (2D SLM) (Hamamatsu X8267, 768 × 768 pixels), in which a 2D checkerboard diffractive phase pattern has been specified. The four beams diffracted in the first-order directions are collimated by a lens, generating the BOXCAR geometry. The four beams are spectrally dispersed by a grating and focused by a cylindrical lens onto a second identical 2D SLM, such that the frequency components of each beam are dispersed across a distinct horizontal section of the SLM. As a function of frequency, the 2D SLM can shape the phase and amplitude profiles of each beam individually. This enables a 2D FT OPT measurement to be controlled entirely by the SLM, with complete phase coherence among all the beams that is not lost when time delays between pulses are changed. In addition to pulse delays and phase cycling, as previously utilized,16,17,20,22 the 2D SLM generates the tailored pulse shapes necessary for coherent control. A vertical sawtooth pattern is applied down the entire 2D SLM display, the four dispersed beams in the first direction order are picked off by a mirror, and their shaped frequency components are recombined at the grating. Finally, the four beams are focused onto the GaAs QW sample, which is cooled in a coldfinger cryostat to 10 K. Three of the beams, in directions ka, kb, and kc, are used to generate signals in the fourwave mixing (FWM) direction, ka + kb − kc, overlapped with the fourth beam, which is used as a local oscillator (LO). The time ordering of the pulses in different beams depend on the type of 2DS obtained. The superposed signal and LO beams are sent to a spectrometer to determine the full complex FWM signal field using spectral interferometry. The sample structure consists of ten layers of 10 nm thick GaAs sandwiched between 10 nm thick Al0.3Ga0.7As. In Figure 1a, the relative energy orderings are shown of the ground state, g, the single-exciton heavy-hole, H, and light-hole, L, states, and the

many-body effects, giving rise to peaks due to different biexciton states, unbound two-exciton correlations between various combinations of single-exciton states, and even triexciton states. All of these multiexciton coherences may interfere with each other and single-exciton peaks. The many-body effects also produce distortions in the line shapes of 2DS. Despite these complications, selective amplification of final emission pathways and signals from biexciton states are demonstrated using DP and PW shapes, respectively. The effects of the shaped pulses on the line shapes of target peaks are considered by comparison with experimental 2DS using Gaussian pulses and calculated 2DS using optical Bloch equations. The paper is organized as follows. In the Methods section, the experimental setup that was used to obtain rephasing and twoquantum 2D spectra with and without tailored pulse shapes is discussed. The Methods section also details the mechanisms of the DP and PW shapes as well as the selection of parameters needed to selectively enhance peaks in 2D spectra. The theoretical methods to calculate the phase shifts caused by the PW shapes are also described in the Methods section. In the Results and Discussion section, rephasing and two-quantum 2D spectra with and without tailored pulse shapes are discussed. The selective enhancement of different emission peaks using the DP shapes is demonstrated in rephasing and two-quantum 2D spectra and the selective enhancement of two-quantum peaks using the PW shapes is demonstrated in two-quantum 2D spectra.

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METHODS Experimental Setup. The experimental setup has been described in detail elsewhere.21 Transform-limited 100 fs pulses from a 92.5 MHz Ti:sapphire oscillator are focused onto the B

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strengths, energies, and line widths, the emission from one state may dominate over that from another, obscuring the spectroscopic observation of some states. In the Feynman pathways in Figure 1, the final emission is represented by the final curvy arrow on the left side of all the pathways. In both rephasing and two-quantum pathways, there are several emission pathways that may overlap. For example, the rephasing pathways A1 and A2 are generally well separated because the energy difference between the two emission pathways, given by the energy difference between the H and L excitons, is large relative to the line widths of the peaks. However, the emission from the twoquantum pathways B1 and B3 are strongly overlapped because the energy difference between the two emission pathways is only given by the binding energy of the HH biexciton state, which is on the order of the line widths of the peaks. Selection for different emission pathways can be accomplished using phase-controlled pulse pairs or longer sequences that induce constructively or destructively interfering coherent oscillations in the system, a technique that is ubiquitous in the coherent control literature.23−25 In particular, using a DP for the final field interaction of a third-order optical spectroscopy measurement can enable selective enhancement and suppression of different emission pathways.9 A DP, as shown in Figure 2a, is

biexciton states composed of two H excitons, HH, two L excitons, LL, and a H and L exciton, HL. Depending on the biexciton state, the biexciton energy is reduced by a binding energy relative to the combined two-exciton energies. Polarization selection rules may be used to isolate specific exciton and biexciton resonances, but because we are interested in demonstrating the selectivity of the DP and PW pulse shapes, we utilize only collinear (all parallel polarizations) or cross-linear (vertical and horizontal polarizations for the first two fields and for the third and LO fields, respectively) for all experiments. In both polarization configurations, all exciton and biexciton states may be excited. Additionally, in the collinear polarization configuration, unbound two-exciton correlations can also give rise to peaks in the 2DS at energies equal to the sum frequencies of the two excitons, a larger energy than the biexciton peaks by the biexciton binding energies. In the cross-linear polarization excitation scheme, the unbound correlations are suppressed. Depending on the time ordering of the electric fields, rephasing or two-quantum 2DS are recorded. In rephasing 2DS, an electric field Ec in the conjugate beam, with wavevector kc, excites coherences in the sample that oscillate during a time period τ1. After τ1, the fields in the nonconjugate beams, Ea(ka) and Eb(kb), interact with the sample, creating coherences with phase evolution of opposite sign to the coherences during τ1. These final coherences emit a third-order electric field that is superposed with the LO and detected by a spectrometer. By correlation of the emitted frequencies to frequencies of modulations in the spectral interference pattern as a function of τ1, a complex rephasing 2D spectrum can be recorded. Feynman pathway diagrams showing the coherent excitation of exciton states in a rephasing scan are shown in Figure 1b. The first and third fields can excite coherences of the same exciton state (pathways A1 and A1′) or different exciton states (pathways A2 and A2′), resulting in diagonal and off-diagonal peaks, respectively. Additionally, red-shifted shoulders to the peaks A1, A1′, A2, and A2′, due to emission from the various biexciton states to single-exciton states, are anticipated in Figure 1b as pathways A3, A3′, A4, and A4′. In a two-quantum 2DS, the nonconjugate beams, Ea(ka) and Eb(kb), excite the sample first, creating two-quantum coherences that oscillate during τ2 before the third field, Ec(kc), excites coherences in the sample that emit a third-order electric field in the signal direction. The signal is superposed with the LO and a complex 2D spectrum is recorded by varying the delay of Ec(kc) relative to the first two fields. Feynman pathway diagrams in Figure 1c describe the coherent excitation of biexciton states. After excitation of a biexciton state, the biexciton can be converted into a single exciton coherence with the ground state, as in pathways B1, B1′, B2, and B2′, leading to emission from H or L, or a coherence between the biexciton and a single-exciton state can be excited, as in pathways B3, B3′, B4, and B4′, leading to red-shifted shoulders to the first four pathways. Different Feynman pathways contributing to the rephasing and two-quantum 2D spectra can be selectively enhanced using the DP and PW shapes, as discussed in the next two subsections. The DP can be used to selectively enhance between singlequantum coherences, such as the final emission pathways in either the rephasing or two-quantum 2D spectra. The PW pulse shapes can be used to selectively enhance between different twoquantum coherences in the two-quantum 2D spectra. Double Pulses. In 2D spectra, as in many nonlinear optical experiments, emission from different excited states may contribute to the observed spectrum. Depending on the dipole

Figure 2. (a) Single Gaussian pulse shaped into a double pulse with a well-defined pulse delay, τDP, and phase difference, ϕDP. (b) Two nonresonant excitation pathways (left) normally interfere destructively but can be made to interfere constructively by placing a phase window in the excitation fields (right), enhancing the two-quantum signal.

defined as two Gaussian shaped pulses that are separated by a time difference, τDP, and a relative phase difference between the two pulses, ϕDP. In this case, the third-order density matrix of the system can be written as ρ(3)(t ) −i = 3 ℏ

∞

∞

∫0 ∫0 ∫0

∞

dτ3 dτ2 dτ1 E(t − τ3 − τ2 − τ1)

× E(t − τ3 − τ2)[Ei(t − τ3 + τDP/2) + Eii(t − τ3 − τDP/2) exp( −iϕDP)]R(3)(τ1 ,τ2 ,τ3)

(1)

where the polarization and wave vector dependence of the fields has been suppressed and R(3) is the third-order response function. After the first pulse in the DP, an induced coherence accumulates a phase during τDP. The coherence can be made in or out of phase with a second coherence induced by the second pulse through control over ϕDP, amplifying or suppressing the C

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first field interaction detuned to the red from ωig (path 1 in Figure 2b) and one where the first field interaction is detuned to the blue from ωig (path 2 in Figure 2b). In both nonresonant pathways, the second interaction must be detuned from ωfi such that the total energy of the two-photon process is still given by ωig + ωfi. Although nonresonant pathways can contribute to ρ(2), the two nonresonant pathways are π out of phase: as apparent from eq 2, the contributions have opposite signs and the net contribution to ρ(2) is zero. However, applying a π/2 phase offset between ωig and ωfi in the two excitation fields, as shown on the right side of Figure 2b, results in a net phase factor of π in only path 2 so that the two nonresonant pathways contribute constructively to ρ(2). As shown by Dudovich and co-workers, the two-photon signal amplitudes in rubidium gas can be increased 7-fold compared to the amplitudes induced by ordinary Gaussian pulses. By application of a PW shape in the first two electric fields of a two-quantum scan, different two-quantum coherences can be selectively enhanced in two-quantum 2DS. To demonstrate the selectivity of the PW shape, the parameters of the PW were chosen to enhance the HL biexciton by applying a π/2 phase shift over the range of frequencies between the two transition energies needed to excite the HL biexciton in the Feynman pathways B2, B2′, B4, and B4′ shown in Figure 1c. Specifically, a π/2 phase shift is applied between the frequency of the H exciton, ωH, and the frequency needed to excite the H exciton to the HL biexciton, ωHL − ωH = ωL − δHL. For coherent measurement of two-quantum states, not only is the amplitude of ρ(2) important but also the phase. The work by Dudovich and co-workers demonstrated the increase in twophoton absorption by incoherent emission from a neighboring high-energy single-photon state. Consequently, modifications to the phase of ρ(2) due to the PW in the excitation fields were not characterized. As discussed below, the effect of the PW on the phase of the two-quantum coherence is a calculable constant phase factor for each of the peaks of interest. Because 2D FT OPT allows direct observation of the two-quantum coherence phase, we are able to experimentally verify this result, as shown below. Theoretical Modeling. To simulate the effects of pulse shaping on the 2DS, and in particular to understand possible phase distortions caused by the PW shape, 2DS are modeled incorporating Gaussian shapes in the calculations. There are several approaches used to model 2DS that describe the time evolution of all density matrix elements of a model Hamiltonian.15,26,27 Here, differential equations for each density matrix term are derived (i.e., Bloch equations28) and solved numerically. As shown in the Supporting Information, the Bloch equations can be derived starting with a three-level Hamiltonian representing the ground, exciton (H or L), and biexciton (HH, HL, or LL) states and including one-photon dipole-allowed transitions between the ground and exciton states and between the exciton and biexciton states. The Hamiltonian is inserted into the Louiville equation, giving differential equations for all density matrix terms. To isolate the signal in the third-order direction, giving the four-wave mixing signal in which each of the three incident excitation beams provides one field interaction, the density matrix terms are expanded into spatial Fourier components and the FWM signal is separated.29 Although phenomenological density-dependent terms are often added to simulate many-body interactions in QWs,15 these terms were neglected in this work because the primary goal of the

peak in the 2DS corresponding to this specific coherence. If there are two neighboring peaks, selective amplification of one peak and suppression of the other can be achieved by selecting τDP such that the phases accumulated at the frequencies of the two coherences are π out of phase with each other and selecting ϕDP to enhance the desired peak. Shaping the third electric field to be a DP can selectively enhance Feynman pathways with different emission frequencies. A full theoretical description of the selectivity of different Feynman pathways using DP shapes is given in the Supporting Information. Similar strategies are possible with any of the incident fields. To demonstrate the selectivity of the DP technique, the final electric field in rephasing and two-quantum scans were shaped into a DP. For the rephasing scans, the parameters of the DP were chosen to discriminate between the Feynman pathways in Figure 1 that emit out of the H exciton coherence with the ground state (A1 and A2′) and the Feynman pathways in Figure 1 that emit out of the L exciton coherence with the ground state (A2 and A1′). For the two-quantum scans, the parameters of the DP were chosen to discriminate between the two-quantum Feynman pathways in Figure 1 that emit out of the H exciton (B1) and the Feynman pathways in Figure 1 that emit out of the HH biexciton coherence with the H exciton (B3). Phase Window Pulse Shapes. Two-quantum coherences that are measured in two-quantum 2DS are often obscured by other two-quantum coherences that overlap. For example, three biexciton states exist in GaAs QWs that may overlap in the twoquantum frequency domain: the HH biexciton (excited in the pathways B1 and B3 in Figure 1), the HL biexciton (excited in the pathways B2, B2′, B4, and B4′ in Figure 1), and the LL biexciton (excited in the pathways B1′ and B3′ in Figure 1). In GaAs QWs, the spectral crowding is made worse because of unbound two-exciton coherences and free carrier scattering caused by many-body interactions.2,16,17 Dudovich and co-workers have shown that selectivity in the excitation of two-quantum coherences, assuming a resonant intermediate state, may be achieved by applying a PW shape to the two electric fields used in the two-photon excitation.10 A PW shape, shown on the right side of Figure 2b, is defined by a π/2 phase offset over a range of frequencies in the electric field. The selective enhancement is based on coherent addition of contributions to the two-quantum coherence that is produced by the two field interactions. Considering a three-level ladder system, shown on the left of Figure 2b, with one-photon transitions allowed from the ground state, g, to the intermediate state, i, and from the intermediate to the final state, f, the secondorder density matrix can be written as10 ρfg(2) =

⎡ −1 ⎢iπE(ωig ) E(ωfg −ωig ) μ μ iℏ2 fi ig ⎢⎣ ∞

+ CPV

∫−∞ dω

E(ω) E(ωfg −ω) ⎤ ⎥ ωig − ω ⎥⎦

(2)

where E(ω) is an excitation field that interacts twice with the sample, μmn and ωmn are the dipole and frequency of the transition from state n to state m, and CPV is the Cauchy principle value. The first term on the right-hand side of eq 2 describes a resonant process, where the components of the fields resonant with ωig and ωfi contribute to the density matrix. The second term describes two nonresonant pathways: one with the D

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relative amplitudes of the laser spectrum, shown in Figure 3a, at the two peak frequencies) due to the relative transition dipole strengths of the H and L states, are found along the diagonal. The shared ground state leads to off-diagonal peaks between the H and L states. Red-shifted shoulders to all the peaks are attributed to excited-state absorption to the biexciton state (pathways A3, A3′, A4, and A4′ in Figure 1b). Two-quantum 2DS of GaAs QWs are shown in Figure 3c,d for cross-linear and collinear polarization conditions and are consistent with previous 2DS of the system.18 Peaks appear corresponding to the excitation of various biexcitons and emission out of H or L as labeled by the Feynman pathway diagrams in Figure 1c. The peak corresponding to HL emitting out of H is obscured slightly by a vertical stripe corresponding to scatter from free carriers into the H exciton state,30 which is not completely suppressed in the cross-linear polarization. Selective Enhancements Using DP Shapes. The relative intensities of peaks can be easily manipulated by replacing a single Gaussian pulse with a DP. To demonstrate this capability, the peaks in a rephasing spectrum that appear due to emission from H (pathways A1 and A2′ in Figure 4b) and emission from L

calculations was to simulate the changes in signal phase and amplitude due to due to pulse shaping using the simplest approach possible, not to simulate all the contributions to 2DS. As seen below, neglecting the many-body interactions does not prevent accurate prediction of the phase shifts caused by the PW pulse shapes.

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RESULTS AND DISCUSSION 2DS without Selective Enhancements. A rephasing 2D spectrum of GaAs QWs is shown in Figure 3b, consistent with previous 2DS of the system.15 Peaks corresponding to the H and L states, with an intensity ratio of 9:1 (after normalization for the

Figure 4. (a) Rephasing 2D spectrum recorded under the same conditions as Figure 3b except a double pulse is used in the last field to enhance H emission. (b) Same as (a) except the double pulse parameters are selected to amplify L emission. (c) Two-quantum 2D spectrum recorded under the same conditions as Figure 3c except a double pulse is used to enhance pathway B1. (d) Same as (c) except the double pulse is used to enhance pathway B3.

(pathways A2 and A1′) were selectively enhanced or suppressed. In Figure 4a, τDP = 170 fs and ϕDP = 2πωHτDP. In Figure 4b, τDP is the same as in Figure 4a but ϕDP has been shifted by π. The ratio of the diagonal peak amplitudes, corresponding to H and L emission pathways, was switched from 31:1 to 1:1, compared to the ratio of 9:1 using the normal pulse sequence shown in Figure 3a. The control shown in Figure 4a,b can be understood by considering the interference between the different Feynman pathways that contribute to the 2DS. As with the rephasing 2DS without spectral enhancements, three electric fields along ka, kb, and kc are sent to the sample. The conjugate beam, Ec(kc), and

Figure 3. (a) Spectral intensity of optical fields used to excite the sample. (b) Rephasing 2D spectrum taken in the collinear polarization configuration, without double pulse or phase window shaped pulses. (c) Two-quantum 2D spectrum taken in the cross-linear polarization configuration, without double pulse or phase window shaped pulses. (d) Same as (c) except electric fields are in the collinear configuration. Features in the spectra are labeled according to the Feynman pathway diagrams in Figure 1b,c. The color bar in (d) is used for all figures in this paper. E

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one of the nonconjugate beams, Ea(ka), are normal Gaussian pulses but the third electric field, Eb(kb), is shaped into a DP. In a subset of the excitons initially excited by Ec(kc) and Ea(ka), the first pulse in the DP initiates coherences oscillating at either the H exciton frequency ωH, pathways A1 and A2′, or the L exciton frequency ωL, pathways A2 and A1′. After the delay between the two pulses in the DP, the phase difference between the excitons in H and L emission pathways is given by Δϕ = 2π(ωL − ωH)τDP. The value of τDP = 170 fs is chosen so that Δϕ = π. To enhance the H emission pathways, the second pulse in the DP must excite coherences that are in phase with the coherences started by the first pulse in the DP; i.e., the phase of the second pulse is chosen to match the phase of the H emission pathways started by the first pulse, ϕDP = 2πωHτDP. Simultaneously, the L emission pathways are suppressed due to the selection of τDP. It is important to note that the signal is detected in the four-wave mixing direction so that each of the three beams incident on the sample at wavevectors ka, kb, and kc contributes exactly one field interaction with the sample. Thus a subset of the excitons generated by the first two fields is acted upon by the first pulse in the Eb DP, and another subset of the excitons is acted upon by the second pulse in the Eb DP. It is the interference between the different subsets excited by the different pulses in the DP that gives the enhancement or suppression of the signal. To suppress the H emission pathways and amplify the L emission pathways, ϕDP is flipped by π. Selective enhancement between two peaks using the DP is useful even if the target peaks overlap, such as the two peaks given by Feynman pathways B1 and B3 in Figure 1c, which are separated only by the HH biexciton binding energy in the emission dimension. To selectively enhance either of these peaks, the third electric field in a two-quantum scan, Ec(kc), is shaped into a DP with τDP = 700 fs and ϕDP = 2πωHτDP to enhance pathway B1 or ϕDP = 2πωHτDP + π to enhance pathway B3. Because the energy separation between these two transitions, given by the binding energy of HH, is much smaller than the energy separation between the H and L transitions to the ground state, τDP = 700 fs is much longer than τDP used to distinguish between H and L emission. Even in this case of overlapping peaks, the ratio of the amplitudes of the two pathways can be switched from 3:4 to 7:4. A high energy tail appears in the emission of Figure 4d. Because there are no peaks expected at this position, the high energy tail is most likely due to constructive interference of the emission from the high energy side of the H exciton line width in pathway B1. The selectivity between two overlapping peaks is reduced compared to two peaks that are well separated because, to reach a π phase shift between these two coherences, τDP must be comparable to their dephasing times and the coherences excited by the first pulse in the DP decay during τDP. Assuming two peaks of equal line widths, γ, and an energy separation of δω, the DP selectivity, measured by the relative change in ratio of the two peak amplitudes, decays as exp(−πγ/2δω) if 2πδωτDP = π. The DP is therefore a useful approach to selectively amplify or suppress neighboring peaks within the limitations determined by γ and δω. In a two-quantum 2D spectrum, γ is due to both homogeneous and inhomogeneous broadening. However, in a rephasing scan the inhomogeneity of a sample is isolated along the diagonal of the 2D spectrum so that the DP technique is only limited by the homogeneous broadening. Furthermore, because the 2D rephasing spectrum can be thought of as the sum of Feynman pathways with an inhomogeneous distribution of absorption and emission energies,1 the parameters τDP and ϕDP

can be chosen in a rephasing scan to selectively enhance different contributions of the inhomogeneous distribution. Such selectivity could be accomplished by choosing a portion of the inhomogeneous distribution to be enhanced and a portion to be suppressed. The central frequencies of these two portions could then be used to choose τDP and ϕDP to selectively enhance a portion of the inhomogeneous distribution. However, the GaAs QW sample used in this paper has comparable inhomogeneous and homogeneous line widths, as seen by the nearly round peaks in the rephasing spectra in Figure 3a, so the DP shapes used in this paper do not discriminate between different contributions to the inhomogeneous distribution. Selective Enhancements Using PW Shapes. The PW is able to selectively enhance two-quantum peaks, even in the presence of overlapping signals. As an example, the HL peak is selectively enhanced in a two-quantum 2D spectrum despite the fact that it overlaps with an elongated peak due to free-carrier scattering in the H state. To enhance the HL peak, a π/2 phase shift is applied to fields Ea(ka) and Eb(kb), between the H → HL transition frequency, ωHL−H, and the g → H transition frequency, ωH. The magnitude 2D spectrum obtained using PW pulses is shown in Figure 5. Relative to the magnitude of the HH peak, the

Figure 5. Two-quantum 2D spectrum taken under the same conditions as Figure 3d except that a phase window is applied to the first two excitation fields.

magnitude of the HL peak increases nearly 3 times, corresponding to an enhancement in the electric field intensity by 9 times, similar to the enhancements measured in Rb gas.10 A binding energy of 1.7 meV is measured for the HL peak, in agreement with previous results.17 Because the PW is only in the spectral region between the two transition energies, ωHL−H and ωH, the resonant contribution to eq 2 is unaffected by the PW shape. However, the nonresonant contribution is greatly enhanced by constructively interfering the pathways that are blue-detuned and red-detuned from ω H . The selective amplification of the HL peak is demonstrated despite the presence of many-body interactions that generate overlapping signal, specifically the peak representing free-carrier scattering into the H exciton state. The PW is tailored to specifically enhance two-photon transitions with single-photon energies tuned to the HL transitions. The PW also causes a phase shift in the signal. The real parts of the two-quantum 2DS generated with and without PW pulse shapes are shown in Figure 6, which differ from the usual dispersive line shape due to phase shifts caused by many-body interactions.15 The imaginary parts were also determined, and the phase of the 2DS at any frequency was found from the real and imaginary parts of the 2DS at that frequency. A cut of the phase of the 2DS along the two-quantum energy at the energy of F

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particularly relevant for systems in which overlapping or weak features prevent clear elucidation of coupling mechanisms or multiexciton peaks. More generally, the present demonstration suggests the use of other types of shaped waveforms in thirdorder and higher-order multidimensional spectroscopy.

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ASSOCIATED CONTENT

S Supporting Information *

Full theoretical descriptions of the double pulse and phase window shapes, including the equations used to calculate phase shifts in the two-dimensional spectra. This material is available free of charge via the Internet at http://pubs.acs.org/.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The work was supported as part of the Excitonics Center, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DESC0001088. The authors thank S. Cundiff for the QW sample.

Figure 6. (a) Real part of the two-quantum 2D spectrum shown in Figure 3b. (b) Real part of the two-quantum 2D spectrum shown in Figure 5. (c) Calculated and experimental phase shifts in 2DS due to phase windows of two different spectral widths in the optical fields.

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the HL biexciton is used to find the change in phase at that same cut when a PW is applied, as shown in Figure 6c for two different PW widths. As can be seen, the change in phase depends on the width of the window used. A shift of approximately π/2 at the HL peak emitting out of H is recorded for a PW width of 5.5 meV, as used in Figure 6b. For a PW width of 2.6 meV, a phase shift of about π/3 is observed. Although the phase of the 2DS is altered using PW pulses, the change in phase is easily calculated using the spatially expanded optical Bloch equations, which yield results within 5% of the experimentally measured phase shifts. The phase change dependence on the PW width can be understood by considering the limiting cases of a PW of zero width, which gives zero change in phase because no phase pattern is applied, and a PW of infinite width, which gives a phase shift of π because each field contributes a shift of π/2 for all frequencies to the 2DS. Even in the presence of many-body interactions that alter the phases of the 2DS features, the calculations demonstrate that, although the PW pulse shape alters the phase of the 2DS, the real and imaginary parts of the 2DS can still be determined. The phase shifts in the signals could be used for relative enhancements or suppression of one feature relative to another by appropriate selection of the PW frequency range and the LO heterodyne phase.

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CONCLUSIONS Pulse sequences have been designed and applied to selectively enhance exciton and biexciton peaks in GaAs QW 2D spectra. The peaks are enhanced by using two pulse sequences and phase window profiles in the excitation fields. Despite overlapping peaks and complex many-body interactions, both methods are able to selectively enhance or suppress selected peaks. It is found that the PW causes a phase shift in the 2DS that can be calculated by using the optical Bloch equations. In theory, the pulse shapes should be applicable to other systems by adapting a small number of control parameters (e.g., DP time delays, phase shifts, and PW frequency limits) appropriately. The parameter values can be calculated with simple third-order perturbation theory. Selective amplification of resonances in 2D FT OPT is expected to be G

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