Diagonal Slice Four-Wave Mixing: Natural Separation of Coherent Broadening Mechanisms
DDiagonal Slice Four-Wave Mixing: NaturalSeparation of Coherent Broadening Mechanisms
Geoffrey M. Diederich , Travis M. Autry , and Mark E.Siemens Department of Physics and Astronomy, University of Denver,2112 East Wesley Avenue, Denver, Colorado 80208, USA National Institute of Standards and Technology, 325 Broadway,Boulder, Colorado 80305, USA * Corresponding author: [email protected] 23, 2018
Abstract
We present an ultrafast coherent spectroscopy data acquisition schemethat samples slices of the time domain used in multidimensional coherentspectroscopy to achieve faster data collection than full spectra. We deriveanalytical expressions for resonance lineshapes using this technique thatcompletely separate homogeneous and inhomogeneous broadening contri-butions into separate projected lineshapes for arbitrary inhomogeneousbroadening. These lineshape expressions are also valid for slices takenfrom full multidimensional spectra and allow direct measurement of theparameters contributing to the lineshapes in those spectra as well as ourown.
Multidimensional Coherent Spectroscopy (MDCS) is a powerful spectro-scopic tool for measuring dephasing and coherent dynamics of electronic andvibrational resonances on ultrafast timescales [1–5]. MDCS was first imple-mented in nuclear magnetic resonance (NMR) experiments using radio frequen-cies [6–9], was then used to study vibrational coupling in the IR [10], and hasrecently been extended to electronic resonances in the visible and phonons insolids at terahertz frequencies [11–13].An advantange of MDCS is that the coupling between resonances becomesobvious and seperable because they are spread across multiple frequency di-mensions; a result not available in linear spectroscopy. Because of this, MDCSprovides a clear visualization of the qualitative dynamics of the system andrich quantitative information on material properties that are accessible throughanalysis of the resonance lineshapes present in the spectra. For example, homo-geneous and inhomogeneous mechanisms broaden the resonance in orthogonal1 a r X i v : . [ phy s i c s . i n s - d e t ] O c t irections in a MDCS spectrum, allowing straightforward separation and iden-tification of these contributions [14].While powerful and intuitive, it is difficult to quantify material propertiesdue to the increased complexity of multidimensional lineshapes. One dimen-sional lineshape analysis has historically enabled measurement of the oscilla-tor strength [15], excitation lifetime [16, 17], homogeneous and inhomogeneouslinewidths [18–20], and chemical shifts [21]. The most common approach to opti-cal multidimensional data sets employs quasi one-dimensional lineshape analysisof frequency-domain slices [22], although other approaches exist [23–25]. Thatmethod required taking data slices from the frequency domain for lineshapefitting. The frequency slice approach can be used to measure changes in thehomogeneous response of an inhomogeneous distribution [14, 26, 27]. However,all extracted slices depend on both homogeneous and inhomogeneous broaden-ing. This problem can be mitigated by simultaneous fitting of diagonal andcross-diagonal lineshapes or fitting entire MDCS spectra [22, 23]. However, toour knowledge, no MDCS lineshape analysis has completely separated homoge-neous and inhomogeneous broadening.In this letter, we present diagonal slice four-wave mixing (DS-FWM), a dataacquisition scheme for rapidly measuring material properties accessible in aMDCS spectrum without acquiring a full MDCS data set.Critically, our analysis determines both a time-domain basis and frequency-domain analytic functions that are orthogonal with respect to the broadeningmechanisms found in MDCS by utilizing time slices from the rephasing pulse se-quence in either the diagonal or cross-diagonal directions. We derive analyticalexpressions for the complex signal response in which different broadening mecha-nisms are decoupled along different time axes, using perturbative solutions to theoptical Bloch equations (OBEs). Analytical expressions for frequency domainprojections are derived by applying the projection-slice theorem to the time-domain slices. The resulting frequency-domain expressions completely separatethe homogeneous and inhomogeneous broadening, fit simulated resonances andexperimental data from GaAs quantum wells (QWs), and demonstrate excellentagreement with previously used lineshape analysis.The Projection-Slice Theorem states that the Fourier transform of a slicein two dimensions is equivalent to a projection onto that axis in the Fourierdomain [24]. Mathematically, P ( k x ) = F [ S ( x )] = (cid:90) ∞−∞ S ( x ) e − i πk x x dx, (1)where S ( x ) denotes a slice in any arbitrary x direction and P ( k x ) denotes aprojection onto the same k x axis in the Fourier domain. We consider only therephasing pulse sequence for a sample with inhomogeneous broadening. Thissystem will exhibit a photon-echo at t = τ , where τ, T = 0 , t are the inter-pulse delays between the first-second ( τ ), second-third ( T ), and third-fourth ( t )pulses. Note, in the present analysis the signal is assumed to be a third ordercoherence mapped onto a fourth order population by a fourth pulse in contrastto heterodyning schemes [28, 29]. 2he perturbative time-domain solution to the OBEs for an inhomogeneouslybroadened ensemble of two-level systems interacting with this pulse sequence is; s ( t, τ ) = s , e − ( γ ( t + τ )+ iω ( t − τ )+ σ ( t − τ ) / )Θ( t )Θ( τ ) , (2)where s , is the signal amplitude at time zero, ω is the center frequency of theresonance, γ and σ are the homogeneous and inhomogeneous dephasing rates,Θ denotes a unit step function that enforces causality between the pulses andthe signal, and τ ( t ) is the time delay corresponding to absorption (emission)processes. In the frequency domain, ω τ ( ω t ) is the frequency axis for absorp-tion (emission) processes. Recent work has extended the formalism of spectralanalysis to include non-delta function pulses [30], pulses with chirp [31], andnon-Gaussian responses [32]. However, in this study these considerations areexcluded in that delta function pulses and Gaussian inhomogeneous broadeningare assumed.We can now rewrite Eq. (1) in terms of physical parameters relevant toMDCS data, P ( ω t + ω τ ) = (cid:90) ∞−∞ S ( t + τ ) e i π ( t + τ )( ω t + ω τ ) d ( t + τ ) . (3)The two orthogonal time axes t (cid:48) = ( t + τ ) √ and τ (cid:48) = ( t − τ ) √ (shown in Fig.1) correspond to the diagonal and cross-diagonal directions in the MDCS timedomain, and allow Eq. 2 to be rewritten in this new basis. The signal normalizedto s , in this new basis is s ( t (cid:48) , τ (cid:48) ) = e − ( √ γt (cid:48) + i √ ω τ (cid:48) + σ τ (cid:48) )Θ( t (cid:48) − τ (cid:48) )Θ( t (cid:48) + τ (cid:48) ) . (4)This form of the signal simplifies the contribution of homogeneous and inhomo-geneous broadening to the diagonal ( t (cid:48) ) and cross-diagonal ( τ (cid:48) ) linewidths, atthe cost of adding complexity to the step functions involved.The Fourier transform of time domain slices provides simplified expressionsin both the time and frequency domains.A slice along t (cid:48) , at τ (cid:48) = 0, gives S ( t (cid:48) , τ (cid:48) = 0) = e −√ γt (cid:48) Θ ( t (cid:48) ) (5)and a slice along τ (cid:48) , at a fixed t (cid:48) = t (cid:48) , results in the expression s ( t (cid:48) = t (cid:48) , τ (cid:48) ) = e − ( √ γt (cid:48) + i √ ω τ (cid:48) + σ τ (cid:48) )Θ( t (cid:48) − τ (cid:48) )Θ( t (cid:48) + τ (cid:48) ) , (6)where t (cid:48) is the intercept of the slice on the t (cid:48) axis. We note here that t (cid:48) must be greater than zero to retrieve any meaningful information from a dataset thatdoes not extend to negative delays. As shown by the purple dashed line in Fig. 1[e.), h.), i.)], a slice along t (cid:48) = 0 gives a delta function in time that when Fouriertransformed becomes a constant in the frequency domain. Likewise, a slice at t (cid:48) < .) P ( ω τ ' ) ωτ ' S ( τ ' ) t ' t' =0t' >0 τ ' S ( t ' ) t ' τ t ω t ω ττ t ω t ω τ ω t't' τ ' P ( ω t ' ) t' =0 t' >0 a.) b.)c.) d.)e.)h.) i.) Figure 1: Diagram showing the rotated coordinate system in the MDCS time[a.), e.)] and frequency domains [b.), f.)], along with the slices [c.), h.)], andassociated projections [d.), i.)], that are the focus of this work. The dashed linesin a.) and e.) represent the data slices in the time domain and arrow tippedlines in b.) and f.) represent bins that are integrated in the projections. Allplots show the simulated rephasing amplitude.4ourier transforming Eq. 5 gives P ( ω t (cid:48) ) = 1 √ π (cid:0) √ γ − iω t (cid:48) (cid:1) , (7)with an absorptive (dispersive) Lorentzian component for the real (imaginary)part of the lineshape. The absolute value of this complex lineshape is a squareroot Lorentzian, with a full-width at half-maximum (FWHM) of √ γ . TheFourier transform of the τ (cid:48) slice gives P ( ω τ (cid:48) ) = e −√ t (cid:48) γ e − (cid:18) √ ω − ωτ (cid:48) σ (cid:19) σ (cid:18) Erf (cid:34) t (cid:48) σ + i (cid:0) √ ω − ω τ (cid:48) (cid:1) σ (cid:35) + Erf (cid:34) t (cid:48) σ − i (cid:0) √ ω − ω τ (cid:48) (cid:1) σ (cid:35) (cid:19) (8)which does have a γ dependent term, but only as a constant scaling factorthat does not affect the lineshape. Here Erf denotes an error function. Theexpression in Eq. 8 has a Gaussian lineshape with a FWHM of 4 (cid:112) ln (2) σ .The expressions in Eqs. 7 and 8 are analytical projections onto the ω t (cid:48) and ω τ (cid:48) axes, respectively. The use of the projection-slice theorem to arrive atthese expressions is similar to the treatment of NMR spectra in [24] with theimportant distinction that we use only the t (cid:48) and τ (cid:48) directions to take advantageof their isolated γ and σ dependent behavior.These expressions show the advantage of using the frequency projection asthe basis for lineshape analysis: complete separation of the inhomogeneous andhomogeneous broadening via their independent axes. These expressions thusimprove upon previous analysis [22, 23] that resulted in coupled expressions forthe different broadening mechanisms.The disadvantage of DS-FWM is that taking a projection removes the in-dividual information content providing only an average material response [33].Thus projections onto the frequency axis may be a more natural basis for consid-ering single resonances or ensemble responses as a whole, while slices along thefrequency axis are better suited to studying the response of individual oscillatorsin an ensemble.To validate the derived expressions we use them to fit simulated and ex-perimental data. The experimental setup used is described in [34, 35], withthe signal collected via photoluminescence (PL). This signal choice providesa direct analog to optical Ramsey spectroscopy in atomic physics. Briefly, apulsed Ti:Sapphire oscillator (Spectra-Physics Tsunami ), with 90 f s pulses anda bandwidth of 30 meV is split into four identical copies, each with a preciselycontrolled delay via mechanical translation stages. Each beam is passed throughan acousto-optic modulator (AOM) (Isomet ) where it is given a uniquecarrier frequency shift with a distinct radio frequency (RF). The resulting pulse5 ħ ω τ ( m e V ) τ ( p s ) t (ps) ħω t (meV) b.)a.) Figure 2: Experimental MDCS data from
GaAs
QWs in the a.) Time and b.)frequency domain corresponding to the DS-FWM data shown in Figs. 3 and 4.Both spectra are absolute value rephasing spectra.train undergoes dynamic, pulse to pulse, phase cycling that averages out un-wanted signal contributions and forces the signal PL to beat at RF frequenciesspecific to the desired quantum pathway.This signal choice requires that all data must be collected in the time domain,point by point. This can significantly increase the number of points, and hencethe acquisition time to acquire a MDCS spectrum as compared to measurementsusing a spectrometer. However, by only collecting data along the t (cid:48) and τ (cid:48) directions, we can measure the homogeneous and inhomogeneous linewidths in asimilar amount of time that a coherently detected MDCS experiment could withno ambiguity or mixing of the broadening contributions. DS-FWM providesa greatly simplified analysis to extract many of the most important physicalparameters accessible with MDCS. We realize data collection along the t (cid:48) and τ (cid:48) axes by simultaneously stepping the stages that control t and τ time delaysin our experiment and collecting data at each point. This treatment is similarto the radial sampling used in some NMR experiments to reduce data collectiontime [36], with the difference that we do not reconstruct full MDCS datasetsfrom our projections.To verify the derived DS-FWM expressions, we fit both simulated and ex-perimental data to our complex functions. The sample used in this experimentconsists of four GaAs
QWs surrounded by Al . Ga . As barriers. It has beenpreviously shown that strained bulk GaAs [37] and ’natural quantum dots’ [38]can be present in QW samples, in addition to the light hole (LH) exciton thatis present in the sample. Here, we isolate the resonance of the heavy-hole (HH)exciton in the well by checking its frequency in a PL spectrum (Ocean Optics
USB2000 ) and tuning the excitation frequency to the HH PL frequency, with aslittle overlap with the LH exciton as possible. The sample is kept at a tempera-ture below 10K by a recirculating liquid Helium cryostat (Montana Instruments6 A m p li t ude ( a . u . ) A m p li t ude ( a . u . ) ħω t' ħω t' (meV) t' (ps) t' AbsRealImagAbsRealImag AbsRealImagDataAbsRealImagSim a.) b.)d.)c.)
Figure 3: Time [a.), c.)] and frequency [b.), d.)] domain DS-FWM data for asimulated resonance [a.), b.)] and
GaAs
QWs [c.), d.)]. All data was taken inthe τ (cid:48) direction and fit using Eq. 7. Dots show the data and solid lines showthe function with best fit parameters ( γ = 0 . meV ).7 A m p li t ude ( a . u . ) A m p li t ude ( a . u . ) ħω τ ' ħω τ ' (meV) τ ' (ps) τ ' AbsRealImagAbsRealImag AbsRealImagDataAbsRealImagSim a.) b.)d.)c.)
Figure 4: Time [a.), c.)] and frequency [b.), d.)] domain DS-FWM data for asimulated resonance [a.), b.)] and
GaAs
QWs [c.), d.)]. All data was taken inthe τ (cid:48) direction and fit using Eq. 7. Dots show the data and solid lines showthe function with best fit parameters ( ω = 1548 . , σ = 0 . meV ). Cryostation ). For simulated data of a single resonance with σγ ≈
3, both thediagonal, and cross-diagonal, projection is fit with r ≈ . t (cid:48) and τ (cid:48) directions. The experimental ω t (cid:48) pro-jection was fit to our expressions with r ≈ . ω τ (cid:48) projections fitthe data with r ≈ . GaAs
QW time-domain data in Figs. 2 and 3. The LH exciton of the quantum wellhas a small signal that is present in the DS-FWM spectra but is much weakerthan the HH resonance and should not significantly contribute to the projectedlineshapes. Furthermore, we note that we have normalized the amplitude of ourprojections to unity in our treatment of the data. Normalizing the data in thisway forfeits our ability to extract oscillator strengths from our fits, as we arefocused on the broadening contributions of the resonance.We note that this procedure may be performed directly by the experimentin the time domain by only collecting data in the t (cid:48) and τ (cid:48) directions or in theanalysis of a full MDCS spectrum by extracting data slices from a larger data8et. We have performed our analysis using each of these procedures and seen nosignificant difference in the results. We also note that any MDCS experimentthat collects all data in the time domain, such as those that detect a populationwith a fourth readout pulse as opposed to an emitted electric field or those thatuse a fourth pulse for heterodyne detection on a photodiode, is already set upto take DS-FWM spectra, although the potential of a simplified experimentalsetup that collects data slices only along t (cid:48) and τ (cid:48) in the MDCS time domainis very promising. Such an experiment could access critically relevant mate-rial parameters ( γ, σ, ω ) without the data collection times and experimentalcomplexity of many MDCS experiments.In this letter, we have presented a data collection scheme for ultrafast co-herent spectroscopy and an associated lineshape analysis that is applicable toMDCS spectra as well as DS-FWM spectra. We have derived analytical ex-pressions for slices along the t (cid:48) and τ (cid:48) axes in the MDCS time domain, as wellas expressions for the associated frequency domain ω t (cid:48) and ω τ (cid:48) projections andshown that these projections completely separate the homogeneous and inhomo-geneous broadening mechanisms to the lineshape. We have fit these expressionsto both simulated and experimental data and shown excellent agreement. Thetechnique presented here offers a deeper insight into the nature of lineshapebroadening in coherent spectroscopy as well as a protocol for faster data collec-tion to find key material parameters.National Science Foundation (NSF) (1511199, 1553905).The authors thank Christopher Smallwood and Matthew Day for advice onphotoluminescence detection and for sharing their design of a custom amplifiercircuit, as well as Samuel Alperin and Jasmine Knudsen for helpful discussionsconcerning the manuscript. T.M.A. acknowledges support from a National Re-search Council (NRC) Research Associate Program (RAP) award at the Na-tional Institute of Standards and Technology (NIST).c (cid:13) References [1] Steven T Cundiff. Coherent spectroscopy of semiconductors.
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