Coherent Multi-Dimensional Spectroscopy: Experimental Considerations, Direct Comparisons and New Capabilities

Abstract

Optical Coherent Multidimensional Spectroscopy (CMDS) has been developed to probe the electronic states of a diverse range of complex systems. The great advantage of CMDS over linear spectroscopy is the ability to separate and quantify different types of interactions. To do this, multiple carefully controlled femtosecond laser pulses drive a non-linear response in the sample. A specific component of this non-linear response is selected and its amplitude and phase measured. There are several challenges for the experimental realization of optical CMDS, yet there have been several different approaches developed, each with their own advantages and limitations. Identifying the best approach then becomes dependent on the sample and the information being sought. Here we review the various experimental considerations and different approaches that have been developed. We consider the advantages and limitations of each of these, specifically in the context of experiments on solid state systems such as semiconductor nanostructures and 2D atomically thin materials. Two important considerations that are difficult to compare independently of other extraneous factors are the stability and sensitivity of the system. Here, we describe the experimental implementation of two different approaches that experience otherwise identical conditions and present an unbiased comparison of the stability and sensitivity. Furthermore, we demonstrate that by merging these two approaches we are able to combine the advantages of both into a single experiment.

Introduction

Quantifying the dynamics and interactions of electronic systems is crucial for understanding the mechanisms that drive functionality. From light harvesting in photosynthesis, to semiconductor based devices, and much in-between, dynamics on the femtosecond timescale play an important role(citation not found: Zewail_2000) Agarwal 2000, Jimenez 1994, van Grondelle 1994, Polivka 2004, Shah 1999, Cundiff 1996, Burgess 2016, Hao 2016). The development of femtosecond lasers has enabled an array of ultrafast spectroscopy techniques, which have been applied with great success for over 3 decades(Prasankumar 2011, Shah 1999, Fleming 1986). Different experiments have been used to probe population dynamics (e.g. time-resolved photoluminescence pump-probe/transient absorption) and coherent dynamics (e.g. coherent control, four-wave mixing (de Boeij 1995, Koch 1992, Cundiff 1996, Borri 2001, Agarwal 2000, Joo 1996, Wegener 1990)), yielding insight into energy and charge transfer(Xu 1999, Agarwal 2000, Asbury 2001, Jimenez 1994, van Grondelle 1994, Polivka 2004), excitonic interactions(Bott 1993, Birkedal 1996, Mayer 1994, (citation not found: Davis_2006), quantum coherence(Borri 2001, Davis 2007) and more. Coherent multi-dimensional spectroscopy (CMDS) is an extension of four-wave mixing (FWM) and transient absorption, both of which probe the third-order response of the sample to the applied electric field(Hybl 1998, Jonas 2003, Hybl 2001, Tian 2003, Keusters 1999, Aue 1976, Brixner 2005, Brixner 2004, Mukamel 1995, Shah 1999). In both, there are effectively three light-matter interactions. For third order CMDS, each of these interactions are carefully controlled in time and the phase and amplitude of the third order signal is measured. This provides an ability to separate the response into a 2 (or higher)-dimensional spectrum, which in-turn, provides simpler access to details that can be inaccessible in linear spectroscopy. The success of this approach has enabled greater insight into photosynthetic light-harvesting(Brixner 2004, Turner 2012, Brixner 2005, Zigmantas 2006, Wong 2012), interactions between excitons(Nardin 2014, Tollerud 2014, Tollerud 2016), multi-particle correlations, (Turner 2010, Almand-Hunter 2014, Bristow 2009, Stone 2009) and many-body effects in semiconductor nanostructures (Karaiskaj 2010, Nardin 2014, Turner 2012, Li 2006, Borca 2005), fundamental properties of atomically thin 2D materials (Moody 2015, Hao 2016, Hao 2016a) and more.

This type of approach was originally developed in Nuclear Magnetic Resonance (NMR) experiments, where radio-frequency pulses excite and probe nuclear spin transitions(Aue 1976, Ernst 1987, Mukamel 2000, Mukamel 2009). In NMR, the interactions between nuclear spins can provide information on molecular structure and is frequently used to do so. Over the 50 years since 2D-NMR was first demonstrated, many different schemes, with different pulse sequences, have been established to access specific information(Sattler 1999, Wüthrich 2003, Keeler 2010). The extension of these approaches to higher and higher order (e.g. modern pulse sequences can have hundreds of pulses) has led to the use of multidimensional for revealing the structure of large complex molecules and proteins(Wüthrich 2003, Keeler 2010).

The development of the analogous techniques for optical spectroscopy is enticing but limited by various experimental challenges and by the nature of electronic interactions(Mukamel 2000, Mukamel 2009). Chief among the experimental challenges is the requirement to measure the signal phase and maintain phase stability between all pulses. At RF and IR wavelengths this is relatively straightforward(Asplund 2000, Shim 2007, Woutersen 2002, Khalil 2003, Deflores 2007), but for optical fields, the higher frequency makes this more challenging. Nonetheless, many different approaches have been developed and applied successfully. Each of these approaches has advantages and drawbacks; none are able to realise all desirable capabilities. The choice of which technique to use is then very dependent on the sample and the information sought. In this review we describe the most commonly used approaches, their advantages and their limitations.

For the purpose of determining what are the most important capabilities, we concentrate on the needs as related to solid state systems, including semiconductor nanostructures and the increasing array of atomically thin materials. In these systems, it is frequently the coherent dynamics and the interactions that are of interest and which can be uniquely accessed by these approaches. Over the past decade, development in optical CMDS and the data analysis has been used to measure homogeneous broadening and radiative linewidths in the presence of substantial inhomogeneous broadening(Borca 2005, Siemens 2010, Moody 2011, Bristow 2011, Singh 2013, Nardin 2015); to identify and quantify coherent coupling between different types of excitation(Moody 2014, Nardin 2014, Li 2009, Davis 2011, Hall 2013, Tollerud 2014, Tollerud 2016a, Moody 2011); to reveal and identify many body effects such as excitation induced shifts and excitation induced dephasing(Karaiskaj 2010, Nardin 2014, Turner 2012, Li 2006, Borca 2005); to reveal and quantify weakly interacting type-II excitons and parity forbidden excitons(Tollerud 2016b); and to identify multi-body correlations between two and three excitons, (but not between four excitons)(Turner 2010, Turner 2009, Stone 2009); to measure the coherent dynamics of and interactions between excitons, trions and intervalley coherences in monolayer transition-metal dichalcogenides (Singh 2014, Moody 2015, Dey 2016, Moody 2016, Jakubczyk 2016, Hao 2016a, Singh 2016); and to measure the interactions, correlations, and coupling between exciton polaritons(Wilmer 2015, Wen 2013, Albert 2013, Takemura 2015)

The development of new materials with new applications such as 2D semiconductors(Mak 2010), 2D topological insulators(Peleckis 2006, Sushkov 2013), Weyl semi-metals(Lv 2015), polariton and exciton condensates(Kasprzak 2006, Byrnes 2014, Perali 2013), and other strongly correlated materials(Giannetti 2016) is continuing to transform condensed matter physics and needing new ways to measure and understand different types of interactions. CMDS has been developed with the aim of measuring interactions, and together with continued extensions of the capabilities provides an excellent experimental tool for understanding many of these systems.

In this review we focus on the experimental aspects of CMDS and how they can be used to access the important information. We begin with a brief overview of CMDS and a discussion of some of the different types of multiD spectra that can be obtained, and the information they provide. We then detail the experimental considerations and the different approaches to overcoming the various challenges. One important consideration that is difficult to compare across different techniques is the stability and sensitivity. This is challenging not only because there is no established standard method used to determine the relevant quantities, but also because these measurements are so dependent on the laboratory environment and may not represent the intrinsic stability or sensitivity of a specific approach. In the second half of this review we directly compare two approaches, one regarded to be among the most stable and one among the least. By merging these two approaches we also demonstrate a hybrid experiment that is able to combine the advantage of both.

CMDS of electronic transitions

CMDS of electronic transitions has been developed over the last two decades, with several different types of experiments being realized. However, due to technical challenges and inherent limitations (which will be discussed in detail in section \ref{SecTech}), CMDS has been limited primarily (with a few exceptions) to third order experiments. Nonetheless, these approaches have been able to provide significant insight into a range of different phenomena across a range different sample types. In this section, we briefly describe the nature of the nonlinear signal and the most common types of 2D spectra that have been realized to date.

Generation of the nonlinear signal

The excitation pulses used in CMDS experiments are typically interpreted as a perturbation of the system Hamiltonian, and the nonlinear signal is thus represented as a perturbative expansion of the light-matter interaction up to the order of the experiment. We briefly describe the salient elements of this interpretation necessary for a general understanding, while more detailed general (Hamm 2005) and rigorous (Mukamel 1995) explanations can be found elsewhere.

Each pulse acts on the system Hamiltonian with an electric field of the form:

$$\label{Eq1} \label{Eq1}\mathbf{E_{j}}(\mathbf{r},t)=E_{j}(t)e^{+(i\mathbf{k_{j}}\mathbf{r}-i\omega t+\phi_{j})}+E_{j}(t)e^{-(i\mathbf{k_{j}}\mathbf{r}-i\omega t+\phi_{j})}\\$$

where j is the pulse label, $$E_{j}$$ is the pulse envelope function, $$k_{j}$$ is the wavevector, and $$\omega$$ is the carrier frequency. The response of the system is given by the Liouville variation of the Schrodinger equation:

$$\label{Eq1.1} \label{Eq1.1}\dot{\rho}(k,t)=[\textbf{H},\rho(k,t)]\\$$

where the density matrix $$\rho$$ varies as a function of time and maintains the wavevector dependence of the electric fields. The Hamiltonian, H, is given by:

$$\label{Eq1.2} \label{Eq1.2}\mathbf{H}=\mathbf{H_{0}}+\mathbf{H_{int}}+\mathbf{H_{dis}},\\$$

with $$H_{0}$$ the system Hamiltonian, $$H_{int}$$ describes the interaction of the light field with the system, and $$H_{dis}$$ describes the dissipation of the excitation. Ignoring dissipation, the third order signal11This same process can be generalized for a signal of arbitrary order. See Ref —(Mukamel 1995)— for details. generated in the experiment is proportional to the third order polarization, which can be expressed as:

\begin{aligned} P^{(3)}(\mathbf{r},t)= & \int_{0}^{\infty}dt_{3}\int_{0}^{\infty}dt_{2}\int_{0}^{\infty}dt_{1}S^{(3)}(t_{3},t_{2},t_{1})\cdot \\ & E_{I3}(\mathbf{r},t-t_{3})E_{I2}(\mathbf{r},t-t_{3}-t_{2})E_{I1}(\mathbf{r},t-t_{3}-t_{2}-t_{1})\nonumber \\ \end{aligned}

where $$E_{I1}$$, $$E_{I2}$$, and $$E_{I3}$$ represent the electric fields that interact with the sample at time $$t_{1}$$, $$t_{2}$$ and $$t_{3}$$, respectively (each of which can be provided by any of the pulses used in the experiment). $$S^{(3)}$$ is the third order response function of the sample and is given by:

\begin{aligned} S^{(3)} & (t_{3},t_{2},t_{1})=\left(\frac{i}{\hslash}\right)^{3}\theta(t_{1})\theta(t_{2})\theta(t_{3})\cdot \\ & \left<\check{\mu}(t_{3}+t_{2}+t_{1})\left[\check{\mu}(t_{2}+t_{1}),\left[\check{\mu}(t_{1}),\left[\check{\mu}(0),\rho(-\infty)\right]\right]\right]\right>\nonumber \\ \end{aligned}

where $$\theta$$ is the Heaviside step function, and $$\mu$$ is the transition dipole moment. When the commutators in Eq. \ref{Eq3} are expanded, they generate a series of terms that each represent a ‘quantum pathway’ – that is, a well defined series of interactions with the system that leads to a signal photon. Each of those quantum pathways can be generated by any combination of the pulses used in the experiment, and all linear combinations of $$k_{j}$$ and $$\phi_{j}$$ are possible. A specific linear combination can be selected, for example, by having a well-defined pulse ordering and giving each pulse different wave vectors or different, controllable phase (these allow the use of phase-matching and phase cycling, respectively, to isolate specific signal contributions, and are discussed in Section \ref{SubSechSignalSeparating}.)

The linear combinations typically detected in a third-order experiment involve the conjugate of one pulse (and transition dipole), giving a signal with $$\mathbf{k_{sig}}=-\mathbf{k_{1}}+\mathbf{k_{2}}+\mathbf{k_{3}}$$ and $$\phi_{sig}=-{\phi_{1}}+{\phi_{2}}+{\phi_{3}}$$, for example. The order in which the different pulses arrive then determines the type of quantum pathway being probed. There are thus three different pulse orderings: rephasing, non-rephasing, and double-quantum, as shown in Fig. \ref{FigPulseOrderings}. 2D spectra are generated by recording the amplitude and phase of the signal as a function emission frequency and of one of the delays. The delay that is scanned, together with the pulse ordering, determines the type of 2D spectrum. These can be classified as single-quantum (1Q) when the delay between the first two pulses is scanned, zero-quantum (0Q), when the delay between the second and third pulses is scanned (in rephasing or non-rephasing pulse-ordering) and double-quantum (2Q) when the delay between the second and third pulses is scanned in the double quantum pulse-ordering. Each of these scans provides different information, which will be discussed in the following sections.

Third-order 2D Spectroscopy

\label{FigPulseOrderings}Three common pulse orderings used in third-order 2D/3D spectroscopy. The pulse with the asterisk indicates the pulse that provides a conjugate contribution. The bars below the sequences indicate the delays that are scanned for different types of multidimensional spectra.

1Q rephasing and non-rephasing

The most common type of 2D spectrum is the 1Q spectrum, in which $$t_{1}$$ (the coherence time) is scanned for a fixed $$t_{2}$$ (the population time) with the conjugate pulse arriving first (rephasing) or second (non-rephasing). The spectra produced are correlation maps of $$E_{1}$$ (the Fourier transform of the data as a function of $$t_{1}$$) and $$E_{3}$$ (the directly measured FWM spectrum or a Fourier transform with respect to $$t_{3}$$). $$E_{1}$$ can be thought of as the absorption energy, as it is the energy with which the coherent superposition excited by the first pulse oscillates, and $$E_{3}$$ is the emission energy. This correlation of absorption and emission provides a powerful and intuitive way to understand broadening mechanisms, measure homogeneous linewidths, identify coupling between states and measure energy transfer dynamics.

To understand the terms rephasing and non-rephasing, we must consider the evolution of the signal phase as a function of the $$t_{1}$$ and $$t_{3}$$. The sign of the signal evolution as a function of $$t_{1}$$ depends on whether the first pulse is the conjugated one or not. If the conjugate pulse arrives first, the phase of the signal evolves as a function of $$t_{1}$$ in the opposite direction to the evolution as a function of $$t_{3}$$; if the non-conjugate pulse arrives first, the phase evolves in the same direction as a function of $$t_{1}$$ and $$t_{3}$$.

When the conjugate pulse arrives first, inhomogeneous dephasing (caused by excitations that have a distribution of oscillation frequencies evolving to be out of phase) that occurs during $$t_{1}$$ is reversed in $$t_{3}$$, with the out of phase oscillators “rephasing” to produce a signal at $$t_{3}=t_{1}$$. This is known as the “photon echo” effect (Abella 1966) and is analogous to the previously discovered spin echo effect (Hahn 1950). In the non-rephasing pulse ordering the phase evolution is not reversed and there is no photon echo. In contrast to inhomogeneous dephasing, homogeneous decoherence (whereby the phase of an excitation is changed stochastically through, for example, elastic scattering) is not reversible. Thus, the decay of the signal in a rephasing pulse ordering is limited by decohorence and the homogeneous linewidth.

The effects of rephasing manifest in the shape of the 2D peaks in 1Q spectra . For transitions that include both homogeneous and inhomogeneous broadening, the rephrasing, whereby the signal occurs at $$t_{1}=t_{3}$$, produces a peak that is elongated along the diagonal of the 2D spectrum (i.e. along the line $$E_{3}=-E_{1}$$11In this review we plot 1Q rephasing using negative frequencies for the $$E_{1}$$ axis to signify that the reversal of the $$t_{1}$$ phase evolution in $$t_{3}$$. This is a convention that is not universally adopted in the field, but more common in publications studying semiconductor nanostructures.. A more detailed fit is required to get the precise values (Siemens 2010), but – roughly speaking – the width of the peak along the diagonal is the inhomogenous linewidth and the cross-diagonal width is the homogeneous linewidth.

When the non-conjugate pulse arrives first and the phase evolves in $$t_{1}$$ and $$t_{3}$$ in the same direction, the inhomogeneous dephasing is not recovered and the signal does not rephase. This produces symmetrical diagonal peaks whose width is set by the combined (homogeneous and inhomogeneous) linewidth.

Cross-peaks in 1Q spectra

Quantum pathways that include interactions between states with different transition energies produce peaks away from the $$E_{3}=-E_{1}$$ diagonal line, which are typically referred to as cross-peaks (CPs). The position of the CPs can be used to identify which states are interacting, and the presence (or absence) of CPs in a 2D spectrum can provide insights into the nature of the electronic system. For example, a system which produces multiple diagonal-peaks but no cross-peaks can be effectively modelled as a set of isolated two-level systems. Each pair of excited states can produce a cross-peak above the diagonal ($$|E_{1}|<E_{3}$$) and below the diagonal ($$|E_{1}|>E_{3}$$). Rarely, however, are the above and belpow diagonal CPs of equal magnitude, which can be used to help understand the origin of the signal and nature of the interactions. CPs can arise from incoherent pathways (where the interactions occur between populations and don’t rely on coherent interactions) or coherent pathways, where the coherent evolution of two states affect each other. The former includes processes like population transfer, whereby below diagonal CPs can be produced by relaxation of the excited state population to a lower state, while above diagonal peaks can be produced by ’uphill’ population transfer provided the thermal energy is large enough to bridge the energy gap. CPs can also arise where a common ground state can affect the ability of another state to absorb light (know as ground state bleach (GSB)); or where absorption to a higher excited state is enabled by being in the first excited state (known as excited state absorption (ESA)).

Transitions that are coherently coupled can produce both above and below diagonal peaks, which – in the simplest case – should be perfectly symmetrical. These CPs arise from the excitation of a coherent superposition of excited (or ground) states by the first two pulses, as depicted in Fig. 2a.

The precise quantum pathway that produces CPs in a given sample cannot always be determined easily, because multiple overlapping pathways may contribute to the cross-peaks. In real systems, these various pathways may combine in non-intuitive ways to modify the amplitude, phase, and shape of the CPs, producing asymmetrical 2D spectra. For example, many-body interactions in semiconductor nanostructures can shift signals, effect dephasing times and alter the real peak shape of CPs (Li 2006, Nardin 2014, Turner 2012, Bristow 2009). Furthermore, incoherent population relaxation and activation can play a significant role even in coherently coupled systems, further confounding exact identification of quantum pathways. To some degree, these different contributions can be separated by measuring the dynamics of the CPs as a function of $$t_{2}$$. Population relaxation and activation can be identified as signals that grow in as a function of $$t_{2}$$. Ground state bleach, excited state absorption and Raman-like pathways that involve a coherent superposition of excited states during $$t_{2}$$ can be seen even when $$t_{2}=0$$. In the latter case (which is considered definitive evidence of coherently coupled transitions) the phase of the CP evolves as a function of $$t_{2}$$ at a frequency characteristic of the energy difference between the coupled states.

The shape of cross-peaks in rephasing spectra can also provide information about the interactions and broadening mechanisms. For example, structural disorder may result in correlated (or anti-correlated) inhomogeneous broadening of a pair transitions. In rephasing spectra, this correlation results in a tilted shape of the CP. A tilt towards the diagonal line indicates that the inhomogeneous broadening is correlated , and the detailed peakshape depends on the linewidths of the associated transitions and the degree of correlation . A symmetric, untilted peak indicates that there is no correlation. This information about the broadening mechanisms provides insight regarding the nature of the coupling that produces the CP, and can also provide important clues for identifying states in a complex electronic system (Tollerud 2016b). Further information can be obtained by looking at the real part of the 2D spectrum, for example, Cundiff et al. have demonstrated the ability of these techniques to identify and quantify different many-body effects by comparing the phase of the real valued 2D spectrum (Borca 2005, Li 2006, Bristow 2009, Nardin 2014).

\label{PathwaysqW}(a) two possible pathways for a 1Q spectrum of a coupled three level system in a rephasing pulse ordering. (b) A cartoon 1Q 2D spectrum in which the broadening of transition 1 is homogeneous and inhomogenous, while the broadening of transition 1’ is purely homogeneous. The inhomogeneous broadening of 1 results in a peak shape tilted along the diagonal, in which the width along the diagonal (cross-diagonal) is roughly the inhomogeneous (homogeneous) linewidth. The shape of the cross-peaks depends on the nature of the broadening. Fig. (c)-(f) show cartoons of cross peak shapes for different pairs of transitions. (c) Both homogeneous. (d) homogeneous of differing linewidths. (e) inhomogeneous, uncorrelated (orange), correlated (red) and anticorrelated (red-dashed). (f) inhomogeneous of two different linewidths, uncorrelated (green), correlated (blue) and anticorrelated (blue-dashed). The left hand side of the figure provides a key to the symbols used in Figs 2-5.

0Q

In zero-quantum 2D spectra, $$t_{2}$$ is scanned instead of $$t_{1}$$. As described above, Raman-like excited state coherences can be induced as a function of $$t_{2}$$ if the first and second pulses interact with different states (Yang 2008). These types of pathways can be identified by their phase evolution during $$t_{2}$$. The phase will evolve such that the real and imaginary parts of the signal will oscillate at a frequency set by the energy difference between the two states in superposition. Pathways that involve a population in $$t_{2}$$, on the other hand, will exhibit no phase evolution. As a result, in 0Q spectra, any signals that involve an excited (or ground) state coherent superposition over the waiting time can be identified as peaks that appear away from $$E_{2}=0$$, while those that involve a population – which typically constitutes the bulk of the pathways – appear at $$E_{2}=0$$ (see Fig.3). This type of 0Q spectra can then be particuarly useful and suited to identifying coherent superpositions .

\label{Fig0QCartoon}A cartoon 0Q spectrum and two possible pathways, with the pulse ordering depicted top left. Population pathways appear at $$E_{2}=0$$ as there is no phase evolution over the $$t_{2}$$ time period. In contrast, pathways involving a coherent superposition of excited (or ground) states will generate signal with a phase that varies as the delay $$t_{2}$$ is scanned over the phase evolution of the coherent superposition, leading to peaks away from $$E_{2}=0$$.

2Q

Another type of non-radiative coherence can be detected in the double-quantum pulse ordering (in which the conjugate pulse arrives last), as depicted in Fig. 4. The first two pulses generate a coherence between the ground state and a doubly-excited state (also called a double-quantum coherence). The third pulse then converts this double-quantum coherence back into a single-quantum coherence, which radiates as the signal. Typically, the first two pulses arrive at the same time ($$t_{1}=0$$), and $$t_{2Q}$$ is scanned. The doubly excited states can then be identified based on where the signals appear on the double quantum spectrum. Quantum pathways in which the first two excitation pulses interact with the same transition appear as peaks along the $$E_{2Q}=E_{3}$$ line. These peaks can therefore be used to understand interactions between excitations of the same type. Quantum pathways in which the first two excitation pulses interact with different transitions appear as cross-peaks (peaks that do not appear on the $$E_{2Q}=E_{3}$$ line). These peaks can be used to study interactions between excitations of different types, or to excite higher energy states through a two-photon process. 2Q spectroscopy has been used primarily to study two-exciton correlations and biexcitons in semiconductor quantum wells (Stone 2009, Karaiskaj 2010, Turner 2009, Tollerud 2016a, Wen 2013, Tollerud 2016b), and has been shown to be very sensitive to excitation induced dephasing of exciton correlations (Tollerud 2016a). More recently, 2Q spectroscopy has been used to study interactions between polaritons and excitons in micro-resonators (Wen 2013a, Wilmer 2015, Takemura 2015), and layered semiconductors (Wilmer 2016, Dey 2015).

\label{Pathways2Q}A cartoon 2Q spectrum and two possible pathways, with the pulse ordering depicted top left. The first two pulses generate the 2Q coherence, with the conjugate pulse arriving last. The diagonal in the 2Q-2D spectrum runs along the 2:1 line. Peaks on this diagonal correspond to 2Q coherences involving identical excitons, whereas cross-peaks involve two different exciton transitions with different energy.

Fifth-order and seventh-order experiments

So far, we have focused on third-order experiments as they are experimentally the simplest and most common, but the conceptual frameworks can be extended to five, seven or more interactions. The additional time/frequency dimensions open up a range of new phenomena to study. For example, three-exciton correlations have been detected in semiconductor quantum wells using fifth- and seventh- order CMDS experiments (Turner 2010). Correlations of this type tend not to arise from simple few body interactions, but rather develop because of many-body effects. Identifying, quantifying and characterizing correlations can then become a useful tool to probe many-body effects in simple semiconductor systems, as well as in more complex states and complex solid state materials, such as exciton or polariton BECs(Wen 2013a, Wilmer 2015, Takemura 2015), dropletons(Almand-Hunter 2014), topological insulators, Weyl semimetals, high-Tc superconductors(New_perspectives_in_the_ultrafast_spectroscopy_of_many-body_excitations_in_correlated_materials_2016), and other strongly correlated materials.

In a different fifth-order implementation, the ability to add a second population time enabled the direct detection of multi-step energy transfer (Zhang 2012, Zhang 2013, Zhang 2015). This approach could be very useful for elucidating relaxation pathways and transfer times involving multiple states in complex systems. This has been implemented for the photosystem-II complex, which is involved in photosynthesis, where a previously identified pathway was shown explicitly, rather than being inferred from the dynamics (Zhang 2015). The limitation of this approach is the time required to acquire a full 5D data set, incorporating all of the population times, which can stretch into days. This may be able to be overcome by spectrally shaping one or more of the excitation beams, as has been demonstrated for 3rd order experiments, and as is used extensively in multidimensional NMR(Sattler 1999, Keeler 2010). The application of this type of approach could then extend beyond biomolecular systems and beyond 5th order experiments to higher order implemetations.

Another application of higher order CMDS is to measure the interplay between electronic and vibrational degrees of freedom. Spencer et. al recently implemented a fifth-order Raman/electronic experiment which correlates vibrational modes with electronic states, and used it to explore the vibronic nature of the states in a laser dye(Spencer 2017). In this approach, a non-resonant pre-pulse (which arrives before three other excitation pulse) impulsively generates a ground state vibrational coherence via a Stokes Raman process. The typical third-order spectra (generated by the three excitation pulses which are resonant with the electronic transitions) can then be correlated with the phase evolution of the vibrational coherence generated by the pre-pulse. This approach promises a powerful ability to explore the interplay between electronic and vibrational states. Similar approaches to measure interactions between electronic and vibrational dynamics have been realised by combined visible-IR experiments  (Lewis 2016, Gaynor 2016, Lewis 2016a, Courtney 2015, Courtney 2015a)

These examples only scratch the surface of what is possible with higher order techniques, but many of the experimental challenges of CMDS are exacerbated as the order of the experiment increases. Nonetheless, we regard this as an important future direction for optical CMDS and the ability of the different experimental implementations to extend to these higher order experiments is one of the aspects that we consider in our comparisons.

\label{FifthOrderPulseOrdering}A fifth-order rephasing pulse ordering. The bars below indicate the two time delays scanned for the fifth-order 3D spectrum that is the conceptual extension of a third order 1Q 2D spectrum. In this case, there would be three frequency axes that correspond to 1Q coherences, or optical coherences. These are one of many possible fifth-order pulse orderings/types of 3D spectra. The pathway below the pulses is a multi-step population relaxation pathway (as in Ref. (Zhang 2015)).

3D Spectroscopy

A third frequency axis can be added by scanning two time delays instead of one. The benefits of going from two frequency dimensions to three are similar to the benefits of going from 1D to 2D: Signals that otherwise overlap in lower-dimensional spectra can be clearly separated. The phenomena that can be studied depends on how the 3D spectroscopy is implemented (Cundiff 2014).

Rephasing 3D spectroscopy

A third-order 3D spectrum separates the signal along all possible frequency axes, so it represents a nearly complete separation of the third-order quantum pathways. All of the same quantum pathways that can be resolved in rephasing 1Q or 0Q 2D spectra can also be resolved in rephasing 3D spectra, and both 1Q and 0Q 2D spectra can be generated by integrating the 3D spectra along the $$E_{2}$$ and $$E_{3}$$ axes, respectively. Extending these signals across a third frequency axis can also make it easier to fully isolate different contributions – signals that partially overlap in a 1Q and/or 0Q spectrum can sometimes be fully separated in 3-dimensions. (Hall 2013, Tollerud 2014, Hayes 2011, Cundiff 2014, Li 2013, Tollerud 2016)

While 3D spectra can be cumbersome to collect, analyse, and represent, a lot of information can be gleaned through projections and slices of the spectrum along different axes. For example, by isolating a specific pathway in the 3D spectrum the 1D, 2D, and 3D peak-shapes for that pathway can be analysed without overlapping peaks. This can provide a much clearer picture of the interactions and correlations (Tollerud 2014, Hall 2013, Li 2013, Tollerud 2016). It has also been shown that a careful analysis of a complete 3D spectrum can yield a full reconstruction of the density matrix upto third order (Li 2013, Hayes 2011).

Double-quantum 3D spectroscopy

Third-order 3D specroscopy in the double quantum pulse ordering can also clarify 2Q signals (Turner 2009). For example, partially overlapping peaks can be separated, as was demonstrated for the correlated 2-exciton and biexciton contributions in a semiconductor QW, allowing more accurate determination of the biexciton binding energy (Turner 2009). Double-quantum 3D spectroscopy can also be used to separate signals based on the pathway by which mixed double-quantum excitations are generated. In a simple picture, the dynamics of a mixed 2Q coherence shouldn’t depend on which of the stats was excited first, however, recent work has revealed that significant insights can be obtained by separating the two pathways. This includes being able to better identify the nature of interactions and revealing additional pathways involving weaker interactions that are otherwise hidden (Tollerud 2016a)

Fifth-order 3D spectroscopy

A fifth-order experiment comprises a total of five possible frequency dimensions, so a 3D spectrum does not contain the full fifth-order response. Furthermore, many different 3D spectra can be imagined, in which varying population and coherence times can be scanned. Fifth-order 3D spectroscopy was first explored in in the IR using three pulse (Ding 2007) and five-pulse (Garrett-Roe 2009) experiments. Fidler et al. implemented fifth-order 3D electronic spectroscopy by using a single-shot technique for the acquisition of 2D spectra, which vastly reduces data acquisition time (Fidler 2010). More recently, a modified verison of this single-shot method was used again to generate a 4D spectrum in a 4-pulse experiment consisting of a non-resonant Raman pulse followed by three resonant pulses (Hutson 2016, Spencer 2017). While they present their data in the form of 2D slices of the 4D spectrum, it clearly illustrates the power of higher-dimensional spectra.

Another pioneering development in fifth-order 3D spectroscopy of electronic transitions was recently performed by Zhang et al (Zhang 2015, Zhang 2012). They used a five pulse experiment, which consists of three time domains in which the system is in a single-quantum coherence ($$t_{1}$$, $$t_{3}$$, $$t_{5}$$), and two time domains in which the system is in a population ($$t_{2}$$, $$t_{4}$$). They collected 3D spectra with axes $$E_{1}$$, $$E_{3}$$, $$E_{5}$$, for different values of the population times $$t_{2}$$ and $$t_{4}$$. By looking at the dynamics of different 3D cross-peaks as a function of the two time delays, they were able to directly identify multi-step energy relaxation processes and measure the dynamics. This measurement can be thought of as the logical extension to fifth-order of a standard 2D rephasing experiment, in which population transfer between multiple states can be identified. While the data is presented as 3D spectra, exploring population dynamics of peaks in the 3D spectrum requires 4D or 5D datasets, which can be very time consuming to collect, as – in general – the acquisition time scales with the power of the number of delays scanned.

Experimental considerations and implementations

\label{SecTech}

In contrast to NMR, where RF fields are used, optical pulses excite transitions in the 300-700 THz range, $$\sim 10^{6}$$times larger than nuclear spin transitions. At these frequencies many of the experimental considerations are vastly different and present a raft of experimental challenges. The time evolution of the electric fields cannot be directly measured at these frequencies using conventional detectors and electronics, so alternate methods must be used. This typically involves some form of interferometry, the precise details of which vary depending on how there experiments are implemented and the approach used to isolate the signal of interest. The phase of the signal reflects the phase of the excitation pulses, and so it becomes important to maintain phase stability between all pulses. The rate of signal phase evolution as a function of inter-pulse delay is set by the energy of the coherence/s that is/are evolving over that time period. In many types of multidimensional measurements, the energy of this coherence is the transition energy from the ground state to the state of interest. For these 300-700 THz oscillations, there is a full phase rotation every 1-3 fs. The inter-pulse delays must therefore be scanned with a sampling frequency much shorter than this phase rotation period to generate artefact-free multidimensional spectra, so the method used to control the inter-pulse delays must have $$\leq$$1fs precision, and have sub femtosecond timing jitter (although this requirement can be relaxed with rotating frame excitation). It then also becomes clear that at these frequencies instability of opto-mechanical mounts and different beam paths for each pulse can cause fluctuations in the relative phase of the excitation pulses (and hence signal) that can lead to reduced sensitivity and errors in the detected signal. Phase stability of one one hundredth of the period is typically required to obtain reasonable quality data, and is one of the fundamental challenges of CMDS. Several different approaches have been developed to achieve this, and form the basis of the different experimental implementations detailed below in Section \ref{SubSecExpImplimentations}. In addition to this primary concern there are several other important considerations, which can affect the choice of stabilization method.

The non-linear signal of interest in CMDS is often much weaker than the background (scattered light from the excitation pulses, linear signals, other spurious non-linear signals), thus requiring some means of removing the background. One (or more) of several different techniques are used to isolate the desired signal: phase-matching, phase-cycling, amplitude modulation (choppers), and phase modulation. Each of these has advantages and disadvantages and the options available can be limited the beam geometry and detection method, as detailed below.

Another stark difference when comparing to NMR is the dynamics and by corollary the linewidths of electronic transitions. Pulses of $$\sim$$10-100 fs duration pulses are required to resolve the electronic/excitonic coherence dynamics in condensed phase systems which vary from 100’s ps down to as short as 10 fs . This is much shorter than lifetimes in NMR and as result the linewidths are much larger. This is often further complicated by spectral diffusion and inhomogeneous broadening, which lead to peaks that overlap to a large extent. In NMR, to further separate peaks the magnetic field can just be increased, however, no such approach is possible for electronic transitions. Interpretation can thus become difficult, which is where other techniques to separate out specific pathways can be particularly useful, as detailed in Section \ref{SubSecSelectiveTechniques}.

The different implementations of CMDS and how the salient information is accessed depends greatly material system in question and the type of dynamics one wants to measure. For semiconductor systems some specific considerations become relevant and we will, in part, assess each of the different techniques and approaches in this context. For example, at low temperature coherence times can be quite long, so it is important that the scannable range is long enough. Similarly, the ability to achieve high spatial resolution can be important particularly when trying to access details from nanostructures. Finally, interactions between excitons in semiconductor systems can significantly change the signal detected, even when the additional excitations are not generating signal. Being able to control what is excited and to operate at low excitation density thus become important. The ability to operate at low excitation density is dependent on sensitivity and stability. Not only does this vary across different techniques, but it can also vary from laboratory to laboratory, so it becomes difficult to give precise quantitative assessments of the stability and sensitivity of different techniques. This is an issue we address in Section \ref{SecHybridComparison} where we detail our implementation of two different approaches, and quantitatively compare their performance for the same set of samples.

Techniques for separating signal from background

\label{SubSechSignalSeparating}

Phase-matching

Phase-matching relies on the fact that the nonlinear signals of interest are emitted with a wave-vector that is well-defined relative to the wave-vectors of the excitation pulses, as required for momentum conservation. By choosing a beam geometry such that the wave-vectors of the excitation pulses lead to the non-linear signal of interest being emitted in a direction that does not overlap with the excitation beams, the signal can be, ideally, background-free. In practice, however, other techniques, such as phase cycling or amplitude modulation are often required to further isolate the signal and enhance SNR.

Amplitude modulation

Modulation of the amplitude of the pump beam by an optical chopper in pump-probe measurements has been used for decades to isolate the effect on the probe beam that is induced by the pump (Hilton 2012). Indeed, similar approaches to signal processing have been used to isolate signal from background in many different fields(Hilton 2012). The extension to CMDS is then straightforward. In fully non-collinear geometries, where the three excitation beams are separable, a dual chopper scheme is frequently used to further enhance the selection of the signal and rejection of the background (Heisler 2014). The chopper frequencies are commonly chosen so that the frequency of one is exactly half the frequency of the other. The faster of the two is ideally synchronized to the laser repetition rate11It has been shown that the background noise level is lowest if the data is recorded shot by shot from the laser and normalized to the tracked laser intensity —(Heisler 2014, Kearns 2017, Brazard 2015). Spectra are then recorded continuously in sequences of four different types of spectra which (when combined appropriately) can be combined to remove the background, scatter, and spurious nonlinear signals. This approach also has the additional benefit that one of the four spectra that are measured is the pump-probe signal, which means that the data required for phasing is naturally recorded alongside the 2D spectrum. The speed of chopping and potential for single shot readout can allow relatively fast acquisition of 2D spectra, which can further reduce noise. The down sides of this approach are that there only one in four shots contains the signal, which can an issue if photodamage is a problem for the sample, and there remains one scatter pathway that is not removed with only two choppers. This could be eliminated by introducing a third chopper into the third excitation beam, but this would reduce the number of signal shots to one in eight.

Phase cycling

Phase cycling can be used to isolate the desired non-linear signal by exploiting the fact that the phase of the coherent non-linear signal depends on the phase of all the excitation pulses (i.e. a phase rotation of any of the excitation pulses yields a phase rotation in the signal) (Tian 2003, Vaughan 2007, Yan 2009, Tan 2008). In phase cycling schemes, a series of spectra are recorded in which the phase of the excitation pulses are independently rotated by controlled amounts. The different interferograms can then be combined to remove any signals that do not depend on the phase of all three excitation pulses in the manner expected. The specific phase cycling procedure that is used depends on the beam geometry and the order of the nonlinear signal – some specific examples are given in the following sections. The phase rotations are often made as part of some pulse shaper apparatus and can be slow compared to the laser repetition rate, however, they do have the advantage of measuring and keeping the signal from each shot.

Phase modulation

Phase modulation techniques are conceptually similar to phase-cycling techniques in that they exploit the connection between the signal phase and the phase of the excitation pulses. Instead of recording multiple spectra for specific combinations of the excitation beam phases, a continuous phase oscillation is applied to each of the beams (Borri 2001, Langbein 2006, Kasprzak 2010, Kasprzak 2012, Jakubczyk 2016, Mermillod 2016, Fras 2016, Mermillod 2016a, Tekavec 2007, Lott 2011, Widom 2013, Karki 2014, (citation not found: Nardin2013) Nardin 2015, Aeschlimann 2011). This is usually achieved by acousto-optic modulators (AOMs) in each beam with the frequency of phase oscillation set to be different for each beam. The signal phase is thus modulated at the specific linear combination of the three frequencies that matches the signal pathway of interest. Interfering the 3rd order signal with another beam then leads to oscillations in intensity as a function of time which can be detected and referenced to the AOM frequencies with a lock-in amplifier. This approach can be very good at rejecting other signal contributions, but because the signal is detected in a single channel detector, it requires two pulse delays to be scanned in order to obtain a 2D spectrum, which can slow the data acquisitions.

Different Experimental Geometries

Fully non-collinear beam geometry

In fully non-collinear geometries, each light-matter interaction that contributes to the signal is provided by a different excitation pulse with a unique wave-vector. The signal is thus fully separated from the excitation beams and most other non-linear signals, as required by phase-matching. A common fully non-collinear geometry is the BOXCARS geometry, in which the excitation pulses make up three corners of a square with the signal emitted on the fourth corner (Hybl 1998, Hybl 2001). A reference beam is usually directed on this fourth corner to allow heterodyne detection and recording of the complex nonlinear signal as a spectral interferogram. The significance of this geometry is that momentum and energy can be conserved at the same time, which is not always true for non-collinear geometries (Hybl 2001, Butenhoff 1993). The box geometry for the excitation and reference pulses can be obtained by simple beam splitters (Hybl 2001, Zhang 2005), a spatial mask in an expanded beam (Vaughan 2007, Zhang 2013a) 11This is not an ideal way to generate the beams: it is inefficient as much of the light does not reach the sample, the non-Gaussian beam profiles will result in aberration of peak shapes, and diffraction from the spatial mask creates additional stray light., a 2D diffractive optic (Brixner 2004) or a controllable beam shaper, such as a spatial light modulator (SLM) (Turner 2011). Although the signal direction in this type of non-collinear geometry is ideally background free, significant scatter can still make it to the detector and obscure the signal, particularly if it is an especially weak signal. For this reason, other techniques, such as phase cycling or amplitude modulation are required.

Partially non-collinear beam geometry

Beam geometries in which two or more of the light-matter interactions are provided by pulses with the same wavevector are classified as partially non-collinear. A common partially non-collinear geometry used in CMDS is the pump-probe geometry(Grumstrup 2007, Tian 2003, Myers 2008, Deflores 2007). In this case the first two pulses travel collinearly along the same beam path (the pump) and the third pulse travels along a different beam path (the probe). The signal in this case is emitted in the same direction as the ’probe’ pulse, which thus serves as both the third excitation pulse and as a reference for spectral interferometry. This removes the requirement of a separate reference pulse and ensures the time delay between the final excitation pulse and the reference is locked at zero. The signal detected is thus the purely absorptive contribution and there is no ambiguity regarding the global phase of the signal (Fuller 2015) (see Section \ref{SubSecPhasing}). The two collinear ’pump’ photons that contribute to the signal are usually generated by a pulse shaper, often one based on an acousto-optic programmable dispersive filter (AOPDF). The phase and wavevector of these two pulses have opposite signs, so the momentum and phase provided by one exactly cancel out and the signal is emitted with the same wave-vector as the probe. Rephasing and non-rephasing signals are thus both present and the detected third order signal is a combination of both. This same geometry has also been used for fifth-order experiments where the pump provides four light-matter interactions, with timings that are independently controllable (Zhang 2012a, Zhang 2013, Zhang 2015). For all approaches with this geometry, phase-cycling must be employed to fully disentangle the signal from the probe pulse even in ideal circumstances.

Collinear geometry

Experiments in which all the pulses have the same wave-vector are classified as collinear. In this geometry, the nonlinear signal of interest spatially overlaps all of the excitation pulses, and a wide variety of other spurious nonlinear signals. This can present challenges for isolating the signal, but the collinear geometry also enables high spatial resolution, which can be of significant benefit. The approaches successfully used to isolate the desired signal in a collinear geometry all rely on phase modulation and some form of lock-in detection . This can involve interferometry of the third- (or higher-) order signal with an additional reference beam (Borri 2001a, Langbein 2006, Kasprzak 2010, Kasprzak 2012, Jakubczyk 2016, Mermillod 2016, Fras 2016, Mermillod 2016a); or the absorption of a fourth pulse to drive the system into an excited or ground state population (which is modulated according to the relative phases of the four pulses)(Tekavec 2007, Lott 2011, Widom 2013, Karki 2014, Nardin 2013, Nardin 2015, Aeschlimann 2011). The modulated excited state population is then read out by some other means, for example, photocurrent, photoluminescence or photoelectrons. These approaches all work well, obtaining good signal to noise, but have the downside that the signal is not spectrally resolved and thus the delay of the fourth pulse needs to be scanned in addition to the other delays.

Selective techniques

\label{SubSecSelectiveTechniques}

So far in the discussion of separating quantum pathways using CMDS, we have implicitly considered only pulse sequences in which all of the pulses are identical and broadband. One of the major advantages of this approach is that it simultaneously excites and probes many quantum pathways, which provides a rich variety of information simultaneously in a single scan. This ceases to be advantageous if the signals of interest overlap with or are obscured by other pathways in the 2D or 3D spectra (Tollerud 2014). Even worse, for some semiconductor systems excitation of states not directly involved in generating the signal can actually affect the signal being detected. For example, the excitation of free carriers in a quantum well can scatter with the excitons being probed and thus decrease the measured decoherence times (Cundiff 2008, Cundiff 2012, Shah 1999). One further limitation of broadband excitation is that, because everything is initially coherently excited, it can be difficult to identify the coherent evolution of some specific coherent superposition, caused by adiabatic relaxation of the excited state, for example. Suppressing these undesired and unnecessary signals can thus reveal otherwise hidden signals or signal pathways (Tollerud 2014, Tollerud 2016a), significantly reduce the data acquisition time (Senlik 2015), and simplify interpretation (Yuen-Zhou 2014). Several different approaches to isolate or enhance particular signals have been utilised: control of the polarization of each excitation pulse (Woutersen 2000, Zanni 2001, Zanni 2001a, Khalil 2003, Dreyer 2003, Karaiskaj 2010, Bristow 2009, Read 2007, Singh 2016a, Turner 2009, Stone 2009, Wilmer 2016), modifying the temporal shape of one or more of the excitation pulses (Prokhorenko 2009, Prokhorenko 2011, Wen 2013a), or by using sequences of pulses which are spectrally shaped to be resonant with specific transitions (Myers 2008, Tollerud 2014, Senlik 2015, Tollerud 2016a, Yuen-Zhou 2014). These types of approaches are commonly applied in multidimensional NMR, where the pulse spectrum, timing and area are carefully controlled for each of the hundreds of pulses typically applied in order to select specific quantum pathways. These specific combinations are then used to help determine the structure of complex macromolecules. In this case, collection of the complete multidimensional spectrum which would involve hundreds of dimensions is completely infeasible and unnecessary, because most of the multidimensional space is empty.

Selection using pulse polarization

One way to isolate or enhance specific contributions to 2D spectra is by controlling the polarization of the excitation pulses and the reference beam. The polarization sequences use the selection rules of the states to enhance or suppress certain signals, so the usefulness of different sequences depends on the sample. For example, using co-circular polarization can be used to suppress bi-exciton related signals in semiconductor quantum wells (Bristow 2009, Turner 2009, Stone 2009, Wilmer 2016). Cross-linear polarization, on the other hand, can enhance biexciton signals, making them much more readily identifiable (Karaiskaj 2010, Bristow 2009), suppress signals resulting from many-body interactions (Bristow 2009), and a provide a clear demonstration of the bosonic nature of the excitons (Singh 2016a). In molecular systems, where the transition dipoles are often linearly polarized, polarization pulse sequences involving different combinations of linearly polarized pulses have been devised following on from similar experiments in 2D IR spectroscopy (Woutersen 2000, Zanni 2001, Zanni 2001a, Khalil 2003, Dreyer 2003). For example, linear polarization sequences have been used to enhance cross-peak signals (Read 2007), and to determine details of the molecular structure by determining the relative angle of coupled transition dipoles (Ginsberg 2011).

Selection by shaping spectral amplitudes

Another approach to isolating specific quantum pathways is to use excitation pulses that don’t all share the same spectrum. The spectral amplitudes of each pulse can then be shaped to be resonant only with a particular transition (or group of transitions), and pulse sequences are then chosen to selectively excite a specific pathway (or pathways) while suppressing others. This can be thought of as a filtering of the signal in the spectral domain, which is useful in that it reduces the ambiguity in interpreting 2D spectra in both 1Q and 2Q pulse orderings (Tollerud 2014, Tollerud 2016a). In this way, a 2D spectrum acquired with selective pulse sequence can isolate some of the signals that would otherwise require a 3D spectrum to isolate. This results acquisition times that can be orders of magnitude shorter, which makes the measurement of a specific signal/s as a function of some other parameter (e.g. as a function of excitation density or temperature) more feasible (Tollerud 2014, Senlik 2015).

This approach to pathway selection, like polarization control, can also be useful for revealing very weak features (Tollerud 2014). The presence of strong features can increase the noise level across the 2D spectrum, bringing it up above the amplitude of the weak features, thereby making them difficult to identify. Similarly, the spectral tails of these strong peaks can obscure the much weaker features. It is also important to note that for overlapping signals it is the full complex signals that are summed, not simply the magnitudes, which can lead to difficulty in interpretation. For example, if the features are $$\pi$$ out of phase the stronger peak can appear split, or for other phase differences phase twists can significantly modify the shape of the weak feature, making it’s precise spectral position and shape difficult to determine. Suppressing the stronger feature alleviates these deleterious effects, and enables much more incisive analysis of weaker signals (Tollerud 2014).

Selective excitation by spectral shaping not only suppresses other signal pathways, but it also suppresses the population of other excitations, which can affect peak shapes and amplitudes through various interactions, despite not directly contributing to the signal. This is particularly relevant for semiconductor systems, where the interactions between excitons and free-carriers play an important role in determining their properties (Cundiff 2008, Cundiff 2012, Shah 1999).

Further refinement and filtering of the signal detected can be achieved by adding in control of the delays. This has recently been demonstrated for pathway selective 2Q spectroscopy, where the faster decoherence time of free carriers can be used to remove their contribution while maintaining the contribution from excitons (Tollerud 2016a). This temporal filtering is applied by choosing a value for the delay between the first two pulses ($$t_{1}$$) that is longer than the free carrier coherence time but shorter than the exciton coherence time. The 2Q 2D spectrum, which plots $$\omega_{2}$$ versus $$\omega_{3}$$ is, however, not affected because the data is still acquired over the full range of delays for $$t_{2Q}$$.

The experimental approach used to realise pathway selectivity by spectral shaping depends on the degree of selectivity desired and the specific pathway of interest. For example, if it is a specific population transfer pathway that is of interest, it is possible to use two separate OPAs to generate the two spectrally distinct pulses. This is possible because for population pathways, phase stability over the $$t_{2}$$ time period, in which there is no phase evolution, is not required (Myers 2008, Woutersen 1997, Manzoni 2006, Tekavec 2009). On the other hand, if the pathway of interest involves a 0Q coherence or a 2Q signal, phase stability is required between all four pulses. It is then necessary that the pulses all come from the same source. With a spectrum broad enough to cover all of the relevant transitions, the output from a NOPA can be spectrally shaped by a pulse shaper to tailor the spectrum of each pulse (Myers 2008, Senlik 2015, Yuen-Zhou 2014, Tollerud 2014). The downside of this approach is that in spectrally narrowing the pulses, the duration of the pulses increase and thus the time resolution of the experiment gets worse and it becomes a balance between time resolution and spectral selectivity.

Finally, pathway selectivity by spectral shaping is an important part of the procedures developed for using CMDS for quantum state and quantum process tomography. Where all transitions are well-separated, this may not be necessary, but for the many examples where there are overlapping peaks, this type of pathway selectivity is required for the full tomography. This approach has been demonstrated for a simple molecular system, but there remain many opportunities to further enhance and explore the capabilities for full quantum state and quantum process tomography using pathway selective CMDS.

Selection by temporal phase shaping

Concepts developed in coherent-control experiments can also be used to enhance or suppress particular signals by modifying the electric field of the excitation pulses (Brif 2010). A pulse shaper modifies the spectral phase and amplitude of one (or more) of the pulses such that the pulse electric field has some arbitrary shape that leads to quantum pathways interfering constructively or destructively. So far, only somewhat simple implementations of this approach have been demonstrated: Wen et al. shaped one of the excitation pulses into double pulse to enhance coupling in a semiconductor quantum well (Wen 2013a), and Prokhorenko et al. used an arbitrary waveform to study energy transfer in Rhodamine 101 (Prokhorenko 2009, Prokhorenko 2011).

Coherent-control schemes often rely on adaptive algorithms to find the ideal waveform (Brif 2010), but in both of the pioneering applications of coherent-control to 2D spectroscopy, no adaptive algorithm was used. Wen et al. used a relatively simple waveform whose effects on the 2D spectrum could be predicted. Prokhorenko et al. used a more complex waveform, but one that was previously optimized iteratively using a coherent-control pump-probe experiment instead of 2D spectroscopy. The implementation of such iterative optimization algorithms directly to 2D spectroscopy is a promising method for isolating specific signals or enhancing weak features but would require faster acquisition of the 2D spectra than most current experimental techniques provide.

Phasing

\label{SubSecPhasing}

Ideally, the real part of a 2D spectrum should correspond to the purely absorptive part of the third order nonlinear susceptibility. In fully non-collinear beam geometries that rely on heterodyne detection, the 2D spectrum has an additional phase rotation called the global phase offset ($$\Phi_{G}$$), which is a consequence of the unknown relative phase of the excitation beams ($$\Phi_{1}$$, $$\Phi_{2}$$, $$\Phi_{3}$$) and the LO ($$\Phi_{LO}$$). In BOXCARS geometry the global phase is $$\Phi_{G}=-\Phi_{1}+\Phi_{2}+\Phi_{3}-\Phi_{LO}$$. The real part of the 2D spectrum corresponds to the purely absorptive part of the signal, only when $$\Phi_{G}=0$$, so $$\Phi_{G}$$ must be determined and used to rotate the data after acquisition (in a processes known as ‘phasing’), or added to one of the excitation pulses (or the LO) during the acquisition. Several different approaches for phasing of 2D spectra have been developed. One common approach is to exploit the projection-slice theorem which states that the projection of complex data in one domain is equivalent to the data at zero in the conjugate Fourier domain(Nagayama 1978, Hybl 2001). The effect of this is that the projection of the real part of the combined nonrephasing and rephasing 2D spectra onto the $$\omega_{3}$$ axis at a given waiting time ($$t_{2}$$) should be equivalent to the pump-probe signal (where $$t_{1}$$ and $$t_{3}$$ are automatically zero) at the same $$t_{2}$$ delay. A fitting procedure can then be used to determine how much phase rotation is required for the projected 2D data and the pump-probe data to coincide (Hybl 2001). This process can be tricky for samples which exhibit weak pump-probe signals, and cannot be used to phase double-quantum signals.

Alternatively, an all-optical approach can be used, in which the phase of different pairs of beams can be determined by imaging the interference pattern at the sample (Bristow 2008). The difficulty with this approach is that the reference pulse is ideally several orders of magnitude weaker than the excitation beams and delayed by the neutral density filter used to reduce its intensity. Nonetheless, these challenges can be overcome in some circumstances.

A transient grating method has also been developed which is conceptually similar to the pump-probe approach (Milota 2013). Instead of the traditional BOXCARS geometry, this approach uses a slightly different 5-beam ‘W’ geometry, which includes two LO beams. To phase the signal, the transient grating signal is detected in two symmetric directions. They show that when the two signals are perfectly out of phase, the combined transient grating signal is equivalent to the pump-probe signal and hence the projection of the 2D spectra. This requires a slightly more complex experiment, but has the additional benefit that rephasing and non-rephasing signals can be detected simultaneously. The other downside of this experiment is that energy and momentum are not both perfectly conserved in this geometry, which can cause changes to peak shapes (Butenhoff 1993, Hybl 2001, Mercer 2009).

Sampling requirements

In most CMDS techniques, multidimensional data is acquired by scanning the inter-pulse delay with a constant sampling frequency (or conversely a constant delay step size), and then using a discrete Fourier transform to represent the data in the frequency domain. The rate at which the signal phase evolves as a function of the scanned delays imposes a minimum sampling rate for the inter-pulse delays. The specific sampling requirements depend on details of the sample, and the type of spectrum being acquired. For example, when acquiring a 1Q 2D spectrum and delaying the beams using conventional beam geometries, the signal phase changes with $$\omega_{1}t_{1}$$ where $$t_{1}$$ is the delay induced by the movement of the stage and $$\omega_{1}$$ is the energy of the coherence excited by the first pulse. The $$t_{1}$$ step-size, $$\Delta t_{1}$$ needs to be sufficiently small to properly sample the field oscillations (as defined by the Nyquist limit). THe maximum value of $$\Delta t_{1}$$ is then inversely proportional to $$\omega_{1}$$. Satisfying these sampling requirements necessitates high precision (sub-cycle, ideally sub-fs) control of the inter-pulse delays, and leads to long acquisition times to sample the complete dynamics - especially for transitions with long coherence times.

To ease these sampling requirements, several approaches have been devised to allow larger delay step sizes ($$\Delta t_{1}$$). One approach is to delay the pulses in a rotating frame by keeping the phase fixed for a specific frequency (using a pulse-shaper) or keeping the phase fixed across the entire excitation spectrum (in pairwise delay stage geometry), or using birefringent prism pairs, which intrinsically apply delays in a partially rotating frame (Preda 2016). Applying delays in these different rotating frames significantly reduces the rate at which the phase of the signal evolves as a function of the inter-pulse delays, which relaxes the sampling requirements. Alternatively, the delay step size can also be increased by changing how the inter-pulse delay is sampled (Roeding 2017, Dunbar 2013, Spencer 2016, Sanders 2012, Almeida 2012). For example, the inter-pulse delay sampling can be randomized (instead of linear). Compressed sensing algorithms can then be used instead of discrete Fourier transforms to represent the randomly sampled data in the frequency domain. Fewer data points are required in this approach because a wider range of frequencies are inherently captured for the number of data points. Systematic undersampling can also be used if some assumptions can be made about the signal that is being collected (e.g. that the phase and amplitude evolve in a predictable way) (Dunbar 2013).

Experimental implementations for phase stability and delay control

\label{SubSecExpImplimentations}

The most important challenge in performing CMDS measurements is how to maintain phase stability between pulses and vary the delays. Without the ability to control the delays between pulses while maintaining phase stability, all of the other experimental considerations are irrelevant. To overcome this problem a significant number of different approaches have been developed. These can be grouped broadly into four different categories: active stabilization methods, whereby nanoscale changes to mirror positions are made to compensate for any phase fluctuations; passive stabilization approaches, which are based on the principle of all beams hitting common optics; compensation of phase fluctuations so that there is no nett phase change of the detected spectral interferogram even in the presence of fluctuations; and methods by which the phase between pulses is not stable, but the phase changes are measured and utilised in analysing the signal. In this section, we provide a brief discussion of these approaches and the specific implementations that have been developed, including their advantages and limitations, and how they have been used to study semiconductor nanostructures. While this may not be a comprehensive list, we have tried to add a description for most of the experimental approaches utilised.

Actively phase stabilized beam paths

The simplest approach for controlling delays is to have each beam travel separate beam paths so that the lengths of each can be controlled. This is the standard approach used for most transient optical spectroscopy, where path lengths are controlled by retro-reflectors mounted on linear translation stages. The problem with this is that each beam is incident upon different opto-mechanical elements which naturally introduce instability in the relative phase of the excitation pulses, reducing the stability in the phase of the nonlinear signal and introducing artefacts in (or flat out preventing the acquisition of) multidimensional spectra. This can be overcome by actively stabilizing the pathlengths.

To actively stabilize the phase of the excitation pulses, the path length drift of different optical paths must be measured and compensated. To this end, a continuus-wave (CW) reference beam is added to co-propagate with the pulsed excitation beam into the experimental setup, passing through all the same optical elements (beam splitters, mirrors, delay lines, etc.) as the excitation beam. In the implementation developed by the Cundiff group (Zhang 2005, Bristow 2009) the CW reference beams that have travelled the four different paths of the excitation beams are reflected using a dichroic mirror just before the sample. For the references, this now looks like two Michelson inferferometers inside a larger Michelson interferometer, and the interference of the CW beams can provide precise measurement of path-lengths. The first two interferometers measure the path-length difference of different pairs of beams, and the final interferometer measures the interference of all four copies of the CW reference. The error signal from these interferometers is then used as part of a feedback loop to make small changes to the path-lengths of different beams using the delay lines and thereby compensate for any phase drift. For more details on actively stabilized CMDS, see Ref. (Bristow 2009).

Actively stabilized translation stage based experiments have, for example, been used to measure the homogeneous linewidth of excitons in quantum wells (QWs) and quantum dots(Singh 2016, Singh 2013, Moody 2011, Moody 2011a, Bristow 2011, Suzuki 2016). Homogeneous linewidths can only be accurately measured if the inter-pulse delay range of the experiment allows the full decoherence decay to be recorded. For that reason, the long ($$\geq$$100 ps) delay range of translation stage based experiments is ideal for measuring the homogeneous linewidth of excitons in quantum wells and quantum dots, which can be as narrow as a $$\sim$$0.1 meV (Bristow 2011).

These approaches have also been used to definitively identify signatures of biexcitons in polarization dependent single-quantum spectroscopy (Bristow 2009a) and double-quantum spectroscopy (Karaiskaj 2010). This type of experiment has then allowed identification and characterization of biexciton contributions in self-assembled quantum dots (Moody 2013), bulk semiconductors (Dey 2015, Webber 2016, Wilmer 2016), and in microcavities (Wen 2013, Bristow 2016).

This type of implementation of CMDS has also been used in establishing the capabilities of CMDS to investigate many-body effects and biexcitons in semiconductor QWs (Karaiskaj 2010, Nardin 2014, Turner 2012, Li 2006, Borca 2005). These experiments compare the peak shapes of real valued spectra in single-quantum and double-quantum spectroscopy to simulations that phenomenologically include many-body effects. The ability to scan long coherence times – which provides precise peak-shapes – is crucial for this type of peak-shape analysis, and phase stability between all four pulses is required for the 2Q experiments. Both of these requirements are satisfied by the actively phase stabilized approach

Coherent coupling and interactions between excitons in different InGaAs quantum wells(Nardin 2014), between excitons and trions in CdTe quantum wells (Moody 2014) and alternatively between pairs of atoms in an atomic gas(Dai 2012, Li 2013) have also been identified using this implementation to generate 0Q 2D spectra and 3D spectra. In the latter case, the 3D spectrum was used to directly determine the Hamiltonian of the system (Li 2013). More recently, these approaches have been applied to reveal coherent dynamics in atomically thin 2D semiconductors (Moody 2015, Hao 2016, Hao 2016a). Even though the spatial resolution achievable is not optimal for studying these samples, substantial insight has been gained.

Tracking and Referencing Relative Phases

Several groups have implemented collinear and non-collinear CMDS experiments using phase-modulation and referencing of the signal phase (Borri 2001a, Langbein 2006, Kasprzak 2010, Kasprzak 2012, Jakubczyk 2016, Mermillod 2016, Fras 2016, Mermillod 2016a, Tekavec 2007, Lott 2011, Widom 2013, Karki 2014, Nardin 2013, Nardin 2015, Aeschlimann 2011). In these implementations the four pulses are delayed using translation stages on different delay lines, each of which has an acousto-optic modulator (AOM) in the beam path, which labels each beam with a characteristic phase modulation frequency. The signal will then have a characteristic phase modulation depending on the specific combination of excitation beams, which can be referenced and used for detection. This referencing can automatically account for any fluctuations in the pathlength of the delay lines, as they will appear in both the reference signal and the FWM signal. Within this class of technique there have been different referencing and detection schemes developed, each with their own advantages.

Population detection

Instead of using the fourth pulse as a reference to interfere with the third-order polarization, it possible to use this pulse to drive the system into a population state(Aeschlimann 2011, Widom 2013, Karki 2014, Nardin 2013). In this case, it is the interference between the fourth pulse and the third-order polarization in the sample that allows determination of the phase information. The readout for these experiments is then based on the population, which will depend on the phase of the third-order polarization relative to the fourth pulse. In these experiments where the phase of the excitation pulses are modulated, the population after the fourth pulse will also be modulated at a frequency well-defined by the AOMs. The population can be detected in a number of ways and converted to an electronic signal, which is run through a lock-in amplifier to isolate the desired third order signal. The reference for the lock-in detection is obtained by interfering CW reference beams that have co-propagated with the excitation beams. This referencing not only accounts for any fluctuations in the pathlengths of the delay lines, thereby providing a signal that is effectively phase locked, but it also enables intrinsic phasing of the signal: The interference of the four excitation beams precisely determines the point where all four beams are in phase. This is automatically incorporated into the lock-in detection and so any relevant phase shifts in the signal will appear as a phase shift relative to the reference in the response measured by the lock-in amplifier.

In principle, any measurement technique that is sensitive to populations can be used, for example, the initial experiments using this approach measured the photoluminescence from the sample to determine the population (Tekavec 2007, Lott 2011, Widom 2013). Measurement of photocurrent is another means of detecting the population that has been utilised(Nardin 2013, Karki 2014). Nardin et al. used this approach to measure the coherent dynamics of quantum well excitons in a P-I-N diode (Nardin 2013), and Karki et al. measured 2D spectra of a PbS quantum dot solar cell (Karki 2014). One of the great advantages of photocurrent detection is that it directly reports on the properties related to the intended use of these devices; namely, to convert optical energy into electrical current.

Another experiment with population detection used photoemission electron microscopy (Aeschlimann 2011) to provide spatial resolution far below the diffraction limit. In that case they used phase cycling rather than phase modulation for their signal extraction, but were able to collect 2D spectra for localized ($$\sim$$100 $$nm$$) sized excitations in a corrugated silver film. The combination of $$\sim$$50 $$nm$$ resolution with CMDS has the potential to provide significant insight into a range of processes, for example, in the coherent properties of excitons in individual nanostructures.

The detection of the population means there is no advantage to having a non-collinear geometry and so collinear approaches are typically used for their ability to achieve high spatial resolution through the use of high NA microscope objectives to focus the beam into the sample. One of the limitations of these approaches, however, is that they utilise single channel detection, which means an additional delay needs to be scanned to obtain 2D spectra and increases the time required for each experiment. This overcome to some extent by hyperspectral imaging, as described below.

Hyper-spectral Imaging

Langbein et al (Langbein 2006) developed a slightly different collinear phase modulation approach (based on earlier heterodyne-detected collinear pump-probe/FWM experiments (Hofmann 1996, Mecozzi 1996, Borri 2001a, Borri 2001)) in which the emitted third-order signal is measured using heterodyne detection instead of populations detection. As in the collinear approaches described above, the excitation beams are sent down different delay lines – which each include an AOM operating at a unique frequency – and are then recombined.

In addition to the two/three excitation pulses, a reference pulse (the 0-order beam from one of the AOMs) is sent down a similar path to the excitation, but angled slightly so that it is focused to a spot on the sample about 10$$\mu$$m away from the excitation beams (Langbein 2006). The reference beam and the signal/excitation beams are then focused onto an additional AOM which is used to spatially overlap the signal and the reference. The frequency of the final AOM is chosen so that the angle between the two beams matches the diffraction angle, and so that the phase shift of the reference beam matches the expected phase shift of the signal beam, based on the appropriate combination of the phase shifts induced in the excitation beams. The portion of the reference beam that is diffracted into the direction of the signal then gains a phase shift equal to that of the signal (Langbein 2006).

This approach ensures that the phase of the signal and the phase of the reference evolve uniformly so that the phase difference is ideally fixed over the course of a measurement. Interferograms between the signal and the reference can therefore be detected. The detection is very slow compared with the AOM induced phase evolution, so spurious interferograms involving the excitation beams, which have phases that evolve over the course of the measurement, are inherently washed-out. The non-interferometric background signals, however, remain, which means the ratio of the signal to the background is rather small, and can limit the sensitivity of a measurement. Furthermore, the signal and the reference aren’t inherently phase locked, so there is some slow drift in their relative phase. This must be corrected by using a resonance with a robust and well-defined phase evolution as a phase reference.

The spatial resolution of hyperspectral imaging experiments has been exploited to perform 2D spectroscopy on small samples – including individual semiconductor nanostructures – that would otherwise be ensemble averaged in other experimental implementations (Kasprzak 2010, Kasprzak 2012, Jakubczyk 2016, Mermillod 2016, Fras 2016, Mermillod 2016a). For example, this approach has been used to measure the coherent and population dynamics of excitons and biexcitons in individual quantum dots (Kasprzak 2012, Mermillod 2016a). By collecting 2D spectra for many individual interfacial quantum dots, Kasprzak et al. were able to identify coupling of quantum dots separated in-plane by up to micron distances (Kasprzak 2010). Jakubczyk et al used hyperspectral imaging to study the coherent and incoherent dynamics of monolayed MoSe$${}_{2}$$ (Jakubczyk 2016). Unlike other measurements of monolayer transition metal dichalcogenides, the spatial resolution (¡1$$\mu m$$) is much smaller than the sample size ($$\sim 30\mu m$$), so variations in the dynamics can be spatially resolved. They find significant variation across the sample, emphasizing the importance of the local environment and suggesting that micrometer resolution experiments are required to accurately quantify these dynamics.

This approach has also been used to obtain 2D spectra from dots embedded in a photonic waveguide antenna (Mermillod 2016). The shape of the waveguide antenna significantly enhanced the nonlinear processes and enabled them to measure the exciton and biexciton dynamics of a single quantum dot and identify coupling between dots embedded in the same antenna. Extending this approach they were able to demonstrate control of the state of the dot by controlling the pulse area, in the same way pulses are controlled in NMR.

Cancellation of jitter with pairwise delays

\label{SubSec_PairWise}

An alternative approach that maintains the simplicity of scanning delays with translation stages, without the complexity of active stabilization of phase modulation relies on the cancellation of phase variations in the measured interferogram (Selig 2008, Selig 2010, Cowan 2004, Heisler 2014, Gellen 2016). This is based on a beam geometry whereby the beams are split in pairs so that any phase shifts caused by path length instabilities are common across the pair and have opposite effects on the measured interferogram. In other words, for third order experiments, each pair must have one of $$k_{1}$$ or $$k_{LO}$$ and one of $$k_{2}$$ or $$k_{3}$$. To enable collection of the data of interest, $$t_{1}$$ and $$t_{2}$$ must be able to be controlled. This is achieved by splitting and recombining the four beams (in a box geometry) twice so that different combinations are paired. This enables independent control of each of the relevant delays. Further, because this can be arranged with all reflective optics, the response is roughly wavelength independent, which has led to this approach being used for CMDS measurements extending into the UV(Selig 2010).

While the phase of the interferogram doesn’t change as phase fluctuations are introduced, the same is not true for the scatter. In other words, the scatter is not phase-stabilised, and so there are fluctuations in the interferograms measured, which cannot be removed by phase cycling or any other slow signal isolation. Instead, approaches using double chopping are often applied to isolate the signal from the scatter. While perhaps the simplest of the approaches detailed here, this is regarded to be among the least sensitive because of this problem and because of greater phase instability compared to other approaches. However, detailed comparisons have not been previously made under the same conditions. In Section \ref{SecHybridComparison} we make these quantitative comparisons and show that while it may not be as good as some approaches, for samples with minimal scatter, very good signal to noise can still be obtained. Regardless, this approach has been used extensively, for example, to reveal the interactions between chromophores and their solvent environment and the effect on structure (Camargo 2015, Moca 2015, Camargo 2015, Bizimana 2015); to unveil the effects of vibronic coupling on exciton coherences in a molecular dimer(Halpin 2014), and to reveal the role of spin–orbit coupling in the non-adiabatic dynamics in a transition metal compound(Carbery 2017)

A schematic implementation of this pairwise beam geometry is shown in Fig. \ref{Fig_Pairwise}. The beams are in a box geometry and are split off and delayed in pairs by four delay lines (DL1 – DL3, FD), which each go via retro-reflectors on translation stages (ST1, ST2) or a fixed delay line. ST1 controls the timing of $$k_{2}$$ and $$LO$$ (from DL1) and in the opposite direction $$k_{1}$$ and $$k_{2}$$ (from DL2). ST2 controls the timing of $$k_{3}$$ and LO (from DL3). A fixed delay FD for $$k_{1}$$ and $$k_{3}$$ ensures the pathlengths for all pulses are the same (each having been through two delay lines) for some position of ST1 and ST2. In this particular implementation, $$t_{1}$$ can be scanned with ST1 and $$t_{2}$$ can be scanned with ST2. To understand how this arrangement leads to a passively stabilized signal phase, let us consider some change in the optical pathlength along DL1, $$\Delta t_{DL1}$$, which may be either due to intentional movement of the translation stage or due to opto-mechanical instabilities. The signal phase is given by

$$\label{EqPhisig} \label{EqPhisig}\Phi_{LO}=–\Phi_{k_{1}}+\Phi_{k_{2}}+\Phi_{k_{3}}-\Phi_{LO}\\$$

where both $$\Phi_{k_{2}}$$ and $$\Phi_{LO}$$ include $$\Delta t_{DL1}$$:

$$\label{EqPhik2} \label{EqPhik2}\Phi_{k_{2}}=\tilde{\Phi}_{k_{2}}+\omega\Delta t_{DL1}\\$$ $$\label{EqPhiLO} \label{EqPhiLO}\Phi_{LO}=\tilde{\Phi}_{LO}+\omega\Delta t_{DL1}\\$$

and $$\tilde{\Phi}_{k_{2}}$$ and $$\tilde{\Phi}_{LO}$$ are the relative phases of $$k_{2}$$ and $$LO$$ absent experiment instabilities. When we substitute Eqs \ref{EqPhik2} and \ref{EqPhiLO} into Eq. \ref{EqPhisig}, we find that $$\Delta t_{DL1}$$ cancels out and thus any changes in the optical path length along this delay line will have a no net contribution to the signal phase (assuming that the vibrations affect $$\Phi_{k_{2}}$$ and $$\Phi_{LO}$$ equally). The same is true of the DL2, DL3 and FD. It is also important to note that $$\Delta t_{DL1}$$ includes both incidental vibrations and intentional movement of the stage, so the phase of the signal does not change when TS1 is moved to acquire a multidimensional spectrum. In other words, the delays can be moved in a completely rotating frame, which significantly increases the minimum sampling frequency (i.e. maximum delay step size), and reduces acquisition time.

The frequency $$\omega_{cf}$$ for the rotating frame is equal to the detection frequency and thus varies across the detected spectrum. Any measured evolution of the signal phase must then be a result of phase-altering signal pathways or processes in the sample, rather than due to the changing phase of the excitation pulses. In this ‘rotating frame’ DPs exhibit no phase evolution as a function of $$t_{1}$$, and CPs have a phase that evolves with a frequency set by its separation from the diagonal line along $$E_{1}$$ (which we will call $$\Delta\omega_{1}$$). The minimum sampling frequency is thus proportional to the bandwidth of the excitation pulses, rather than the transition energy.

With this layout, increasing $$t_{1}$$ can be achieved simply by translating TS1 (to shorten the pathlength of $$k_{1}$$), $$k_{2}$$ will be unshifted in time becasue it goes via both sides of ST1, while the pathlength of the LO will increase by an equal amount. This can be an advantage for measurement of inhomogeneously broadened transitions with narrow homogeneous linewidths, as it maintains a constant delay between the LO and the photon-echo signal, which ensures spacing between fringes in the interferogram remains constant and optimal. For a purely homogeneously broadened transition, the delay between LO and signal will vary as $$t_{1}$$ is scanned, which could potentially affect the ability to resolve fringes in the interferogram, leading to a spurious additional decay of the signal. However, if the homogeneous linewidth is so narrow that the decoherence time is greater than $$\sim$$ 10 ps 11For a typical setup using a high quality imaging spectrometer with resolution $$\sim$$0.1 nm, the fringes will start to overlap when the delay between signal and LO is $$\sim$$10 ps then the resolution of the spectrometer will be a greater limitation. For most applications, if the peaks are that narrow then details of peak shapes from CMDS measurements are not likely to provide much insight22if these peak shapes do need to be resolved, then a higher resolution spectrometer would be required and would remedy the limitations in both $$t_{1}$$ and $$\omega_{3}$$., although identifying interactions between states via cross-peaks may still be useful, and will be unaffected by the resolution limitations.

\label{Fig_Pairwise}Beam layout of a pairwise delay experiment. $$t_{1}$$ can be scanned with ST1, $$t_{2}$$ can be scanned with ST2.

Passively phase stabilized

\label{SubSub_PassivelyStable}

A range of other passively phase stabilized experimental implementations have been developed by eliminating translation stage based delay lines so that the beams are incident upon mostly or entirely the same optics. For approaches that utilise a non-collinear geometry, diffractive elements must be used to generate the beam pattern in order to maintain the use of common optics. With passive phase stabilization the physical pathlengths of the different beams are locked, which intrinsically stabilizes their relative phase, but also necessitates alternate methods of controlling the inter-pulse delays. This has been achieved in different implementations using glass wedges or pulse shapers. The challenge with each of these approaches is the limited delay range that can be achieved. In some implementations a delay stage is introduced to allow the ’population’ delay to be scanned to larger values. In this case, phase stability between the first two pulses and between the third pulse and reference is maintained, but between the pairs it is lost. For rephasing and non-rephasing scans this is typically not a problem because the system is in a population state with no phase evolution, or coherent superposition with slow phase evolution, during this time period. It does, however, limit the extension of the technique to 2Q or higher order spectroscopies.

Wedge pairs

A pair of 5°glass wedges placed into one (or more) of the beam-paths can be used to control the inter-pulse delays (Brixner 2004, Brixner 2004a, Nemeth 2009, Nemeth 2010, Turner 2012a, Turner 2011a, West 2012). The added glass changes the optical path length, thus delaying the pulse by an amount proportional to the thickness of the wedge. The wedges must be used in pairs to avoid changing the beam pointing. One of the wedges is mounted to a translation stage which controls how far the wedge is inserted into the beam, and hence the timing of that pulse. A BOXCARS beam pattern generated by a diffractive optic element is generally used, which allows for chopping of one or two beams (Heisler 2014) and independent control of the polarization of each beam (Ginsberg 2011, Read 2007).

This approach intrinsically provides phase stability and timing control on the order of an attoseconds (Brixner 2004, Turner 2011a)11Each wedge pair has unique thickness variations, so careful calibration of the delay induced as a function of prism insertion is required. The stability and timing precision makes this approach particularly suitable for measurements of molecular samples with very fast coherent dynamics. However, the limited thickness of the wedge pairs limits the inter-pulse delays that can be achieved to a few hundred femtoseconds (Brixner 2004, Nemeth 2010). For this reason, and the fact that population dynamics are often orders of magnitude larger than this, a delay stage is usually introduced in these implementations to allow long population times to be scanned. As discussed above this, removes the phase stability between the first and second pair of pulses, which limits the flexibility of this approach and makes 2Q or higher order experiments impossible. The other issue with these approaches is that dispersion in the wedges can cause the pulse duration to change as the delay is scanned. This is particularly a problem for very broad bandwidth experiments, and can impact on peak shapes and interpretations.

Nonetheless, this approach has been very successful and led to a lot of high quality research(Collini 2009, Wong 2012, Turner 2012a, Turner 2011a, Turner 2013, Collini 2010, Cassette 2015, Turner 2011a, Kim 2009, Hayes 2011, Panitchayangkoon 2011, Read 2007, Milota 2013, Christensson 2010, Nemeth 2010, Zigmantas 2006, Augulis 2011, Dostál 2012, Romero 2014, Westenhoff 2012). Most applications of this approach have been for molecular and biomolecular systems, although there have been some studies of other materials with coherence times that are short enough to be fully captured within the delay range of the wedges. For example, wedge based experiments have been used to study nanplatelets at room temperature and determine the excitonic decoherence time and identify coherent interactions between separate excitonic states, both of which require interpulse delay range of $$\sim$$100 fs (Cassette 2015).

Diffraction based pulse-shaping

\label{SubSecPS_Diff}

Pulse-shapers provide another powerful tool for controlling inter-pulse delays. There are a range of different ways pulse-shaper based CMDS setups can be implemented, which can be largely separated into two types: 1. diffraction based pulse shaper using a fully non-collinear geometry in which all beams are controlled by the same pulse-shaper and 2. a two-beam partially non-collinear pump-probe geometry in which one of the beams is sent through a pulse-shaper.

In the diffraction based approach (Turner 2011, Tollerud 2014), all of the excitation beams and the reference beam are sent to the pulse shaper. A beam geometry is chosen such that each of the beams hits a different vertical region of a 2D spatial light modulator (SLM) and is spectrally dispersed in the horizontal direction, allowing the spectral phase of each beam to be controlled independently. This, among other things, allows the pulses to be compressed and the timing of each pulse to be varied – a linear spectral phase gradient corresponds to a shift in time. In addition, a vertical sawtooth pattern is applied to the SLM so the shaped beams are angled down slightly relative to the unshaped beams, enabling them to be picked off and sent to the sample, while also ensuring that just the first order diffraction in time is selected. By varying the depth of the vertical sawtooth pattern across the spectrum, the spectral amplitude of the pulses can also be modified. The pulse shaper can also be used for phase-cycling schemes by adding a series of phase offsets to the excitation beams and the reference beam. The diffraction based pulse shaper thus provides independent control of the spectral phase and amplitude for each of the excitation pulses and the LO, enabling pathway selection schemes based on temporal shaping (Wen 2013a) and shaping of the spectral amplitudes (Tollerud 2014, Yuen-Zhou 2014, Tollerud 2016a, Rodriguez 2015).

The most significant disadvantage of this approach is the delay dependent modulation of the pulse amplitude which results from the pixelation of the SLM and the finite spectral resolution of the pulse-shaper. This effect imposes a limit on the range of interpulse delays that can be applied which, like in wedge-pair based experiments, prevents the acquisition of long dynamics. In addition, the pulse amplitude can also be altered as a function of delay at shorter delays, which can induce artifacts into the multidimensional spectra even for samples with dynamics shorter than the delay range . The exact form of this delay dependent amplitude depends on the spacing of the diffraction grating and the focal length of the cylindrical lens used in the pulse shaper, but the delay range is typically limited to about $$\pm$$5 ps. To compensate for the amplitude modulaion of this delay range, the height of teh vertical grating can be adjusted as the delay is scanned to ensure the intensities of all beams remain equal. The consequences of this amplitude modulation and limited delay range for CMDS spectra are discussed in more detail in Section \ref{SecHybridComparison}.

On the other hand, the major benefit of diffraction based pulse shapers is that they afford a great deal of flexibility and stability: Many different beam geometries can be used including BOXCARS, pump-probe and the more complex geometries required for higher-order experiments (Turner 2010). Phase-cycling sequences can be easily modified to suit different samples and experiments. Selective and non-selective measurements can be performed sequentially with no changes to the optical setup, allowing the resultant spectra to be compared quantitatively. Pulses can be delayed in a rotating frame, which provides a significant reduction in sampling frequency (and hence experiment time) required to capture the signal phase evolution. Finally, experiments based on this approach are intrinsically very stable: there are no moving parts, and the only required optic 11In polarization dependent measurements, waveplates placed in individual are also required. that is not shared by all of the excitation beams is the ND filter that attenuates the LO.

These advantages have seen this approach used to perform experiments and provide insights that other techniques cannot achieve. For example, pulse sequences that isolate individual pathways by shaping the spectral amplitude of different beams have been used to identify weak coherent interactions between excitons localized in different, well-separated, quantum wells (Tollerud 2014), and between excitons localized in a QW with excitons in the barriers (Tollerud 2016b). The flexibility of this approach has been used to conduct 5th order and 7th order experiments that have identified the presence of thee-exciton correlations, but absence of four-exciton correlations (Turner 2010, Wen 2013).

The stability and sensitivity of this approach appears to be amongst the best available, although it is difficult to quantitatively compare different implementations across different laboratories, a point that we address in Section \ref{SubSec_Stabilty}. This sensitivity has been exploited in double-quantum spectroscopy to measure coherent interactions between excitons at very low excitation density  – revealing broadening mechanisms and natural peak shapes that are hidden at higher densities. This approach has also been used to identify and characterise a range of ’dark” QW excitons whose oscillator strengths are small enough that they cannot be seen in linear spectroscopy and are $$\leq 0.01$$ of those for direct, allowed QW transitions .

Pulse shaper in pump-probe geometry

\label{SubSecPS_PP}

Pulse-shapers can also be used to perform CMDS in a pump-probe geometry (Grumstrup 2007, Tian 2003, Myers 2008, Deflores 2007). In this approach, the pump-pulse is routed through a pulse-shaper, and the probe pulse is sent along another path which typically includes a mechanical delay line. The pump-pulse is then split into two (or more) collinear pulses by the pulse shaper (Grumstrup 2007).

This approach has proven to be very successful as it is essentially a pump-probe experiment with the addition of a pulse-shaper in the pump beam, and it is thus fairly straightforward to modify existing pump-probe experiments so that they can perform CMDS (Shim 2008). As in other pump-probe experiments, the detected quantity is the interference between the third-order signal and the probe pulse itself, which intrinsically is the absorptive part of the third order signal. Thus, unlike non-collinear experiments, no phasing is required to determine the absorptive part of the signal. Because the first two pulses (i.e. the pump pulses) are indistinguishable, the rephasing and non-rephasing signals are both present, but can be separated by phase cycling.

Unlike the diffraction based experiments, only a single beam is shaped, so 1D SLMs or AOM based pulse-shapers can be used. Like the diffraction based pulse-shapers, this method produces excellent phase stability between the shaped pulses, however, the pump–probe phase difference is not stabilized. As a result, this type of experiment performs well for collection of rephasing and non-rephasing spectra, but the lack of complete phase stability prevents some types of experiments (for example double-quantum spectroscopy). There are, however, many examples where the capabilities of this approach provide good match(Myers 2008, Tekavec 2009, Myers 2010, Lewis 2012, Fuller 2014, Senlik 2015, Zhang 2012, Wells 2013, Wells 2014, Enriquez 2015, Zhang 2015, Akhtar 2017). This is particularly the case for biomolecular systems, which typically have short coherence dynamics, but significant energy transfer, population transfer, and coupling between states. For example, this approach has been particularly successful in probing photosynthetic light harvesting complexes and recently has been able to identify vibronic coupling between exciton states and localised vibration modes (Fuller 2014).

Most of the pump-probe geometry experiments have been used to measure the third-order response of the system, but it can also be used for higher order experiments by splitting the pump pulse into more than two pulses. Several fifth order experiments have been reported, including 3D spectroscopy of a biological light harvesting complex, LHCII, in which a multi-step relaxation process is measured (Zhang 2015). However, expanding to higher order experiments in this geometry has the intrinsic limitation that many signals overlap, and the number of phase cycling steps required increases rapidly.

One of the advantages of AOM based pulses shapers is that they can change the pulse timings and perform the phase cycling much faster than, for example, SLMs. This in turn leads to shorter acquisition times, the potential for shot to shot changes in the applied pulses and the prospect for lower noise. However, the signal-to-noise ratio and sensitivity in third-order pump-probe geometry experiments is expected to be lower than for an equivalent non-collinear experiments (Fuller 2014a, Fuller 2015). The reason is that probe pulse provides both the third interaction and the reference, which makes it impossible to attenuate the reference so that its intensity is similar to the signal, which has been shown to give the best signal to noise. In principle phase cycling should be able to remove the background contribution, but (as we will discuss later) the amplitude and stability of this background signal and the limited dynamic range of detectors place a limit on the sensitivity of the experiment. Polarization sequences can be used to suppress much of the probe background (Myers 2008, Xiong 2008), but this is not possible in all applications.

Recently, Ogilvie et al, have modified their pump-probe geometry pulse shaper approach to allow background free detection by using a diffractive optic to spilt both the pump and probe into two beams (Fuller 2014a). This overcomes the problem of not being able to control the amplitude of the reference independently of the third interaction and significantly improves performance, with a 20$$\times$$ improvement in SNR. This approach was recently applied, together with spectral amplitude shaping of the two pump pulses, to selectively excite coherence pathways in chlorophyll-a (Senlik 2015).

Further demonstrations of enhanced SNR in pulse-shaper pump-probe geometries has been demonstrated by Zanni et al.(Kearns 2017). They utilise a supercontinuum source with 100 kHz repetition rate and single shot pulse shaping and detection to acquire 2D spectra over a 200 fs delay range in 1 ms. The single shot readout has already been shown to to be advantageous for SNR(Heisler 2014, Brazard 2015), but with the higher repetition rate and by phase cycling and changing delays on a shot-by-shot basis, further enhancement is achieved because the full scan range can be explored more rapidly, and within the self-correlation time of the laser source.

Birefringent wedge pairs

\label{BiWedgePairs}

Another way to perform CMDS in a pump-probe geometry is by placing a series of birefringent optics (a plate and two pairs of wedged windows) into the pump beam (Réhault 2014, Brida 2013). The pulse polarization is initially 45°relative to the optical axis of the the first birefringent plate, which converts the single input pulse into two orthogonally polarized pulses separated by a fixed time delay. A pair of birefringent wedges (cut so that the optical axis is perpendicular to the first plate) are then used to precisely control the time delay by changing the amount one of the wedges is inserted into the pump beam. A second set of birefringent wedges (cut so that the optical axis is oriented along the propagation direction) move anti-symmetrically to the first pair so as to maintain a constant amount of material in the beam, thus ensuring that the delay between one of the pump pulses and the probe pulse doesn’t and that there is no change to the group-delay dispersion of the pump pulses as the wedges are moved (Réhault 2014). Finally, a polarizer with the optical axis at 45°projects both of the pump pulses onto a single polarization.

Like the pulse-shaper based pump-probe geometry experiments, this wedge pair produces excellent phase stability between the first and second pulses, but not between the pump and the probe pulses (Réhault 2014). Unlike pulse-shaper based experiments, phase-cycling cannot be used to extract the signal and the rephasing and non-rephasing cannot be separated. On the other hand, cross-polarized pump-pulses can be generated by omitting the final polariser, and the pulses are naturally delayed in a partially rotating frame (Preda 2016, Brida 2013) which relaxes sampling requirements. The only limitation on the spectral range accessible with this approach is the material chosen for the birefringent optics. 2D spectroscopy using this technique has recently been demonstrated in the infrared (Rehault 2014) and in the ultraviolet (Borrego-Varillas 2016)

Other types of approaches

Single Shot (GRAPES)

The approaches described above all involve “slow” detection that requires many measurements to record a 2D spectrum using point-by-point sampling. Any phase or amplitude variation over that time will lead to a artefacts in the 2D spectrum and for many samples, particularly biological samples, the large number of incident laser pulses can lead to sample degradation over the course of the experiment. This has motivated the development of a single-shot 2D spectroscopy technique called ‘Gradient-Assisted Photon Echo spectroscopy’, which uses one pulse with a tilted wavefront to encode the inter-pulse delay to be scanned onto a spatial coordinate . In this scheme, one of the beams is reflected off of a vertically tilted mirror, and a cylindrical lens is used to focus the beams onto the sample. An imaging spectrometer with a 2D CCD is used to detect the signal. The horizontal axis of the CCD corresponds to the spectral characteristics of the signal, while the dynamics are encoded into the vertical dimension. A 2D spectrum can then be generated by a Fourier transform as a function of the vertical axis of a single image acquired by the spectrometer. The other delay, usually the population delay is varied by a delay stage, as for wedges and the pump-probe geometry approaches. For this approach, phase stability between the pulses is actually not needed because the complete dataset for a 2D spectrum can be acquired in a single shot. The instantaneous phase between the pulses will affect the phase of the signal, but as with other non-collinear approaches, the global phase can be determined by comparison with the pump-probe signal.

This approach is particularly useful in circumstances where the source or sample is unstable, which can be the case for many biomolecular systems. This has allowed measurement of 2D spectra from live cells (Dahlberg 2013), light harvesting complexes(Fidler 2014) and nanoparticles (Harel 2012). Additionally, the ability to measure many 2D spectra quickly allows the use statistical averaging and analysis beyond the capabilities of other CMDS experiments (Dahlberg 2013). GRAPES is, however, less flexible than most other techniques in that the sampling rate and delay ’scanned’ are determined by the beam geometry and focusing/detection optics, and are not easily changed. It also requires samples that are homogeneous over relatively large dimensions, which can present some challenges. Similarly, it is more sensitive to the spatial mode quality than most other CMDS techniques, and the effects of distortions in the beamshapes must be compensated for in the detected signals (Harel 2010, Spokoyny 2015).

Recent developments by the Harel group have enhanced the capabilities and sensitivity(Spencer 2016, Spencer 2015, Spokoyny 2014) and pushed GRAPES into higher order spectroscopies, including 5th order, single shot 2D spectra(Hutson 2016), and mixed 2D-electronic/2D Raman spectroscopy, which is able to identify correlations between electronic states and vibrational modes(Spencer 2017). These developments further enhance the capabilities and utility of the GRAPES implementation.

Incoherent light

While the bulk of CMDS experiments use coherent light produced by fs laser systems, measurements can also be made using incoherent light. The pioneering example of this approach used ns duration amplified spontaneous emission from a non-modelocked oscillator in a traditional wedge-pair experiment (Turner 2013, Ulness 2015). Given that the sample dynamics are much shorter than the pulses, the interpretation of the results is slightly different than results from a traditional CMDS experiment with coherent light (Turner 2012a). This type of approach could allow for CMDS experiments in wavelength ranges where coherent fs pulses cannot be easily achieved.

Summary of techniques

As can be seen, many different implementations have been developed, all of which have both benefits and drawbacks. There is no single approach that can do it all, thus, the best option will vary depending on the type of measurement, the sample, and the information being sought. Table \ref{TechniqueTable} summarizes the capabilities of the most common experimental implementations. The capabilities included are by no means exhaustive, and for some applications others will be more important (for example, the ability to cope with large bandwidths is important for samples with broad spectral absorption). Here we focus on the capabilities that are most important for semiconductor nanostructures and which will enable greater levels of control and specificity, which we see as being an important future direction for CMDS.

• Control of available pathways – Spectral amplitude/phase shaping and polarization selection allow specific signals to be isolated, identified, and characterised. This has been used primarily in cases where there are many overlapping pathways (see Section \ref{SubSecSelectiveTechniques}), but we also see this as being important for identifying weak signals, and quantifying interactions in semiconductor nanostructures and some of the exciting new materials that are being developed, where interactions are intrinsically linked to their macroscopic properties.

• Delay range – many semiconductor nanostructures and crystalline materials have long coherence times at low temperature and thus require much longer delay ranges than is possible with some implementations.

• Spatial resolution – recent experiments show that there is a lot to be gained when individual nanostructures can be studied, revealing variations that are washed out in ensemble measurements. The spatial resolution achievable will be determined, in large part, by the beam geometry and will limit the applicability of some approaches for certain types of sample.

• Rotating-wave delays and intrinsic phasing - these are experimental characteristics that simplify and speed up data acquisition, allowing for the design of more complex measurements, which can provide more detailed information from complex samples.

• Phase stability between all pulses and scalability to higher order - These enable incisive experiments that can’t be done otherwise (such as 2Q, 3Q and measurements of multi-step energy transfer) that are particularly useful for studying multi-particle correlations and many body effects in solid state systems.

Two key characteristics are missing from the table: phase stability and sensitivity, both of which may play an important role in limiting the capabilities of each approach. Phase stability measurements are included in many experimental publications (including this one), but a direct quantitative comparison is not trivial as phase stability values are often quoted for a wide range of different time durations. Furthermore, other experimental characteristics (e.g. the type of light source used, the quality of the optomechanical components, or environmental stability) can greatly impact the phase stability. There are also no established standards for what types of samples should be used, or how the data should be analysed.

Similarly, there are no established standards or methods for characterizing experimental sensitivity or dynamic range. While these characteristics are both related to phase stability, they depend on many other experimental details. A wide variety of different light sources (oscillators, amplified systems, NOPAs, supercontinuum generators, etc) and detectors can be used interchangeably with different CMDS techniques, which – along with the previously mentioned environmental factors – will strongly impact the dynamic range and sensitivity of the device.

In order to benchmark measurements across different laboratories and different implementations, it is desirable to have standard calibration sample (or samples) that can compare and quantify sensitivity, precision and accuracy of the measurement system. To date, this has not been realized. Part of the reason for this is that different implementations operate at different wavelengths, but even within a specific wavelength range there is no readily accessible and reproducible sample that allows for quantitative calibrations. When establishing a CMDS apparatus, the first experiment is often with a laser dye. In the visible, this is often cresyl-violet, and in section 4 of this work, for the near-IR we use IR812. These provide a good reference that can be used to determine phase stability, and could, through detailed concentration dependent studies provide a measure of sensitivity. However, because the linewidths are typically broad, tests of precision and accuracy using these samples are less robust. Crystalline samples and nanostructures often give spectrally narrower features, but these are very sensitive to the specific size, shape, defect density and environment, making it difficult to get an independent calibration. Perhaps the best option would be an atomic vapor at a well-defined pressure which gives well-defined, localized peaks, both on the diagonal and off-diagonal (Li 2013). However, atomic vapour cells are not standard equipment in most spectroscopy laboratories. Independent calibration of CMDS aparatus has thus remained difficult and similarly, comparisons of the performance of different implementations are limited.

A direct in-situ comparison of different techniques where these inherent variables are controlled would be useful for identifying the most suitable technique for a particular measurement, especially given the importance of stability/sensitivity/dynamic range on the quality of the resultant spectra. In the following section we show that two different techniques (pair-wise delay stages and a diffraction based pulse-shaper) can be combined into a single experimental apparatus such that many of the inherent variables (laser source, environment, detection optics, sample, and the majority of the optomechanical components) are controlled. In the following section we describe our implementation of two different approaches: diffractive pulse-shaping and pairwise delays, and quantitatively compare their performance. These two approaches have complementary strengths and our motivation for combining them is to achieve a system that can realize most of the desired capabilities (see the last row of Table \ref{TechniqueTable}). Our comparison of these two approaches also demonstrates how they can be combined to realize a more functional CMDS apparatus with nearly all the ideal characteristics for measurements of semiconductor nanostructures.

\label{TechniqueTable}A summary of several implementations of CMDS outlining some of the capabilities relevant for the study of semiconductor nanostructures and other solid state materials.

Hybrid CMDS apparatus: direct comparison and new capabilities

\label{SecHybridComparison}

\label{FigLayout}Layout of the hybrid CMDS experiment. POM: pick off mirror, SLM: Spatial light modulator, SM: Spatial mask, CL: Cyllindrical Lens, LD: Laser Dye, OD: Optical Density, LO: Local Oscillator, QW: Quantum Well. The circles labelled A-F represent he beam profiles at the corresponding points on the layout.

Experimental apparatus

The layout of the hybrid experimental apparatus is shown in Fig.\ref{FigLayout}. Pulses generated by a mode-locked Ti:Sapph oscillator (KMLabs Collegiate) are split into four beams in a rotated box geometry using an SLM-based Fourier beam-shaper. The four beams are then routed to a pulse shaper using 4-F geometry via two 50 cm achromatic lenses, passing through a focus immediately above a pickoff mirror (POM). Due to the rotated box beam geometry, each of the beams is focused onto a different vertical region of the 2D SLM. The pulse shaper is aligned so that the beams are reflected directly back through the optical setup. A vertical sawtooth phase grating is applied to the SLM. The first order diffraction from this phase grating is then picked off at POM, so that the shaped beams can be separated from the unshaped incoming beams.

After the POM, the beams are collimated and then either routed to the sample in 4-F geometry (we will refer to this as the pulse-shaper only path), or go down a path in which a pair of translation stages are used to control the inter-pulse timings (we will call this the delay stage path, although the pulse shaper can still also be used to control pulse timings down this path). Both paths re-unite just before the lens that focuses the beams onto the sample, where filters and polarization optics may be placed so that they can be used for either beam path. The pairwise delay stage path is designed as in Ref. (Selig 2008), and laid out as a folded version of the schematic depicted in Fig. \ref{Fig_Pairwise}.

\label{PhaseStab}(a) Long term phase stability measurements.(b) Best case short term phase stability measurements with an enclosed setup. (c) Short term phase stability with the experimental setup uncovered.

Stability, sensitivity, and dynamic range

\label{SubSec_Stabilty}

To characterize the phase stability of the FWM signal (a useful baseline metric for a CMDS experiment), we repeatedly measure an 8-step phase-cycled interferogram generated in a sample of IR-812 laser dye in methanol at $$t_{1}=t_{2}=0fs$$. This measurement is performed for both the stage path and the pulse shaper path. Figure \ref{PhaseStab} (a) shows the signal phase deviation over 17 hours, yielding a standard deviation of $$\lambda/507$$ ($$\lambda/192$$) for the pulse shaper (stage) path. The average standard deviation of a 30 minute moving window – the typical acquisition time for a long scan – is $$\lambda/909$$ ($$\lambda/213$$), and the best standard deviation of a single 30 minute window was $$\lambda/1052$$ ($$\lambda/287$$). Figure \ref{PhaseStab} (b) shows the best case traces of the phase deviation over 30 minutes. Figure \ref{PhaseStab} (c) shows a typical trace of the signal phase deviation when the experiment is uncovered, yielding a roughly 10 fold increase in the standard deviation. This demonstrate that air movement in the optical setup is a major contributor to instability in the signal phase, and that properly enclosing CMDS experiments is very important.

Assessing the sensitivity and dynamic range of this interferometric technique is not so straightforward, as they depend on the relative amplitude and stability of the signal, the LO, and scatter from the excitation beams. In the absence of scatter, weak signals can be detected by reducing the amplitude of the LO, and the dynamic range can then be optimized by increasing the detector gain or integration time to match the dynamic range.

In practical terms, however, scatter puts a much more stringent limit on the experimental sensitivity than the signal phase stability. Scatter only signals ($$E_{s}^{2}$$) can saturate the detector and limit the degree to which the gain or integration time can be increased (and hence limits the degree to which attenuating the LO is advantageous), and the phase or amplitude of signal interferograms ($$E_{s}E_{LO}$$) can change over the course of the measurement. Changes in the scatter between different steps of the phase cycling will result in uncompensated scatter contributions to the phase-cycled signal interferogram, obscuring the signal or introducing noise. Thus, the sensitivity can also be limited by changes to the scatter between the phase-cycling steps, and a qualitative understanding of the sensitivity can be understood by comparing the ratio of the signal amplitude to the average amplitude of the scatter that “gets through” the phase cycling process. Optimal sensitivity and SNR are thus achieved by acquiring data as fast as possible to limit the amount drift in the scatter. We therefore do not perform any averaging at the individual (1D) spectral level, but instead acquire (and then average) many complete 2D spectra. In addition to improving the SNR, this also allows us to better quantify the uncertainties in peak-widths and amplitudes. Using these approaches with our pulse shaper setup we have been able to acquire 2D spectra with $$\leq$$ 10,000 photons per pulse, giving $$\leq$$ 10 signal photons per pulse.

This interpretation of the main sources of noise (and how they are mitigated) also has implications for the comparison of these two experimental methods for controlling inter-pulse delays. In both pathways, the signal phase is intrinsically stabilized (as demonstrated by our phase stability measurements), and the entirely common optics in the pulse shaper path ensure that the phase of the scatter relative to the phase of the phase of the LO is also stabilized (as well as, or better than the phase of the signal). The pair-wise delay stage method, on the other hand, does not ensure that the the phase of the scattered light is stable, so changes in the phase of the scatter over the course of a measurement are much more likely to contaminate the signal than when all common optics are used. Thus, the SNR of 2D spectra collected using the pulse-shaper path should be superior to that of 2D spectra collected using the translation-stage path, even if the phase stability of the signal interferogram were the same. This is particularly true when the scatter amplitude is comparable to or larger than the signal amplitude and/or when long scan times are required. The two methods provide similar SNR for samples with strong FWM signals and low scatter, but the pulse-shaper based approach is far superior for high-scatter samples, low excitation density measurements necessitating long integration times, and for weak signals. This can be seen in the CPs in Fig. \ref{FigCP}; the SNR of the pulse-shaper based measurement is clearly superior to that of the translation stage measurement.

Direct comparison of 2D spectra collected with delay stage and pulse-shaper

In this section, we present 2D spectra that have been collected using the pulse shaper and the delay stages. We focus on three different samples with widely varying coherence times. First we show 2D spectra of an excitonic state in a molecular laser dye at room temperature which has a coherence time only slightly longer than the pulse width. Second, at the opposite end of the coherence time range, we show 2D spectra from a high quality MBE grown quantum well exciton which at 10K has a coherence time that vastly exceeds the delay range of the pulse-shaper. Excitation density dependent measurements, and polarization dependent measurements demonstrate the utility of using delay-stage based measurement for this sample. Finally, we present measurements of an exciton in a more dissordered quantum well grown by MOCVD. This shows how the pulse-shaper based measurements can result in abberations in the 2D spectrum even when the coherence time is comparable to or slightly shorter than the pulse-shaper delay range.

IR812 Laser dye

Fig. \ref{FigLD} shows absolute value 1Q rephasing 2D spectra of IR812 laser dye using the (a) delay stage and (b) pulse-shaper paths. Both delay methods produce almost identical spectra, with a slight red shift of the emission spectrum relative to the excitation spectrum due to fast relaxation. The similarity of the spectra tells us that the aberrations induced by the pulse-shaper do not play a significant role when the decoherence time is much shorter than the delay range accessible by the pulse shaper.

\label{FigLD}Absolute value 1Q rephasing measurements of IR812 laser dye measured via the (a) stage path (b) pulse-shaper path.

High quality InGaAs/GaAs QW

We now consider an excitonic transition with very long coherence time: the heavy-hole exciton transition in a high quality (i.e. low disorder) InGaAs/GaAs single quantum well. The decoherence time of the excitonic state in this sample at low temperature (10K) and moderate photon density ($$\approx 10^{11}cm^{-2}$$) is much longer than the decay range of the pulse-shaper . We can get a better handle on the differences between the two delay methods by varying the excitation density, which significantly alters the dynamics of the QW excitons (Shah 1999).

The decoherence of the heavy-hole exciton as a function of the coherence time ($$t_{1}$$) is shown in Fig. \ref{FigTimeDomain} for four different excitation densities. There are clear differences between the delay stage based measurement and the pulse shaper based measurement. Most notably, the measurements using the pulse shaper appear to decohere much more rapidly. Furthermore, the coherence decays measured using the delay stage are single-exponentials, while the decays measured with the pulse shaper have a more complicated shape. While some difference between the different excitation densities can be detected using the pulse-shaper based delays, it is not nearly as clear as when the delay stages are used to control the delays.