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Dual laser frequency combs can rapidly measure high-resolution linear absorption spectra. However, one-dimensional linear techniques cannot distinguish the sources of resonances in a mixture of different analytes, nor can they separate inhomogeneous and homogeneous broadening. Here, we overcame these limitations by acquiring high-resolution multidimensional nonlinear coherent spectra with frequency combs. We experimentally differentiated and assigned the Doppler-broadened features of two naturally occurring isotopes of rubidium atoms (87Rb and 85Rb) according to the placement of their hyperfine energy states in a two-dimensional spectrum.

The motions of electrons in solids may be highly correlated by strong, long-range Coulomb interactions. Correlated electron-hole pairs (excitons) are accessed spectroscopically through their allowed single-quantum transitions, but higher-order correlations that may strongly influence electronic and optical properties have been far more elusive to study. Here we report direct observation of bound exciton pairs (biexcitons) that provide incisive signatures of four-body correlations among electrons and holes in gallium arsenide (GaAs) quantum wells. Four distinct, mutually coherent, ultrashort optical pulses were used to create coherent exciton states, transform these successively into coherent biexciton states and then new radiative exciton states, and finally to read out the radiated signals, yielding biexciton binding energies through a technique closely analogous to multiple-quantum two-dimensional Fourier transform (2D FT) nuclear magnetic resonance spectroscopy. A measured variation of the biexciton dephasing rate indicated still higher-order correlations.

Multi-dimensional coherent spectroscopy (MDCS) has become an extremely versatile and sensitive technique for elucidating the structure, composition, and dynamics of condensed matter, atomic, and molecular systems. The appeal of MDCS lies in its ability to resolve both individual-emitter and ensemble-averaged dynamics of optically created excitations in disordered systems. When applied to semiconductors, MDCS enables unambiguous separation of homogeneous and inhomogeneous contributions to the optical linewidth, pinpoints the nature of coupling between resonances, and reveals signatures of many-body interactions. In this review, we discuss the implementation of MDCS to measure the nonlinear optical response of excitonic transitions in semiconductor nanostructures. Capabilities of the technique are illustrated with recent experimental studies that advance our understanding of optical decoherence and dissipation, energy transfer, and many-body phenomena in quantum dots and quantum wells, semiconductor microcavities, layered semiconductors, and photovoltaic materials.

Multidimensional coherent spectroscopy (MDCS)1,2 is a powerful method that enables the measurement of homogeneous linewidths in inhomogenously broadened systems, many-body interactions and coupling between excited resonances, all of which are not simultaneously accessible by any other method. Current implementations of MDCS require a bulky apparatus and suffer from resolution and acquisition speed limitations that constrain their applications outside the laboratory3–5. Here, we propose and demonstrate an approach to nonlinear coherent spectroscopy that utilizes three slightly different repetition-rate frequency combs. Unlike traditional nonlinear methods, tri-comb spectroscopy uses only a single photodetector and no mechanical moving elements to enable faster acquisition times, while also providing comb resolution. As a proof of concept, a multidimensional coherent spectrum with comb cross-diagonal resolution is generated using only 365 ms of data. These improvements make MDCS relevant for systems with narrow resonances and it has the potential to be field deployable for chemical-sensing applications.

We present multidimensional coherent spectroscopy that utilizes frequency combs and multi-heterodyne detection. We demonstrate its capability to measure collective hyperfine resonances in atomic vapor induced by long-range dipole-dipole interactions.

Optical arbitrary waveform generation will allow waveforms to be synthesized at optical frequencies but with the flexibility currently available at radiofrequencies. This technique is enabled by combining frequency comb technology, which produces trains of optical pulses with a well-defined frequency spectrum, with pulse shaping methods, which are used to transform a train of ultrashort pulses into an arbitrary waveform. To produce a waveform that fills time, the resolution of the shaper must match the repetition rate of the original pulse train, which in turn must have a comb spectrum that is locked to the shaper. Here, we review the current efforts towards achieving optical arbitrary waveform generation and discuss the possible applications of this technology.

We stabilized the carrier-envelope phase of the pulses emitted by a femtosecond mode-locked laser by using the powerful tools of frequency-domain laser stabilization. We confirmed control of the pulse-to-pulse carrier-envelope phase using temporal cross correlation. This phase stabilization locks the absolute frequencies emitted by the laser, which we used to perform absolute optical frequency measurements that were directly referenced to a stable microwave clock.

Optical 2D Fourier transform spectroscopy (2DFTS) provides insight into the many-body interactions in direct gap semiconductors by separating the contributions to the coherent nonlinear optical response. We demonstrate these features of optical 2DFTS by studying the heavy-hole and light-hole excitonic resonances in a gallium arsenide quantum well at low temperature. Varying the polarization of the incident beams exploits selection rules to achieve further separation. Calculations using a full many-body theory agree well with experimental results and unambiguously demonstrate the dominance of many-body physics.

Measuring the 1s–2p splitting of direct and indirect excitons in van der Waals heterostructures allows their binding energy and dynamics to be determined.

Optical spectroscopy using ultrafast light pulses has undergone a revolution in the last decade due to the introduction and development of multidimensional coherent techniques. These methods build on established coherent spectroscopic techniques, such as photon echoes, but go beyond them by correlating the coherent dynamics during two time periods. The resulting multidimensional spectra have a number of advantages including disentangling congested spectra, revealing coupling between resonances, and removing the effects of inhomogeneous broadening. Similar ideas are currently being used for imaging. The papers in this 'focus on' collection document some of the current areas of activity in these fields.