We extend our theoretical framework for time-resolved nonlinear microspectroscopy [M. Cho, J. Chem. Phys. 162, 124201 (2025)] to coherent four-wave-mixing spectroscopy using paraxial Laguerre-Gaussian (LG) beams. Unlike pump-probe or transient absorption techniques, photon echo is highly sensitive to the spatial phase structure of LG beams. This sensitivity arises because coherence evolution in inhomogeneously broadened absorbers depends on the radial and azimuthal indices of the LG modes used in the write-and-read processes within the photon echo configuration. Recent advances in spatial light modulators, metasurfaces, and ultrafast laser techniques have significantly improved spatial and temporal control over quantum materials. These innovations enable new approaches to studying heterogeneous systems, developing multidimensional microspectroscopy, and exploring alternative quantum information storage methods. In this work, we investigate photon echo signals generated by LG beams and derive analytical expressions for rephasing and non-rephasing photon echoes. Our results reveal how beam parameters influence nonlinear spatiotemporal responses, capturing spatial variations in pulse amplitudes, phases, and inhomogeneity-induced dephasing and rephasing. We show that customized ultrafast pulses and structured spatial light fields can enhance the spatial separation of photon echo signals and increase the density of stored quantum information. This work advances nonlinear molecular spectroscopy and quantum information science by leveraging structured light fields and ultrafast optics.
We extend our theoretical framework for time-resolved nonlinear microspectroscopy [M. Cho, J. Chem. Phys. 162, 124201 (2025)] to coherent four-wave-mixing spectroscopy using paraxial Laguerre-Gaussian (LG) beams. Unlike pump-probe or transient absorption techniques, photon echo is highly sensitive to the spatial phase structure of LG beams. This sensitivity arises because coherence evolution in inhomogeneously broadened absorbers depends on the radial and azimuthal indices of the LG modes used in the write-and-read processes within the photon echo configuration. Recent advances in spatial light modulators, metasurfaces, and ultrafast laser techniques have significantly improved spatial and temporal control over quantum materials. These innovations enable new approaches to studying heterogeneous systems, developing multidimensional microspectroscopy, and exploring alternative quantum information storage methods. In this work, we investigate photon echo signals generated by LG beams and derive analytical expressions for rephasing and non-rephasing photon echoes. Our results reveal how beam parameters influence nonlinear spatiotemporal responses, capturing spatial variations in pulse amplitudes, phases, and inhomogeneity-induced dephasing and rephasing. We show that customized ultrafast pulses and structured spatial light fields can enhance the spatial separation of photon echo signals and increase the density of stored quantum information. This work advances nonlinear molecular spectroscopy and quantum information science by leveraging structured light fields and ultrafast optics.
We present a theoretical framework for time-resolved nonlinear microspectroscopy using structured light to probe spatial coherence transfer in condensed-matter systems. Unlike traditional photon echo (PE) spectroscopy, our method employs Laguerre-Gaussian (LG) modes to encode and track quantum coherence migration spatially. By expanding the diffusion propagator in LG modes, we reveal radial mode mixing and conservation of orbital angular momentum during coherence migration. Analytical formulas for heterodyne-detected PE signals and two-dimensional spectra show distinct signatures of spatial coherence transfer in decay and line shapes. This technique allows differentiation between coherent and incoherent transfer, enabling measurement of exciton coherence diffusion constants in materials such as organic semiconductors, perovskites, and two-dimensional materials and provides new insights into coherence dynamics for quantum materials and energy transport applications.
A new method of picosecond and femtosecond transient spectroscopy yielding information about ultrafast relaxation processes without requiring ultrashort light pulses is presented. It is based on the present theoretical analysis of the resonant degenerate four-wave-mixing process excited by two temporally incoherent light beams with wave vectors ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{1}$ and ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{2}$ which are originated from a single beam at frequency $\ensuremath{\omega}$ but have mutual time delay $\ensuremath{\tau}$. Under the assumption that the incoherent light field has Gaussian random complex amplitude and that the resonant material consists of the usual two-level atoms, the statistically averaged intensity of the output light field with ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{3}=2{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{2}\ensuremath{-}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{1}$ at $\ensuremath{\omega}$ is calculated as a function of $\ensuremath{\tau}$. Even with the light having a much longer duration than both T1 (the longitudinal relaxation time) and T2 (the transverse relaxation time), the correlation trace, i.e., output intensity versus $\ensuremath{\tau}$, represents a decay profile determined mainly by T2 for both homogeneously and inhomogeneously broadened transitions, as long as the light correlation time ${\ensuremath{\tau}}_{c}$ is much shorter than the relaxation times. The correlation trace does not always represent a single-exponential decay but is sometimes slightly deformed by the T1 effect. However, it does not cause a significant error in the determination of T2. Moreover, as $\frac{{T}_{2}}{{T}_{1}}\ensuremath{\rightarrow}0$, the trace becomes a single-exponential decay curve determined only by T2. The feature of the results obtained by the present method is similar to that obtained by the conventional coherent transient spectroscopy with short pulses, such as the photon echo. The time resolution in the present method, however, is limited only by ${\ensuremath{\tau}}_{c}$ much shorter than the light duration. By regarding the incoherent light as a series of random ultrashort pulses, the present four-wave-mixing process is also interpreted as the ensemble of numerous transient four-wave-mixing processes caused by various combinations of these pulses.
Rare-earth-ion doped materials are studied for a quantum information protocol, controlled reversible inhomogeneous broadening, for storing single photons. Coherence properties of Er-doped silicate optical fibers and Er-doped LiNbO3 waveguiding structures are investigated. Spectral hole burning spectroscopy is performed in Erbium and Neodymium doped crystals and spectral tailoring is demonstrated in Nd³?:YVO4. The preservation of information encoded into the relative phase and amplitudes of optical pulses during storage and retrieval in an optical memory based on stimulated photon echo is demonstrated. Furthermore photon echoes stored in spatially separated optical memories in several temporal modes, are retrieved and then combined at a beam splitter. They show almost perfect interference demonstrating both phase preserving storage and indistinguishability of photon echoes. Finally, we report an interference experiment of spontaneous emission of light from two distant solid-state ensembles of atoms and the resulting high fringe visibility implies that the observed spontaneous emission is coherent.
A systematic theoretical and computational investigation of the microscopic factors which determine the phase of the signal field in time-resolved quasidegenerate three-pulse scattering experiments is presented. The third-order phase-matched response is obtained by density-matrix perturbation theory using a Green-function formalism for a system composed of two well-separated sets of closely spaced energy levels. Equations for calculating the electric field of four-wave-mixing signals generated by path-length delayed pulses are given. It is found that the phase of the signal field is determined by the excitation pulse phases, the dynamics of the nonlinear polarization decay, the product of four transition dipole matrix elements, and by a pulse-delay-dependent phase modulation at the frequency of the first dipole oscillation in the four-wave-mixing process. Analytic results for a two-level Bloch model show the phase shift from rapid nonlinear polarization decay. The product of dipole matrix elements is real and positive for three-level processes (bleached ground-state absorption and excited-state emission), but can be real and negative for some four-level Raman processes. The pulse-delay-dependent phase modulation treated here is closely related to the interferometric pulse-delay-dependent amplitude modulation observed in some collinear experiments, and plays a role in producing photon echos in inhomogeneously broadened samples. Numerical calculations of phase-resolved electric fields for finite duration pulses using a Brownian oscillator model appropriate for condensed-phase dynamics are presented. The ability of pulse-delay-dependent phase modulation to encode the frequency of the initially excited dipole onto the phase of the signal field can be exploited to examine energy-level connectivity, reveal correlations hidden under the inhomogeneous lineshape, and probe relaxation pathways in multilevel systems.
For decades, ultrafast light-matter interactions have provided a powerful means to observe electrons in motion across a wide range of physical systems. This dissertation builds new theoretical frameworks to make sense of the emergent phenomena that arise when light drives matter into the strong-field regime, spanning atomic, molecular, and condensed-matter systems. The resulting dynamics leave their fingerprints on a variety of observables, such as photoionization spectra, high-harmonic generation, and nonlinear spectroscopic signals, each of which provides a different window onto the underlying electronic motion. The work is organized into two parts.The first part focuses on the electric-field observables of nonlinear optical signals. Because both the amplitude and phase of the emitted field are experimentally accessible through techniques such as spectral interferometry and electro-optic sampling, the full field itself serves as a rich observable for characterizing nonlinear responses in atomic, molecular, and condensed-matter systems. Leveraging this, the simulations presented here uncover novel and largely unexplored physical phenomena encoded in these field observables. The second part turns to optical spectroscopy with quantum light. Although nonlinear optical processes are conventionally modeled within a semiclassical framework, treating the matter quantum mechanically while the driving ultrafast fields remain classical, the resulting signals can nonetheless exhibit intrinsically non-classical properties. Building on this insight, a framework for interpreting experimental data is introduced and used to show that four-wave mixing driven by ultrafast femtosecond laser pulses generates bright squeezed light. Beyond squeezed light, strong field-matter interactions can also produce attosecond entangled photons, a distinct form of quantum light. Several strategies are proposed to enhance the efficiency of their generation, with the developed theory predicting rates of energy-time entangled biphoton production orders of magnitude beyond those achievable with conventional spontaneous parametric down-conversion (SPDC) sources. The studies presented here are initial steps toward investigating quantum properties of these systems through quantum phynonlinear spectroscopy, which is not achievable with classical light.
We study theoretically ultrafast modulation spectroscopy in Doppler-broadened cascading three-level and four-level systems. The pump beams consist of two frequency components that drive two-pathway excitation. The four-wave mixing (FWM) signal exhibits damping oscillation as the relative time delay τ between the two pump beams is varied, from which the resonant frequencies between energy levels can be deduced. Furthermore, Doppler-free precision in the measurement of the resonant frequencies can be achieved when the pump beams have either narrow-band or broadband linewidths. We study two-photon resonant FWM induced by time-delayed correlated fluctuating fields in a Doppler-broadened cascading three-level system. It is related intrinsically to the sum-frequency trilevel echo when the pump beams have broadband linewidths. However, unlike the corresponding incoherent-light stimulated photon echo, the maximum of the FWM signal is shifted from τ≃0. If we consider the fluctuations of the pump beams, the FWM signal exhibits damping oscillation when the laser frequency is off resonant from the two-photon transition.