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arXiv:2610.18712

PULSUS: Direct Frequency-Domain Multidimensional Spectroscopy Beyond the Impulsive Rotating-Wave Approximation

Alfonso Castillo-Gonzalez, Eric R. Bittner

quant-phSubmitted Oct 13, 2026• Announced Oct 14, 2026• Revised Oct 14, 2026 (v2)
v1 · Oct 13, 2026v2 · Oct 14, 2026
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Abstract

We introduce PULSUS, a direct frequency-domain framework for multidimensional spectroscopy of open quantum systems beyond the impulsive rotating-wave approximation. Coherence-frequency intervals are represented through Liouvillian resolvents, while finite Gaussian pulses are incorporated through operator-valued functions of the field-free Liouvillian. This enables direct evaluation on Cartesian grids or arbitrary spectral coordinates while retaining molecular evolution during the pulse envelopes and, when required, the complete light--matter interaction. The implementation is validated against analytical and independent time-domain benchmarks and supports targeted spectral sampling that reduces the number of evaluated coordinates when the response is spectrally localized. A controlled hierarchy of finite-pulse, rotating-wave, short-pulse, and impulsive models separates the errors introduced by each approximation. For a coupled electronic dimer, the total impulsive-RWA error is nonmonotonic in pulse duration: an intermediate regime provides the highest accuracy, whereas ultrashort pulses expose coupling-enabled beyond-RWA pathways and longer pulses develop substantial finite-pulse corrections. These results show that the impulsive and rotating-wave approximations cannot be assumed to improve simultaneously and should be tested when quantitatively interpreting multidimensional spectra. PULSUS provides a direct frequency-domain framework for carrying out that comparison.

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We introduce PULSUS, a direct frequency-domain framework for multidimensional spectroscopy of open quantum systems beyond the impulsive rotating-wave approximation. Coherence-frequency intervals are represented through Liouvillian resolvents, while finite Gaussian pulses are incorporated through operator-valued functions of the field-free Liouvillian. This enables direct evaluation on Cartesian grids or arbitrary spectral coordinates while retaining molecular evolution during the pulse envelopes and, when required, the complete light--matter interaction.
The implementation is validated against analytical and independent time-domain benchmarks and supports targeted spectral sampling that reduces the number of evaluated coordinates when the response is spectrally localized. A controlled hierarchy of finite-pulse, rotating-wave, short-pulse, and impulsive models separates the errors introduced by each approximation. For a coupled electronic dimer, the total impulsive-RWA error is nonmonotonic in pulse duration: an intermediate regime provides the highest accuracy, whereas ultrashort pulses expose coupling-enabled beyond-RWA pathways and longer pulses develop substantial finite-pulse corrections. These results show that the impulsive and rotating-wave approximations cannot be assumed to improve simultaneously and should be tested when quantitatively interpreting multidimensional spectra. PULSUS provides a direct frequency-domain framework for carrying out that comparison.
We present perturbative analytical solutions to the optical Bloch equations at third order, with finite duration Gaussian pulse envelopes. We find that a given double-sided Feynman diagram in this approximation can be conveniently described in the frequency domain as a product of the expression in the impulsive limit and a finite-pulse factor. Finite-pulse effects are Feynman-diagram-dependent, however, and include nontrivial phase corrections that can occur even in the case of transform-limited pulses. The results constitute a practical framework for modeling phenomena in multidimensional coherent spectroscopy that cannot easily be captured in the impulsive limit, including the roles of bandwidth, resonance, and pulse chirp.
Coherent multidimensional spectroscopy enables detailed investigations of vibronic effects in molecular and solid-state systems. We present explicit analytical expressions for multidimensional nonlinear response functions in the presence of Herzberg-Teller (non-Condon) coupling, within the displaced harmonic oscillator model. The formulation applies to electronic systems with an arbitrary number N of electronic states and to response functions of arbitrary order M in the light-matter interaction. We show that Herzberg-Teller coupling introduces additional oscillatory factors in the time-domain response functions, leading, upon Fourier transformation, to replicas of the Franck-Condon multidimensional spectra shifted by integer multiples of the vibrational frequencies. The present results provide a general analytical framework for the interpretation of non-Condon effects in coherent multidimensional spectroscopies. A Python code that implements the present approach and simulates multidimensional spectra for N-level systems is available on GitHub.
We present a general and efficient approach to compute phase-resolved multidimensional spectra of anharmonic molecular polaritons, based on a semiclassical evolution of the molecular Hamiltonian and cavity field in the large-N\mathcal{N}N limit of many molecules coupled to a confined photonic mode. By systematically expanding the response in both amplitudes and phases of the input fields, our method enables a transparent and computationally simple construction of phase-cycled two-dimensional single- and double-quantum polariton spectra from the underlying nonlinear signal components. Here, phase cycling acts as an analogue of phase matching with oblique pulses, allowing for the isolation of the contributing nonlinear pathways in Liouville space. We specialize to vibrational polaritons and benchmark the method through direct comparison with experimentally measured single-quantum spectra, providing an explanation for the longstanding puzzle of the polariton bleach effect observed at short waiting times. Further, we show how the imprint of various types of anharmonicities on the double-excitation manifold can be directly probed and analyzed through double-quantum coherence spectroscopy. Taken together, our results establish a practical and powerful framework for the modeling and interpretation of nonlinear spectroscopic experiments on strongly coupled light-matter platforms and for guiding the design of cavity-enhanced molecular platforms.
Abstract Developing an analytical theory for atomic coherence driven by ultrashort laser pulses has proved to be challenging due to the breakdown of the rotating wave approximation (RWA). In this paper, we present an approximate analytical solution that describes a two-level atom under the excitation of a far-off-resonance, few-cycle pulse of arbitrary shape without invoking the RWA. As an example of its applicability, a closed-form solution for Gaussian pulses is explicitly given, and the result is used to analyse the impact of carrier envelope phase on atomic population ratios. Comparisons with numerical solutions validate the accuracy our solution within the scope of the approximation. Finally, we outline an alternative approach that can lead to a more accurate solution by capturing the nonlinear behaviors of the system. The work lays out feasible theoretical paths toward analytically describing two-level atoms driven by ultrashort pulses.
Multidimensional spectroscopy has become an established tool for disentangling microscopic processes that remain unresolved in conventional linear spectroscopy. Its extension to the terahertz regime now enables analogous questions to be addressed for low-energy collective excitations in quantum materials and synthetic many-body systems. Unlike isolated few-level systems, however, these collective modes generally cannot be described within the standard framework of nonlinear spectroscopy based on a few-level description. Their response is instead governed by their extended nature, interplay of interactions, fluctuations, disorder, dissipation, and coupling to additional degrees of freedom. This thesis develops a theoretical framework for computing and interpreting multidimensional nonlinear spectroscopy in quantum many-body systems. Using real-time nonequilibrium field theory, we formulate nonlinear susceptibilities diagrammatically in terms of propagators, interaction vertices, and self-energies, with particular emphasis on the third-order susceptibility relevant to infrared-active collective excitations. This approach places the established concepts and capabilities of multidimensional spectroscopy within a systematic many-body field-theoretical framework. Furthermore, this formulation makes it possible to move beyond the established probing capabilities, extending multidimensional spectroscopy beyond few-level descriptions and revealing new signatures and functionalities specific to interacting many-body systems. At the same time, it provides a controlled framework for constructing approximations that preserve the underlying symmetries and conservation laws of the system while remaining amenable to both analytical and numerical calculations. By incorporating quantum and thermal fluctuations within the random-phase approximation, we demonstrate that two-dimensional spectra can reveal the growth of fluctuations, identify field-theoretical bound states, and characterize their coupling to many-body continua. We further develop a diagrammatic description of rephasing in extended bosonic systems in the presence of static disorder. We identify the disorder vertex corrections responsible for echo formation and show how echo spectroscopy separates disorder-induced broadening from intrinsic relaxation. This framework is extended to coupled modes and strongly interacting regimes, where interactions can either enhance or suppress rephasing. A further result is the identification of a nonrephasing four-wave-mixing signal that is critically enhanced near a phase transition, in contrast to simpler nonlinear protocols like third-harmonic generation. We apply this framework to electromagnons in NiI2_22​, Josephson plasmons in La2−x_{2-x}2−x​Srx_xx​CuO4_44​, and the low-energy collective excitations in coupled one-dimensional Bose-Einstein condensates. Across these systems, we show how multidimensional spectroscopy can disentangle disorder from intrinsic relaxation, reveal the effects of interactions and fluctuations, identify coupling between collective modes and continua, and extract microscopic information that remains obscured in conventional spectroscopy. Taken together, the results of this thesis provide a diagrammatic framework for calculating and interpreting multidimensional nonlinear spectroscopic signals in many-body systems, while establishing this technique as a selective probe of collective dynamics across a broad range of quantum materials and synthetic platforms.
Liouville pathways provide the conceptual foundation for interpreting multidimensional spectroscopies and are commonly represented as independent contributions to the nonlinear response. In open quantum systems, however, the effective reduced Liouvillian need not preserve this observational pathway decomposition. Environmental interactions can transport amplitude between pathway sectors during the free-evolution intervals, generating measurable signatures in the nonlinear spectroscopic response.} We develop a geometric framework in which pathway transport is governed by a Liouvillian connection, its associated curvature, and the resulting observational holonomy. The framework applies to open quantum systems in which the environment selects a pointer basis distinct from the observational basis used to construct the spectroscopic response. This basis incompatibility induces transport among Liouville pathways, generating characteristic spectral distortions and a nontrivial Liouvillian curvature. Using a Duhamel expansion of the Liouvillian propagator, we derive a reconstruction procedure that identifies the transport operators responsible for the observed redistribution of pathway weight, accurate throughout the full range of basis misalignment. This perspective reframes spectral features as determined not only by which pathways exist but by how amplitude is transported among them. Spectral distortions, peak shifts, and otherwise-forbidden pathway contributions become geometric signatures of a curved Liouville-space manifold rather than phenomenological broadening corrections, identifying pathway geometry as a complementary layer of organization in nonlinear spectroscopy.