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Frontiers in Optics

Phase Cycling Beyond Perturbative Limits: Exact Solutions for Nonlinear Response

Rishabh Tripathi, Krishna Kumar Maurya, Pradeep Kumar, Bhaskar De, Rohan Singh

Published Jan 1, 2025• 0 citations
Publisher

Abstract

We present exact simulations of nonlinear optical signals under high-intensity excitation using phase-cycling, validating with experiments and showing accurate modeling beyond perturbative limits without pulse shape or order truncations.
DOI
10.1364/fio.2025.jtu5a.29

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The calculation of the coherent nonlinear response of a system is essential to correctly interpret results from advanced techniques such as two-dimensional coherent spectroscopy. Usually, even for the simplest systems, such calculations are either performed for low-intensity excitations where perturbative methods are valid and/or by assuming a simplified pulse envelope, such as a δ-function in time. Here, we use the phase-cycling method for the exact calculation of the nonlinear response without making the aforementioned approximations even for high-intensity excitation. We compare the simulation results to several experimental observations to prove the validity of these calculations. The saturation of the photon-echo signal from excitons in a semiconductor quantum well sample is measured. The excitation-intensity dependent measurement shows nonlinear contributions up to twelfth order. Intensity-dependent simulations reproduce this effect without explicitly considering higher-order interactions. In addition, we present simulation results that replicate previously reported experiments with high-intensity excitation of semiconductor quantum dots. By accurately reproducing a variety of phenomena such as higher-order contributions, switching of coherent signals, and changes in photon-echo transients, we prove the efficacy of the phase-cycling method to calculate the coherent nonlinear signal for high-intensity excitation. This method would be particularly useful for systems with multiple, well-separated peaks and/or large inhomogeneities.
The multiple-linear-time-scale method is used to construct a perturbation theory for NNN-level quantum systems subjected to time-dependent perturbations. The secular and small-denominator terms which plague conventional time-dependent perturbation theory are avoided; consequently, the theory is useful for treating long-term behavior and resonant interactions. The introduction of a self-energy operator allows the shifts in energy levels to be displayed explicitly. When the perturbation is the dipole interaction with an electromagnetic field, the successive approximations yield the well-known rotating-wave approximation, corrections due to counter rotating terms, and the Bloch-Siegert frequency shift. The formalism is applicable to arbitrary pulse shapes, provided that $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}\ensuremath{\ll}\ensuremath{\hbar}\overline{\ensuremath{\omega}}$, where $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}$ represents the inter-action strength and $\overline{\ensuremath{\omega}}$ is a characteristic frequency of the field. Rabi oscillations induced by multiphoton resonances are automatically included, and this effect is demonstrated in the case of a square pulse with frequency approximately one half the resonance frequency for a transition between two vibrational levels of a molecule. These calculations are compared to an exact numerical solution in order to find the limits of validity of the approximation. Even at high intensities ($I\ensuremath{\sim}{10}^{14}$ W/cm2{\mathrm{cm}}^{2}cm2) the shifted resonance frequency is quite accurate; however, the approximate and exact solutions are slightly out of phase, so that the approximate solution can only be trusted for a few Rabi cycles. For much lower intensities ($I\ensuremath{\sim}{10}^{10}$ W/cm2{\mathrm{cm}}^{2}cm2), the approximate solution is valid for many thousands of Rabi cycles.
A central challenge in nonlinear spectroscopy is the order-selective readout of weak higher-order responses that spectrally overlap with dominant lower-order signals. This bottleneck is particularly severe in two-dimensional (2D) spectroscopy, where extending conventional phase-cycling schemes to higher orders rapidly increases measurement and analysis complexity. Here we introduce a computation-assisted strategy that combines rotating-frame acquisition with a frame-shift tracking algorithm to separate signals by their frame-dependent spectral shifts. In a rubidium vapor experiment, we use this approach to isolate a 7th-order nonlinear contribution from coexisting 3rd-order components, enabling direct access to higher-order quantum-coherence dynamics without sacrificing operation at comparatively high pulse intensities. The method is broadly compatible with multidimensional spectroscopy platforms and provides a practical route to probing many-body and collective ultrafast dynamics beyond third order.
Controlling stimulated emission is of importance because it competes with absorption and fluorescence under intense laser excitation. We performed resonant nonlinear optical spectroscopy measurements using femtosecond pulses shaped by π- or π/2-step phase functions and carried out calculations based on density matrix representation to elucidate the experimental results. In addition, we compared enhancements obtained when using other pulse shaping functions (chirp, third-order dispersion, and a time-delayed probe). The light transmitted through the high optical density solution was dominated by an intense stimulated emission feature that was 14 times greater for shaped pulses than for transform limited pulses. Coherent enhancement depending on the frequency, temporal, and phase characteristics of the shaped pulse is responsible for the experimental observations.