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arXiv:2404.10768 · Physical Review A

Generalized linear response theory for pumped systems and its application to transient optical properties

Amir Eskandari-asl, Adolfo Avella

Published Oct 25, 2024• 9 citations
arXivHTMLPDF

Abstract

We derive the two-time linear response theory for out-of-equilibrium pumped systems, generic pump-probe delays, and probe frequencies. Such a theory enormously simplifies the numerical calculations, for instance, of the optical conductivity with respect to the actual procedure, which requires computing the effect of the probe pulse for each time delay with respect to the pump pulse. The theory is given for a generic observable and pumped Hamiltonian and then specialized for a system with a quadratic Hamiltonian and its transient optical properties, exploiting the dynamical projective operatorial approach (DPOA). The theory is complemented by a set of crucial numerical guidelines that help perform actual calculations in a computationally affordable way. The optical response (differential transient reflectivity and absorption) of a prototypical three-band (core, valence, and conduction) model in the extreme ultraviolet regime is analyzed in detail to illustrate the theory and its application. Using some generalizations of the density of states, we provide a systematic approach to exploring the optical properties in terms of the system band-structure features and the pump parameters. Such an analysis can be extremely helpful in understanding the actual results of experimental optical measurements. Moreover, we study the effects of interband and intraband transitions, the local dipole coupling, and single and multiphoton processes. The latter is further investigated by varying the central frequency of the pump pulse to have different regions of the first Brillouin zone in resonance with it. We also study the effect of varying the pump pulse intensity. Finally, we study and analyze the transient optical properties in the probe pulse regime of the infrared and visible.

DOI
10.1103/physreva.110.043520

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We derive the two-time linear response theory for out-of-equilibrium pumped systems, generic pump-probe delays, and probe frequencies. Such a theory enormously simplifies the numerical calculations, for instance, of the optical conductivity with respect to the actual procedure, which requires computing the effect of the probe pulse for each time delay with respect to the pump pulse. The theory is given for a generic observable and pumped Hamiltonian and then specialized for a system with a quadratic Hamiltonian and its transient optical properties, exploiting the dynamical projective operatorial approach (DPOA). The theory is complemented by a set of crucial numerical guidelines that help perform actual calculations in a computationally affordable way. The optical response (differential transient reflectivity and absorption) of a prototypical three-band (core, valence, and conduction) model in the extreme ultraviolet regime is analyzed in detail to illustrate the theory and its application. Using some generalizations of the density of states, we provide a systematic approach to exploring the optical properties in terms of the system band-structure features and the pump parameters. Such an analysis can be extremely helpful in understanding the actual results of experimental optical measurements. Moreover, we study the effects of interband and intraband transitions, the local dipole coupling, and single and multiphoton processes. The latter is further investigated by varying the central frequency of the pump pulse to have different regions of the first Brillouin zone in resonance with it. We also study the effect of varying the pump pulse intensity. Finally, we study and analyze the transient optical properties in the probe pulse regime of the infrared and visible.
We develop a general theoretical framework for computing the time-resolved magneto-optical Kerr effect in ultrafast pump-probe setups, formulated within the Dynamical Projective Operatorial Approach (DPOA) and its application to the generalized linear-response theory for pumped systems. Furthermore, we exploit this formalism to express the post-pump optical conductivity and consequently the Kerr rotation in terms of the time-evolved single-particle density matrix (SPDM), providing a transparent and computationally efficient description of photo-excited multi-band systems. This extension, in addition to its lower computational cost, has the advantage of allowing the inclusion of phenomenological damping. We illustrate the formalism using both (i) a two-band tight-binding model, which captures the essential physics of ultrafast spin-charge dynamics and the Kerr rotation, and (ii) weakly spin-polarized germanium, as a realistic playground with a complex band structure. The results demonstrate that, by exploiting DPOA and/or its SPDM extension, one can reliably reproduce both the short-time features under the pump-pulse envelope and the long-time dynamics after excitation, offering a versatile framework for analyzing time-resolved magneto-optical Kerr effect experiments in complex materials. Moreover, this analysis clearly shows that the Kerr rotation can be used to deduce experimentally the relevant n-photon resonances for a given specific material.
Recent experimental advancements in ultrafast pump-probe setups drive a growing need for efficient and insightful theoretical methods. This study will delve into the intricacies of our newly developed approach, the Dynamical Projective Operatorial Approach (DPOA). DPOA stands out as a highly efficient model-Hamiltonian method capable of calculating a wide range of single/multi-time single/multi-particle properties, such as excitation populations, out-of-equilibrium Green's functions, TR-ARPES signals, and various response functions. When dealing with transient optical properties, utilizing the DPOA formulation and capitalizing on the weak intensity of the probe pulse, we efficiently compute the linear response of the pumped system to a generic probe pulse, significantly accelerating calculations. Alongside the theoretical framework, we present noteworthy results that experimentalists can utilize to establish connections between observed effects and their underlying physical phenomena.
The physical interpretation of time-resolved photoabsorption experiments is not as straightforward as for the more conventional photoabsorption experiments conducted on equilibrium systems. In fact, the relation between the transient photoabsorption spectrum and the properties of the examined sample can be rather intricate since the former is a complicated functional of both the driving pump and the feeble probe fields. In this work, we critically review the derivation of the time-resolved photoabsorption spectrum in terms of the nonequilibrium dipole response function $\ensuremath{\chi}$ and assess its domain of validity. We then analyze $\ensuremath{\chi}$ in detail and discuss a few exact properties useful to interpret the transient spectrum during (overlapping regime) and after (nonoverlapping regime) the action of the pump. The nonoverlapping regime is the simplest to address. The absorption energies are indeed independent of the delay between the pump and probe pulses and hence the transient spectrum can change only by a rearrangement of the spectral weights. We give a close expression of these spectral weights in two limiting cases (ultrashort and everlasting monochromatic probes) and highlight their strong dependence on coherence and probe envelope. In the overlapping regime, we obtain a Lehmann-type representation of $\ensuremath{\chi}$ in terms of light-dressed states and provide a unifying framework of various well-known effects in pump-driven systems. We also show the emergence of spectral substructures due to the finite duration of the pump pulse.
Abstract The quantum dynamics of a system of Rb atoms, modeled by a V -type three-level system interacting with intense probe and pump pulses, are studied. The time-delay-dependent transient-absorption spectrum of an intense probe pulse is thus predicted, simulating pump-probe experiments in which this is preceded or followed by a strong pump pulse. Numerical results are interpreted in terms of an analytical model based on interaction operators , which quantify the transformation undergone by the system under the action of an intense pulse. The oscillating features of the resulting transient-absorption spectra, due to the coupling of several excited states, are thus interpreted in terms of the atomic population and phase changes imposed by the pump and probe pulses. Strong-field-induced phases and their influence on the resulting transient-absorption spectra are thereby investigated for different values of pump and probe intensities and frequencies, focusing on the atomic properties which are encoded in the absorption line shapes for positive and negative time delays.
The quantum dynamics of a system of Rb atoms, modeled by a V-type three-level system interacting with intense probe and pump pulses, are studied. The time-delay-dependent transient-absorption spectrum of an intense probe pulse is thus predicted, when this is preceded or followed by a strong pump pulse. Numerical results are interpreted in terms of an analytical model, which allows us to quantify the oscillating features of the resulting transient-absorption spectra in terms of the atomic populations and phases generated by the intense pulses. Strong-field-induced phases and their influence on the resulting transient-absorption spectra are thereby investigated for different values of pump and probe intensities and frequencies, focusing on the atomic properties which are encoded in the absorption line shapes for positive and negative time delays.
We employ the Dynamical Projective Operatorial Approach (DPOA) to investigate the ultrafast optical excitations of germanium under intense, ultrashort pump pulses. The method has very low resource demand relative to many other available approaches and enables detailed calculation of the residual electron and hole populations induced by the pump pulse. It provides direct access to the energy distribution of excited carriers and to the total energy transferred to the system. By decomposing the response into contributions from different multi-photon resonant processes, we systematically study the dependence of excited-carrier density and absorbed energy on key pump-pulse parameters: duration, amplitude, and photon energy. Our results reveal a complex interplay between these parameters, governed by resonant Rabi-like dynamics and competition between different multi-photon absorption channels. For the studied germanium setup, we find that two-photon processes are generally dominant, while one- and three-photon channels become significant under specific conditions of pump-pulse frequency, duration, and intensity. This comprehensive analysis offers practical insights for optimizing ultrafast optical control in semiconductors by targeting specific multi-photon pathways.