(b) Write out the equations for the time dependence of ρ11, ρ22, ρ12 and ρ21 assuming that a light field interaction V = -μ12Ee iωt + c.c. couples only levels |1> and |2>, and that the excited levels exhibit spontaneous decay. (8 marks) (c) Under steady-state conditions, find the ratio of populations in states |2> and |3>. (3 marks) (d) Find the slowly varying amplitude ̃ ρ 12 of the polarization ρ12 = ̃ ρ 12e iωt . (6 marks) (e) In the limiting case that no decay is possible from intermediate level |3>, what is the ground state population ρ11(∞)? (2 marks) 2. (15 marks total) In a 2-level atom system subjected to a strong field, dressed states are created in the form |D1(n)> = sin θ |1,n> + cos θ |2,n-1> |D2(n)> = cos θ |1,n> sin θ |2,n-1>

The Quantum Theory of Light

Annual review of condensed matter physics

Journal of Physics Condensed Matter: Preface

This self-contained, comprehensive book describes the fundamental properties of soft x-rays and extreme ultraviolet (EUV) radiation and discusses their applications in a wide variety of fields, including EUV lithography for semiconductor chip manufacture and soft x-ray biomicroscopy. The author begins by presenting the relevant basic principles such as radiation and scattering, wave propagation, diffraction, and coherence. He then goes on to examine a broad range of phenomena and applications. The topics covered include EUV lithography, biomicroscopy, spectromicroscopy, EUV astronomy, synchrotron radiation, and soft x-ray lasers. He also provides a great deal of useful reference material such as electron binding energies, characteristic emission lines and photo-absorption cross-sections. The book will be of great interest to graduate students and researchers in engineering, physics, chemistry, and the life sciences. It will also appeal to practicing engineers involved in semiconductor fabrication and materials science.

X-Rays and Extreme Ultraviolet Radiation: Principles and Applications

When intense light interacts with an atomic gas, recollision between an ionizing electron and its parent ion creates high-order harmonics of the fundamental laser frequency. This sub-cycle effect generates coherent soft X-rays and attosecond pulses, and provides a means to image molecular orbitals. Recently, high harmonics have been generated from bulk crystals, but what mechanism dominates the emission remains uncertain. To resolve this issue, we adapt measurement methods from gas-phase research to solid zinc oxide driven by mid-infrared laser fields of 0.25 volts per angstrom. We find that when we alter the generation process with a second-harmonic beam, the modified harmonic spectrum bears the signature of a generalized recollision between an electron and its associated hole. In addition, we find that solid-state high harmonics are perturbed by fields so weak that they are present in conventional electronic circuits, thus opening a route to integrate electronics with attosecond and high-harmonic technology. Future experiments will permit the band structure of a solid to be tomographically reconstructed.

/pdf/linking-high-harmonics-from-gases-and-solids-4rtfjbqvr5.pdf

Linking high harmonics from gases and solids

We extend the semi-classical trajectory description for the high-order harmonic generation~(HHG) from solids by integrating the effect of electron-scattering. Comparing the extended semi-classical trajectory model with a one-dimensional quantum mechanical simulation, we find that the multi-plateau feature of the HHG spectrum is formed by Umklapp scattering under the electron-hole acceleration dynamics by laser fields. Furthermore, by tracing the scattered trajectories in real-space, the model fairly describes the emitted photon energy and the emission timing of the HHG even in the higher plateau regions. We further consider the loss of trajectories by scattering processes with a mean-free-path approximation and evaluate the HHG cutoff energy as a function of laser wavelength. As a result, we find that the trajectory loss by scattering causes the wavelength independence of the HHG from solids.

Role of electron scattering on the high-order harmonic generation from solids

Optical cavities confine light on a small region in space which can result in a strong coupling of light with materials inside the cavity. This gives rise to new states where quantum fluctuations of light and matter can alter the properties of the material altogether. Here we demonstrate, based on first principles calculations, that such light-matter coupling induces a change of the collective phase from quantum paraelectric to ferroelectric in the SrTiO$_3$ groundstate, which has thus far only been achieved in out-of-equilibrium strongly excited conditions[1, 2]. This is a light-matter-hybrid groundstate which can only exist because of the coupling to the vacuum fluctuations of light, a "photo-groundstate". The phase transition is accompanied by changes in the crystal structure, showing that fundamental groundstate properties of materials can be controlled via strong light-matter coupling. Such a control of quantum states enables the tailoring of materials properties or even the design of novel materials purely by exposing them to confined light.

The Ferroelectric Photo-Groundstate of SrTiO$_3$: Cavity Materials Engineering

Recent experiments have demonstrated that light can induce a transition from the quantum paraelectric to the ferroelectric phase of SrTiO_{3}. Here, we investigate this terahertz field-induced ferroelectric phase transition by solving the time-dependent lattice Schrödinger equation based on first-principles calculations. We find that ferroelectricity originates from a light-induced mixing between ground and first excited lattice states in the quantum paraelectric phase. In agreement with the experimental findings, our study shows that the nonoscillatory second harmonic generation signal can be evidence of ferroelectricity in SrTiO_{3}. We reveal the microscopic details of this exotic phase transition and highlight that this phenomenon is a unique behavior of the quantum paraelectric phase. 

http://link.aps.org/pdf/10.1103/PhysRevLett.129.167401

Simulating Terahertz Field-Induced Ferroelectricity in Quantum Paraelectric 
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Nonequilibrium electron dynamics in solids is an important subject from both fundamental and technological points of view. The recent development of laser technology has enabled us to study ultrafast electron dynamics in the time domain. First-principles calculation is a powerful tool for analyzing such complex electron dynamics and clarifying the physics behind the experimental observation. In this article, we review the recent development of the first-principles calculation for light-induced electron dynamics in solids by revising its application to recent attosecond experiments. The electron dynamics calculations offer an accurate description of static and transient optical properties of solids and provide physics insight into light-induced electron dynamics. Furthermore, the microscopic decomposition of transient properties of nonequilibrium systems has been developed to extract microscopic information from the simulation results. The first-principles analysis opened a novel path to analyze the nonequilibrium electron dynamics in matter and to provide the fundamental understanding complementarily with the sophisticated experimental technique.

First-principles calculations for attosecond electron dynamics in solids

Various nonlinear characteristics of solid states, such as the circular photogalvanic effect of time-reversal symmetric insulators, the quantized photogalvanic effect of Weyl semimetals, and the nonlinear Hall effect of time-reversal symmetric metals, have been associated with the Berry curvature dipole (BCD). Here, we explore the question of whether the Berry curvature dipole and multipoles of time-reversal symmetric insulators can be traced in the nonlinear optical responses. We performed real-time time-dependent density functional theory calculations and examined the second harmonic generation susceptibility tensors. The two-band term of the susceptibility tensor is sharply proportional to the interband BCD, dominating over the Hall response once the cancellation effect of the multiple reflection symmetries is lifted. We suggest that the nonlinear Hall component of the second-harmonic spectra of insulators can also be utilized as an effective tool to extract the band structure geometry through Berry curvature dipole and possibly multipoles. 

/pdf/second-harmonic-hall-responses-of-insulators-as-a-probe-of-hd0q249h.pdf

Shunsuke A. Sato

Papers

Role of electron scattering on the high-order harmonic generation from solids

The Ferroelectric Photo-Groundstate of SrTiO$_3$: Cavity Materials Engineering

Simulating Terahertz Field-Induced Ferroelectricity in Quantum Paraelectric <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi>SrTiO</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>

First-principles calculations for attosecond electron dynamics in solids

Second harmonic Hall responses of insulators as a probe of Berry curvature dipole