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Physical Review A

Complete Keldysh theory and its limiting cases

Howard R. Reiss

Published Aug 1, 1990• 160 citations
Publisher

Abstract

An investigation of the foundations of strong-field theories of the Keldysh type is undertaken. It is found first that the Volkov solution, upon which the Keldysh approximation depends, exists unambiguously only in the relativistic case. Accordingly, a fully relativistic theory of atomic photoionization is formulated from first principles. A strong-field approximation (SFA) is expressed, based only on the proposition that the photoionizing field dominates the atomic potential in the final state. The nonrelativistic limit of the SFA gives exactly one of the known Keldysh-type theories, but the original Keldysh theory itself is found to neglect some fraction of the strong-field effects. It is shown on a variety of grounds that ${\mathbf{A}}^{2}$ terms must be fully retained in Keldysh-type theories. Important in this matter are the proper application of asymptotic conditions, the interchangeability of A\ensuremath{\cdot}p and ${\mathbf{A}}^{2}$ terms, and the consistent reduction to the nonrelativistic limit.

DOI
10.1103/physreva.42.1476

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Reply to ‘‘Comment on ‘Complete Keldysh theory and its limiting cases’ ’’

Howard R. Reiss

Milonni [preceding Comment, Phys. Rev. A 45, 2138 (1992)] raises objections to my work [Phys. Rev. A 42, 1476 (1990)] on Keldysh-like methods in photoionization. His results are shown to follow from procedures that are inappropriate to the strong-field environment of the Keldysh method. These procedures lead to physical inconsistencies. My technique is verified to be rigorous and unique.

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An investigation of the foundations of strong-field theories of the Keldysh type is undertaken. It is found first that the Volkov solution, upon which the Keldysh approximation depends, exists unambiguously only in the relativistic case. Accordingly, a fully relativistic theory of atomic photoionization is formulated from first principles. A strong-field approximation (SFA) is expressed, based only on the proposition that the photoionizing field dominates the atomic potential in the final state. The nonrelativistic limit of the SFA gives exactly one of the known Keldysh-type theories, but the original Keldysh theory itself is found to neglect some fraction of the strong-field effects. It is shown on a variety of grounds that A2{\mathbf{A}}^{2}A2 terms must be fully retained in Keldysh-type theories. Important in this matter are the proper application of asymptotic conditions, the interchangeability of A\ensuremath{\cdot}p and A2{\mathbf{A}}^{2}A2 terms, and the consistent reduction to the nonrelativistic limit.
It is shown that strong-field atomic stabilization can occur at any frequency, that analytical methods exist that can describe all essential features of stabilization, that relativistic effects enhance the stabilization phenomenon, and that a simple physical picture exists that explains these properties. A necessary prelude is to show that the frequency properties of the three methods often conjoined by the KFR (Keldysh-Faisal-Reiss) label are quite different. Applicability of the SFA (Strong-Field Approximation) to stabilization at any frequency is shown, and verified by exhibiting close correspondence to numerical predictions by Popov et al. that also span both low and high frequencies.
Formal time-independent scattering theory is applied to multiphoton ionization of atoms in intense electromagnetic fields. The quantized-field version of the Volkov solution makes this approach possible. With the electron-photon interaction in a monochromatic photon field, it is found that, in the nonrelativistic and large-photon-number limits, the final scattering state exists only in the special case in which the ponderomotive potential per unit photon energy is an integer; otherwise the final state vanishes. In the integer case the corresponding wave function reduces to a single Volkov function, multiplied by an overlap factor. A simple interpretation of this result is given, and some other consequences of this work are discussed.
A formulation of quantum electrodynamics is given that applies to atoms in a strong laser field by perturbation theory in a non-relativistic regime. Dipole approximation is assumed. The dual Dyson series, here discussed by referring it to the Birkhoff theorem for singularly perturbed linear differential equations, can be applied and a perturbation series obtained transforming the Hamiltonian by a Pauli-Fierz transformation. But, if just few photons are present high-order harmonics cannot be generated. So, it is proven that odd high-order harmonics only appear when the laser field is intense and one can substitute the creation and annihilation operators by the square root of the mean number of photons taken to be huge, the field retaining its coherency property as observed experimentally for harmonics. In this case, the Hamiltonian for perturbation theory comes to the Kramers-Henneberger form. The theory has a dipolar contribution when the free-electron quiver motion amplitude is larger than the atomic radius. For a Coulomb potential one has that the outer electron is periodically kicked, and so a prove is given that the same should happen to Rydberg atoms in intense microwave fields. The distribution representing the kicking has a Fourier series with just odd terms. Using a modified Rayleigh-Schrödinger perturbation theory, it is shown that under the same condition of validity of the quiver motion amplitude to atomic radius ratio, the atomic wave function is only slightly modified by the laser field due to the way the energy levels rearrange themselves. This gives a prove of stabilization in the limit of laser frequency going to infinity. Then, perturbation theory can be applied when the Keldysh parameter becomes smaller with respect to the shifted distance between the energy levels of the atom.
When matter is exposed to strong fields, electrons are ionized and a number of processes such as high harmonic generation (HHG), above threshold ionization (ATI), non-sequential ionization happen. These processes are usually investigated semiclassically, i.e. matter is treated by quantum mechanics, and radiation is treated classically. In particular HHG which is the generation of ultrashort coherent extreme ultraviolet (XUV) pulses, has enabled a wealth of novel ultrashort spectroscopic methods in atomic, molecular, and condensed matter physics. Further, ionization has applications in micro-machining. Strong field physics continues to have numerous open questions. As an illustration, the mechanisms underlying dominant HHG pathways in solids remain insufficiently understood. It is not clear which mechanism dominates HHG in various materials. Possible pathways are the interband and intraband contributions. Furthermore, many of these processes in strong field physics have been described by single active electron (SAE) approximation. This approach relies on the assumption that the interaction between a single electron and the intense laser field dominates all other interactions. However, this is clearly not the case especially in solids where interactions with the lattice and other electrons can be important. In addition, quantum optical aspects of strong field physics have generated increasing interest, yet they remain partially explored. This chapter involves treating electromagnetic fields using the formalism of second quantization. Quantum optical properties, such as squeezing, entanglement and the negativity of the Wigner function, are of fundamental importance for the field of quantum information and quantum computation. The central theme of my thesis is the development of quantum optical and statistical theoretical frameworks for describing intense laser field processes such as ionization and HHG in atomic and molecular gases and in solids. The second chapter focuses on gaining a detailed understanding of the mathematical steps involved in the Keldysh theory of ionization to establish a solid theoretical foundation. Through this analysis we identified an additional factor of two compared to Keldysh’s original derivation of atomic ionization rates. The third chapter addresses the inadequately understood mechanisms that govern dominant high harmonic generation (HHG) pathways in solids. One approach to clarify these mechanisms involves introducing real (resonant) and virtual processes. We developed the strong field adiabatic following (SFAF) formalism which is based on Dyson expansion using the von-Neumann equation of density matrix. Using SFAF a diagnostic method is obtained to separate virtual and resonant channels. Through this separation, and by comparing with experimental results, we identified the need to incorporate many-body effects. The fourth chapter explores the fact that a solid is a complex many body system in which the SAE approximation is very crude. The electron interacts with holes, other electrons, collective excitations such as plasmons, and phonons. Our work is based on the idea that all theses effects can be treated as a quantum statistical heat bath of bosonic harmonic oscillators. We apply this to our SFAF formalism. In the ionization case, this work has settled a long standing issue which is the fact that simple phenomenological approaches such as the relaxation time approximation 𝑇2 result in nonphysical enhancement of ionization. In chapter five, we have investigated ways to transfer quantum optical properties on the otherwise classical high harmonic radiation. The goal is to use HHG to scale quantum sources to smaller wavelengths. Quantum properties can be imprinted by perturbing HHG with a quantum field, such as bright squeezed vacuum (BSV). In this chapter, the theory of quantum sideband high harmonic generation (QSHHG) in atoms and solids is derived to find ways by which to transfer quantum properties from the perturbative BSV to the harmonic sideband. The theoretical framework is a quantum generalization of the semi-classical Lewenstein model of HHG. It gives closed-form solutions for the HHG and QSHHG wavefunctions. Knowing the wavefunction, we can identify the quantum properties of QSHHG. The additional photons absorbed and emitted from the quantum perturbation, here BSV, create entanglement between individual harmonic sidebands and between the harmonic sidebands and the BSV. We show how this entanglement can be used to create a variety of non-classical states commonly used in quantum information science, such as high-purity single-photon states, Schrödinger cat states, and photon-added squeezed vacuum states. Some of these non-Gaussian states with negative Wigner functions have been shown to provide a quantum computational advantage over their Gaussian counterparts. Additionally, they play a significant role in quantum metrology, enhancing precision measurements beyond classical limits. This chapter has opened a path to quantum engineering of HHG.
A non-relativistic quantum theory of matter-photon interaction is formulated, within the framework of the dipole approximation, by employing a canonical transformation to diago­ nalize the A' term plus the free photon Hamiltonian. The matter-photon interaction is there­ by expressed rigorously in a renormalized A·p form. Using this form for the Dicke model of two-level atoms interacting with a single-mode radiation field, we examine a possibility of the occurrence of stationary coherent states which could arise for a photon and atomic polarization as the ground state of the system, deducing a condition for such coherent states. The condition, expressed as an inequality which imposes a stable nontrivial Bloch angle of the uniform atomic polarization, is identical with that for the occurrence of a second order phase transition, and indicates that the non-existence of a second order phase transition pointed out by Rza2ewski et al. io; in fact a consequence of the present treatment of the A'­ term renormalization. However, a fulfillment of the condition is recovered, if exchange­ type atom-atom interaction is taken into account in the matter system. We also present a method to construct an effective Hamiltonian of atom-atom interaction which is equivalent to the above Dicke model in the thermodynamic limit.