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Scientific Reports

Distribution of absorbed photons in the tunneling ionization process

Igor A. Ivanov, Kyung Taec Kim

Published Feb 17, 2021• 3 citations
Publisher

Abstract

We describe a procedure that allows us to solve the three-dimensional time-dependent Schrödinger equation for an atom interacting with a quantized one-mode electromagnetic field. Atom-field interaction is treated in an ab initio way prescribed by quantum electrodynamics. We use the procedure to calculate probability distributions of absorbed photons in the regime of tunneling ionization. We analyze evolution of the reduced photon density matrix describing the state of the field. We show that non-diagonal density matrix elements decay quickly, as a result of the decoherence process. A stochastic model, viewing ionization as a Markovian birth-death process, reproduces the main features of the calculated photon distributions.

DOI
10.1038/s41598-021-83453-0

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The ionization of a model atom in a strong single-mode non-classical electromagnetic field is studied beyond the perturbation approach using the model of ``one discrete level + continuum'' for the atom. Explicit analytical expressions for the time-dependent probability of atomic ionization are obtained for different initial states of radiative field: Fock state, coherent wave packet, squeezed vacuum state. The results are compared to the ionization decay of discrete atomic level in a classical light field, and are analyzed for different mean number of radiative photons. The time-dependent distribution of the quantized field over photon number states is examined.
We consider ionization of an atom by the electromagnetic field in a Fock state $|N\ensuremath{\rangle}$: An eigenvector of the operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{N}}_{\mathbit{k},\ensuremath{\lambda}}={\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{\mathbit{k},\ensuremath{\lambda}}^{\ifmmode\dagger\else\textdagger\fi{}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{\mathbit{k},\ensuremath{\lambda}}$ of the number of photons in the mode $\mathbit{k},\ensuremath{\lambda}$. We treat the electromagnetic field in a quantized way, which allows us to obtain the photon picture of the strong field ionization. We obtain, in particular, probability distributions for the number of absorbed photons in the process of strong field ionization for atomic systems governed by short-range and Coulomb potentials.
Using the analytically obtained solution of the Schrödinger equation for an electron in a quantized electromagnetic field, the ionization of an atom has been considered in the framework of a simple one-dimensional model. Expressions for the ionization rate and photoelectron spectrum have been derived and analyzed. We discuss a difference between the results obtained in the model in which the electromagnetic field is described classically and the results obtained in the model of quantized electromagnetic field. In particular, in the latter model the photoelectron spectrum can have a minimum at a certain number of absorbed photons. The minimum is a consequence of destructive interference between the contributions of two parts of the interaction operator.
We consider the quantized atom-field model and for the regime ; where , and respectively represent the self-Hamiltonians of the environment, the system, and the interaction between the system and the environment. Considering a single-mode quantized field we obtain the time evolution operator for the model. Using our time evolution operator we calculate the time-dependent pointer states of the system and the environment, which are characterized by their property of not entangling with states of another subsystem. We obtain a closed form expression for the off-diagonal element of the reduced density matrix of the system and study the decoherence of the central system in our model. We show that for the case where the system initially is not prepared in one of its pointer states, the off-diagonal element of the reduced density matrix of the system decays with a decoherence time that is inversely proportional to the square root of the mass of the bosonic field particles.
Double ionization is studied by direct numerical integration of the time-dependent Schrödinger equation for a model two-electron system in the field of an electromagnetic wave in the case when the photon energy exceeds the energy required for the removal of both electrons of the atom. The probabilities of single-electron and double ionization are obtained as functions of the radiation intensity. The energy spectra and double pulse distributions of photoelectrons are analyzed. It is shown that single-photon ionization plays a significant role only in the limiting cases of weak and strong fields. The obtained results are used in an analysis of the contributions from different channels to the double-ionization process (in particular, for clarifying the role of the electron-electron interaction). The results of numerical calculations are compared with the analytic model of the phenomenon.