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Physical Review A

Dynamic hyperpolarizability and two-photon detachment in the presence of a strong static electric field: Application to H^{-}

M. V. Frolov, Nikolai L. Manakov, Anthony F. Starace

Published Jul 1, 2001• 7 citations
Publisher

Abstract

Our recent analysis [J. Phys. B 33, R141 (2000)] of the problem of ${\mathrm{H}}^{\ensuremath{-}}$ in both laser and strong static electric fields, which treated effects that are linear in the laser intensity I (e.g., the dynamic polarizability and the single-photon detachment cross section), is extended here to treat effects of higher order $(\ensuremath{\sim}{I}^{2})$ in the laser intensity (e.g., the dynamic hyperpolarizability, the two-photon detachment cross section, the linear in I corrections to the single-photon detachment rate, etc). We introduce the concept of the dynamic hyperpolarizability of an atom in the presence of a strong electric field based on the complex quasienergy approach with properly normalized, quasistationary quasienergy wave functions. Our analysis of the general structure of the dynamic hyperpolarizability tensor of an atom in a nondegenerate $(S$-) state is performed for arbitrary field geometries and laser polarizations. The connection of the hyperpolarizability $\ensuremath{\gamma}(\mathcal{F};\ensuremath{\omega})$ to the complex quasienergy and to atomic ionization rates is established. Analytic results (in terms of Airy functions) are obtained for five irreducible components of a hyperpolarizability tensor (that are independent of the laser polarization and the field geometry) for the case of a weakly bound electron in a three-dimensional, zero-range potential. These results are used for the analysis of the frequency, field geometry, and laser polarization dependence of the two-photon detachment rate as well as of the linear in laser intensity corrections to the single-photon detachment rate. It is shown that the oscillatory behavior of the frequency dependence of both the real and the imaginary parts of $\ensuremath{\gamma}(\mathcal{F};\ensuremath{\omega})$ exists for both the coplanar and orthogonal field geometries, and that this behavior is qualitatively different for frequencies below and above the single-photon detachment threshold. The threshold behavior of the two-photon detachment rate is analyzed in detail and the static electric-field-induced modification of Wigner's threshold law for a short-range potential is discussed.

DOI
10.1103/physreva.64.023417

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Comment on “Photodetachment in combined static and dynamic electric fields”

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The theoretical analysis of C. Rangan and A. R. P. Rau [Phys. Rev. A 61, 033405 (2000)] on the process of photodetachment of ${\mathrm{H}}^{\ensuremath{-}}$ in a strong static electric field, which calls into question the predictions of B. Gao and A. F. Starace [Phys. Rev. A 42, 5580 (1990)] and also of M. Q. Bao et al. [Phys. Rev. A 58, 411 (1998)], is shown in this Comment to be incorrect. First, we point out that a number of assumptions of Rangan and Rau's analysis rest on tenuous theoretical grounds. Second, we adduce two completely independent and different analyses of the problem which precisely confirm the results of Gao and Starace. These independent analyses also provide the interpretation that Gao and Starace's predicted strong-field effects are due to the exact account of the influence of the static electric field, not only on the final state of the detached electron, but also on the initial bound-electron state.

Physical Review A•Aug 8, 2001•8 citations
Our recent analysis [J. Phys. B 33, R141 (2000)] of the problem of H\ensuremath−{\mathrm{H}}^{\ensuremath{-}}H\ensuremath− in both laser and strong static electric fields, which treated effects that are linear in the laser intensity I (e.g., the dynamic polarizability and the single-photon detachment cross section), is extended here to treat effects of higher order (\ensuremath∼I2)(\ensuremath{\sim}{I}^{2})(\ensuremath∼I2) in the laser intensity (e.g., the dynamic hyperpolarizability, the two-photon detachment cross section, the linear in I corrections to the single-photon detachment rate, etc). We introduce the concept of the dynamic hyperpolarizability of an atom in the presence of a strong electric field based on the complex quasienergy approach with properly normalized, quasistationary quasienergy wave functions. Our analysis of the general structure of the dynamic hyperpolarizability tensor of an atom in a nondegenerate (S(S(S-) state is performed for arbitrary field geometries and laser polarizations. The connection of the hyperpolarizability \ensuremathγ(F;\ensuremathω)\ensuremath{\gamma}(\mathcal{F};\ensuremath{\omega})\ensuremathγ(F;\ensuremathω) to the complex quasienergy and to atomic ionization rates is established. Analytic results (in terms of Airy functions) are obtained for five irreducible components of a hyperpolarizability tensor (that are independent of the laser polarization and the field geometry) for the case of a weakly bound electron in a three-dimensional, zero-range potential. These results are used for the analysis of the frequency, field geometry, and laser polarization dependence of the two-photon detachment rate as well as of the linear in laser intensity corrections to the single-photon detachment rate. It is shown that the oscillatory behavior of the frequency dependence of both the real and the imaginary parts of \ensuremathγ(F;\ensuremathω)\ensuremath{\gamma}(\mathcal{F};\ensuremath{\omega})\ensuremathγ(F;\ensuremathω) exists for both the coplanar and orthogonal field geometries, and that this behavior is qualitatively different for frequencies below and above the single-photon detachment threshold. The threshold behavior of the two-photon detachment rate is analyzed in detail and the static electric-field-induced modification of Wigner's threshold law for a short-range potential is discussed.
We present a detailed theoretical formulation of the problem of an electron moving in a static electric field, a laser field, and an atomic potential. Our formulation treats the electron-atom interaction in the zero-range potential approximation and employs both the quasienergy approach and an analytic expression for the Green's function describing electron propagation in a combination of static and laser electric fields. Our formulation is applied to one- and two-photon detachment of ${\mathrm{H}}^{\ensuremath{-}}$ in a strong static electric field and takes into account all final-state interactions of the detached electron with the static and laser fields and with the atomic core. Our results show that rescattering effects are small in the case of one-photon detachment, where our results are close to those obtained previously by Gao and Starace [Phys. Rev. A 42, 5580 (1990)], who ignored rescattering effects but who found a strong-field treatment of the laser field to be important, even in the limit of weak laser fields, owing to interference terms involving both the laser and static electric fields. Our results for two-photon detachment of ${\mathrm{H}}^{\ensuremath{-}}$, on the other hand, show that rescattering effects are very significant. In the presence of a strong static electric field, moreover, the two-photon detachment cross section is found to be very sensitive to the magnitude of the static field.
An analytic momentum-space wave function for an electron in both laser and static uniform electric fields is presented. It is used to obtain analytic multiphoton detachment cross sections in a static, uniform electric field which include effects of static-field-induced electron-photon interactions. These general results are not restricted to weak laser intensities or to weak static-field strengths and depend only on (a) the electric dipole approximation and (b) the approximation that final-state electron-atom interactions are ignored. Four specific predictions of our general formulas for the most interesting case of linearly polarized light polarized along the static-field direction are presented for weakly bound electrons initially in an s state. First, the effects of a weak static electric field on N-photon detachment cross sections near threshold are shown to be described by two modulation factors, one for odd N and one for even N, which depend only on a scaled energy. Second, for photodetachment in a static electric field, effects of static-field-induced electron-photon interactions are demonstrated. Third, the lifetime against field ionization is presented. Fourth, the cross section for electric-field-induced stimulated emission is presented. Numerical results for the latter three effects are presented for the ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ ion. The simpler case of circularly polarized light directed along the static-electric-field direction is treated briefly. In particular, we show for this case that the static- and laser-field effects are uncoupled and that the near-threshold, weak static-electric-field modulation factors for the N-photon detachment cross sections are dependent on N.
On the basis of the generalized Sturm expansion of the radial part of the Coulomb Green function, a computational method is proposed and numerical results are presented for the dynamic hyperpolarizability γ and the corrections E (4) (quadratic in the light intensity) to the quasi-energy of the ground and excited states of hydrogen with principal quantum numbers n≤5 in a monochromatic light field. In this approach, the problem is reduced to the summation of well-convergent double series of the hypergeometric kind, which ensures reliable numerical results both for states with a large n, and in a wide range of field frequencies ω, including the above-threshold frequency range of ℏω≪|E n | (|E n | is the ionization potential of the state |nlm〉 under investigation). We consider the frequency dependence of γ and E (4), their differences for the cases of linear and circular polarizations of the field, and the relation between their real and imaginary parts, which determine the laser field-induced corrections to the position and width of energy levels. For n=5, the significant role of mixing the |nlm〉 states with different values of l by a laser field in the region of resonances on intermediate bound states is demonstrated. The linear (in intensity) corrections to the photoionization cross section for excited states are analyzed and the threshold intensity corresponding to the onset of atomic level stabilization is estimated for a number of states with n=3 and n=5.
In the first part of this thesis, a new variational principle is presented for the calculation of perturbative transition amplitudes of any order but free from any singularities. Applications are made to the multiphoton ionization and harmonic generation for atomic hydrogen. Its formulations in hyperspherical coordinates as well as in Hartree-Fock representation are also presented. In the second part of this thesis, theories for multiphoton detachment in a static electric or magnetic field are developed. They are based on the analytic solutions of an electron in the combined field of a laser and a static electric or magnetic field, and, therefore, are nonperturbative in both the laser intensity and the static electric or magnetic field strength. Heretofore unexpected cyclotron resonances, which are due to the infinite range character of the electron-photon coupling induced by the external magnetic field, are shown to exist under certain conditions.
The cross sections for resonant two-color, two-photon detachment of ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ with excitation of the degenerate H(2s) and H(2p) levels are calculated within a semiempirical adiabatic hyperspherical representation. The first photon, with energy ${\mathrm{\ensuremath{\omega}}}_{1}$=0.4017 a.u., is resonant with the well-known Feshback 1^{1}1Po{\mathit{P}}^{\mathit{o}}Po resonance below the H(n=2) threshold. The second photon, with energy ${\mathrm{\ensuremath{\omega}}}_{2}$\ensuremath{\ge}0.126 05 a.u., scans the energy region above the H(n=2) threshold over which long-range dipole-field-induced cross-section oscillations are predicted to occur. Such Gailitis-Damburg oscillations have not yet been observed experimentally. Results for various pairs of light polarization for the two photons are presented. Our resonant two-color, two-photon detachment cross sections are 8--9 orders of magnitude greater than the corresponding nonresonant, single-color, two-photon detachment cross sections obtained by Liu, Du, and Starace [Phys. Rev. A 43, 5891 (1991)]. Unmistakable evidence of long-range dipole-field effects is presented over the 5-meV energy range above the H(n=2) threshold. Furthermore, the differential cross sections for right- and left-circularly polarized, copropagating photons and especially the circular dichroism differential cross sections are shown to have nearly a full cycle of a greatly enhanced dipole-field-induced oscillation extending over the region from threshold to \ensuremath{\approxeq}34 meV above.
The theoretical analysis of C. Rangan and A. R. P. Rau [Phys. Rev. A 61, 033405 (2000)] on the process of photodetachment of ${\mathrm{H}}^{\ensuremath{-}}$ in a strong static electric field, which calls into question the predictions of B. Gao and A. F. Starace [Phys. Rev. A 42, 5580 (1990)] and also of M. Q. Bao et al. [Phys. Rev. A 58, 411 (1998)], is shown in this Comment to be incorrect. First, we point out that a number of assumptions of Rangan and Rau's analysis rest on tenuous theoretical grounds. Second, we adduce two completely independent and different analyses of the problem which precisely confirm the results of Gao and Starace. These independent analyses also provide the interpretation that Gao and Starace's predicted strong-field effects are due to the exact account of the influence of the static electric field, not only on the final state of the detached electron, but also on the initial bound-electron state.