Kramers-Henneberger Frame Research Articles

Dynamically assisted tunneling in the Floquet picture

We study how tunneling through a potential barrier V(x) can be enhanced by an additional harmonically oscillating electric field E(t)=E0cos(ωt) or a similar oscillating force. To this end, we transform into the Kramers-Henneberger frame and calculate the coupled Floquet channels numerically. We find distinct signatures of resonances when the incident energy E equals the driving frequency ω=E, which clearly shows the breakdown of the time-averaged potential approximation in this regime. As a simple model for experimental applications (e.g., in solid-state physics), we study the rectangular potential, which can also be benchmarked with respect to analytical results. Finally, we consider the truncated Coulomb potential relevant for nuclear fusion. Published by the American Physical Society 2024

Diatomic molecule in a strong infrared laser field: level-shifts and bond-length change due to laser-dressed Morse potential

We present a general mathematical procedure to handle interactions described by a Morse potential in the presence of a strong harmonic excitation. We account for permanent and field-induced terms and their gradients in the dipole moment function, and we derive analytic formulae for the bond-length change and for the shifted energy eigenvalues of the vibrations, by using the Kramers–Henneberger frame. We apply these results to the important cases of H2 and LiH, driven by a near- or mid-infrared laser in the 1013 W cm−2 intensity range.

Phase-dependent cross sections of deuteron-triton fusion in dichromatic intense fields with high-frequency limit

We investigate the influence of strong dichromatic laser fields (i.e. 1{\omega}-2{\omega} and 1{\omega}-3{\omega}) with high-frequency limit on the cross sections of deuteron-triton(DT) fusion in Kramers-Henneberger(KH) frame. We focus on the transitions of phase-dependent effects depending on a dimensionless quantity n_{d} , which equals the ratio of the quiver oscillation amplitude to the geometrical touching radius of the deuteron and triton as defined in our previous research. Theoretical calculations show that the angle-dependent as well as phase-dependent Coulomb barrier penetrabilities can be enhanced in dichromatic intense fields, and the corresponding angle-averaged penetrabilities and the fusion cross sections increase significantly compared with field-free case. Moreover, we find that there are twice shifts of the peak in the cross sections whenever the frequency becomes sufficiently low or the intensity sufficiently high. The reason for the first shift is the angle-dependence effects for sub-barrier fusion, while the second shift is due to the accumulation of over-barrier fusion, these mechanisms are analyzed in detail in this paper.

Bogolyubov’s averaging theorem applied to the Kramers–Henneberger Hamiltonian

We apply Bogolyubov’s averaging theorem to the motion of an electron of an atom driven by a linearly polarized laser field in the Kramers–Henneberger frame. We provide estimates of the differences between the original trajectories and the trajectories associated with the averaged system as a function of the parameters of the laser field and the region of phase space. We formulate a modified Bogolyubov averaging theorem based on the Hamiltonian properties of the system, and show that this version is better suited for these systems. From these estimates, we discuss the validity of the Kramers–Henneberger approximation.

Strong-field approximation for high-order harmonic generation in infrared laser pulses in the accelerated Kramers-Henneberger frame

The strong-field approximation for high-harmonic generation in near-infrared and infrared laser pulses is formulated in the accelerated Kramers-Henneberger frame. The accompanying physical picture is discussed and the nature of the leading-order term is contrasted with that of the three-step model following the strong-field-approximation formulation in the length or velocity gauges. The theory is illustrated by high-harmonic generation spectra for atomic hydrogen.

Time-Dependent Unitary Transformation Method in the Strong-Field-Ionization Regime with the Kramers-Henneberger Picture.

Time evolution operators of a strongly ionizing medium are calculated by a time-dependent unitary transformation (TDUT) method. The TDUT method has been employed in a quantum mechanical system composed of discrete states. This method is especially helpful for solving molecular rotational dynamics in quasi-adiabatic regimes because the strict unitary nature of the propagation operator allows us to set the temporal step size to large; a tight limitation on the temporal step size () can be circumvented by the strict unitary nature. On the other hand, in a strongly ionizing system where the Hamiltonian is not Hermitian, the same approach cannot be directly applied because it is demanding to define a set of field-dressed eigenstates. In this study, the TDUT method was applied to the ionizing regime using the Kramers-Henneberger frame, in which the strong-field-dressed discrete eigenstates are given by the field-free discrete eigenstates in a moving frame. Although the present work verifies the method for a one-dimensional atom as a prototype, the method can be applied to three-dimensional atoms, and molecules exposed to strong laser fields.

Intensity-dependent angular distribution of low-energy electrons generated by intense high-frequency laser pulse.

By solving the three-dimensional time-dependent Schrödinger equation, we investigate the angular distributions of the low-energy electrons when an intense high-frequency laser pulse is applied to the hydrogen atom. Our numerical results show that the angular distributions of the low-energy electrons which generated by the nonadiabatic transitions sensitively depend on the laser intensity. The angular distributions evolve from a two-lobe to a four-lobe structure as the laser intensity increases. By analyzing nonadiabatic process in the Kramers-Henneberger frame, we illustrate that this phenomenon is attributed to the intensity-dependent adiabatic evolution of the ground state wavefunction. When the laser intensity further increases, the pathway of nonadiabatic transition from the ground state to the excited state and then to the continuum states is non-negligible, which results in the ring-like structure in the photoelectron momentum distribution. The angular distributions of the low-energy electrons provide a way to monitor the evolution of the electron wavefunction in the intense high frequency laser fields.

On Laser-Modified Rutherford Scattering

We present a theoretical analysis of a charged-particle scattering by a Coulomb potential in the presence of laser radiation. The effect of a laser field is studied using our recently developed nonperturbative parabolic quasi-Sturmian approach for solving the system of coupled Lippmann–Schwinger–Floquet equations in the Kramers–Henneberger frame. We calculate the ratio of multiphoton differential cross sections to the Rutherford cross section in the case of a laser-assisted electron-proton scattering process. Our results are compared with predictions of the Bunkin–Fedorov, Kroll–Watson, and Coulomb–Volkov analytical approximations: marked discrepancies are found for different net numbers of exchanged photons and different orientations of the laser-field polarization vector. Our findings clearly demonstrate deficiencies of those well-known approximations for describing laser-modified Rutherford scattering processes.

Laser-modified Coulomb scattering states of an electron in the parabolic quasi-Sturmian-Floquet approach

Electron scattering states in combined Coulomb and laser fields are investigated with a nonperturbative approach based on the Hermitian Floquet theory. Taking into account the Coulomb-specific asymptotic behavior of the electron wave functions at large distances, a Lippmann-Schwinger-Floquet equation is derived in the Kramers-Henneberger frame. Such a scattering-state equation is solved numerically employing a set of parabolic quasi-Sturmian functions which have the great advantage of possessing, by construction, adequately chosen incoming or outgoing Coulomb asymptotic behaviors. Our quasi-Sturmian-Floquet approach is tested with a calculation of triple differential cross sections for a laser-assisted $(e,2e)$ process on atomic hydrogen within a first-order Born treatment of the projectile-atom interaction. Convergence with respect to the number of Floquet-Fourier expansion terms is numerically demonstrated. The illustration shows that the developed method is very efficient for the computation of light-dressed states of an electron moving in a Coulomb potential in the presence of laser radiation.

Kramers–Henneberger Form of Strong Field Theory with the Correction of Dipole Approximation**Supported by the National Natural Science Foundation of China under Grant Nos 11274149 and 11304185, and the Program of Shenyang Key Laboratory of Optoelectronic Materials and Technology under Grant No F12-254-1-00.

We show that the breakdown of dipole approximation can be adopted to explain the asymmetry structure in the photoelectron momentum distributions along the beam propagation direction, which is defined as the photoelectron longitudinal momentum distributions (PLMD), in tunneling regime (γK ≪ 1), based on the strong field approximation theory. The nondipole Hamiltonian for photoelectrons interacting with laser fields from a hydrogen-like atom is transformed into the Kramers–Henneberger frame in our model. To introduce the correction of dipole approximation, the spatial variable is kept in a vector potential A(r, t), demonstrating that the breakdown of dipole approximation is the major reason for the shift of the peak in PLMD. The nondipole effects are apparent when circularly polarized lasers are adopted to ionize the atoms, and clear tendency to increase offsets is found for increasing laser intensities.

Spatial transport of electron quantum states with strong attosecond pulses

This work follows up the work of Dimitrovsky, Briggs and co-workers on translated electron atomic states by a strong field of an atto-second laser pulse, also described as creation of atoms without a nucleus. Here, we propose a new approach by analyzing the electron states in the Kramers–Henneberger moving frame in the dipole approximation. The wave function follows the displacement vector . This allows arbitrarily shaped pulses, including the model delta-function potentials in the Dimitrovsky and Briggs approach. In the case of final-length single-cycle pulses, we apply both the Kramers–Henneberger moving frame analysis and a full numerical treatment of our 1D model. When the laser pulse frequency exceeds the frequency associated by the energy difference between initial and final states, the entire wavefunction is translated in space nearly without loss of coherence, to a well defined distance from the original position where the ionized core is left behind. This statement is demonstrated on the excited Rydberg states (n = 10, n = 15), where almost no distortion in the transported wave functions has been observed. However, the ground state (n = 1) is visibly distorted during the removal by pulses of reasonable frequencies, as also predicted by Dimitrovsky and Briggs analysis. Our approach allows us to analyze general pulses as well as the model delta-function potentials on the same footing in the Kramers–Henneberger frame.

An electron of helium atom under a high-intensity laser field

We scrutinize the behavior of eigenvalues of an electron in a helium (He) atom as it interacts with electric field directed along the z-axis and is exposed to linearly polarized intense laser field radiation. To achieve this, we freeze one electron of the He atom at its ionic ground state and the motion of the second electron in the ion core is treated via a more general case of screened Coulomb potential model. Using the Kramers–Henneberger (KH) unitary transformation, which is the semiclassical counterpart of the Block–Nordsieck transformation in the quantized field formalism, the squared vector potential that appears in the equation of motion is eliminated and the resultant equation is expressed in the KH frame. Within this frame, the resulting potential and the corresponding wave function are expanded in Fourier series and using Ehlotzky’s approximation, we obtain a laser-dressed potential to simulate intense laser field. By fitting the more general case of screened Coulomb potential model into the laser-dressed potential, and then expanding it in Taylor series up to , we obtain the solution (eigenvalues and wave function) of an electron in a He atom under the influence of external electric field and high-intensity laser field, within the framework of perturbation theory formalism. We found that the variation in frequency of laser radiation has no effect on the eigenvalues of a He electron for a particular electric field intensity directed along z-axis. Also, for a very strong external electric field and an infinitesimal screening parameter, the system is strongly bound. This work has potential application in the areas of atomic and molecular processes in external fields including interactions with strong fields and short pulses.

Hydrogen atom in a laser-plasma

We scrutinize the behaviour of the eigenvalues of a hydrogen atom in a quantum plasma as it interacts with an electric field directed along θ = π and is exposed to linearly polarized intense laser field radiation. We refer to the interaction of the plasma with the laser light as laser-plasma. Using the Kramers–Henneberger (KH) unitary transformation, which is the semiclassical counterpart of the Block–Nordsieck transformation in the quantized field formalism, the squared vector potential that appears in the equation of motion is eliminated and the resultant equation is expressed in the KH frame. Within this frame, the resulting potential and the corresponding wavefunction have been expanded in Fourier series, and using Ehlotzky’s approximation we obtain a laser-dressed potential to simulate an intense laser field. By fitting the exponential-cosine-screened Coulomb potential into the laser-dressed potential, and then expanding it in Taylor series up to , we obtain the eigensolution (eigenvalues and wavefunction) of the hydrogen atom in laser-plasma encircled by an electric field, within the framework of perturbation theory formalism. Our numerical results show that for a weak external electric field and a very large Debye screening parameter length, the system is strongly repulsive, in contrast with the case for a strong external electric field and a small Debye screening parameter length, when the system is very attractive. This work has potential applications in the areas of atomic and molecular processes in external fields, including interactions with strong fields and short pulses.

Multiphoton ionization and stabilization of helium in superintense xuv fields

Multiphoton ionization of helium is investigated in the superintense field regime, with particular emphasis on the role of the electron-electron interaction in the ionization and stabilization dynamics. To accomplish this, we solve ab initio the time-dependent Schr\"odinger equation with the full electron-electron interaction included. By comparing the ionization yields obtained from the full calculations with corresponding results of an independent-electron model, we come to the somewhat counterintuitive conclusion that the single-particle picture breaks down at superstrong field strengths. We explain this finding from the perspective of the so-called Kramers-Henneberger frame, the reference frame of a free (classical) electron moving in the field. The breakdown is tied to the fact that shake-up and shake-off processes cannot be properly accounted for in commonly used independent-electron models. In addition, we see evidence of a change from the multiphoton to the shake-off ionization regime in the energy distributions of the electrons. From the angular distribution it is apparent that correlation is an important factor even in this regime.

Atoms and molecules in intense attosecond fields: beyond the dipole approximation

The exact non-dipole minimal-coupling Hamiltonian for an atomic system interacting with an explicitly time- and space-dependent laser field is transformed into the rest frame of a classical free electron in the laser field, i.e., into the Kramers-Henneberger frame. The new form of the Hamiltonian has been used to study the non-dipole dynamics of atoms and molecules in intense XUV laser pulses. The time-dependent Schrödinger equation is solved without any simplifications.

Siegert-state expansion in the Kramers-Henneberger frame: Interference substructure of above-threshold ionization peaks in the stabilization regime

The Siegert-state expansion approach is applied to the solution of the time-dependent Schr\"odinger equation describing a model one-dimensional laser-atom interaction problem in the Kramers-Henneberger frame. Our method is mathematically rigorous and numerically exact even though a very restricted spatial box is considered, since the use of Siegert states as a basis in the expansion eliminates unphysical reflections from the boundary of the box and the Kramers-Henneberger frame enables us to fully take the interaction with the laser field into account. The method is demonstrated by calculations of above-threshold ionization spectra generated by strong high-frequency laser pulses. We found an oscillating substructure of multiphoton peaks caused by an interference of photoelectron wave packets produced at different times during the pulse which becomes especially pronounced in the stabilization regime. An interpretation of this effect in terms of the high-frequency Floquet theory is given.

Exact Nondipole Kramers-Henneberger Form of the Light-Atom Hamiltonian: An Application to Atomic Stabilization and Photoelectron Energy Spectra

The exact nondipole minimal-coupling Hamiltonian for an atom interacting with an explicitly time- and space-dependent laser field is transformed into the rest frame of a classical free electron in the laser field, i.e., into the Kramers-Henneberger frame. The new form of the Hamiltonian is used to study nondipole effects in the high-intensity, high-frequency regime. Fully three-dimensional nondipole ab initio wave packet calculations show that the ionization probability may decrease for increasing field strength. We identify a unique signature for the onset of this dynamical stabilization effect in the photoelectron spectrum.

Floquet scattering and classical-quantum correspondence in strong time-periodic fields

We study the scattering of an electron from a one dimensional inverted Gaussian atomic potential in the presence of strong time periodic electric fields. Using Floquet theory, we construct the Floquet Scattering matrix in the Kramers-Henneberger frame. We compute the transmission coefficients as a function of electron incident energy and find that they display asymmetric Fano resonances due to the electron interaction with the driving field. We find that the Fano resonances are associated with zero-pole pairs of the Floquet Scattering matrix in the complex energy plane. Another way we "probe" the complex spectrum of the system is by computing the Wigner-Smith delay times. Finally we find that the eigenphases of the Floquet Scattering matrix undergo a number of "avoided crossings" as a function of electron Floquet energy that increases with increasing strength of the driving field. These "avoided crossings" appear to be quantum manifestations of the destruction of the constants of motion and the onset of chaos in classical phase space.

Linear-least-squares-fitting procedure for the solution of a time-dependent wave function of a model atom in a strong laser field in the Kramers-Henneberger frame

We present a linear-least-squares-fitting method to solve the time-dependent Schrodinger equation of a one-dimensional model atom in an intense laser field in the Kramers-Henneberger frame. In comparison with calculations in the laboratory frame, it is shown that in the Kramers-Henneberger frame, the method is more stable and larger time steps in the propagation can be used. The fitting procedure also allows for easy change of basis functions during the time propagation. The method is applied to study the stabilization of a model atom in an intense laser field where the electron is bound initially by a one-dimensional soft Coulomb potential.

Partial rates in the high-frequency regime of intense-field photoionization of hydrogen

Complex rotation in the exterior scaling version is applied to the solution of the coupled equations of Floquet theory for ground state hydrogen submitted to an external intense oscillating linearly polarized electric field. The calculations are done in the Kramers–Henneberger frame. The fluxes are calculated in the asymptotic real part of the integration axis. The total and 1-photon rates are used to reassess the performances of a high-frequency two-pole approximation previously developed (J Phys B 1999, 32, 3271). For overthreshold frequencies, the partial rates with definite values of n, the number of absorbed photons, and l, the angular quantum number of the ejected electron, show oscillations which are erased when summing over l. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001