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Abstract
In intense laser fields, electrons of atoms will follow the laser field and undergo quiver motion just like free electrons but still weakly bound to the atomic core, thus forming a set of specific dressed states named Kramers-Henneberger (KH) states, which comprise the KH atoms. In a focused laser beam, in addition to Ponderomotive (PM) force, KH atoms will experience KH force, which is unique to KH atoms. We examine both PM and KH forces as well as corresponding velocity gain of hydrogen and helium atoms in a focused laser field with circular polarization. We work out laser parameters which can be used in experimental confirmation of circularly polarized KH atoms.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
There are many unique properties for atoms and molecules subjected to intense laser fields. The most striking one is a paradoxical stabilization, in which the ionization rate decreases as the laser intensity increases [1–4]. This counterintuitive phenomenon arises from the new effective binding potential between nucleus and the rapid oscillating electrons driven by the intense laser fields. The spacial structure and photoionization lifetime of the stabilized atoms are changed dramatically by the laser field [3–6]. This type of stabilization even extends to some simple molecules which are completely different from their field-free counterparts [7–12].
When an atom is exposed to an intense laser field, electrons will follow the laser field and undergo quiver oscillations, which are very similar to free electrons but still weakly bound to the nucleus core. A successful non-perturbative method dealing with this situation is the Kramers-Henneberger (KH) transformation [13–15], which transforms from lab frame to KH frame. In KH frame, electrons are at rest and nucleus quivers instead. According to High-Frequency Floquet Theory (HFFT), the electrons will feel an averaged Coulomb potential, named KH potential, which is capable of supporting many bound states. Those bound states are called KH states and the corresponding atoms are called KH atoms [3,4]. An excellent overview of the KH atom and its evolution can be found in a recent thesis by Maria Richter [16]. The existing of stable bound KH state atoms is a perspective of high-frequency limit (UV range) with zero-order approximation of KH expansion when higher order harmonics of oscillating interaction can be neglected. But now KH atoms are believed to exist even in low-frequency lasers in the range from visible to infrared [17–21]. There are two types of KH atoms: linear and circular, depending on laser polarization direction. For a KH atom in linearly polarized laser fields, KH potential is similar to a coulomb potential with nuclear charge smeared along a line segment of twice the length $\alpha _0=\sqrt {I}/\omega ^2$; for a KH atom in circularly polarized laser fields, KH potential is similar to a coulomb potential generated by a circular charge with radius $\alpha _0=\sqrt {I/2}/\omega ^2$, where $I$ and $\omega$ are intensity and frequency of the laser field.
Inspired by their unique properties, extensive study of KH atoms has lasted for decades [1–12,16–28]. Nowadays, the concepts of KH atoms are also used to explain or predict phenomena in strong laser fields: from slow electrons generation to higher-order Kerr effect [20,29], from laser filamentation to high-harmonic generation [21,30]. Although there are proposals of using photoionization spectra to image spacial structure of KH atoms directly [19], experimental confirmation has still lagged behind. The first experimental demonstration of such stabilization was performed more than two decades ago using circularly polarized lasers with neon atoms prepared in Rydberg states [31–33]. Since then, more experiments relating to inhibition of ionization in strong laser fields have emerged [34–39]. However, direct evidence of existence of KH atoms was not found until recently. In an experiment of neutral atoms acceleration of helium and neon by linearly polarized IR laser, it was believed that the accelerated atoms in intense laser field are in Rydberg states and can only feel ponderomotive (PM) force for acceleration [35]. But the observed velocities lie somewhat above theoretical prediction. Later, it was found that treating the accelerated atoms in KH states rather than Rydberg states gives much better fit to experimental results [40,41]. The reason is in addition to PM force, KH atoms will feel another force named KH force which is small but significant. Including KH force brings the theoretical velocities to a closer match to experimental results [41]. This can be regarded as a direct confirmation of KH atoms of linear polarization.
Circularly polarized KH atoms have their unique properties which are different from those of linear polarization. They are equally important in understanding the true mechanism of atomic stabilization in intense laser field and have attracted lots of attention [1,3,4,24]. Therefore, in this paper, we study the dynamics of singly excited KH state helium and hydrogen atoms in circularly polarized laser beams. We evaluate PM force and KH force, as well as velocity gain induced by both forces. Laser parameters we used are experimentally accessible today. We hope our study will invite experimental measurements which can lead to confirmation of KH atoms of circular polarization.
Unless specified, we use atomic units (a.u.), with Planck constant, the electron charge and its mass all set to unity $(\hbar =e=m_e=1)$. Thus, 1 a.u. for energy is a hartree unit (27.2 eV), for distance a bohr unit (0.529 Å), for laser intensity $3.51\times 10^{16}$ $\text {W}/\text {cm}^2$ and for force $8.24\times 10^{-8}$ N.
2. Singly excited KH state atoms
For a one-electron atom in laser field, the time-dependent Schr$\ddot {\mathrm {o}}$dinger equation in lab frame is:
Here, we assume that the quantum state of the KH atom evolves adiabatically, which keeps the initial state when the field amplitude varies slowly during the laser pulse [3,4]. Then at high-frequency limit, the time-dependent potential $V\left (\textbf {r}+{\boldsymbol \alpha }\left (t\right )\right )$ can be replaced by a time-averaged effective one. Finally, we can get the following structure equation [3]
Figure 1 shows ground state wavefunctions (in KH frame) of KH state helium atoms in $x-y$ plane (polarization plane) for different quiver amplitudes. The KH state atomic shape and structures are completely different from those of field-free case. This is because in circularly polarized laser field, the dressed potential $V_0$ is similar to a circular charge with radius of $\alpha _0$, which will certainly reshape the atom. At small quiver amplitude, $\alpha _0=5$ a.u., the wavefunction is more localized with peak at center. When $\alpha _0$ increases, the KH state electron is more and more delocalized and spreads outward till reaches the "circular charge". As a result, the size of the atom is in the scale of $\alpha _0$ and its shape is like a ring, particularly at large quiver amplitude ($\alpha _0 \ge 50$ a.u.).
Fig. 1. Ground state wavefunctions (in $x-y$ plane) of KH state helium atoms in a circularly polarized laser field for $\alpha _0$ = 5, 10, 20, and 50 a.u. The laser is polarized in $x-y$ plane.
Besides its shape and size, binding energy of the KH state electron also changes dramatically. Figure 2(a) displays the ground state energy $\epsilon _{0}$ of the KH hydrogen and helium atoms as a function of $\alpha _0$ in circularly polarized laser fields. With the increasing $\alpha _0$, the magnitude of $\epsilon _0$ decreases sharply at first and thereafter the curve becomes flatter and flatter. For example, when $\alpha _0$ changes from 0 to 10 a.u., -$\epsilon _{0}$ decreases from 0.9 a.u. to 0.1 a.u., whereas, when $\alpha _0$ increases further to 100 a.u., -$\epsilon _{0}$ decreases from 0.1 a.u. to 0.011 a.u.. Since -$\epsilon _{0}$ decreases monotonically with $\alpha _0$, the ground state KH atom is a low-field seeker. The slope of $\epsilon _{0}$ curve, ${\mathrm {d}\epsilon _{0}}/{\mathrm {d}\alpha _0}$ can be calculated numerically and is displayed in Fig. 2(b), which will be used in the next section.
Fig. 2. Ground state energy $\epsilon _{0}$ (left panel) and slope $\mathrm {d}\epsilon _{0}/\mathrm {d}\alpha _0$ (right panel) as functions of the quiver amplitude $\alpha _0$ for KH state hydrogen (black dash) and helium (red solid) atoms in a circularly polarized laser field.
3. Acceleration of KH atoms
3.1 PM and KH forces
A circularly polarized laser pulse with Gaussian spatial intensity distribution can be specified in cylindrical coordinates,
In a focused laser beam, the ponderomotive potential is no longer a constant, thus the oscillating electrons will experience a non-negligible ponderomotive force, governed by the gradient of the spatial intensity distribution and its temporal dependence:
The KH force is arising from the fact that the eigenenergies of KH atom is spacial dependent inside the laser focusing region, which is governed by the gradient of binding energy of the KH state electron. For ground state, it can be specified by
Here we only consider ground state, because its gradient is much larger than other states. For the circularly polarized intense laser field, the radial KH force can be expressed asFigure 3 shows spacial distribution of radial and longitudinal forces exerted on a KH state helium atom located in $x-z$ plane when laser intensity reaches maximum at $t = 0$. Spacial locations of PM and KH forces differ dramatically. PM forces are more spacially concentrated in inner area of the laser focusing region, whereas KH forces are distributed more at outer area. For instance, both radial and longitudinal PM forces concentrate in the region of $r< w_0$ and $|z|\le 0.02 z_0$; however, radial KH forces are located in outer region of $w_0<r<2.5w_0$ and $|z|<0.03 z_0$; longitudinal KH forces are located in the region of $r<2w_0$ and $0.02z_0<|z|<0.04z_0$. In the $x-z$ plane, $F_{PM}$ peaks at ($x=\pm 0.5w_0$, $z=0$) for radial force and ($x=0$, $z=\pm 0.01z_0$) for longitudinal force. Nevertheless, $F_{KH}$ peaks at ($x=\pm 2w_0$, $z=0$) and ($x=0$, $z=\pm 0.035z_0$) for radial and longitudinal forces, respectively.
Fig. 3. Contour plot of radial (a) and longitudinal (b) forces on KH state helium atoms located in $x-z$ plane of a circularly polarized laser field propagating along $z$ direction when laser intensity reaches maximum at $t = 0$. Upper, middle and lower panels are for PM, KH, and total forces, respectively. Laser parameters used are: $I_0=2.8\times 10^{15}\;\mathrm {W/cm}^2$; $w_0$ = 17.5 $\mathrm {\mu} \textrm{m}$; $\lambda$ = 407 nm.
Frequency plays an important role in generating both PM and KH forces. Figure 4(a) exhibits wavelength-dependent radial forces exerted on ground KH state helium atoms located at $x$ axis ($z=0,y=0$) of a laser pulse when laser intensity reaches maximum at $t = 0$. For $\lambda =400$ nm, $F_{PM}^R$ is dominant and much larger than $F_{KH}^R$. With increasing frequency, both PM and KH radial forces decrease since both are explicitly proportional to $\omega ^{-2}$ (see Eq. (11) and (14)). But $F_{PM}^R$ decreases much faster than $F_{KH}^R$. E.g, when wavelength changes from 400 nm to 200 nm, maximum $F_{PM}^R$ decreases from $5.6$ to $1.5 \times 10^{-6}$ a.u., whereas, maximum $F_{KH}^R$ decreases only from 2.0 to $1.6 \times 10^{-6}$ a.u.. The reason is $F_{KH}^R$ has one more factor term of ${\mathrm {d}\varepsilon _{0}}/{\mathrm {d}\alpha _0}$, which in this case increases with increasing frequency. As a result, when $\lambda =200$ nm, $F_{KH}^R$ becomes as large as $F_{PM}^R$ and will become the dominant force if frequency increases further. The maximum $F_{PM}^R$ always appears at half beam waist ($x=\pm w_0/2$) and never changes with frequency, but location of maximum $F_{KH}^R$ is frequency-dependent and moves towards focal axis with increasing frequency. Similar properties of longitudinal forces can be observed for atoms located at $z$ axis, which is plotted in Fig. 4(b).
Fig. 4. (a) Wavelength-dependent radial forces exerted on ground KH state helium atoms located at $x$ axis ($z=0,\;y=0$) of a circularly polarized pulsed laser field when laser intensity reaches maximum at $t = 0$. (b) The same as (a) but for longitudinal forces exerted on ground KH state helium atoms located at $z$ axis ($r=0$). Other laser parameters used are: $I_0=2.8\times 10^{15}\;\mathrm {W/cm}^2$; $w_0$ = 17.5 $\mathrm {\mu} \textrm{m}$. Panels pertain to force types: upper panel for PM force; middle panel for KH force; lower panel for total force.
3.2 Velocity gain by KH atoms
Here we deal with intense lasers with pulse duration less than a few hundreds femtoseconds. In such a short time, the atom’s spacial shift is negligible small, thus we simply assume the atoms do not move during the laser pulse. Then the velocity gain is simply proportional to the momentum transferred to the atoms within laser pulse duration,
where M is the atomic mass, $F$ is the force. $t_0$ is the time when the atom is excited to KH state; $t_\text {end}$ is the time when the atom quits the KH state (either gets ionized or returns to normal state).For acceleration in radial direction, maximally accelerated atoms are those stay in ground KH state for the longest time. For simplicity, we assume the maximally accelerated atoms are in KH state for the whole duration of laser pulse, then maximum radial velocity gain from PM force can be obtained analytically:
whereFig. 5. Contour plot of velocity gained in the radial (a) and longitudinal (b) directions during the laser pulse to a $^4$He atom located in the $x-z$ plane within the laser beam. Upper panel depicts contribution from PM force; middle panel from KH force; lower panel from total force. Laser parameters used are: $I_0=2.8\times 10^{15}\;\mathrm {W/cm}^2$; $w_0$ = 17.5 $\mathrm {\mu} \textrm{m}$; $\mathrm {\tau} _{\textrm{FWHM}}$ = 170 fs; $\lambda$ = 407 nm.
In order to check frequency effect in atomic acceleration, we plot in Fig. 6 the radial velocity gain during the laser pulse to a $^4$He atom at the focal plane ($z=0$) as a function of its radial location and laser wavelength. The range of wavelength (from 200 nm to 400 nm) is the same as shown in Fig. 4. At low frequency ($\lambda =400$ nm), the maximum velocity gain from total force is close to that from the PM force (at $r = w_0/2$), well beyond that from the KH force (at $r=1.7w_0$). Increasing the frequency decreases the PM velocity more rapidly than the KH velocity. For instance, when $\lambda$ decreases from 400 nm to 200 nm, maximum PM velocity decreases from 13 m/s to 3.5 m/s, whereas maximum KH velocity decreases only from 9 m/s to 7 m/s. Also, the radial location of maximum PM velocity remains the same, whereas that of the KH velocity shifts outward markedly. As a result, when $\lambda = 200$ nm, KH velocity becomes dominant and the maximum total velocity appears at $r=1.3 w_0$ instead of $r = w_0/2$.
Fig. 6. Wavelength-dependent velocity gained in the radial direction during the laser pulse to a $^4$He atom as a function of its radial location and laser wavelength. Panel (a) depicts contribution from PM force; (b) from KH force; (c) from total force. Other laser parameters used are: $I_0=2.8\times 10^{15}\;\mathrm {W/cm}^2$; $w_0$ = 17.5 $\mathrm {\mu} \textrm{m}$; $\mathrm {\tau} _{\textrm{FWHM}}$ = 170 fs.
4. Discussion and prospects
This study arises from a set of remarkable experimental works about neutral atom acceleration by focused intense laser beams [35–39]. The basic idea is that neutral atoms with electrons excited to Rydberg states will experience the PM force and result in atomic acceleration, which can be measured experimentally. This renewed our interest in the KH atoms and led us to the proposal of confirming KH atoms with neutral atoms acceleration. KH force is unique to KH atoms, thus compared with Rydberg atoms, KH atoms will have an extra velocity gain during the laser pulse. Since the role of KH force is spacial dependent, if we can measure the velocity gain of the atoms from regions of the laser focusing spot where KH force is dominant, it can be regarded as a direct confirmation of KH atoms. However, it seems difficult to track the accelerated atoms in experiment. For current experimental setup (see Ref. [35] for the details), the most practical way for confirmation of KH atoms is by measuring maximum radial velocity gain in the focal plan. The maximum radial velocity gain of Rydberg atoms and KH atoms are different and this difference can be measured experimentally. Figure 7 plots average acceleration (defined as $\mathrm {\Delta} \textrm{V}^{\textrm{R}}/\mathrm{\tau} _{\textrm{FWHM}}$) along radial direction for both Rydberg atoms (blue) and KH atoms (red) as a function of laser peak intensity $I_0$. The difference between the two is the acceleration from KH force. If the measured acceleration matches that of KH atoms, then it can also be regarded as a direct confirmation of KH atoms. For the purpose of comparison, results of linear polarization are also plotted in Fig. 7. The PM acceleration of circular polarization is the same as that of linear polarization. However, KH acceleration of circular polarization is slightly smaller than that of linear polarization, which makes it even more difficult to detect the extra velocity gain and thus confirm the existance of circularly polarized KH atoms.
Fig. 7. Average acceleration along radial direction, defined as $\mathrm {\Delta} \textrm{V}_{\textrm{PM}}^{\textrm{R}}/\mathrm{\tau} _{\textrm{FWHM}}$ (blue) and $\mathrm {\Delta} \textrm{V}_{\textrm{Tot}}^{\textrm{R}}/{\mathrm{\tau} _{\textrm{FWHM}}}$ (red) as a function of laser peak intensity $I_0$. Solid and dashed lines are for the results of circular and linear polarization, respectively. Note lines of $\mathrm {\Delta} \textrm{V}_{\textrm{PM}}^{\textrm{R}}/{\mathrm{\tau} _{\textrm{FWHM}}}$ for circular polarization overlap with those of linear polarization. Left panel is for $^4$He atoms; right panel is for H atoms. Other laser parameters used are: $w_0$ = 17.5 $\mathrm {\mu} \textrm{m}$; $\lambda$ = 400 nm and 200 nm.
In most experiments about neutral atom acceleration, laser wavelength of $\lambda =800$ nm was used. However for experimental confirmation of KH atoms, higher frequency is preferred, particularly for circular polarization. Otherwise, the maximum velocity gain from KH force will be outweighed by that from PM force, which makes it difficult to distinguish KH atoms from Rydberg atoms. That is the main reason why we stick to laser wavelength between $\lambda =200$ nm and $\lambda =400$ nm, which are relatively high frequencies but still accessible in practice.
In order to enhance the role of KH force thus can be detected by extra velocity gain, lower laser intensity is preferred (see Fig. 7). But KH atoms only exist in strong laser fields therefore laser intensity cannot be too low. Since most neutral atom acceleration experiments were performed with laser intensity around $10^{15} \;\mathrm {W/cm}^{2}$ [34–36] and linearly polarized KH atoms are also believed to exist in this intensity [35,40,41]. Hence an intensity range of $10^{15} \sim 10^{16}$ $\mathrm {W/cm}^{2}$ as shown in Fig. 7, is an appropriate choice for experiment.
We note that, for laser intensities used in this paper, the effect of magnetic field has already been incorporated in the PM model. Since the PM force comes from the quiver motion of the KH state electron and the quiver motion is the combined effect of electric and magnetic field of the laser. That means the magnetic field will not have an extra influence on the quiver motion of the electron so long as the PM model is valid. Actually, the effectiveness and validity of PM model have been studied extensively in literature [45,46]. The PM model becomes invalid only when the speed of the electron is close to the speed of light. Now the intensity range of $10^{15} \sim 10^{16}$ $\mathrm {W/cm}^{2}$ lies well within the nonrelativistic regime, thus the extra effect of magnetic field on the quiver motion of the electron can be neglected.
Our study of atomic acceleration by considering KH force contribution brings out the prospect of experimental confirmation of KH atoms. In a broad context, it also opens up the perspective of enhancing the scope of molecular reactive dynamics by PM and KH forces.
Funding
National Natural Science Foundation of China (11974113); Natural Science Foundation of Shanghai (17ZR1402700).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data that support the findings of this study are available from the corresponding author upon reasonable request.
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