Robust Strategies for Affirming Kramers-Henneberger Atoms

TL;DR

This paper proposes experimental strategies using bichromatic pump-probe techniques and geometric phase analysis to confirm the existence of Kramers-Henneberger atoms, which are deformed atomic states under intense laser fields.

Contribution

It introduces novel experimental methods combining pump-probe schemes and geometric phase analysis to robustly verify KH atom states, accessible with current laser technology.

Findings

Double-slit photoelectron momentum distribution maps KH states.

Characteristic momentum drift observed in tunneling ionization.

Geometric phase approach achieves spin flipping, confirming KH states.

Abstract

Atoms exposed to high-frequency strong laser fields experience the ionization suppression due to the deformation of Kramers-Henneberger (KH) wave functions, which has not been confirmed yet in experiment. We propose a bichromatic pump-probe strategy to affirm the existence of KH states, which is formed by the pump pulse and ionized by the probe pulse. In the case of the single-photon ionization triggered by a vacuum ultra-violet probe pulse, the double-slit structure of KH atom is mapped to the photoelectron momentum distribution. In the case of the tunneling ionization induced by an infrared probe pulse, streaking in anisotropic Coulomb potential produces a characteristic momentum drift. Apart from bichromatic schemes, the non-Abelian geometric phase provides an alternative route to affirm the existence of KH states. Following specific loops in laser parameter space, a complete spin flipping transition could be achieved. Our proposal has advantages of being robust against focal-intensity average as well as ionization depletion, and is accessible with current laser facilities.

Equations

$$i\frac{\partial }{\partial t}\psi (\mathbf{r},t)=\left[\frac{1}{2}{\mathbf{p}}^2+V(\mathbf{r}+\mathbf{\beta })\right]\psi (\mathbf{r},t),$$

$$\Psi (t)=\sum _N{C}_N(t)e^{-i{\int }^{t}d\tau E_N(\tau )}{\varphi }_N({\mathbf{\beta }}_0),$$

$$\begin{array}{c}{\dot{C}}_N={\sum }_M⟨N|\left(-\frac{\partial {\mathbf{\beta }}_0}{\partial t}\frac{\partial }{\partial {\mathbf{\beta }}_0}-i{H}_I\right)|M⟩\\ \times e^{-i{\int }^{t}d\tau (E_M-E_N)}{C}_M.\end{array}$$

$$⟨N|\frac{\partial }{\partial {\mathbf{\beta }}_0}|M⟩=⟨N|\frac{\partial H_0}{\partial {\mathbf{\beta }}_0}|M⟩/(E_M-E_N),$$

$$|\pm \left(\theta ,\varphi \right)⟩=\exp (-i{J}_z\varphi )\exp (-i{J}_y\theta )|\pm \left(\theta =0,\varphi =0\right)⟩.$$

$$\begin{array}{c}{\mathfrak{A}}_{\theta }=\frac{1}{2}\left(\begin{array}{cc}0& -i\kappa \\ i\kappa & 0\end{array}\right),{\mathfrak{A}}_{\varphi }=\frac{1}{2}\left(\begin{array}{cc}\cos \theta & -\sin \theta \kappa \\ -\sin \theta \kappa & -\cos \theta \end{array}\right).\end{array}$$

$$W=\left(\begin{array}{cc}\cos \left(\frac{\kappa {\varphi }_0}{2}\right)+i\sin \left(\frac{\kappa \pi }{2}\right)\sin \left(\frac{\kappa {\varphi }_0}{2}\right)& -i\cos \left(\frac{\kappa \pi }{2}\right)\sin \left(\frac{\kappa {\varphi }_0}{2}\right)\\ -i\cos \left(\frac{\kappa \pi }{2}\right)\sin \left(\frac{\kappa {\varphi }_0}{2}\right)& \cos \left(\frac{\kappa {\varphi }_0}{2}\right)-i\sin \left(\frac{\kappa \pi }{2}\right)\sin \left(\frac{\kappa {\varphi }_0}{2}\right)\end{array}\right).$$

$$\begin{array}{c}\frac{\partial C_N}{\partial {\mathbf{\beta }}_0}\approx -{\sum }_{E_M=E_N}⟨N|\frac{\partial }{\partial {\mathbf{\beta }}_0}|M⟩C_M\end{array}$$

Full text

††preprint: APS/123-QED

Modern light-matter interaction researches date back to Einstein’s explanation on the photoelectric effect, in which ionization happens only if the absorbed photon energy is larger than the binding energy. The advent of laser technologies has boosted light-matter interaction researches into new eras, where novel nonperturbative phenomena are discovered, for examples, strong-field tunneling ionization TI07 , above threshold ionization ATI , high-harmonic generation HHG1 ; HHG2 ; HHG3 , nonsequential double ionization NSDI , low energy structures LES1 ; LES2 , and photoelectron holography holo . Among these fascinating scenarios, stabilization of atoms in intense laser fields, i.e., the counterintuitive decreasing of ionization probability with the increasing of driving laser intensities, attracts attention of the ultrafast community Gav02 ; stabs02 ; Richter .

Two mechanisms are known for stabilization. One is interference stabilization sta1 , in which the released electron wave packets from populated Rydberg states interfere destructively. The other is adiabatic stabilization, in which the multiphoton ionization is suppressed due to the deformation of Kramers-Henneberger (KH) wave functions Pont ; stab1 , which are defined to be the eigenstates of a time-averaged Hamiltonian KH0 .

Though theoretically predicted for decades, the experimental confirmation of adiabatic stabilization is obscure due to ionization depletion and the focal-intensity average of lasers. In real experiments, the fine structure related to the stabilization may be smeared out after integrating all ionized fragments driven by different laser intensities. Furthermore, while the field strength in the focused center reaches the threshold of stabilization, the lower intensity around the focusing spot may completely ionize the target. The target might also be completely ionized before the laser field reaches its peak intensity in the time domain DV . Up to now, there is only tantalizing indirect experimental evidence for the adiabatic stabilization. For example, in Ref. evidemce , a large acceleration of neutral atoms was reported and regarded as a signal of stabilization Wei . However, this evidence is not convincing enough as frustrated ionization FTI , in which the ionized electrons get recaptured by the parent nuclei, has a similar output.

There are vast researches on adiabatic stabilization Gav02 ; stabs02 ; Richter . However, only a few attempted to directly identify KH states. Kulander et al. suggested that the appearance of the even order of high-harmonic generation KHHHG is a manifest of KH states. Morales et al. identified specific fine structures in photoelectron momentum distribution contributed by excited KH states Smirnova . Jiang et al. suggested that the photoelectron momentum distribution carrying dynamical interference structures provides information on adiabatic stabilization Jiang . However, these proposals are sensitive either to the laser intensity or to the pulse envelope and are not robust against ionization depletion. Thus, the experimental realization is still challenging.

In this letter, we proposed to detect KH states using a bichromatic pump-probe strategy, in which the KH state is formed by the pump pulse and ionized by the probe one. By detecting the photoelectron momentum distribution, one is able to extract the dichotomic structure of the target, and thereby affirm the existence of KH states. The spin flipping for atoms following a loop in the laser parameter space provides an alternative route.

Our start point is the three-dimensional time-dependent Schrödinger equation (TDSE) in the KH frame KH0 (atomic units are used throughout unless stated otherwise)

[TABLE]

where $\boldsymbol{\beta}$ is the time-dependent electron displacement $\boldsymbol{\beta}=\boldsymbol{\beta}_{0}\sin(\omega_{0}t)$, with $\boldsymbol{\beta}_{0}=\beta_{0}f(t)\mathbf{e}_{x}$ and $\beta_{0}=\frac{E_{0}}{\omega^{2}}$ if the driving laser field is linearly polarized. The corresponding laser field is given by $\mathbf{E}(t)=-\frac{\partial^{2}}{\partial t^{2}}\boldsymbol{\beta}$. We use the envelope $f(t)=\cos^{2}(\pi t/L)\ (-L/2<t<L/2)$ throughout this paper, where $L$ stands for the pulse duration. The ground state of Eq. (1) is obtained using the imaginary time method imag , and the split-operator method is adopted to propagate the wave function in real time. By Fourier transforming the ionized wave function, we obtained the photoelectron momentum distribution. Using the hydrogen atom as the prototypical target, we calculated the ionization probability as a function of $\beta_{0}$, as shown by the black solid line in Fig. 1 (a). Here, the laser pulse has a frequency of $\omega_{0}=3$ a.u., and a total duration $L$ of sixty cycles. The “death valley” DV structure is clearly shown.

Researches on different aspects of high-frequency-laser ionization scatter in references dich ; omg2 ; KHT2 ; DI1 ; DI2 ; Rost17 ; Jiang2 ; Toyota ; Rost18 ; Wei18 , for our purpose here, we summarize the main conclusions with a special emphasis on the role played by KH states. We expanded the oscillating Coulomb potential as $V(\textbf{x}+\boldsymbol{\beta})\approx\sum_{\text{n}}V_{\text{n}}(\textbf{x};\boldsymbol{\beta}_{0})e^{-i\text{n}\omega_{0}t}$ enve ; Rost15 . $V_{0}(\textbf{x};\boldsymbol{\beta}_{0})$ provides a laser-dressed adiabatic potential, while the nonzero harmonic components $V_{\text{n}}(\textbf{x};\boldsymbol{\beta}_{0})$ induce photon absorption/emission. The Hamiltonian in Eq. (1) can now be regrouped into two parts, i.e., the adiabatic term $H_{0}=\frac{\textbf{p}^{2}}{2}+V_{0}(\textbf{r};\boldsymbol{\beta}_{0})$ and the remaining part $H_{I}=\sum_{\text{n}}V_{\text{n}}(\textbf{x};\boldsymbol{\beta}_{0})e^{-i\text{n}\omega_{0}t}$. The time-dependent electron wave packet can be expanded as

[TABLE]

in which $\phi_{N}(\boldsymbol{\beta}_{0})$ is the instantaneous eigenstate of $H_{0}$ and satisfies the governing equation $H_{0}\left|\phi_{N}(\boldsymbol{\beta}_{0})\right\rangle=E_{N}(\boldsymbol{\beta}_{0})\left|\phi_{N}(\boldsymbol{\beta}_{0})\right\rangle$. Inserting Eq. (2) into the Schrödinger equation yields

[TABLE]

The term $\left\langle N\right|H_{I}\left|M\right\rangle$ is responsible for photons absorption/emission, and the skew-hermitian matrix $\left\langle N\right|\frac{\partial}{\partial\boldsymbol{\beta}_{0}}\left|M\right\rangle$ provides the nonadiabatic coupling KHT2 ; Toyota ; Rost18 and the geometric phase geo0 . As indicated by Eq. (3), KH states are of central importance here. The deformation of the KH wave function $\phi_{N}(\boldsymbol{\beta}_{0})$ leads to the suppression of $\left\langle N\right|V_{\text{n}}\left|M\right\rangle$, which is the fundamental reason for adiabatic stabilization Pont ; stab1 . The phase accumulation due to the distorted KH state Wei leads to the dynamic interference DI1 ; DI2 ; Rost17 ; Jiang2 . Furthermore, KH states determine the strength of nonadiabatic coupling $\left\langle N\right|\frac{\partial}{\partial\boldsymbol{\beta}_{0}}\left|M\right\rangle$.

With Eq. (3) in hand, we explored scenarios of ionization shown in Fig. 1 by dividing the ionization probability curve into three stages marked by A, B, and C. In stage A, $\beta_{0}$ is small and so is the changing rate $\frac{\partial\boldsymbol{\beta}_{0}}{\partial t}$, which means the deformation of KH wave functions and the nonadiabatic coupling are negligible. We extracted the single-photon ionization fragment from the total ionization spectra and presented it by the red dashed curve in Fig. 1. The single-photon-ionization probability overlaps with the total ionization probability as $\beta_{0}<1$ a.u., which suggests that the ionization can be well described by the conventional first-order perturbation theory and the nonadiabatic coupling is negligible. In stage B, the total ionization probability decreases due to the significant deformation of KH wave functions. The norm of $\left\langle N\right|V_{\text{n}}\left|M\right\rangle$ is suppressed with an increasing $\beta_{0}$. The dichotomic characteristic of KH states, i.e., the dimensionless number $\frac{Z}{\beta_{0}I_{p}}$, serves as a measure of the deformation of wave functions. For the ground state hydrogen atom, the nuclear charge $Z=1$ a.u. and the ionization energy $I_{p}=0.5$ a.u.. $\frac{Z}{\beta_{0}I_{p}}\approx 1$ roughly corresponds to the point where the second order derivative of the laser-dressed ground state energy curve vanishes supp . $\frac{Z}{\beta_{0}I_{p}}<1$ implies the deformation of the wave function is significant. In stage C, the nonadiabatic ionization becomes more and more important KHT2 ; Toyota ; Rost18 . There is no distinct boundary between B and C. The nonadiabatic coupling is determined by the product of $\left\langle N\left|\frac{\partial}{\partial\boldsymbol{\beta}_{0}}\right|M\right\rangle$ and $\frac{\partial\boldsymbol{\beta}_{0}}{\partial t}$, which implies an envelope dependence. Similar to the excited-state tunneling ionization STI , in the nonadiabatic ionization, the atom first transits to excited states, which are then mediated to continuum states. The effective ionization potential decreases with the increasing of $\beta_{0}$, as shown in the supplementary file supp , and thus the nonadiabatic ionization is more important for a larger $\beta_{0}$. Ionization from excited states dominates since the ionization potential gets smaller in this situation.

With these understandings about the central role played by KH states in the $\omega_{0}\gg I_{p}$ regime, we now make the following proposal to experimentally affirm KH states. The strategy is basically a pump-probe scheme: a linearly or circularly polarized high-frequency strong laser pulse is used to irradiate on a prototypical hydrogen atom, which is to be ionized by another circularly polarized probe pulse. When the electron is released from the KH hydrogen atom, who plays the role of a double-slits, the photoelectron momentum distribution will inherit the double-slit interference structure. In principle, this proposal can already be performed on advanced laser facilities LF1 ; LF2 ; LF3 . In this strategy, the probe pulse should be strong enough to trigger noticeable ionization, but not so strong that the formed KH state gets destroyed. This imposes a constraint $I_{2}\ll(\frac{\omega_{2}}{\omega_{1}})^{4}I_{1}$, where $I_{1}$ and $I_{2}$ are intensities of the pump and probe pulses. Note that the subscript 0 is preserved for the case of using only one pulse. Besides that, laser frequencies of the pump and probe pulses should be proper so that the photoelectron momentum distributions induced by the pump and probe pulses do not overlap. $\omega_{2}$ should be sufficiently large to avoid interfering with very low energy electron produced by the nonadiabatic coupling KHT2 ; Toyota .

The upper row of Fig. 2 shows the above-threshold-ionization (ATI) containing the fragments released by absorbing $n\omega_{1}$ photons and $m\omega_{2}$ photons, where $n$ and $m$ are integers. Though the probability of absorbing $\omega_{2}$ is small due to the relatively weak intensity of the probe pulse, the ionization by the probe pulse contributes distinguished angular distribution and non-overlapping photoelectron energy with the ionization fragments induced by the pump pulse. Thus one can easily separate one-probe-photon ionization from the dominating pump-photon ionization, as shown in the lower row in Fig. 2. The laser parameters for the three columns are presented in the caption. The ionization amplitude of the formed KH states, with the field parameters used in (d), is proportional to $J_{n}(p_{x}A_{1}/\omega_{1})J_{m+1}(\sqrt{p_{x}^{2}+p_{y}^{2}}A_{2}/\omega_{2})+J_{n}(p_{x}A_{1}/\omega_{1})J_{m-1}(\sqrt{p_{x}^{2}+p_{y}^{2}}A_{2}/\omega_{2})$, where $J_{n}$ is the $n$-th order Bessel function supp . All panels in the lower row present clearly angular nodal structures, which are the manifestation of double-slit interference DoubleS . The number of nodes is determined by $\beta_{1}\sqrt{2E_{\textbf{k}}}$ with $E_{\textbf{k}}$ the photoelectron energy. Inversely, $\beta_{1}$ can be extracted from the angular distribution of the photoelectron induced by the one-probe-photon ionization. The comparison of (a), (b) and (c) shows that a larger $\omega_{1}$ is more convenient for separating the ionization events from the pump and probe pulses. A larger frequency $\omega_{1}$ is also better for avoiding the ionization depletion omg .

The unavoidable focal-intensity average in real experiments must be taken into account to judge the feasibility of the above proposal. By assuming that the intensity of the laser pulses has the spatially Gaussian distribution intavg , we plotted the focal-intensity-averaged photoelectron angular distribution in Fig. 3. The frequencies are $\omega_{1}=1$ a.u. and $\omega_{2}=3.5$ a.u., corresponding to the parameters used in Fig. 2 (a)(d). The laser intensities used for the three columns from left to right are $I_{1}=3.5\times 10^{16}$ W/cm2 ( $\beta_{1}=1$ a.u. ), $I_{1}=1.4\times 10^{17}$ W/cm2 ( $\beta_{1}=2$ a.u. ), and $I_{1}=3.2\times 10^{17}$ W/cm2 ( $\beta_{1}=3$ a.u. ), respectively. The photoelectron from unperturbed hydrogen atoms should be rotational invariant in the laser polarization plane, hence, the anisotropic ionization probability, as shown in all these panels, confirms the existence of dichotomic distribution of the KH hydrogen atom, which in turn provides the evidence of adiabatic stabilization. The onset of adiabatic ionization implies that ionization is mainly contributed from small $\beta_{1}$ supp , which explains why interference structures in photoelectron momentum distributions are not as distinct as that in Fig. 2. As the number of nodes is determined by $\beta_{1}\sqrt{2E_{\textbf{k}}}$, a larger $\omega_{2}$ is favored to produce distinctive angular distributions. This strategy works for diverse laser parameters and is robust against focal-intensity average. Moreover, the pump-probe delay can be tuned thus the probe pulse can contribute noticeable ionization before the KH atom is depleted.

Instead of the single-photon ionization triggered by the high-frequency probe pulse, the KH atom may be tunneling ionized by an infrared probe pulse. Figure 4 shows the focal-volume-averaged photoelectron momentum distributions for the probe laser intensity (a) $I_{2}=3\times 10^{14}$ W/cm2 and $\beta_{1}=2$ a.u. and (b) $I_{2}=2\times 10^{14}$ W/cm2 and $\beta_{1}=4$ a.u.. Here, He+ in the KH state is prepared and used as the target, and the two cases have the same Keldysh parameter. In contrast to the one-probe-photon ionization, signals from large $\beta_{1}$ are dominating in the tunneling regime supp .

The streaking of the photoelectron momentum distribution in the anisotropic Coulomb field produces a tilt angle, which is a function of $\beta_{1}$ and $I_{2}$ tilt1 ; tilt2 . The existence of $\beta_{1}$ can thus be mapped into the streaking tilt angle. We point out that it is also possible to use the laser induced electron diffraction LIED1 ; LIED2 or charge resonance enhanced ionization CRI to reconstruct $\beta_{1}$ supp .

Besides utilizing the double-slit interference structure and the anisotropic angular streaking, the electron spin-flipping provides another route to affirm KH atoms. The physical principle is based on the non-Abelian geometric phase. Neglecting the ionization of KH atoms for a moment, the nonadiabatic coupling among those non-degenerate states, i.e.,

[TABLE]

can be suppressed in the adiabatic limit. However, situations are different for systems with energy degeneracies, where non-Abelian geometric phases play a role geo1 . Denoting the polarization axis of the laser pulse by $\textbf{n}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$, we have degenerate KH states $\left|\pm\left(\theta,\phi\right)\right\rangle$ satisfying $\textbf{n}\cdot\mathbf{J}\left|\pm\left(\theta,\phi\right)\right\rangle=\pm 1/2\left|\pm\left(\theta,\phi\right)\right\rangle$, where $\mathbf{J}$ is the addition of orbital angular momentum $\mathbf{L}$ and spin angular momentum $\mathbf{S}$. $\left|\pm\left(\theta,\phi\right)\right\rangle$ are connected to $\left|\pm\left(\theta=0,\phi=0\right)\right\rangle$ via rotations

[TABLE]

Components of non-Abelian connection 1-form $\mathfrak{A}=i\left\langle M|\textbf{d}|N\right\rangle$, where d is exterior derivative, are thus given by

[TABLE]

Here $\kappa=2\left\langle+\left(\theta=0,\phi=0\right)\right|J_{x}\left|-\left(\theta=0,\phi=0\right)\right\rangle$ is a real valued function of $\beta_{0}$, up to a trivial phase factor. We have $\kappa=1$ when $\beta_{0}\rightarrow 0$. In the limit $\beta_{0}\rightarrow\infty$, $\kappa=0$ if we choose $\left|+\left(\theta=0,\phi=0\right)\right\rangle=\left|L_{z}=1,S_{z}=-1/2\right\rangle$ and $\left|-\left(\theta=0,\phi=0\right)\right\rangle=\left|L_{z}=-1,S_{z}=1/2\right\rangle$. Such a choice is possible here, as the off-diagonal elements of spin-orbital coupling matrix vanish.

Consider the situation that the laser field is linearly polarized along the $z$ axis. One adiabatically rotates the polarization axis by $\theta_{0}=\pi/2$ in the $z-x$ plane, then rotates it by $\phi_{0}$ in the $x-y$ plane, and finally rotates it back to the $z$ axis. The holonomy $W=P\left\{e^{i\oint{\mathfrak{A}}}\right\}$ for such a closed path is

[TABLE]

By setting $\phi_{0}=\pi/\kappa$, we have the spin flipping transition amplitude $-i\cos\left(\frac{\kappa\pi}{2}\right)$, which is zero when $\kappa=1$ and maximal when $\kappa$ approaches zero.

The deformation of KH states is crucial here. When the laser field is not very strong, i.e. $\beta_{0}\ll 1$ a.u., the dynamical evolution of the TDSE is determined by the dipole coupling matrix $\left\langle N\right|\textbf{r}\left|M\right\rangle$. Typically, only a minor fraction of spin flipping could be achieved nondipole . However, with an increasing $\beta_{0}$, couplings with the highly oscillating laser pulses are suppressed, which reduces Eq. (3) into

[TABLE]

in the adiabatic limit.

The spin flipping in KH atom due to the adiabatically rotating electric field is isomorphic to that of in diatomic molecule due to the rotating molecule axis geo2 . The nontrivial value $\kappa\neq 1$ manifests the breaking of atomic isotropic symmetry thus the existence axial symmetric KH atom. In this strategy, ensuring the adiabaticity implies that $\frac{\partial\boldsymbol{\beta}_{0}}{\partial t}$ should be small. Therefore, a pulse with a very long duration is demanded. In this case, in order to avoid ionization depletion, a laser pulse with a large $\omega_{0}$ is required.

To summarize, a pump-probe scheme is suggested to detect the KH state. The pump pulse creates a KH atom, whose dichotomic structure is imprinted on the photoelectron momentum distribution. This strategy is robust against the focal-intensity average and ionization depletion. Alternatively, the spin flipping induced by the non-Abelian geometric phase in the adiabatically changing laser field can also provide the evidence of KH atoms. An ideal implementation of our proposal requires $\frac{\omega_{1}}{I_{p}}>1$ and $\frac{Z}{\beta_{1}I_{p}}<1$. In our strategy, the required laser intensity is about $I_{1}=10^{16}$ – $10^{17}$ $\text{W}/\text{cm}^{2}$, and the laser wavelength is around 10 – 50 nm, which is within the reach of current laser facilities. Relativistic and quantum electrodynamics effects QED1 ; QED2 are negligible for the considered laser parameters. Moreover, we are concerned only about one-probe-photon ionization from KH states, thus the deviation from relativistic theory mainly differs by scaling factors Dirac . We observed that the low energy electron ionized by the nonadiabatic coupling is altered by the nondipole effect Smeenk ; Ludwig . However, this does no harm to our proposal due to the non-overlapping energy. Our schemes not only provide accessible routes for detecting KH states, thus adiabatic stabilization, but are also useful for understanding up-coming high frequency strong laser-matter interaction.

This work was supported by National Key R&D Program of China (2018YFA0404802), Innovation Program of Shanghai Municipal Education Commission(2017-01-07-00-02-E00034), National Natural Science Foundation of China (NSFC) (Grant No. 11574205, 11721091, 91850203), and Shanghai Shuguang Project (17SG10). Simulations were performed on the $\pi$ supercomputer at Shanghai Jiao Tong University.