Subcycle resolved strong-field tunneling ionization:
Identification of magnetic dipole and electric quadrupole effects

Tunneling ionization is the first step of many strong-field phenomena and is one of the cornerstones of strong-field and attosecond physics. Tunneling ionization from atoms and molecules is typically described within the electric dipole approximation, where the laser field is considered homogeneous along its propagation direction, thereby neglecting the photon momentum and the field retardation [1]. The dipole approximation holds well for typical parameters of experimental table-top laser setups. In turn, nondipole effects, which result in asymmetric photoelectron momentum distributions about zero along the field propagation direction, are typically very small.

With advances in the detection technology, such weak nondipole effects have recently become accessible [2, 3]. In 2011, Smeenk et al. [4] experimentally observed the linear momentum transfer for tunneling ionization of argon and neon. Since then, nondipole tunneling effects have attracted considerable attention. It has been found to cause a negative shift in the photoelectron momentum distribution for linear polarization due to its interplay with the Coulomb field [5]. Remarkably, for nonsequential double ionization, the sum of the nondipole momentum shifts of the two electrons is opposite to the case of single ionization and considerably larger [6]. The nondipole momentum shift follows the prediction of the classical model for low photoelectron energies [7], while for high-order above-threshold ionization, the nondipole momentum shift is found to be substantially modified by large-angle rescattering of the electron [8, 9]. Nondipole effects also play an observable role during the under-barrier tunneling ionization [10, 11]. Moreover, nondipole effects induce a modification to the ponderomotive potential thereby shifting the center of the energy rings in the above-threshold ionization [12, 13, 14]. In 2019, the pioneering experimental work of Willenberg et al. [15] studied the subcycle linear momentum transfer with an attoclock protocol [16, 17], and found a counterintuitive local minimum of the transferred momentum at the peak of the laser electric field. Based on earlier theories of nondipole effects [18, 20, 21, 19, 22, 24, 23, 25, 26], we have subsequently developed the theory of subcycle linear momentum transfer [27], which accounts for the interplay between nondipole and nonadiabatic tunneling effects on the sub-optical-cycle time scale.

Different contributions to the linear momentum transfer by nondipole effects can be distinguished. In the time domain, the momentum transfer $\langle p_{z}\rangle$ along the laser propagation direction ($\hat{z}$) can be partitioned as [28, 27]

with $\langle v_{z}\rangle$ the momentum transfer due to tunneling ionization and $\Delta E/c$ the contribution due to the subsequent continuum motion after tunneling with $\Delta E$ denoting the energy gain during the continuum excursion of the electron and $c$ representing the speed of light. An alternative partitioning can be achieved in the energy domain by exploring the influence of different terms of nondipole Hamiltonian to a given order $\alpha$ ($\alpha$ is the fine structure constant) [29, 30, 31] on the linear momentum transfer. Recently, the time-integrated contributions of the electric field gradient and the magnetic field to the final momentum distribution in tunneling ionization have been studied [32].

In the present article, we aim at combining the time domain and spectral domain analysis in order to pinpoint the influence of individual nondipole terms in the Hamiltonian on the subcycle time-resolved momentum transfer within a typical attoclock protocol. Using an elliptically polarized laser pulse, the electrons emitted at different instances of time within an optical cycle are emitted into different directions and backscattering is suppressed, which facilitates the subcycle time resolution of nondipole-induced momentum transfer. In the present study, we consider two frequently employed gauges, the multipole gauge and the radiation gauge. The multipole gauge allows to identify the contributions of the magnetic dipole and electric quadrupole interactions to leading order $1/c$ in the retardation. The radiation gauge, on the other hand, reveals separate contributions linear and quadratic in the laser field to the same order $1/c$. While in exact calculations the results should be invariant under changes of the gauge, approximate treatments may be sensitive to the choice of gauge. A well-known case in point are dipole transitions in weak-field photoionization. Depending on the gauge employed (length, velocity, accelerations), different spatial regions of the wave functions can be sensitively probed. Here, we extend such an analysis to the nondipole strong-field regime. We show that the electric quadrupole (E₂) coupling primarily probes the field-driven continuum motion which, overall, provides the dominant contribution to the linear momentum transfer while the tunneling process is controlled by the magnetic dipole (M₁) coupling, which provides a minor correction for the present parameter settings.

The article is organized as follows. In Sec. II, we briefly review the theoretical framework within which we treat the partitioning of the leading-order nondipole corrections in the Hamiltonian for atom-laser interaction. In Sec. III, the subcycle contributions originating from each nondipole Hamiltonian term are analyzed and approximate analytic expressions within the framework of the strong-field approximation (SFA) are given. We present results for the subcycle contributions of the different nondipole interactions in Sec. IV. The time integral momentum transfer as a function of the laser parameters is explored in Sec. V. Conclusions are given in Sec. VI. Atomic units are used throughout unless noted otherwise.

In this section, we apply the ndSPA and the ndSPANE, as outlined above to the subcycle linear momentum transfer with the goal to identify the contributions from individual nondipole terms in the Hamiltonian [Eqs. (5) and (9)]. We first demonstrate the evaluation for full $H_{\rm ND}$ in the radiation gauge [Eq. (5)]. The analogous evaluation for the alternative decompositions of $H_{\rm ND}$ are given in the Appendix.

We first apply a gauge transformation to Eq. (5) with gauge function

which results in [8, 7, 27]

This gauge transformation implies a time-dependent shift of the origin in the nondipole frame similar to the Kramers-Henneberger gauge. Even when $V(\bm{r})$ is eventually neglected in the SFA, such a frame transformation is needed to retrieve quantities in the lab frame from results obtained in this nondipole frame. The corresponding action entering the ndSPA is

and the saddle-point equation reads

With the substitution $\bm{k}_{\perp}=\bm{p}_{\color[rgb]{0,0,0}\perp}+{\rm Re}\bm{A}(t_{s})$, the imaginary part of the saddle-point equation becomes

Therefore, by choosing the auxiliary perpendicular momentum $k_{\perp}$ in the polarization plane as

the imaginary part of the saddle-point equation [Eq. (25)] is automatically fulfilled for arbitrary pulse shapes. The relation between the auxiliary momentum $k_{\perp}$ [Eq. (27)] and the transverse momentum at the tunnel exit $v_{\perp}$ will be explored below. Expressing the momentum differential ionization probability $W_{\rm ndSPA}(\bm{p})$ [Eq. (14)] in terms of the coordinates ($t_{r}$, $k_{\perp}$, $p_{z}$), the time-resolved initial linear momentum $\langle v_{z}(t_{r})\rangle$ is obtained by evaluating Eqs. (17) and (19), and the final linear momentum $\langle p_{z}(t_{r})\rangle$ follows from Eq. (20).

Analytic expressions for $\langle v_{z}\rangle$ and $\langle p_{z}\rangle$ can be obtained when keeping only lowest-order nonadiabatic corrections using ndSPANE. Inserting $\bm{A}(t_{r}+it_{i})$ [Eq. (21)] into the saddle-point equation [Eq. (16)] and keeping terms up to second order in $t_{i}$ results in the relationship

i.e., the transverse momentum $\bm{k}_{\perp}$ is orthogonal to the attoclock field at the instant of tunneling ionization $t_{r}$. Eq. (28) defines the temporal location $t_{r}$ of the tunnel exit separating tunneling from continuum motion. Remarkably, this condition remains unchanged whether or not nondipole effects are included. For later reference, we note that $k_{\perp}$ determined through Eq. (27) may vary when the ndSPA or ndSPANE are used. These deviations do not significantly affect the numerical results. The real part of the saddle-point equation leads to

with the effective field $\widetilde{F}(t_{r})=\sqrt{F^{2}(t_{r})-\bm{k}_{\perp}\cdot\dot{\bm{F}}(t_{r})}$. The above expression of $t_{i}$ explicitly defines an effective Keldysh parameter $\gamma_{\mathrm{eff}}=\omega t_{i}$, which should be small in the tunneling regime.

The relation between $\bm{k}_{\perp}$ and the momentum at the tunnel exit $(\bm{v}_{\perp},v_{z})$ follows from the Heisenberg’s equations of motion for $H_{\mathrm{ND}}^{\prime}$ [Eq. (23)]

In addition, the gauge transformation [Eq. (22)] implies the frame transformation [Eq. (23)]

Accordingly, we find for the velocities

Using $z\approx 0$ at the tunnel exit, we have

We note that Eq. (35) enters the expression in Eq. (19) (dropping the subscript “lab”).

Ionization rate $\tilde{w}_{\rm ndSPA}$ [Eq. (17)] is now obtained by evaluating the exponential factor

the preexponential prefactor

and the Jacobian

where $F_{d}=k_{\perp}\left[F_{x}(t_{r})F_{y}^{\prime}(t_{r})-F_{x}^{\prime}(t_{r})F_{y}(t_{r})\right]/F^{2}(t_{r})$.

Rearranging these expressions, we find

where

Hence, the mean initial linear momentum in the lab frame can be rewritten as

where we have dropped the subscript “lab”, and the corresponding asymptotic linear momentum follows as

with the ponderomotive energy gain $\Delta E$,

The derivation of analogous expressions for $\langle p_{z}\rangle$ and $\langle v_{z}\rangle$ predicted by the various approximate Hamiltonian operators [Eqs. (10–13)] containing selected nondipole terms is given in Appendix B to E.

The results of the expectation value of the linear momentum in the lab frame for different Hamiltonian are summarized in Table 1 in the Appendix, where we present the asymptotic linear momentum transfer $\langle p_{z}(t_{r},v_{\perp})\rangle$ [Eq. (42)], which is composed of the initial linear momentum at the tunnel exit $\langle v_{z}(t_{r},v_{\perp})\rangle$ [Eq. (41)] and $\Delta E/c$ for the continuum motion after tunneling. From Table 1, it is obvious that

i.e., the sum of the nondipole contributions of the reduced Hamiltonians adds up to that of the full nondipole Hamiltonian within each gauge. This underscores the suitability of the decomposition into reduced Hamiltonians for the observables under consideration.

In this section, we quantitatively investigate and illustrate the effects of individual nondipole Hamiltonian terms, most prominently of the magnetic dipole (${\rm M}_{1}$) and electric quadrupole (${\rm E}_{2}$) contributions for tunneling ionization for helium. Within SFA helium is effectively represented by a zero-range potential with ionization potential $I_{p}=0.903570$. For comparative numerical solutions of the TDSE we use model potential with a Coulomb tail or a short-ranged Yukawa-type potential (see Appendix A).

For the simulation we use a laser pulse with a vector potential

with a number $N$ of cycles, $N=6$, a carrier-envelope phase $\phi_{\mathrm{CEP}}=0$, pulse ellipticity $\epsilon=0.75$, angular frequency $\omega=0.057$ corresponding to a central wavelength of 800 nm, and the vector potential amplitude $A_{0}$ corresponding to a laser peak intensity of $5\times 10^{14}$ W/cm². For these laser parameters, the Keldysh parameter is $\gamma\approx 0.8$ corresponding to the (moderate) tunneling regime. In all figures that follow, solid lines represent the results calculated using the ndSPA method and the dashed lines stand for those obtained by the ndSPANE method.

Fig. 1 shows the subcycle time-resolved linear momentum at the tunnel exit $\langle v_{z}(t_{r})\rangle$ caused by various nondipole Hamiltonian terms. Within the multipole gauge [Fig. 1(a)], we find the remarkable results that the magnetic dipole interaction (M₁) strongly dominates the linear momentum transfer during the under-barrier motion while the electric quadrupole interaction (E₂) leaves only an almost negligible footprint on the longitudinal momentum at the tunnel exit. For illustrative purposes we also give the corresponding results for the nondipole contributions of the field-linear (F₁) and field-quadratic (F₂) terms. Here, we find strong oscillations with opposite sign that result in near cancellation of the two contributions. This observation clearly indicates that a decomposition according to the order in coupling with the laser field, unlike the multipole decomposition, does not provide a well-suited starting point for further approximations as both orders are responsible for contributions of comparable magnitude to order $1/c$. This is closely related to the fact that the gauge transformation [Eq. (6)] contains field coupling to all orders thereby transforming the Hamiltonian in the radiation gauge [Eq. (5)] which contains terms of first and second order in the field into the multipole Hamiltonian [Eq. (9)], which is strictly linear in the field. We also note that the analytic ndSPANE results for all gauges (Fig. 1) slightly differ from the corresponding full ndSPA calculations on the subcycle level due to the neglect of higher-order corrections in the nonadiabatic expansion $\sim t_{i}^{n}$ ($n\geqslant 3$) [Eq. (21)]. Nevertheless, after summation over the corresponding terms within each gauge, the results closely agree with each other.

Turning now to the asymptotic linear momentum $\langle p_{z}\rangle$, which is directly experimentally accessible, we display in Fig. 2 the dependence on both the release time $t_{r}$ (left column) and on the attoclock angle $\phi_{p}=\arctan(p_{y}/p_{x})$ (right column). Most remarkably, while the electric quadrupole term provides only a negligible contribution to the longitudinal momentum transfer during tunneling ionization (Fig. 1), it dominates over the M₁ contribution in the post-tunneling continuum momentum transfer [Fig. 2(a)]. Moreover, the asymptotic momentum transfer originates predominantly from the field-driven motion in the continuum. The $2\omega$ subcycle oscillations of the momentum transfer during tunneling mediated by the M₁ interaction [Fig. 1(a)] is in phase with the momentum transfer in the continuum, mostly caused by the E₂ term [Figs. 2(a,b)]. By contrast, for the decomposition into the field-linear (F₁) and field-quadratic (F₂) terms of the radiation gauge [Figs. 2(c,d)], the two contribution terms are out of phase by $\pi$. Interestingly, the F₂ nondipole term induces a momentum transfer due to the nondipole force $-\left(\bm{A}\cdot\bm{F}\right)/c$ opposed to that of the laser propagation. Time integration of this force leads to a negative momentum transfer of $-A^{2}(t_{r})/2c$. Regardless of partitioning, summation of the respective individual contributions matches the result of the full nondipole Hamiltonian. Here again, the ndSPANE results agree well with full ndSPA calculations.

In this section, we present results for the laser parameter dependence of the time-integrated linear momentum transfer induced by individual nondipole Hamiltonian terms both at the tunnel exit $\langle v_{z}\rangle$ and in the asymptotic region $\langle p_{z}\rangle$ via integration of Eqs. (19) and (20) over $t_{r}$ covering the time span of the entire laser pulse.

Fig. 3(a) (upper row) shows the dependence of $\langle v_{z}\rangle$ on the laser ellipticity $\epsilon$ (column 1), wavelength $\lambda$ (column 2), and peak intensity $I_{0}$ (column 3). Fig. 3(b) (lower row) shows the corresponding dependences for $\langle p_{z}\rangle$. In order to demonstrate the level of accuracy reached by the ndSPA for this observable we also present numerical TDSE results for $\langle p_{z}\rangle$ as a function of $\lambda$ in Fig. 3(b2). Overall, the under-barrier momentum transfer $\langle v_{z}\rangle$ is remarkably insensitive to the variation of laser parameters, also the partitioning into M₁ and E₂ changes only marginally with intensity and wavelength which remains fixed when varying the ellipticity. Throughout, the M₁ term strongly dominates over the E₂ term. However, when separating the F₁ and F₂ contributions within the radiation gauge, strong variations as a function of $\epsilon$ can be observed underscoring that this decomposition is not well suited as a starting point for further approximations based on the order of the atom-field interaction.

The asymptotic momentum transfer $\langle p_{z}\rangle$ (second row) displays a monotonic increase with intensity $I_{0}$, wavelength $\lambda$, and ellipticity $\epsilon$, in line with the expression

where we drop the term $-\langle\bm{v}_{\perp}\cdot\bm{A}\rangle/c$ from the second line. Clearly, the nondipole momentum transfer is proportional to the field intensity $I_{0}$ and $\lambda^{2}$. It increases monotonically with ellipticity $\epsilon$. The saturation near $\epsilon\approx 1$ is due to the replacement of $\langle A^{2}\rangle$ by $\epsilon^{2}A_{0}^{2}$ taken at the peak electric field.

For all laser parameters, the E₂ contribution dominates over the M₁ nondipole contribution to the momentum transfer during the continuum motion and provides, overall, the leading contribution to $\langle p_{z}\rangle$. The difference between $\langle p_{z}\rangle$ induced by the E₂ term and that by the M₁ term remains almost constant with a value of about $2I_{p}/3c$, originating mainly from their difference during the under-barrier dynamical process (c.f., Table 1).

In summary, we have systematically investigated the effect of various distinct nondipole terms in the Hamiltonian describing the coupling of the atomic electron to the radiation field. We consider the subcycle time dependence of strong-field tunneling ionization in an attoclock setting. We include terms to leading order in the field retardation $c^{-1}$. In the multipole gauge, the magnetic dipole (M₁) interaction and the electric quadrupole (E₂) interaction are accounted for. Alternatively, in the radiation gauge, terms linear (F₁) and quadratic (F₂) in the field strength are considered. The contributions of individual nondipole Hamiltonian terms to the initial linear momentum $\langle v_{z}\rangle$ at the tunnel exit from the under-barrier motion as well as the asymptotic linear momentum $\langle p_{z}\rangle$ are studied numerically with the ndSPA method and analytically using the ndSPANE method.

The analysis presented in this work offers novel insights into the nondipole-induced linear momentum on a subcycle time scale. The M₁ magnetic dipole effect plays a dominant role in the under-barrier tunneling process while the E₂ electric quadrupole effect only leaves a small footprint on the initial linear momentum transfer $\langle v_{z}\rangle$ at the tunnel exit. This trend is largely independent of the laser parameters. In stark contrast to $\langle v_{z}\rangle$ at the tunnel exit, the E₂ electric quadrupole effect dominates over the M₁ magnetic dipole effect in the asymptotic linear momentum $\langle p_{z}\rangle$. Each of the nondipole terms induce an oscillation with frequency $2\omega$ in the time-resolved electron emission which arises from the coupling of the nondipole and nonadiabatic tunneling effects. With increasing pulse ellipticity, wavelength, and peak intensity, most of individual nondipole contributions increase.

Regardless of the partitioning and gauge, the sum of individual nondipole contributions equals the results of the full nondipole Hamiltonian, implying a clean separation of individual nondipole effects. The identification of the contributions of individual nondipole terms provides novel insights into their importance in different spatiotemporal domains during the strong-field ionization process.

This work was supported by the National Key R&D Program of China (Grant No. 2018YFA0306303), the National Natural Science Foundation of China (Grant Nos. 11904103, 92150105, 11834004, 12122404, 11974114), the Austrian Science Fund (Grant Nos. M2692, W1243), and the Science and Technology Commission of Shanghai Municipality (Grant Nos. 21ZR1420100, 19JC1412200, 19560745900). Numerical computations were in part performed on the ECNU Multifunctional Platform for Innovation (001).