Introduction

The universal exchange of photon energy and momentum triggers and steers light-matter interactions, resulting in a plethora of intricate physical phenomena. Each photon carries an energy of ℏω and a momentum of ℏω/c, with ω the angular frequency of light and c the light speed in vacuum [atomic units (a.u.) are used throughout unless stated otherwise, where the electron mass me, elementary charge e, and reduced Planck constant ℏ are set to 1 a.u.]. Upon photoionization, the absorbed photon energy and momentum are distributed among the resulting photoelectrons and ions. While the linear momentum of a photon is typically very small and often disregarded under the dipole approximation, there are numerous fundamental physical processes that are driven by photon momentum, such as radiation pressure, laser cooling1, second-harmonic generation2, Compton scattering3, and the Kapitza–Dirac effect4. Understanding the partitioning of the transferred photon momentum among the photoelectrons and ions during light-matter interactions is therefore a critical question to address (Fig. 1).

Fig. 1: Sketch of photon momentum transfer and partitioning.
figure 1

A circular laser pulse ionizes Xe atoms, transferring on average a linear momentum of pe = 2Eγ/c − Up/c to photoelectrons and pi = − Eγ/c + (Ip + Up)/c to ions along the light propagation direction z, where Up is the ponderomotive potential, Ip is the ionization potential, and c is the light speed in vacuum. The photoelectrons and ions are detected coincidently by two detectors at the opposite sides along the y direction. Photoabsorption from the light pulse gives rise to ATI peaks in the photoelectron momentum distribution (PMD), as indicated by the purple concentric spherical shells, which have a common shift of  − Up/c along the light propagation direction z. These concentric shells suggest a definition of the dressed energy \({E}_{\gamma }\equiv \frac{1}{2{m}_{e}}[{p}_{\perp }^{2} + {({p}_{z}+{U}_{p}/c)}^{2}]\), where pz is the momentum in the z direction and \({p}_{\perp }^{2}={p}_{x}^{2}+{p}_{y}^{2}\) is that in the polarization plane. The concentric rings at the bottom and the side represent the cut of the PMD at pz = 0 and px = 0, respectively. The PMD tilts in the forward direction, as indicated by the color distribution along the rings in the side cut, such that photoelectrons gains on average a positive linear momentum.

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With recent advances in laser and detection technologies, the subtle transfer of photon momentum can now be measured with unprecedented precision. Remarkably, it is found that the distribution of the photon momentum between the photoelectron and ion significantly depends on the specific process at play. In 1930, it has already been predicted that, in the high-energy limit of single-photon ionization of an s-shell, the photoelectron acquires a linear momentum of \(\frac{8}{5}\frac{E}{c}\)5,6,7,8,9, with E the photoelectron energy. This prediction has been verified experimentally recently10,11. In the multi-photon limit of tunneling ionization, in contrast, the photoelectron acquires a linear momentum of about \(\frac{E}{c}+\frac{{I}_{p}}{3c}\)12,13,14,15,16,17,18,19,20,21,22,23, with Ip the ionization potential. This leads to the question: What governs the dependence of photon momentum transfer on ionization dynamics? More specifically, why does the slope in front of \(\frac{E}{c}\) vary with the number of photons absorbed from the driving light field? And how to bridge the single-photon and multi-photon limits? To answer these questions, in this work we establish a general framework for photon momentum transfer applicable to an arbitrary number of photons involved.

Above-threshold ionization (ATI)24,25,26 provides a unique platform to explore the distribution of the transferred photon energy and momentum in a single experiment. In order to avoid backscattering close to the multi-photon limit, we utilize circularly polarized light pulses. Given that the photoion is significantly heavier than the photoelectron, the latter typically retains most of the photon energy. Consequently, the energy of the photoelectron can be expressed as \(E=n\hslash \omega -{I}_{p}-{U}_{p}^{{\prime} }\), where \(n\in {\mathbb{N}}\) is the number of photons absorbed, and \({U}_{p}^{{\prime} }=[1+{p}_{z}/({m}_{e}c)]{U}_{p}\) is the ponderomotive potential beyond the dipole approximation17,27,28,29, with Up its dipole counterpart and pz the photoelectron momentum along the laser propagation direction z. The ponderomotive energy Up represents the average quiver energy of a free electron in a laser field. In a long circular laser field, \({U}_{p}={e}^{2}{A}_{0}^{2}/(2{m}_{e})={e}^{2}{F}_{0}^{2}/(2{m}_{e}{\omega }^{2})\) remains constant, with A0 and F0 being the amplitudes of the vector potential and the electric field, respectively. Clearly, the photoelectron energy spectrum exhibits distinct peaks, known as ATI peaks, separated by one photon energy, in accordance with the shells or rings present in the photoelectron momentum distribution (PMD, see Fig. 1). Additionally, by reshaping \(E=n\hslash \omega -{I}_{p}-{U}_{p}^{{\prime} }\) up to order 1/c, it is easy to show that all ATI momentum rings are shifted by a common value of  − Up/c along the laser propagation direction z:

$${E}_{\gamma }\equiv \frac{1}{2{m}_{e}}\left[{p}_{\perp }^{2}+{\left({p}_{z}+\frac{{U}_{p}}{c}\right)}^{2}\right]=n\hslash \omega -{I}_{p}-{U}_{p},$$
(1)

where \({p}_{\perp }^{2}=({p}_{x}^{2}+{p}_{y}^{2})\) is the transverse momentum in the polarization plane and a transverse energy \({E}_{\perp }={p}_{\perp }^{2}/(2{m}_{e})\) can be defined accordingly. Equation (1) motivates a definition of the “dressed energy” Eγ, which is a concept that will enable us to reveal the general rule of the photon momentum transfer. Based on this, we establish a consistent framework for photon momentum transfer covering an arbitrary number of absorbed photons. In addition, we challenge the existing knowledge in the multi-photon limit by showing that the slope is 2 instead of 1 for multi-photon ionization. In other words, with each additional photon absorbed, the photoelectron acquires a momentum that is twice that of a single photon, and the ion gains a recoil momentum anti-parallel to the momentum of the absorbed photon, as schematically shown in Fig. 1.

Results

Photon momentum transfer in above-threshold ionization

Experimentally, we carry out studies using a cold-target recoil-ion momentum spectroscopy (COLTRIMS) setup30. Careful calibration of the zero point in the momentum spectrum, which is crucial for the present study, has been enabled via the formation of a standing wave in the vacuum chamber of the COLTRIMS reaction microscope by using two counter-propagating laser pulses. During the measurement of linear momentum transfer, one of the two counter-propagating lasers is turned off. The coincident detection capability of the COLTRIMS setup enables a precise recording of the PMD. See the Methods section for details.

Figure 2 displays the measured photon momentum transferred to the photoelectron, \({\langle {p}_{z}\rangle }_{\perp }\), for the Xe atom as a function of the transverse energy E⊥ in panel (a) and \({\langle {p}_{z}\rangle }_{\gamma }\) as a function of the dressed energy Eγ in panel (b). While \({\langle {p}_{z}\rangle }_{\perp }^{({{{\rm{Exp.}}}})}\) shows a qualitative agreement with9,23

$${\langle {p}_{z}\rangle }_{\perp }=\frac{{E}_{\perp }}{c}+\frac{{I}_{p}}{3c},$$
(2)

or a better agreement with20

$${\langle {p}_{z}\rangle }_{\perp }=\frac{\Delta {E}_{\perp }}{c}+\delta \frac{{\tilde{I}}_{p}}{3c}=\frac{{E}_{\perp }-{E}_{\perp 0}}{c}+\delta \frac{{I}_{p}+{E}_{\perp 0}}{3c},$$
(3)

where 〈⋅〉⊥ denotes integration along the iso-transverse-energy-E⊥ surface, E⊥0 denotes the initial kinetic energy at the tunnel exit, \({\tilde{I}}_{p}\) represents the effective ionization potential, and δ originates from the prefactor in the ionization rate (see Supplementary Materials for details); \({\langle {p}_{z}\rangle }_{\gamma }^{({{{\rm{Exp.}}}})}\), on the other hand, shows quantitative agreement with

$${\langle {p}_{z}\rangle }_{\gamma }=2\frac{{E}_{\gamma }}{c}-\frac{{U}_{p}}{c},$$
(4)

where 〈⋅〉γ represents integration along the iso-dressed-energy-Eγ surface. Clearly, the slopes of the average linear momentum transfer differ by a factor of 2 with respect to their respective energy.

Fig. 2: ATI spectrum and linear momentum transfer.
figure 2

a Linear momentum transfer \({\langle {p}_{z}\rangle }_{\perp }\) as a function of the transverse energy E⊥. b Linear momentum transfer \({\langle {p}_{z}\rangle }_{\gamma }\) as a function of the dressed energy Eγ [Eq. (1)]. The shaded areas stand for the ATI spectra. The red and purple lines are calculated by TDSE, and the blue lines denote experimental results. The binning size employed is 0.02 a.u., and the results are robust against variations in binning size. In (a), the black dotted line represents \({\langle {p}_{z}\rangle }_{\perp }={E}_{\perp }/c+{I}_{p}/(3c)\) [Eq. (2)] and the orange dashed line denotes \({\langle {p}_{z}\rangle }_{\perp }=\Delta {E}_{\perp }/c+\delta {\tilde{I}}_{p}/(3c)\) [Eq. (3)]. In (b), the black dotted line represents \({\langle {p}_{z}\rangle }_{\gamma }=2{E}_{\gamma }/c-{U}_{p}/c\) [Eq. (4)].

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The comparison above demonstrates that, the observed photon momentum transfer depends not only on the particular ionization mechanism at play but also on different ways to extract its average value. In previous experimental studies, the average photon momentum transfer has been obtained by integration over the PMD along the pz axis, leading to the approximate relation \({\langle {p}_{z}\rangle }_{\perp }=\frac{{E}_{\perp }}{c}+\frac{{I}_{p}}{3c}\) [Eq. (2)], see Fig. 2a. Notably, the transverse energy \({E}_{\perp }={p}_{\perp }^{2}/(2{m}_{e})=({p}_{x}^{2}+{p}_{y}^{2})/(2{m}_{e})\) lacks the pz component along the laser propagation direction and is thus not the full photoelectron energy. In contrast, upon photoabsorption, the full photon energy ω is deposited to the photoelectron and is reflected in the increase in its full energy. Therefore, E⊥/c does not mirror the photon momentum and Eq. (2) does not answer the question how the photon momentum is distributed between the photoelectron and the ion.

In order to pinpoint the origin of the vastly different conclusions of linear momentum transfer from different perspectives, we numerically solve the time-dependent Schrödinger equation (TDSE) for the Xe atom interacting with a circularly polarized laser pulse. See the Methods section for details. Figure 3a shows the calculated PMD across the p⊥ pz plane, where the angular direction in the polarization plane (xy plane) has been integrated over. The overall PMD is centered around p⊥ ≈ A0 = 0.42 a.u., a well-known conclusion of strong-field ionization closely related to the attoclock principle31,32. In addition, the concentric spherical rings arise as ATI peaks.

Fig. 3: Calculated photoelectron momentum distribution (PMD).
figure 3

a PMD calculated by TDSE, where red and purple lines represent iso-transverse-energy-E⊥ and iso-dressed-energy-Eγ surfaces, respectively, and solid and dashed lines denotes the second and fifth ATI peaks, respectively. b linear momentum transfer collected along iso-transverse-energy-E⊥ (red line) and iso-dressed-energy-Eγ (purple line) surfaces for the second ATI peak. c Same as (b), but for the fifth ATI peak. For better visibility, the pz distributions in (b) and (c) have been squeezed by a factor of 100 without changing their peak value.

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Scrutiny into regions surrounding different ATI peaks uncovers how the linear momentum transfer appears differently from different perspectives. Near the second ATI order N = 2, the distribution of linear momentum is obtained by integration along either red (iso-E⊥) or purple (iso-Eγ) solid lines in Fig. 3a, leading to Fig. 3b. To enhance the visibility, the widths of the distributions have been reduced by a factor of 100. Clearly, \({\langle {p}_{z}\rangle }_{\perp } > {\langle {p}_{z}\rangle }_{\gamma }\) here near N = 2. Near the fifth order N = 5, the distribution of linear momentum is obtained similarly, resulting in Fig. 3c. Evidently, it is in stark contrast that \({\langle {p}_{z}\rangle }_{\perp } < {\langle {p}_{z}\rangle }_{\gamma }\) near N = 5, where the order has been reversed. Therefore, as the photoelectron energy increases, the increase in \({\langle {p}_{z}\rangle }_{\gamma }\) is greater than that of \({\langle {p}_{z}\rangle }_{\perp }\), leading to different slopes as illustrated in Fig. 2.

It is clear from Fig. 3 that \({\langle {p}_{z}\rangle }_{\perp }\) obtained from iso-E⊥ integration mixes different ATI orders. This leads to oscillations of \({\langle {p}_{z}\rangle }_{\perp }\) as a function of the transverse energy E⊥, shown as the red solid line in Fig. 2a. This oscillatory behavior becomes smoother after focal volume averaging (see Supplementary Materials for details). In contrast, the energy-___domain \({\langle {p}_{z}\rangle }_{\gamma }\) extracted from iso-Eγ integration, shown as the purple solid line in Fig. 2b, is very smooth and aligns very well with Eq. (4).

Figure 2 shows additionally the distributions of the transverse energy E⊥ and the dressed energy Eγ as shaded areas. Evidently, the distribution of the dressed energy Eγ has smoother and more distinct peaks than that of the transverse energy E⊥, because the former does not mix different ATI peaks. We have also checked that focal volume averaging, albeit leading to smeared-out ATI peaks, does not hamper our conclusions. See the Supplementary Materials for details.

Photon-number-resolved linear momentum transfer

To analyze the photon momentum transfer in more detail, we employ the nondipole strong-field approximation (ndSFA), in which, compared to the full numerical solution using TDSE, the Coulomb interaction in the final state is neglected, simplifying numerical evaluation and theoretical analysis. A direct evaluation of the ndSFA transition amplitude will give the momentum distribution, from which the energy-resolved linear momentum transfer can be extracted. In the multi-photon limit of tunneling ionization, the linear momentum transfer can be obtained by applying an additional saddle-point approximation to ndSFA, which we term ndSPA. There, Eqs. (2)–(4) can be derived fully analytically. See the Methods section for details.

To gain further insights into the physical origin of the photon-number-dependent linear momentum transfer, we apply the long-pulse approximation to ndSFA, leading to the nondipole Keldysh–Faisal–Reiss model (ndKFR) in a photon-number-resolved manner. Thereby, we obtain the photon-number-resolved linear momentum transfer as a key finding of our current work:

$${\langle {p}_{z}\rangle }_{\gamma }=\frac{4\left(n+\frac{\nu }{1-\frac{{U}_{p}}{n\hslash \omega }}\right)}{2n+3}\frac{{E}_{\gamma }}{c}-\frac{{U}_{p}}{c},$$
(5)

where the dimensionless parameter \(\nu=Z/\sqrt{2{I}_{p}/[2{I}_{p}^{({{{\rm{H}}}})}]}\), with the ionization energy Ip = 0.4457 a.u., the ionization energy of the hydrogen atom \({I}_{p}^{({{{\rm{H}}}})}=0.5\) a.u., and the asymptotic charge Z = 1 for the valence shell of Xe. We note that Eq. (5) is valid for an arbitrary number of photons absorbed and can act as a general framework for photon-number-resolved linear momentum transfer to the photoelectron. See the Methods section for derivations.

For the limiting case of single-photon ionization, n = 1 and Up is typically very small and can be neglected. In this scenario, \({\langle {p}_{z}\rangle }_{\gamma }\) reduces to 8E/5c for hydrogen5,6,7,8,9,10,11. For the other limiting case of tunneling ionization, where the involved photon number n is large, \({\langle {p}_{z}\rangle }_{\gamma }\) reduces to 2Eγ/c − Up/c, thereby reproducing Eq. (4). In the present scenario of ATI, the minimal number of photons needed to produce ionization is n0 = ceil[(Ip + Up)/(ℏω)] = 10, fulfilling the large-n requirement to make Eq. (4) a valid approximation.

We define the slope parameter αγ = 4{n + ν/[1 − Up/(nℏω)]}/(2n + 3) to quantify the slope of linear momentum transfer as a function of the number of absorbed photons, which is shown in Fig. 4. The blue line with circles is obtained for Xe interacting with an 800 nm laser at a peak intensity of 4 × 1013 W/cm2. Evidently, the slope parameter αγ is close to 2 within the range shown. The orange line with diamonds corresponds to the case of the H atom interacting with a laser at the same intensity and an angular frequency of ω = 0.6 a.u.. Obviously, αγ = 1.6 when n = 1, and it approaches 2 as the number of absorbed photons n increases.

Fig. 4: Slope parameter αγ as a function of the number of absorbed photons n.
figure 4

for the ATI of the Xe atom (blue line with circles) and the H atom (orange line with diamonds).

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Field dressing and photon momentum partitioning

At last, we would like to elucidate the rationale behind employing the “dressed energy” Eγ as a means to quantify the partitioning of photon momentum transfer. Equation (1) suggests that both the photoelectron energy and linear momentum in the ATI process are simultaneously influenced by the laser dressing, manifested as the Up term. At low laser intensity, Up in Eq. (1) is nearly negligible, and the photoionization process is mainly influenced by the frequency of the laser field. In contrast, under a strong laser field, the photoelectron returns an energy Up to the field, with a concurrent return of linear momentum Up/c to the field. The energy return of Up is evident in the last term of Eq. (1), while the linear momentum return of Up/c is observed as a common shift to the center of all ATI momentum rings. Not surprisingly, these energy and linear momentum exchanges adhere to the photon’s dispersion relation. Within the dipole approximation, it is valid to consider only the influence of the laser field on the energy. However, in the nondipole regime, we must simultaneously analyze the consistent adjustments to both the photoelectron energy and linear momentum. Therefore, it is helpful to define a “dressed energy” Eγ as given in Eq. (1), which represents the center-shifted full photoelectron energy. The virtue of Eγ is that, in the nondipole case, this quantity is quantized analogously to the the quantization of the photoelectron energy in the dipole case.

We consider now the partitioning of the linear momentum between the photoelectron and the ion. Under the influence of the laser field, a total energy of nℏω − Up = Eγ + Ip is deposited to the target, and a corresponding linear momentum of (nℏω − Up)/c = (Eγ + Ip)/c is transferred to the center of mass of the photoelectron and the ion. Both experimental and theoretical analyses have verified that an average linear momentum of 2Eγ/c − Up/c is transferred to the photoelectron. Thus, by momentum conservation, the residual linear momentum of  − Eγ/c + (Ip + Up)/c is transferred to the ion, as illustrated in Fig. 1.

Discussion

We have established a general and consistent framework for photon linear momentum transfer applicable to an arbitrary number of photons absorbed that bridges the single-photon and multi-photon limits, using the prototypical process of above-threshold ionization. For the multi-photon limit, our results suggest that the established understanding needs to be reconsidered, as we conclusively show that, for each additional photon absorbed above the threshold, the photoelectron acquires on average twice of the photon momentum, while the photoion acquires a recoil linear momentum opposite to the light propagation direction. Our findings underscore a perspective of light-matter interaction involving both energy absorption and momentum transfer.

Distinguishing our approach from preceding studies, our research introduces a consistent framework for photon momentum transfer that takes into account the entirety of the photoelectron’s energy. This comprehensive consideration of energy stands in contrast to earlier analyses, which focused on the partial transverse energy of the electron that does not account for the full photon momentum. Our approach, therefore, offers an unprecedented understanding of how photon momentum is partitioned during light-matter interactions that cover the range from single-photon absorption, to the multi-photon and the tunneling regime. This improved understanding will pave the way for further work that harnesses the transfer of photon momentum and further research on light-matter interactions.

Methods

Experimental setup

Experimentally, we adopt the same strategy as described in refs. 29,33,34,35,36. We only give a brief introduction here. We obtain two laser beam pathways of opposite propagation directions by splitting the output (25 fs, 800 nm, 10 kHz) of a Ti:Sapphire laser system (Coherent Legend Elite) using a 50% dielectric beam splitter. In both pathways, a neutral filter and two waveplates of half and quarter wavelengths are used to adjust the intensity and polarization. For this experiment, the two beams are shaped to circular polarizations, followed by focusing into the vacuum chamber of a COLTRIMS reaction microscope30 from two opposite sides using two independent lenses (f = 25 cm) onto the same spot inside a supersonic gas jet of Xe atoms. To minimize the systematic errors during the data collection, two motorized shutters are used to toggle between the two pathways every 3 minutes. This procedure allows us to eliminate most systematical errors since the expected changes of the linear momentum transfer are on the order of 0.001 a.u.

To ensure circular polarization of the laser fields entering the vacuum chamber, we carefully controlled the polarization state using a combination of a half-wave plate and a quarter-wave plate in both pathways. This setup effectively compensates the differences in reflectivity between the s- and p-polarization components of the circularly polarized light. The degree of circular polarization achieved was estimated to be better than 0.9, approaching the ideal value of 1. The peak intensity in the laser focus is estimated to be 4 × 1013 W/cm2 with an uncertainty of  ±20%. The laser intensity has been calibrated from the most probable radial momentum and scaled according to the power measured by a power meter.

The electrons and ions created from single ionization of Xe atoms are guided to two time- and position-sensitive detectors at opposite ends of the spectrometer by applying a static electric field of 29.8 V/cm. The spectrometer consists of acceleration and field-free drift regions for both ions and electrons to realize high momentum resolution. For the electron side, the acceleration and drift regions are 15 and 30 cm, respectively. For the ion side, the acceleration and drift regions are 58 and 108 cm, respectively. We did not employ a magnetic field to guide electrons and ions. Instead, we utilized a μ-metal shield within the vacuum chamber to effectively screen external magnetic fields, particularly the Earth’s magnetic field. The electron (ion) detector is composed of a three-layer (two-layer) stack of multichannel plates followed by a three-layer delay-line anode37.

From the measured times-of-flight and positions-of-impact, the three-dimensional momenta of the electrons and ions are retrieved coincidently. The y-direction is the time-of-flight direction of the COLTRIMS reaction microscope. We employed coincident detection of the Xe ion (Xe+) and the electron (e−). By applying a momentum gate along the polarization direction, we ensured that only the ion and electron pairs originating from the same parent atom were selected, based on momentum conservation. With the above configuration, the single-event momentum resolution for a single electron is 0.003 a.u. in px and pz directions and 0.03 a.u. in the py direction.

To retrieve the expected linear momentum transfer, an integration over an adequate range is necessary. The present experimental setup, however, is limited in the recording range of pz due to the long spectrometer length and small size of the electron detector. Therefore, the peak value of the linear momentum transfer has been used as a replacement for the expectation value. We show that the peak and expectation values are essentially identical. See the Supplementary Materials for details.

The error bars in Fig. 2 represent statistical uncertainties associated with the experimental events. Given that our analysis involves the use of a standing wave, and the final results are derived by subtracting the forward and backward curves, any uncertainties related to the spectrometer calibration effectively cancel out along the light-propagation direction. Consequently, the primary source of uncertainty in our results is statistical error, stemming from the inherent variability of the experimental measurements.

Time-dependent Schrödinger equation

The strong-field ionization of Xe in a circularly polarized laser field is simulated by numerically solving the three-dimensional TDSE under the single-active-electron approximation with the minimal-coupling nondipole Hamiltonian

$$H=\frac{1}{2{m}_{e}}{[{{{\bf{p}}}}+e{{{\bf{A}}}}(\eta )]}^{2}+V({{{\bf{r}}}}),$$
(6)

where the atomic potential \(V({{{\bf{r}}}})=-\!{1}/\sqrt{{r}^{2}+{a}_{0}}\) with a0 the soft-core parameter, and the vector potential of the laser pulse A(η) is expressed as

$${{{\bf{A}}}}(\eta )={A}_{0}{\cos }^{4}(\omega \eta /2{N}_{L})\left[\cos (\omega \eta ){\hat{{{{\bf{e}}}}}}_{x}+\sin (\omega \eta ){\hat{{{{\bf{e}}}}}}_{y}\right],$$
(7)

polarized across the xy plane, where η = t − z/c is the light-cone time, the angular frequency ω corresponds to a central wavelength of 800 nm, the amplitude A0 corresponds to a peak laser intensity of 4 × 1013 W/cm2, and NL = 10 is the number of optical cycles. The corresponding electric field is F(η) = − ∂tA(η).

The vector potential A(η) can be approximated up to order 1/c as

$${{{\bf{A}}}}(\eta )={{{\bf{A}}}}\left(t-\frac{z}{c}\right)={{{\bf{A}}}}(t)+\frac{z}{c}{{{\bf{F}}}}(t),$$
(8)

where A(t) and F(t) represent the vector potential and electric field evaluated at the nucleus z = 0, respectively. The nondipole Hamiltonian can thus be rewritten up to order 1/c as

$$H=\frac{1}{2{m}_{e}}{[{{{\bf{p}}}}+e{{{\bf{A}}}}(t)]}^{2}+\frac{e}{{m}_{e}}\frac{z}{c}[{{{\bf{p}}}}+e{{{\bf{A}}}}(t)]\cdot {{{\bf{F}}}}(t)+V({{{\bf{r}}}}).$$
(9)

Performing a unitary transformation, we may obtain the expression for the transformed nondipole Hamiltonian20,38

$$H=\frac{1}{2{m}_{e}}{\left\{{{{\bf{p}}}}+e{{{\bf{A}}}}(t)+\frac{{\hat{{{{\bf{e}}}}}}_{z}}{c}\left[\frac{e}{{m}_{e}}{{{\bf{p}}}}\cdot {{{\bf{A}}}}(t)+\frac{{e}^{2}}{2{m}_{e}}{A}^{2}\right]\right\}}^{2}+V\left[{{{\bf{r}}}}-\frac{e}{{m}_{e}}\frac{z}{c}{{{\bf{A}}}}(t)\right].$$
(10)

In this transformed Hamiltonian [Eq. (10)], the spatial part and momentum part are separated, enabling straightforward application of the split-operator Fourier method for the solution of the TDSE. It is solved on a spatial grid with 1024 points and a spatial step size of Δx = 0.3 in each of the three dimensions. The time step is Δt = 0.02. Setting the soft-core parameter a0 = 0.03159, the ground state is obtained using the imaginary-time propagation method with an energy E0 = −Ip = −0.4457 a.u., matching the experimental value of the first ionization potential of Xe. Note that we have used the ground s state of the present potential V(r) to represent the valance shell of Xe. For the multi-photon limit of ionization, the specific structure of the initial state plays a negligible role.

The ground-state wave function is then evolved under the laser field using the nondipole Hamiltonian [Eq. (10)]. The outgoing wave function is damped by an absorbing boundary of the form \(1/[1+\exp \{(r-{r}_{0})/d\}]\) with r0 = 138.6 and d = 4 to avoid reflection from the grid border. The absorbed wave function at each time step is cumulatively projected onto nondipole Volkov states39,40

$$\left\vert {\psi }_{{{{\bf{p}}}}}(t)\right\rangle=\left\vert {{{\bf{p}}}}\right\rangle \exp \left\{-\frac{i}{\hslash }{\int}^{t}d\tau \frac{1}{2{m}_{e}}{\left\{{{{\bf{p}}}}+e{{{\bf{A}}}}(\tau )+\frac{{\hat{{{{\bf{e}}}}}}_{z}}{c}\left[\frac{e}{{m}_{e}}{{{\bf{p}}}}\cdot {{{\bf{A}}}}(\tau )+\frac{{e}^{2}}{2{m}_{e}}{A}^{2}(\tau )\right]\right\}}^{2}\right\}$$
(11)

to obtain the PMD W(p). Finally, the expectation value of linear momentum transfer can be calculated as

$$\langle {p}_{z}\rangle=\frac{\int\,{p}_{z}W({{{\bf{p}}}})d{{{\bf{p}}}}}{\int\,W({{{\bf{p}}}})d{{{\bf{p}}}}}.$$
(12)

Nondipole strong-field approximation

The ndSFA neglects the Coulomb interaction in the final state as well as the laser polarization in the initial state, and the corresponding transition amplitude in the length gauge can be written as39

$${M}_{{{{\rm{ndSFA}}}}}({{{\bf{p}}}})=-\frac{i}{\hslash }\int_{-\infty }^{\infty }d\eta \left[e\left(1-\frac{{p}_{z}}{{m}_{e}c}\right)i{{{{\mathbf{\nabla }}}}}_{{{{\bf{k}}}}}{\psi }_{0}({{{\bf{k}}}})\cdot {{{\bf{F}}}}(\eta ){e}^{\frac{i}{\hslash }S(\eta )}\right],$$
(13)

where ψ0(k) is the ground-state wave function in the momentum representation evaluated at \({{{\bf{k}}}}={{{\bf{p}}}}+e{{{\bf{A}}}}(\eta )-(E+{I}_{p}){\hat{{{{\bf{e}}}}}}_{z}/c\) with \(E={p}^{2}/(2{m}_{e})=({p}_{\perp }^{2}+{p}_{z}^{2})/(2{m}_{e})\), and the nondipole action is expressed as \(S(\eta )={\int}^{\eta }dt\{E+[(e/{m}_{e}){{{\bf{p}}}}\cdot {{{\bf{A}}}}(t)+{e}^{2}{A}^{2}(t)/(2{m}_{e})]/[1-{p}_{z}/({m}_{e}c)]+{I}_{p}\}\). Direct evaluation of Eq. (13) will give the momentum distribution, from which the energy-resolved linear momentum transfer can be extracted.

Nondipole saddle-point approximation

Further applying the nondipole saddle-point approximation with nonadiabatic expansion (ndSPA) on top of ndSFA, the transition rate can be written, up to exponential accuracy, as20,41,42

$$\begin{array}{rcl}&{W}_{{{\rm{ndSPA}}}}({{\bf{p}}})\hfill\\=&\exp \left\{-\frac{2{\left[2{I}_{p}+\frac{1}{{m}_{e}}{p}^{2}+\left(1+\frac{{p}_{z}}{{m}_{e}c}\right)\left(\frac{e}{{m}_{e}}2{{\bf{p}}}\cdot {{\bf{A}}}+\frac{{e}^{2}}{{m}_{e}}{A}^{2}\right)\right]}^{3/2}}{3\hslash \sqrt{\left(1+\frac{{p}_{z}}{{m}_{e}c}\right)\frac{e}{{m}_{e}}\left(e{F}^{2}-{{{\bf{v}}}}_{\perp }\cdot \dot{{{\bf{F}}}}\right)}}\right\}\hfill\\=&\exp \left\{-\frac{2}{3\hslash \sqrt{\frac{e}{{m}_{e}}\left(e{F}^{2}-{{{\bf{v}}}}_{\perp }\cdot \dot{{{\bf{F}}}}\right)}}{\left[2{I}_{p}+\frac{1}{{m}_{e}}{v}_{\perp }^{2}+\frac{1}{{m}_{e}}{\left({p}_{z}-\left(\frac{{p}_{\perp }^{2}-{v}_{\perp }^{2}}{2{m}_{e}c}+\frac{2{m}_{e}{I}_{p}+{v}_{\perp }^{2}}{6{m}_{e}c}\right)\right)}^{2}\right]}^{3/2}\right\},\end{array}$$
(14)

where v⊥ = p⊥ + eA(t). Clearly, the iso-transverse-energy-E⊥ expectation value of the linear momentum transfer can be obtained as

$${\langle {p}_{z}\rangle }_{\perp }=\frac{{p}_{\perp }^{2}-{v}_{\perp }^{2}}{2{m}_{e}c}+\frac{2{m}_{e}{I}_{p}+{v}_{\perp }^{2}}{6{m}_{e}c}.$$
(15)

In the adiabatic limit where the average initial transverse momentum 〈v⊥〉 = 0, we have

$${\langle {p}_{z}\rangle }_{\perp }=\frac{{p}_{\perp }^{2}}{2{m}_{e}c}+\frac{{I}_{p}}{3c}=\frac{{E}_{\perp }}{c}+\frac{{I}_{p}}{3c}.$$
(2)

Now, we aim to obtain the iso-dressed-energy-Eγ expectation value of the linear momentum transfer. Along a specific ATI ring, nℏω = Eγ + Ip + Up is a constant, Eq. (14) can thereby be rewritten as

$${W}_{{{{\rm{ndSPA}}}}}({{{\bf{p}}}})= \exp \left\{-\frac{2{\left[2n\hslash \omega+2\left(1+\frac{{p}_{z}}{{m}_{e}c}\right)\frac{e}{{m}_{e}}{{{\bf{p}}}}\cdot {{{\bf{A}}}}\right]}^{3/2}}{3\hslash \sqrt{\left(1+\frac{{p}_{z}}{{m}_{e}c}\right)\frac{e}{{m}_{e}}\left(e{F}^{2}-{{{{\bf{v}}}}}_{\perp }\cdot \dot{{{{\bf{F}}}}}\right)}}\right\}\\= \exp \left\{-\frac{2}{3\hslash \omega }\frac{{\left[2n\hslash \omega -2\left(1+\frac{{p}_{z}}{{m}_{e}c}\right)\frac{e}{{m}_{e}}{p}_{\perp }{A}_{0}\right]}^{3/2}}{{\left[\left(1+\frac{{p}_{z}}{{m}_{e}c}\right)\frac{e}{{m}_{e}}{p}_{\perp }{A}_{0}\right]}^{1/2}}\right\}$$
(16)

for long circular pulses. This ionization rate maximizes when [1 + pz/(mec)]p⊥ peaks, leading to

$${\langle \cos {\theta }_{\gamma }\rangle }_{\gamma }=\frac{\sqrt{2{m}_{e}{E}_{\gamma }}}{{m}_{e}c}.$$
(17)

The iso-dressed-energy-Eγ expectation value of the linear momentum transfer is thereby

$${\langle {p}_{z}\rangle }_{\gamma }=\sqrt{2{m}_{e}{E}_{\gamma }}{\langle \cos {\theta }_{\gamma }\rangle }_{\gamma }-\frac{{U}_{p}}{c}=2\frac{{E}_{\gamma }}{c}-\frac{{U}_{p}}{c}.$$
(4)

Nondipole Keldysh–Faisal–Reiss theory

The SFA is often used as a synonym to the Keldysh–Faisal–Reiss theory (KFR)43,44,45 in the literature, although there are some minor differences46. To distinguish our theoretical approaches in this work, we term the approach with time-___domain integrations as SFA, such as in Eq. (13), and name the approach with energy-___domain summations as KFR, such as in Eq. (18).

In order to gain further insights into the physical origin of the photon-number-dependent linear momentum transfer, we extend the dipole version of KFR47 by applying the long-pulse approximation to ndSFA in the velocity gauge, leading to the ndKFR model for circular pulses

$${M}_{{{{\rm{ndKFR}}}}}({{{\bf{p}}}})=\left(1-\frac{{p}_{z}}{{m}_{e}c}\right)\left(E+{I}_{p}\right){\psi }_{0}\left({{{\bf{p}}}}-\frac{E+{I}_{p}}{c}{\hat{{{{\bf{e}}}}}}_{z}\right) \\ 2\pi i{\sum}_{n}{J}_{n}(\alpha ){e}^{in{\phi }_{{{{\bf{p}}}}}}\delta \left({E}_{\gamma }+{I}_{p}+{U}_{p}-n\hslash \omega \right)$$
(18)

in a photon-number-resolved manner, where \(\tan {\phi }_{{{{\bf{p}}}}}={p}_{y}/{p}_{x}\), α = − eA0p⊥[1 + pz/(mec)]/(meℏω) is a dimensionless parameter, and Jn is the Bessel function. Clearly, the Dirac delta function in Eq. (18) gives rise to the ATI peaks. Along these iso-dressed-energy-Eγ surfaces, on the other hand, the prefactor Jn(α) reshapes how the probability distributes.

Defining pγ as the dressed radial distance, θγ the dressed polar angle and ϕ the azimuthal angle with respect to (0, 0, − Up/c), the expectation value of photon-number-resolved linear momentum transfer is expressed by

$${\langle {p}_{z}\rangle }_{\gamma }=\frac{\int\,d{\theta }_{\gamma }d\phi {p}_{\gamma }^{3}\sin {\theta }_{\gamma }\cos {\theta }_{\gamma }| {M}_{{{{\rm{ndKFR}}}}}({{{\bf{p}}}}){| }^{2}}{\int\,d{\theta }_{\gamma }d\phi {p}_{\gamma }^{2}\sin {\theta }_{\gamma }| {M}_{{{{\rm{ndKFR}}}}}({{{\bf{p}}}}){| }^{2}}-\frac{{U}_{p}}{c}.$$
(19)

We note that the asymptotic part of the bound-state wave function dominates the tunneling ionization, whose Fourier transform is

$${\psi }_{0}({{{\bf{k}}}})\simeq {\left({\kappa }^{2}{\hslash }^{2}\right)}^{\nu+\frac{1}{4}}\frac{1}{{\left({\kappa }^{2}{\hslash }^{2}+{k}^{2}\right)}^{\nu+1}}$$
(20)

with \(\kappa=\sqrt{2{m}_{e}{I}_{p}}/\hslash\). In addition, n > ∣α∣ in the present study, which justifies the approximation45

$${J}_{n}(\alpha )\approx \frac{1}{n!}{\left(\frac{1}{2}\alpha \right)}^{n}.$$
(21)

A direct evaluation of Eq. (19) gives the photon-number-resolved linear momentum transfer

$${\langle {p}_{z}\rangle }_{\gamma }=\frac{4\left(n+\frac{\nu }{1-\frac{{U}_{p}}{n\hslash \omega }}\right)}{2n+3}\frac{{E}_{\gamma }}{c}-\frac{{U}_{p}}{c}.$$
(5)

The angular momentum of the initial state contributes little to Eq. (5) for multi-photon ionization; as we have seen above, it is the exponent part, which is universal, that determines the momentum transfer.