Time- and angle-resolved photoelectron spectroscopy of strong-field light-dressed solids: Prevalence of the adiabatic band picture

In recent years, strong-field physics in condensed matter was pioneered as a potential approach for controlling material properties through laser dressing, as well as for ultrafast spectroscopy via nonlinear light-matter interactions (e.g., harmonic generation). A potential controversy arising from these advancements is that it is sometimes vague which band picture should be used to interpret strong-field experiments: The field-free bands, the adiabatic (instantaneous) field-dressed bands, Floquet bands, or some other intermediate picture. Here, we try to resolve this issue by performing theoretical experiments of time- and angle-resolved photoelectron spectroscopy (Tr-ARPES) for a strong-field laser-pumped solid, which should give access to the actual observable bands of the irradiated material. To our surprise, we find that the adiabatic band picture survives quite well up to high field intensities ($\sim {10}^{12}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$) and in a wide frequency range (driving wavelengths of 4000 to 800 nm, with Keldysh parameters ranging up to $\sim 7$). We conclude that, to first order, the adiabatic instantaneous bands should be the standard blueprint for interpreting ultrafast electron dynamics in solids when the field is highly off resonant with characteristic energy scales of the material. We then discuss weaker effects of modifications of the bands beyond this picture that are nonadiabatic, showing that by using bichromatic fields the deviations from the standard picture can be probed with enhanced sensitivity. In this paper, we outline a clear band picture for the physics of strong-field interactions in solids, which should be useful for designing and analyzing strong-field experimental observables and to formulate simpler semi-empirical models.

Figure 1

(a) Illustration of time- and angle-resolved photoelectron spectroscopy (Tr-ARPES) with strong-field setup. A monolayer of hexagonal boron nitride (hBN) is irradiated by an intense pump pulse that is polarized in the monolayer plane (blue), which dresses the electronic system and potentially causes some band modifications. A delayed extreme ultraviolet (XUV) probe pulse that is polarized transverse to the monolayer (along the $z$ axis) then photo-ionizes electrons into the continuum. The photoemitted electrons propagate up to the detector surface, at which point their momenta and energy is recorded. (b) Ground-state density functional theory (DFT) band structure along the $K^{\prime }$-Γ-$K$ path. The top of the valence band is set to zero energy, and $K$ and $K^{\prime }$ points are indicated with dashed lines. Occupied bands are denoted by red lines, while unoccupied bands with blue lines. (c) Pump-free ARPES spectra that capture the details of the equilibrium occupied bands. The energy of the ARPES spectrum is offset by ${\omega }_{\mathrm{xuv}}$, and dashed white lines indicate the positions of the valence and conduction band edges and Γ point.

Figure 2

Continuum effects in strong-field dressed time- and angle-resolved photoelectron spectroscopy (Tr-ARPES) at a pump-probe delay that corresponds to $\mathbf{A}\left(t_{\mathrm{ion}}\right)=0$. (a) Tr-ARPES spectrum from hexagonal boron nitride (hBN) including full propagation in the continuum. (b) Tr-ARPES spectrum from hBN including only propagation in the continuum within the simulation box, from $z=0$ to $z=w$. The propagation beyond $z=w$ is not performed (see text). (c) Tr-ARPES spectrum from hBN after removing continuum effects according to the two procedures outlined in the text. All Tr-ARPES spectra are calculated in the instantaneous regime (see text) with the parameters $I_0={10}^{12}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$, $\lambda =2000\mathrm{nm}$. A different color scale is used for the case where all continuum effects have been removed to indicate that it represents the approximate energy spectrum of the electrons inside the dressed solid. Dashed white lines indicate the positions of the valence and conduction band edges and the Γ point.

Figure 3

Illustration of the time- and angle-resolved photoelectron spectroscopy (Tr-ARPES) regimes. (a) Instantaneous regime: The probe pulse has a duration much shorter than the pump field's optical cycle. In this exemplary case, the pump-probe delay is set such that the probe overlaps with a momentary zero in the vector potential, i.e., ${\mathbf{A}}_{\mathrm{pump}}\left(t_{\mathrm{ion}}\right)=0$. (b) Floquet regime: The probe pulse duration is like one optical cycle of the pump field. Note that the probe pulse amplitude is many orders of magnitude weaker than the pump amplitude but is plotted here on the same scale for clarity.

Figure 4

Time- and angle-resolved photoelectron spectroscopy (Tr-ARPES) in the instantaneous regime vs pump-probe delay. (a)–(f) Tr-ARPES spectra for different pump-probe delays ranging from ${\mathbf{A}}_{\mathrm{pump}}\left(t_{\mathrm{ion}}\right)=0$ in (a) to ${\mathbf{A}}_{\mathrm{pump}}\left(t_{\mathrm{ion}}\right)=c{E}_0/\omega$ in (f) (indicated in the figure). Tr-ARPES spectra are calculated for pump parameters of $I_0=0.9\times {10}^{12}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$, $\lambda =2000\mathrm{nm}$. The markings in (f) denote a systematic shift of the band center with the pump vector potential, in accordance with the prediction of the adiabatic Houston states (see text). The effects of continuum state dressing have been removed according to the procedure detailed in Sec. 2c. Dashed white lines indicate the positions of the Γ point and valence band edge.

Figure 5

Time- and angle-resolved photoelectron spectroscopy (Tr-ARPES) in the instantaneous regime vs pump power for $\mathbf{A}\left(t_{\mathrm{ion}}\right)=0$. Pump laser powers are indicated over each figure, and the pump wavelength is taken as $\lambda =2000\mathrm{nm}$. The effects of continuum state dressing have been removed according to the procedure detailed in Sec. 2c. Dashed white lines indicate the positions of the Γ point and valence band edge.

Figure 6

Time- and angle-resolved photoelectron spectroscopy (Tr-ARPES) in the instantaneous regime vs pump wavelength for $\mathbf{A}\left(t_{\mathrm{ion}}\right)=0$. The pump wavelengths are indicated in each figure, and the pump power is taken as $I_0=0.9\times {10}^{12}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$. The effects of continuum state dressing have been removed according to the procedure detailed in Sec. 2c. Dashed white lines indicate the positions of the Γ point and valence band edge.

Figure 7

Time- and angle-resolved photoelectron spectroscopy (Tr-ARPES) in the Floquet regime with different levels of theory. (a) Tr-ARPES spectrum within the Floquet regime, where the spectrum is not corrected for effects of continuum propagation up to the detector. (b) Tr-ARPES spectrum obtained from incoherently averaging the individual spectra vs delay in the same conditions as in (a) but where the probe pulse is in the instantaneous regime. Here, the spectra are averaged over a full laser cycle. Continuum effects of propagation of photoelectron up to the detector are not corrected for. (c) Same as in (b) but where the continuum effects are corrected for. The plots are calculated for pump laser conditions of $I_0=0.9\times {10}^{12}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ and $\lambda =2000\mathrm{nm}$. Dashed white lines indicate the positions of the valence and conduction band edges and Γ point.

Figure 8

Time- and angle-resolved photoelectron spectroscopy (Tr-ARPES) in the instantaneous regime with an ω-2ω bichromatic pump [Eq. (12)] for $\mathrm{\Delta }=1$ and $\varphi =\pi /4$. (a) Total vector potential vs time showing that the laser field induces time-reversal and inversion symmetry breaking. Two moments in time where the vector potential is zero are highlighted with dashed lines and denoted by $t_1$ and $t_2$. While the vector potential is instantaneously zero in both moments, the evolution of the field before them is different, and so is the instantaneous electric field. (b) Tr-ARPES spectrum corresponding to a pump-probe delay where the peak ionization occurs at $t_1$. (c) Same as (b) but for $t_2$. Plots calculated with pump parameters $\lambda =4000\mathrm{nm}$ and $I_0=2.5\times {10}^{11}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$. The effects of continuum state dressing have been removed according to the procedure detailed in Sec. 2. (d) ARPES yield traces for a fixed photoelectron energy at the valence band maxima [VBM; along the dashed white lines in (b) and (c)], for the two pump-probe delays, but where for $t_2$ the plot is reflected with $\mathbf{k}\to -\mathbf{k}$. For a monochromatic field, these two curves are identical, whereas the deviation indicates nonadiabatic effects accessible with the two-color approach. Dashed white lines indicate the positions of the Γ point and valence band edge.

Figure 9

Time- and angle-resolved photoelectron spectroscopy (Tr-ARPES) in the instantaneous regime with an ω-2ω bichromatic pump [Eq. (12)] for $\mathrm{\Delta }=\surd 2$ and $\varphi =\pi /2$. (a) Total vector potential vs time showing that the laser field induces time-reversal and inversion symmetry breaking. For these parameters, there are four moments in time where the vector potential is zero, which are highlighted with dashed lines and denoted by $t_{1\text{–}4}$. While the vector potential is instantaneously zero for all moments, the evolution of the field before them is different, and so is the instantaneous electric field. (b)–(e) Tr-ARPES spectra corresponding to a pump-probe delay where the peak ionization occurs at $t_1$, $t_2$, $t_3$, and $t_4$, respectively. Plots calculated with pump parameters $\lambda =4000\mathrm{nm}$ and $I_0=2\times {10}^{11}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$. The effects of continuum state dressing have been removed according to the procedure detailed in Sec. 2c. Note that there is some numerical noise around $K$ and $K^{\prime }$ points for these strong driving conditions causing the vertical lines in the ARPES spectra. Dashed white lines indicate the positions of the Γ point and valence band edge.

Figure 10

Time- and angle-resolved photoelectron spectroscopy (Tr-ARPES) in the instantaneous regime comparing the independent particle approximation (IPA) with a full time-dependent density functional theory (TDDFT) calculation. In both ARPES spectra, the system is driven by a pump field with a wavelength of 2000 nm and $I_0=9\times {10}^{11}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$, and the pump-probe delay is set such that ${\mathbf{A}}_{\mathrm{pump}}\left(t_{\mathrm{ion}}\right)=c{E}_0/\omega$. (a) Full TDDFT calculation. (b) IPA calculation. Dashed white lines indicate the positions of the Γ point and valence band edge.