Introduction

Space-periodic arrangement of lattice ions in crystalline solids defines the k-momentum dispersion of electronic quasiparticle bands. Light can interrogate such bands by stimulating electric dipole transitions. When a time-periodic optical field interaction exceeds all other perturbations, particularly at an optical resonance, however, it can also modify the system eigenstates. Then a quantum system is modified (dressed) by an optical field, it becomes a time crystal with novel, field-dependent electronic properties1. High-optical field control, which ultimately enables the rich physics of high-harmonic and attosecond pulse generation2, then motivates the exploration of the light-wave electronics in solids3,4,5,6,7,8,9. Optical fields with designed electric field strength $${\cal{E}}$$, and frequency ωl can manipulate and control the quasiparticles in solids. Recent goals and achievements include light-induced superconductivity10, high-order nonlinearities3,6, the creation of Floquet topological bands in quantum materials9,11,12,13,14,15,16,17,18,19, and photoinduced phase transitions20,21.

Under perturbative conditions, light fields $${\cal{E}}\left( \tau \right)$$ can excite multiquantum transitions from dispersive Bloch bands that are defined by the crystal structure, with their eigenstates specified by their momentum k. An intense $${\cal{E}}\left( \tau \right)$$ field, however, can perturb the system by causing Rabi flopping to occur between the optically coupled bands at frequencies $$\tilde \Omega _R$$ that can become comparable to that of the excitation field, ωl (ref. 13): the electronic bands are said to be dressed when the optical field amplitude and periodicity shifts their energies, and replicates them by integer photon quanta through the AC Stark and Floquet processes (Fig. 1e)9,11. The optical dressing is thus marked by Autler–Townes (AT) splitting of solid-state bands into a new eigenstates E± (ref. 22) separated by the generalized Rabi frequency

$$\hbar \tilde \Omega _R\left( {\tau ,{\boldsymbol{k}}} \right) = \hbar \sqrt {{\mathrm{\Omega }}_R^2(\tau ) + \Delta ^2({\boldsymbol{k}})},$$
(1)

where $$\hbar \Omega _R(\tau ) = \mu _{{\mathrm{eff}}}{\cal{E}}^n(\tau )$$ is the Rabi frequency of a n-photon transition, and $$\hbar \Delta \left( {\boldsymbol{k}} \right)$$ is the k-dependent detuning of Bloch bands ($$\hbar$$: the Planck constant). Moreover, we assume the dipole moment, $$\mu _{{\mathrm{eff}}}$$, to be independent of k based on approximately symmetric photoemisson intensity distributions (with respect to + and – parallel momentum, k||), and k||-independent linear photoemission spectra of the initial band23.

Here, we apply intense ultrafast optical fields to a metal surface, a solid-state plasma, whose high electron density defines its quantum optical characteristics: attosecond screening and femtosecond electronic dephasing24,25,26. The ground and excited surface electronic bands of several metals like copper exist in projected band gaps that decouple them from stronger bulk interactions25 (Fig. 1d), making them inviting targets for probing their quantum optics by ultrafast multiphoton photoemission (mPP) spectroscopy and many-body theory25,26,27,28,29,30. We dress the surface electronic structure of Cu(111) with ~20 fs IR laser pulses by exciting the three-photon resonant transition from the occupied Shockley surface (SS) to the first image potential (IP1) band, and probe the coherent field interaction by further exciting photoelectrons into the photoemission continuum by interferometric-time-resolved multiphoton photoemission (ITR-mPP) spectroscopy (Fig. 1). The dressing and probing field $${\cal{E}}$$(τ) combines identical, phase correlated pump–probe pulse pairs to excite four-photon photoemission (4PP). Advancing their delay (phase) in Δτ ~ 100 as steps defines the optical fields with subfemtosecond precision, and simultaneously imaging photoelectron energy, Ef, and parallel momentum, k||, distributions records Ef(k||,τ)-movies of coherent polarizations excited in the sample; the movies record the field-induced quasiparticle dressing at the interface between the perturbative and non-perturbative light–matter interactions (Fig. 1a, Supplementary Movie 1). We explore how the optical field modifies the crystal-defined band structure on a sub-optical cycle timescale.

Results

Ultrafast quasiparticle dressing probed by coherent mPP

The $${\cal{E}}$$(τ) field with ħωl = 1.54−eV excites the three-photon resonant transition from the SS to the IP1 band of Cu(111) at k|| = 0 Å−1. Different band dispersions of SS and IP1, which are defined by their band masses $$(m_{SS,}m_{IP1})$$, however, detune the resonant energy by $$\hbar \Delta ( {k_{||}}) = \hbar ^2/2({m_{{\mathrm{SS}}}^{ - 1} - m_{{\mathrm{IP}}1}^{ - 1}})k_{||}^2$$. The time-integrated 4PP spectrum in Fig. 1c confirms the SS and IP1 resonance29,31; such optical phase-integrated measurement, however, obscures their dressing, which, is encoded in the phase-resolved Ef(k||,τ)-movies that reveal the electric field strength-dependent response (Fig. 1a, Supplementary Movie 1). To reveal the dressing, we first extract the $${\cal{E}}\left( \tau \right)$$-dependent 4PP intensity for k|| = 0 Å−1 ($$\hbar \Delta = 0$$ eV) by plotting the 2D Ef(τ) data and, its time profile (1D interferometric two-pulse correlation (I2PC)) showing the photoelectron counts vs. τ at Ef of the IP1←SS resonance (Fig. 1c). If the optical field $${\cal{E}}\left( \tau \right)$$ with approximately a Gaussian time profile generated the 4PP signal by only exciting dipole transitions, the I2PC trace in Fig. 1c would follow $${\cal{E}}$$8(τ) dependence, and therefore, sharply peak at τ = 0 fs (refs. 30,32), but instead, it has a local minimum and retarded maxima at τ ≈ ± 15 fs, which portend the optical field dressing.

Fourier analysis of femtosecond field dressing of resonantly excited surface bands

The Ef(k||,τ)-movie, therefore, incorporates both the $${\cal{E}}\left( \tau \right)$$-dependent band dressing and quantum excitation pathways, which are revealed by its Fourier analysis. Fourier transformation (FT) of the 2D time-domain data in Fig. 1c resolves the induced polarization fields in the sample that oscillate at 1–4ωl frequencies to produce the 4PP signal30; in Fig. 2, we present the 2D-Fourier-filtered time-domain spectra of coherent polarizations oscillating at 2ωl- and 3ωl-frequencies, which reveal the dominant pathways for how these coherences contribute to the 4PP signal, and thereby reveal the dressing. As detailed in Supplementary Note 3, the Fourier filtering involves the FT of the 3D ITR-4PP data to resolve the 1–4ωl frequency components that contribute to the signal, followed by the selective inverse Fourier transformation (IFT) of each.

The 2ωl and 3ωl IFT spectra show peak doubling and tripling, respectively, near τ = 0 fs when the optical field amplitude is the largest (Fig. 2). The 3ωl-polarization represented by brown arrows in Fig. 1e reports how the optical field dresses the resonant bands causing components at 3ωl and $$3{\upomega}_l \pm {\tilde{\mathrm{\Omega }}}_R$$ frequencies to appear; such coherent polarizations produce Mollow-Triplet (MT) structures in gas-phase optical spectra22. By contrast, the nonresonant 2ωl-polarization, represented by red arrows in Fig. 1e, shows how the three-photon IP1SS resonance dresses the interacting bands by revealing their AT doubling, when they are projected into the Ef-continuum. The 2ωl- and 3ωl-spectra in Fig. 2, thus reveal the coherent subfemtosecond responses of the electronic bands to the dressing field. The 2ωl-spectrum shows that when τ = 0 fs and therefore, when $${\cal{E}}\left( \tau \right) = {\cal{E}}_{\mathrm{pump}}\left( t \right) + {\cal{E}}_{\mathrm{probe}}(t + \tau )$$ is maximum, at resonance the AT doublet splitting is $$\hbar {\tilde{\mathrm{\Omega }}}_R\left( \tau \right) \approx 0.3\,{\mathrm{eV}}$$, but as |τ| increases to ~15 fs, the instantaneous $${\cal{E}}$$(τ) field drops to recover the single undressed resonance. The dressing occurs within each optical cycle of 2.7 fs as evidenced by the subfemtosecond splitting of the I2PC fringes near τ = 0 fs (inset of Fig. 1c), when constructive interference maximizes $${\cal{E}}$$(τ). As expected for the AC Stark effect (Eq. (1)), the AT doublet structure disappears when $${\cal{E}}$$(τ) amplitude is reduced by defocusing (Fig. 2b) or increasing τ. Thus, the dressing and undressing follows the instantaneous $${\cal{E}}$$(τ) strength, and is faster than the Cu(111) surface IP1 and SS band dephasing, which occurs by carrier scattering25 on ~20 fs timescale28,29.

Entanglement of in-plane momentum dispersion and photon dressing

A graphic feature that emerges from the ITR-4PP data on Cu(111) is that k||-dispersion and optical dressing are entangled by the generalized Rabi frequency (Eq.(1)), which depends both on $${\cal{E}}$$(τ) and $$\hbar \Delta ({k_{||}})$$. The entanglement is directly evident in the 2ωl-IFT-spectral component, which we extract from the data in Supplementary Movie 1, and examine as a function of k|| and τ in Supplementary Movie 2. Supplementary Movie 2 conveys how the Ef(k||)-dependent polarization spectra, change as τ is advanced in Δτ ~ 100 as steps to define the total $${\cal{E}}$$(τ) field. The selected movie images in Fig. 3, highlight how changing $${\cal{E}}$$(τ) and k||-dispersion affect the AT doublet dressing of the coupled bands. For τ = 65 fs (Fig. 3e), the pump and probe fields do not overlap, so the instantaneous $${\cal{E}}$$(τ) is minimum, and the spectra reproduce the undressed k||-dispersions of the IP1 and SS bands (black lines). By contrast, for $$15\,{\mathrm{fs}} \lesssim |\tau |$$ (Fig. 3b–d), the pump and probe fields add coherently, sufficiently increasing $${\cal{E}}$$(τ) to cause the k||- and τ-dependent dressing, and thereby modify the band dispersions. As $$\hbar \Delta ({k_{||}})$$ increases, however, the bands detune from resonance and therefore dressing diminishes causing them to converge to the undressed ones. This entanglement of band dispersion and dressing causes the quasiparticle band masses to become τ-dependent. Thus, dressing can transform an electron into a hole band, potentially reversing the carrier transport in response to a femtosecond timescale optical field (cf. Supplementary Fig. 4), as could be evaluated by, for example, THz-optical photoemission spectroscopy8. Our coherent multidimensional movies show that light is a vector that can modify the electron mass, and physical properties that derive from it.

Discussion

Besides the dressing and k||-dispersion entanglement and dephasing, however, we must consider the dressing from a solid-state perspective. A quasiparticle in a solid is defined by its many-body interactions, specifically how a solid-state plasma responds to its Coulomb fields on a subfemtosecond screening timescale24,31. Specifically, the IP1 band forms through the many-body screening response of a metal26,32, and therefore, the screening and dressing temporal responses must be interdependent, as further theoretical and experimental scrutiny should elaborate.

Finally, we compare our results to the femtosecond two-color angle-resolved photoemission spectroscopy of Floquet bands in a Bi2Se3 topological insulator (see Supplementary Note 5 for details) and the transition metal dichalcogenide, WS2, in polarization-dependent optical spectroscopy. Gedik and coworkers16,33 demonstrated how time-coincident nonresonant mid-infrared excitation introduces new Floquet bands, and causes Bloch–Siegert band shifts. By contrast, here, we report the optical dressing under three-photon resonant conditions where the Stark shift dominates. We report the optical phase-resolved, subfemtosecond, $${\cal{E}}$$(τ)-dependent, entangled AT dressing through a nonlinear resonance between the equilibrium Bloch SS and IP1 bands. Such resonant dressing has only been reported for the fundamental semiconductor band gaps with all-optical k-integrated techniques12,13,14.

Our 3D coherent photoelectron spectroscopy movies show that electronic bands are no longer defined just by the lattice potential, but applying a time-periodic external field introduces new Floquet bands, and causes shifts of nonlinearly excited resonant electronic bands through the AC Stark effect; specifically, breaking of the time-translation symmetry creates new nonequilibrium electronic structures within the solid state1, and opening of gaps between the time and space-derived bands entangles the space (k||) and time degrees of freedom, introducing frequency responses below that of the driving field. We stress that the interferometric-time- and angle-resolved mPP technique is particularly suited for recording of energy and momentum-resolved real-time movies of the field-dependent dressing of solids. Our multidimensional spectroscopy approach shows that optical dressing dynamics that may be obscured in time-integrated light intensity-dependent spectra, can become conspicuous by probing the induced coherences in electric field-dependent measurements. This experimental approach is applicable to all solid-state materials where momentum-dependent Floquet engineering might cause, for example, a time-dependent change of a material’s electronic band topology9,17. Moreover, the entanglement between dressing and band dispersions provides the means to optically control the physical properties that depend on the quasiparticle band mass on a few femtosecond timescale. Our study demonstrates that quantum optics of the oldest optical material, a metallic surface, along with those of other high electron density solids, no longer reflect the passive coherent responses to optical fields, but can be actively controlled with subfemtosecond precision by nonlinear interactions with judiciously phase and amplitude tailored optical fields.