1 Introduction

The femtosecond frequency comb allows direct control over the optical carrier field in ultrashort laser pulses by virtue of controlling the carrier-envelope phase (CEP) [1]. Hence, with the femtosecond frequency comb the optical carrier wave of ultrashort laser pulses can now be fully controlled, which was so far only possible at much smaller frequencies of the electromagnetic spectrum, such as in the microwave domain. This progress allows new control over processes that are driven by the optical carrier field rather than by the intensity envelope of laser pulses. Prominent examples include high-harmonic and attosecond pulse generation, which are based on strongly driving electrons in gas-phase atoms [2–4].

In this contribution, the recent development of phase-coherent electron control at the surface and in the volume of solids will be discussed. While spectacular control over electrons in atoms and molecules in the gas phase has been obtained in the last one to two decades [5], the work on solids was rather limited. In his Nobel Prize Lecture [6], Theodor Hänsch refers to initial work by Nakajima and Lambropoulos on above-threshold ionization (ATI) of gas-phase atoms in this context [7]. The corresponding process in solids is above-threshold photoemission (ATP) and has been investigated by Theodor Hänsch and colleagues already in 2004 [8]. ATP describes (multi-)photon-driven emission of electrons with a (kinetic) energy larger than the minimally required energy by one and up to many photons. Even though the authors of  [8] used extremely short laser pulses with a duration of only 4 fs, corresponding to 1.6 optical cycles, the carrier-envelope phase-dependent current they could detect was minuscule. This was for one obvious reason, which is spatial averaging over the Gaussian-shaped focal profile on the flat metal surface. But not only does the spatially varying intensity lead to a varying nonlinear order and spatially varying light shifts, also the focal phase may vary transversally for such broadband light fields [9, 10]. It turns out that working with sharp needle tips circumvents these issues: When laser pulses are focused at tips with an apex radius of curvature much smaller than the central laser wavelength, optical field enhancement takes place [11]. The optical field is preferentially enhanced at the apex, so that especially nonlinear effects such as ATP are quite dramatically enhanced at the apex and are thus limited in space to the tip apex region [12–14]. Typical tip radii lie in the range of a few nanometers to around 100 nm, so much smaller than focal spot sizes.

Observing photon orders in electron spectra implies that the coherence of the laser beam is transferred into the electron domain. While early electron spectra recorded from flat metals hinted at above-threshold photoemission [15], it remained open if the coherence of the laser beam was not washed out by a thermal process because photon orders could not be observed. It was therefore insightful that in spectrally resolved photoemission measurements from sharp needle tips above-threshold multi-photon orders were clearly visible [16]. With increasing intensity, these photon orders shifted to smaller energies, evidencing the onset of strong-field effects such as peak shifting and channel closing [16–19]. With only small DC voltages applied to the tips, the characteristic plateau in electron spectra became apparent [20–23], indicating that field-driven electron dynamics is starting to take place. With carrier-envelope phase-stable laser pulses, the laser-field-driven dynamics could be clearly proven [20, 24]. The original work, which is now several years old, has been reviewed and commented on in several occasions to which we refer the interested reader (see, for example, [25–28]).

We could show that electron emission in the energy range of the electron re-collision cutoff can be switched on and off by virtue of the carrier-envelope phase [20]. From there it is only a small step to realize that a light-field driven interface can be built from two tips that can result in an ultrafast, light-field driven switch. With two tips facing each other we have started to work in this direction. In essence, we have so far demonstrated a nanoscale version of a vacuum tube diode [29]. Yet, the demonstration of light-field driven electronic switching in a nanoscale device that allows utilizing the attosecond fast timescales is still elusive.

The physics discussed so far is based on photoemission of electrons from a tip and subsequent (strong-field) dynamics of the electrons inside of the laser field in front of the tip. For this very reason, namely that it is mostly the dynamics of the field-driven electron after it has been born into the laser field that governs its final state, we see such a great similarity between gas-phase strong-field physics spectra and the spectra obtained from solid tips. Examples include the exponentially dropping low-energy part, but most importantly the famous plateau and the similarly famous “10-\(U_p\) cutoff.” For this reason we can explain the physics nicely with models well known from atomic physics [20–22, 30–33], which is also discussed in the original and review papers mentioned above. We now turn to discuss two new phenomena recently observed.

2 Two-color experiments at metal tips

Intriguing effects can arise when a harmonic of the fundamental driving field is superimposed on the fundamental of the optical pulse. A two-color or bichromatic field pulse results, representing a first step (next to CEP control) toward more complex, tailored light fields, with the goal of improving the level of control over reaction or emission pathways, extending electron energies, shaping electron pulses etc. The relative phase between the colors represents an additional control parameter. Investigations of the effects of bichromatic light fields on atomic ATI date back more than two decades [34] and have been utilized to further explore the transition from the multi-photon to the tunneling ionization regime, for example [35]. An extensive review on earlier work of the atomic (strong-field) physics in bichromatic laser fields can be found in [36]. More recently, bichromatic fields have been used to probe molecular strong-field dynamics on the 10-attosecond timescale [37, 38].

We focus the fundamental and the second harmonic of near-infrared laser pulses on a sharp tungsten tip at ambient temperature and vary the temporal delay between the two pulses. Two cases may be considered: In the strong-field regime, the carrier fields add up to yield an asymmetric optical field structure in which the carrier maxima can be enhanced by the second harmonic for one polarity, while for the other the extrema are reduced, for instance. The role of the polarity exactly changes if the second harmonic is delayed by half an optical carrier period, yielding a reversed field structure. This will have direct consequences for the strongly driven electron. On the other hand, for weaker fields, or in the perturbative regime, different electron emission quantum pathways, comprised of sequences of photons of fundamental and second harmonic (with different phases), may constructively or destructively interfere. Here we report on observations in this regime.

Fig. 1
figure 1

Electron current emitted from a 10-nm sharp tungsten tip when irradiated by a two-color laser field as function of time delay between the fundamental and second-harmonic laser pulse. a A large time span between full and hardly no overlapping pulses is shown. b The central region of a, clearly revealing oscillations with the period of the second harmonic (2.6 fs). Data and a sine fit are shown. The large contrast of 94% in this central region is also directly visible. Data points are connected by line segments as a guide to the eye

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We frequency-double a part of the \(\sim 70\)-fs output pulses of an erbium-doped fiber oscillator–amplifier running at a central wavelength of 1560 nm and at 100 MHz repetition rate. With the help of a Mach–Zehnder interferometer with dichroic beam splitters, we can continuously vary the delay (or phase) between the frequency-doubled pulses (“\(2\omega\)”) and the fundamental pulses (“\(\omega\)”). When we focus these \(\omega -2 \omega\) pulses on the tungsten tip and record the emitted electron current as function of time delay, we observe a strong modulation of the emitted current as function of delay, see Fig. 1. Notably, the modulation reaches up to \(94\%\) for near-field intensities of 330 GW / cm\(^2\) of the fundamental and \(6.6\,{\text {GW/cm}}^2\) for the second harmonic. So a weak second-harmonic admixture of \(\sim 1/50\) in relative intensity magnitude suffices to induce such a strong modulation of the output current.

Fig. 2
figure 2

Quantum path interference model we employ to explain the strong modulation of the emission current as function of delay between fundamental and second-harmonic pulses, as shown in Fig. 1. One pathway consists of four photons of the fundamental required to lift an electron from the Fermi level to a bulk state extending to the tip surface. The other pathway consists of two fundamental photons and one photon of the second harmonic (blue). From there the electrons can easily overcome the Schottky-lowered barrier. The full lines show the local density of states, both for the surface (orange) and the bulk (black), as discussed in [39]

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We explain these effects with quantum path interference as indicated in Fig. 2. For the DC-biased tungsten tip, the emission barrier at the (310)-W apex is Schottky-lowered from an unbiased 4.3–3.6 eV. Hence, four fundamental photons of 0.8 eV energy are required to lift an electron to the barrier. Alternatively, two fundamental and one frequency-doubled photon provide the same energy to an electron. We interpret our data in such a way that it is these two quantum pathways that interfere. A much more detailed discussion can be found in [39].

While the discussed two-color effects are already of fundamental interest, they may enable boosting the maximum output current without damaging the tip with the fundamental laser intensity. Furthermore, it will be even more interesting to investigate bichromatic effects at tips in the strong-field regime. With the relatively long laser pulses employed here, the strong-field regime can hardly be reached without damaging the tip, but with shorter fundamental laser pulses at around 1800 nm [40], we expect that this will be easily possible, in particular because of the longer central wavelength as compared the standard 800 nm that have been mostly used so far for strong-field studies at tips.

3 Light-field-induced currents in graphene

The effects discussed in the previous section relate to quantum effects inside of tips, however, in the perturbative regime. In recent years, strong-field physics inside of solids has found much attention, as new effects related to the solid state nature of the samples are expected that do not show up in the atomic and molecular case. So far, large-bandgap materials and/or long-wavelength driving pulses have been used so as not to excite electrons from the valence to the conduction band easily. For example, high harmonic generation has been observed from the large-bandgap semiconductor ZnO (band gap of 3.2 eV), driven with pulses with a central wavelength of 3.2– 3.7\(\,\mu {\hbox {m}}\) [41] or with terahertz frequencies [42]. Reversible attosecond dynamics of strongly driven fused silica glass and silicon have been investigated with the help attosecond transient absorption spectroscopy [43, 44]. The question arises if next to dielectrics and semiconductors also metals can be strongly driven by the optical field of ultrashort laser pulses.

Graphene, a (semi-) metal, is ideally suited for strong-field studies, in particular graphene on SiC. It is a robust material on a robust large-bandgap substrate with good heat conductivity, but first and foremost monolayer graphene stands out for its two-dimensional nature and its cone-shaped Dirac dispersion relation [45]. We have performed laser-induced current measurements and have investigated whether a CEP-dependent current, so again a light-field induced current, can be observed [46].

Fig. 3
figure 3

CEP-dependent current as function of glass thickness in the beam path, leading to a variation of CEP and pulse duration. In-phase (spheres) and quadrature component (squares) of a lock-in measurement are shown, together with fits to the data

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Driven with \(\sim 6\)-fs pulses derived from a Titanium:Sapphire oscillator running at 80 MHz, strongly focused down to \(1.6\,\mu {\hbox {m}}\) (\(1/e^2\) intensity radius) on graphene and leading to peak field strengths of \(0.3\,{\hbox {V}}/\AA\), a carrier-envelope phase-(CEP)-dependent current with a peak magnitude of \(\sim 15\,{\hbox {pA}}\) can be observed, see Fig. 3. Surprisingly, when we vary the peak field intensity, we observe a change in current direction for one and the same CEP setting. While the details are discussed in [46], here we only show the intriguing underlying physical picture that we assume to hold.

Fig. 4
figure 4

a Graphene’s Dirac-cone-shaped dispersion. Here the hyperbolic bands are highlighted that result from the intersection of a plane with a finite \(k_y\)-value with the dispersion cones; note that the K-point lies outside of that plane. Electrons with all initial \(k_y\)-values are driven by the laser field. b Sketch of the dynamics in the two hyperbolic bands shown in a. The laser field is polarized in x-direction. Because of the oscillatory nature of the laser field, the electron is driven repeatedly over the apparent band gap where it can always either undergo a diabatic transition from one band to the other, or stay in the same band (Landau–Zener dynamics). The apparent band gap thus serves as an electron beam splitter. The relative phase between the two arms determines whether the electron ends up in the conduction band (as shown here) or in the valence band. The relative phase depends on the CEP via the electric waveform of the driving field. In order to obtain the resulting CEP-dependent current, an integration over all initial \(k_y\)-values is required

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The semiclassical picture of electron wavepackets in the band structure environment of solids helps to gain insight. Bloch’s acceleration theorem states that an electron is accelerated by an external electric field polarized along x in x-direction only: \(\hbar \dot{\mathbf {k}} = -e {\mathbf {E}} ,\) where \({\mathbf {k}}\) represents the electron momentum, \(- e\) the electron charge and \({\mathbf {E}}\) the externally applied electric field. Because of the cone-shaped nature of graphene’s dispersion, a slice exists in the dispersion relation such that electrons can be driven diabatically across the band crossing even for vanishing field strengths (at \(k_y = 0\), for example), but there also exist slices where the energy between the bands is large so that the electrons remain inside of the band and are just driven within it by the laser field (adiabatically) [47], see Fig. 4. Here, diabatic means that the electron is transferred from the valence to the conduction band, or vice versa, so an interband transition. Conversely, the adiabatic motion implies intraband motion, so an adiabatic exchange of pseudospins labeling the quantum number of the wave functions [45]. The pseudospins are conserved in the diabatic case.

Between these two extrema, a dispersion cone slice exists in which valence band electrons may become excited via Landau–Zener transitions to the conduction band or stay in the valence band, with equal probabilities. This represents a 50 / 50-beam splitter for electrons. It is this beam splitter that is responsible for the CEP-dependent current generation in graphene, see Fig. 4b. Because the optical pulse drives electrons more than once through the energy minimum, at least two beamsplitting events can take place. Much like in an optical interferometer, it is the relative phase between the two arms that determines if interference into a given output port, here the conduction band, for example, is constructive or destructive. By virtue of the CEP we can vary the phase, as it directly influences the propagation of the electrons inside of the interferometer arms and hence the relative interferometer phase. An in-depth discussion including numerical modeling is presented in [46].

To summarize, initial work on strong-field effects at solids showed already a high degree of coherence of the light-field-driven electrons, exemplified in the existence and the complex but well-understood behavior of above-threshold orders. Here we have discussed more recent work on electron emission from needle tips with bichromatic laser fields. We could show that coherent quantum path interference leads to a large variation of the emitted current as a function of delay between fundamental and second harmonic of the driving pulses. The visibility of up to 94% seems to be the highest observed in two-color photoemission experiments over all investigated systems, which is surprising given the metallic nature of the room temperature metal tip as compared to clean gas-phase atomic systems [48], for instance. In the second discussed experiment, we focus on the coherence of electrons inside of the special metal graphene. Here we observe a CEP-dependent current that we explain with a light-field-driven interplay of interband and intraband dynamics, representing another example of laser-field-driven coherent quantum path interference, now inside of a peculiar type of two-dimensional metal—the discovery of which is possible because of the existence of the femtosecond frequency comb.

4 Concluding remarks

The work described here has been done long after the author had finished his PhD thesis in Professor Hänsch’s group at Ludwig-Maximilians-Universität in Munich in 2002, which was already a most enjoyable time. At Stanford University in the group of Mark Kasevich, Yvan Sortais was setting up a femtosecond frequency comb for a precision spectroscopy experiment. At the other end of the optical table, the author had just succeeded in achieving field emission from a single-atom tip in early 2004 (published only much later [49]). From the atomic precision spectroscopy point of view certainly an abuse but probably in line with the Hänsch (and Schawlow) style of doing physics [50, 51], Yvan Sortais and the author decided to rather fire the femtosecond frequency comb laser pulses at the apex of a sharp metal needle tip and so realized an electron source able to generate femtosecond electron pulses from a nanometric emission area [52]—at the time without being able to observe a carrier-envelope phase-dependent current, even though they had tried right away [53]. Equipped with Max Planck Research Group funding, the author went to MPQ to again become a member of the Hänsch department in 2007. It was there where the author’s small group performing Spitzenforschung (translate: tip research) was engulfed and fostered in Theo Hänsch’s group doing Spitzenforschung (translate: top research): From then on until 2013, the author and his group enjoyed a most fruitful and stimulating atmosphere. The weekly Hänsch department seminars provided excellent exchange of scientific and technical ideas, as did the annual highlight of they year, the Hänsch department retreat at Ringberg Castle, resulting in research highlights discussed in the introductory part of this contribution and further including the demonstration of an on-chip Paul trap for electrons as well as the demonstration of laser acceleration of electrons at dielectric structures [54, 55]. A number of joint publications resulted from common experimental interests and requirements, for example on ultrashort pulse generation and single-pass laser amplification [56–60] and on the CEP stability of laser oscillators with various pump lasers [61, 62]. In 2013, with the author’s appointment to Friedrich-Alexander-Universität Erlangen-Nürnberg, his group had to leave the Hänsch department. Theodor Hänsch has profoundly influenced the author’s life and also that of the many students and postdocs who had the pleasure to interact with him. Happy birthday and ad multos annos, lieber Herr Professor Hänsch!