Real-time tracking of coherent oscillations of electrons in a nanodevice by photo-assisted tunnelling

Coherent collective oscillations of electrons excited in metallic nanostructures (localized surface plasmons) can confine incident light to atomic scales and enable strong light-matter interactions, which depend nonlinearly on the local field. Direct sampling of such collective electron oscillations in real-time is crucial to performing petahertz scale optical modulation, control, and readout in a quantum nanodevice. Here, we demonstrate real-time tracking of collective electron oscillations in an Au bowtie nanoantenna, by recording photo-assisted tunnelling currents generated by such oscillations in this quantum nanodevice. The collective electron oscillations show a noninstantaneous response to the driving laser fields with a T2 decay time of nearly 8 femtoseconds. The contributions of linear and nonlinear electron oscillations in the generated tunnelling currents were precisely determined. A phase control of electron oscillations in the nanodevice is illustrated. Functioning in ambient conditions, the excitation, phase control, and read-out of coherent electron oscillations pave the way toward on-chip light-wave electronics in quantum nanodevices.

Interaction of light with metallic nanostructures can lead to a collective oscillation of conduction electrons.
If the frequency of the incident light matches with the intrinsic resonance frequency of the collective electron oscillations (surface plasmons) in nanostructures, such oscillations can be dramatically amplified 1,2 .The resulting strong electromagnetic field arising from the driven collective electron oscillations has now found many applications, ranging from atomic scale nano-optics [3][4][5][6] to single molecule sensing 7 .Moreover, it enables exploring the nonlinear optical response of matter [8][9][10][11][12][13][14][15][16][17][18][19] .On interaction of strong electromagnetic fields with matter, nonlinearity in electron oscillations sets in, implying that the electron motion does not remain harmonic anymore.Anharmonic electronic motion implies that electrons oscillate with many frequencies, which are multiples of the driven frequency, in close analogy to a classical anharmonic (driven) oscillator.
In absence of the capability to directly resolve coherent electron oscillations in the time domain, interaction of light with matter has been studied by spectral measurements (in the UV to infrared range) utilizing the techniques of absorption spectroscopy and transient reflectivity 20,21 .Signatures of nonlinearity in lightmatter interactions in nanostructures have also been studied by spectral measurements, e.g.second and third harmonic generation [22][23][24] .Ultrafast techniques such as time-resolved two-photon photoemission 25,26 (TR-2PPE) and time-resolved scanning near-field optical microscopy 27,28 (TR-SNOM) have been successfully applied to monitor ultrafast plasmon dynamics at the nanoscale.Such measurements can retrieve the ultrafast evolution of the spatially dependent plasmonic electric fields 29,30 , nevertheless, they do not capture the phase information of the frequency-dependent plasmon oscillations.A direct sampling of the ultrafast coherent collective electron oscillations and the resulting local electric field in the time domain has not been reported yet, which is highly desired since it is the key to modern photonic functionalities operating at petahertz frequencies, ultrafast switching, and all-optical signal processing [31][32][33][34][35] .
Here, we demonstrate a novel approach wherein both plasmon oscillations and nonlinear electron oscillations arising from the nonlinear optical response induced by ultrashort laser pulses in a strongly lightinteracting quantum nanodevice can be traced directly in the time domain.Our nanodevice comprises of Au bowtie nanoantennas, with a junction gap of only a few nm.Electron oscillations in the nanodevice were traced by recording photo-assisted tunnelling currents.The plasmonic oscillations shows a noninstantaneous response to the driving electric field of the laser pulses with a decay time of ~ 10 fs.The spectral phase of the plasmonic field is highly dispersive in close agreement with a classical harmonic oscillator driven at its resonance frequency.Furthermore, we show that the contributions of linear and nonlinear electron oscillations in the generated photo-assisted tunnelling currents can be precisely deciphered and sampled in real time.Coherent control of plasmon oscillations directly in the time domain in the quantum nanodevice is also demonstrated.

Real-time sampling of coherent collective electron oscillations
In our experiments, arrays of seven identically designed Au bowtie nanoantennas (Fig. 1a) of ~ 300 nm size (isosceles triangle) with a junction gap of a few nm fabricated on top of a fused silica substrate (see section I in SM for details) were illuminated with two ultrashort laser pulses (pulse duration, τP ~ 7 fs) of slightly different carrier frequencies.Fig. 1b shows a scanning electron microscope (SEM) image of seven Au bowtie nanoantennas.The plasmonic response of these nanoantennas as a function of incident laser wavelength and the spatial distribution of the local field enhancement in the junction were calculated by finite element simulations, as shown in Fig. 1c (see section II in SM for details).Owing to the high field enhancement at the junctions of the bowtie nanoantennas, plasmon oscillations will dominantly be excited when focusing the incident laser pulses in these nanoantenna junctions.In order to time resolve the plasmon oscillations induced by the ultrashort laser pulses, we probe the homodyne beating signal between plasmon oscillations induced by two ultrashort laser pulses with a very small difference in their carrier frequencies.
Here, we briefly explain the technique of homodyne beating, a self-referencing method to measure e.g. the phase of plasmon oscillations that are induced by the laser pulses in the nanoantennas (see section III in SM for details. ).Ultrashort laser pulses (τP ~ 7 fs) coming at a repetition rate of ~ 80 MHz (fr) were passed through an acousto-optic-frequency-shifter (AOFS), which is driven at the frequency of fr + f0 as shown in Fig. 1d, f0 is ~ 700 Hz.The 0 th and 1 st order diffraction beams coming from the AOFS are slightly shifted in their carrier frequencies, by f0.These two laser pulses are then combined and focused in the junction of the nanoantennas, as schematically shown in Fig. 1e.The excited collective electron oscillations in the junction produce a local electric field, which can be expressed as a convolution of the incident laser field Ei(t) and the optical response function (R(t)) of the nanoantennas; where the subscripts, i = 1, 2 denote the two different laser pulses.The net electric field produced by the electron oscillations induced by the two laser pulses at the junction of nanoantennas can then be expressed as;   where nfr is the n th (range from ~ 3.5  10 6 to ~ 6  10 6 ) multiple of the repetition rate of the laser pulses, τ represents the delay between the two pulses.
Due to the highly localized field enhancement in the nanoantenna junction, only the electrons close to the junction can be excited on interaction with photons and tunnel across the junction.A bias voltage is applied on the anoantenna to facilitate the electron tunnelling processes.The mechanism of photo-assisted tunneling in the nanoantenna junction will be discussed later in the text.The single-photon assisted tunnelling of electrons induced e.g. by the plasmon oscillations in the nanoantenna junction is proportional to the linear polarization of the system; A real-time sampling of the plasmon oscillations induced by the laser pulses with total pulse energy of ~ 100 pJ is shown in Fig. 2a.The plasmons undergo an oscillation period of ~ 2.6 fs.Fourier transform of the time trace in Fig. 2a reveals the spectral shape of the plasmonic response of the nanoantennas (Fig. 2b), which closely resembles the spectral shape of the plasmonic response evaluated from the finite element simulations (Fig. 1c and Fig. S2 in SM).A comparison of the spectrum of this plasmonic response with the spectrum of the incident laser pulses reveals significant spectral and temporal shaping of the laser pulses in the nanoantenna junction, as shown in Fig. 2b.A broad plasmonic response of the nanoantennas (Fig. 1c) implies a very fast damping rate of the induced plasmon oscillations, in the range of only a few fs 36 .The decay of the plasmon oscillations can be seen by a long and asymmetric oscillation trace on the positive side of the delay axis in Fig. 2a, as indicated by the black arrows.
Non-resonant excitation of bound electrons in a system is usually instantaneous, i.e. the electrons will oscillate in phase with the driving electric field and the oscillations will fade out as soon as the impetus from the driving field is over 37 .However, when the electrons are excited on resonance, their response is with respect to the driving electric field and it has a much longer decay time.Such delayed response of bound electrons has been earlier reported for a dielectric medium 37,38 as well as for an atom 39 .Furthermore, another key distinctive feature of the resonant electron oscillations compared to the non-resonant case is their phase curve along the frequency axis.The phase curve around the resonance frequency is very dispersive, and the phase difference along the two extrema of the resonance frequency is ~ π radians.
A bound electronic system, such as plasmons in the nanoantenna junction with the restoring force from the ions of the nanoantennas, can be simply modelled as a driven damped harmonic oscillator 40,41 .The electric field of the plasmonic oscillations (EPl) can be expressed as; where R  is the resonance frequency of the plasmons, ELaser is the electric field of the driving laser pulse, and  is the intrinsic damping rate of the plasmon oscillations, which is determined by the bandwidth of the spectrum of the plasmonic resonance.
Fig. 2c shows a comparison of the calculated plasmonic field, considering a resonance frequency ( R  ) at ~ 1.6 eV and a damping rate  of ~ 0.12 fs -1 , with the incident electric field of ~ 7 fs Fourier-limited laser pulse.The local electric field resulting from plasmon oscillations as measured in the experiment is in good agreement with the calculated plasmonic field as shown in Fig. 2c.A long tail associated with the damping of the plasmon oscillations along the positive delay axis can be clearly seen, which decays on the time scale of ~ 10 fs.A direct comparison of the experimentally measured plasmon oscillations, the electric field of the driving laser pulse, and the calculated plasmonic field is shown in Fig. S3 of the SM.The simulation also shows that the peak electric field of the plasmon oscillations is delayed by ~ 3 fs with respect to that of the driving laser pulse (Fig. 2c).
The spectrum and the phase of the plasmon oscillations as captured in the experiment are contrasted with those obtained from calculations (obtained by Fourier transform of EPl) in Fig. 2b.Plasmons undergo a spectral phase shift of nearly π radians across its resonance curve as also consistent with the model.The spectrum of plasmon oscillations as simulated from the model is also in good agreement with the measurement.This consistency further attests the validity of the simple damped harmonic oscillator model describing the plasmon oscillations as measured in the experiments, and transparently demonstrates the capability of our technique to time resolve ultrafast plasmon oscillations.In addition to allowing access into the near-field plasmon oscillations (EPl), an inversion of equation ( 2) also enables a direct measurement of the far field of the driving laser pulse.

Unravelling linear and nonlinear contributions in light-matter interaction
At higher field strength of the incident laser pulses, nonlinear electron oscillations, induced by higher order polarization responses of the nanoantenna junction set in due to the strong plasmon-enhanced light-matter interaction 42 (see section III in SM).Fig. 3a shows the temporal evolution of the local second-order nonlinear oscillation of electrons induced at a total pulse energy of ~ 200 pJ.Photocurrent generated in the nanodevice due to the second-order nonlinear polarization response of the nanoantennas can also be measured with the homodyne beating technique; in detection of the photocurrent signal at twice the offset frequency (2f0) enables temporal sampling of the second-order nonlinear electron oscillations in the junction; entailing its complete phase information.At higher pulse energy, ~ 300 pJ, the third-order nonlinear oscillations of electrons, induced by three-photon absorption, can be recorded as shown in Fig. 3b; measured by performing lock-in detection at 3f0 frequency.
However, the measurements at such high laser pulse energies is not very stable as the nanodevice is prone to physical damage.The oscillation periods of nonlinear electron oscillations for the case of 2 nd and 3 rd order optical responses are ~ 1.3 fs and ~ 0.9 fs, respectively.The spectral responses of 2 nd and 3 rd order nonlinear electron oscillations reveal a significant spectral shaping due to the multi-photon interactions in the nanoantenna junction, as shown in Fig. 3c and 3d.
In the weak-field or perturbative regime, light-matter interaction is usually characterized by a power-scaling experiment, wherein a physically relevant quantity is measured as a function of increasing intensity of the laser pulses.A n th order nonlinearity implies the interaction with n number of photons to be dominated 43,44 .
Nevertheless, n-1 and n-2 photon orders in the light-matter interaction do not cease to exist, but their contributions are harder to access.A technique capable of deciphering the contribution of all photonchannels (linear and nonlinear polarizations) at a particular intensity of the laser pulse in the process of light-matter interaction is considerably sought after.Here, we demonstrate the technique of homodyne beating as a powerful tool to precisely decipher the contributions of different photon channels in the lightmatter interaction.
The variation of the total photocurrent generated by the laser pulses in the junction of the nanoantennas as a function of the increasing intensity is shown in Fig. 3e (black data points), measured by intensity modulation (at ~ 520 Hz) of laser pulses.In a dual logarithmic plot, the scaling of the total photocurrent shows a switch from a linear scaling (slope of 1) at lower intensities to a quadrating scaling (slope of 2) at higher intensities of the laser pulses, indicating the contributions from both linear and nonlinear polarization responses of the nanoantenna junction at higher intensities.In order to disentangle the contributions of the different photon-channels, the variation of the photocurrent signal at zero delay between pulse-1 and pulse-2 (Fig. 1e, see also section I in SM) was measured as a function of intensity of the laser pulses for two different frequencies in the lock-in detection, f0 and 2f0, as shown in Fig 3e .The scaling of the lock-in signal at f0 frequency is similar to the behaviour of the total photocurrent, since this signal can arise from both linear as well as local nonlinear polarization responses of the nanoantenna junction (with a prefactor of 0.5, see section III in SM for details).However, the signal at 2f0 frequency can only arise from the secondorder nonlinear polarization response (with a prefactor of 1/8).Thus, its scaling with respect to the intensity of the laser pulse is purely quadratic (Fig. 3e).This change from linear (dashed green curve) to quadratic (dashed orange curve) behaviour in the scaling of the total photocurrent signal in Fig. 3e occurs at the similar local field strength of the laser pulses where the signal at 2f0 frequency starts emerging, as indicated by a vertical black-dashed curve in Fig. 3e.Thus, demonstrating a direct measurement of the contribution of the 2 nd order light-induced polarization response (nonlinear electron oscillations).We note that the pulse energies were kept below ~ 220 pJ in the measurement in Fig. 3e to avoid damage of the nanoantennas.The contribution of photocurrent at 3f0 frequency is too weak to be reproducibly determined, but in principle, can be measured with our technique.In conclusion, by recording the homodyne beating signal at f0 and its harmonic frequencies (2f0 and higher), we can precisely determine the contributions of linear and nonlinear electron oscillations (polarization responses) in the generated tunnelling currents in the nanodevice.

Photo-assisted electron tunnelling in the nanoantenna junction
In the following, we discuss the mechanism of photocurrent generation in the junction of the Au bowtie nanoantennas.The determination of the contributions of one-and two-photon processes in the power scaling measurements of the photocurrent signal in Fig. 3e excludes the mechanism of laser field-driven tunnelling in the nanoantenna junction, where a much less nonlinear power-dependent behaviour is expected.Besides, the Keldysh parameter for our pulses at the nanoantenna junction is above 4, where the laser field-driven tunnelling effects are virtually absent 44 .In order to understand the mechanism underlying the photocurrent generation, we measured the variation of the photocurrent signal at f0 frequency as a function of the increasing bias voltage applied in the nanoantenna junction, as shown in Fig. 4a.The photocurrent signal shows an extremely nonlinear dependence on the applied bias in the junction.
Therefore, over-the-barrier photoemission 44 can be excluded, as it would be virtually insensitive to the small biases applied in the junction.Moreover, the electrons excited by plasmon oscillations, following photoexcitation by laser pulses, can be up to approximately 1.5 eV above the Fermi level of Au, but, still significantly below the tunnelling barrier of Au (~ 5 eV) and cannot induce photoemission (photocurrent) in the measurement.
Here, we describe photocurrent generation by a simple model accounting for tunnelling of electrons across the nanoantenna junction with an effective Fermi electron distribution 45 that is photo-excited by the laser pulses, as shown schematically in Fig. 4b.In the case of photo-assisted tunnelling 46 , electrons from one side of the junction are photo-excited via one-, two-or three-photon absorption and then tunnel to the other side through a reduced effective tunnelling barrier.The calculated electron tunnelling probability as a function of increasing bias (see section V in SM) for a junction of gap-width of ~ 1.2 nm matches quite well with the experimentally measured nonlinearity of the photocurrent signal (Fig. 4a).The junction (tunnel) gap of ~ 1.2 nm is significantly bigger for any DC tunnelling (below 5 V) but not for photo-assisted tunnelling.It is worth mentioning that the designed junction gap of the nanoantennas is ~ 10 nm.However, at such small dimensions one reaches the limitation of the state-of-the-art lithographic techniques.The actual junction gap can be significantly smaller due to spillage of Au during the fabrication process and electromigration of Au atoms in the nanoantenna junction by the applied DC bias in the device 47 .

Real-time coherent control of localized plasmon oscillations
We illustrate that the collective electron oscillations induced by the laser pulses can be coherently controlled by varying the CEP of the laser pulses, directly in the time domain.A modulation of the CEP of the laser pulse controls the CEP of the laser-induced polarization, which in turn coherently modulates the CEP of the driven plasmon oscillations.As the technique presented in this work is a self-referencing technique, we probe linear plasmon oscillations induced in the junction of the nanoantennas by varying the CEP of one of the laser pulses (pulse-1 in Fig. 1d), while keeping the CEP of the other pulse fixed (pulse-2 in Fig. 1d).
The CEP of the 1 st order diffracted pulse (pulse-1) is controlled by varying the phase of the radio frequency phase-shifter (see section III in SM) driving the AOFS.Fig. 5a shows the temporal evolutions of plasmon (electron) oscillations as a function of the varying CEP of pulse-1.Four representative traces at the CEP of 0, 0.5π, π, and 1.5π are shown in Fig. 5b.Coherent control of plasmon (electron) oscillations as evident by a linear movement of the maxima of the oscillations on change of the CEP can be clearly seen in Fig. 5a and Fig. 5b.

Conclusion
Direct measurement of light-waves can enable the study of quantum properties of ultrashort pulses, e.g.

Nanodevice Fabrication
The devices were fabricated on fused silica substrates using electron beam lithography, evaporation, and lift-off techniques.One device consists of ten arrays, each containing seven bowties with identically designed geometric dimensions and junctions, (see Fig. 1 in the maintext).In the design, the opposing isosceles triangles (triangle height of 300 nm, base of 250 nm) forming the bowtie are separated by gap sizes of 10 nm, 5 nm, 0 nm, and -5nm.The designed sizes of the junctions, e.g. 5 nm, are identical for all bowties in one array.Negative gap sizes indicate merged (overlapping) triangles.Please note that the fabricated gap sizes will differ from the designed values due to the proximity effect and exposure characteristics of the resist.
The electrical connections are fanned out allowing to electrically connect and address every bowtie array individually.In addition, two bowtie arrays are replaced by a rectangle of 1 x 3.5 microns (short cut scenario) allowing for electrical reference measurements.
For nanofabrication, 80 nm or CSAR 62 resist (Allresist) was spin-coated on a fused silica substrate (10 mm x 10 mm) and baked for 60 s at 180° C. To avoid charging during electron exposure with the Raith Eline Plus system, a layer of ESPACER 300Z (Showa Denko, Singapore) was spin-coated (5000 rpm for 60 s ) on top of the CSAR 62 resist.The patterning of the nanostructures (bowties and 30 nm narrow electrical connections, see electron micrograph in figure 1) was performed with an electron beam energy of 20 keV, a current of 0.02 nA and a dose of 130 µC/cm -2 .To fabricate nanometer-sized gaps, corrections for the proximity effect were included in the design process and the movement of the electron beam was optimized.The micro-and millimeter-sized structures (contact pads and electronic connections, see image in Fig. 1, main-text) were patterned in the same exposure step with the same energy and dose, but a larger electron current (9.5 nA).After exposure, the conducting ESPACER was removed in ultrapure water (2 s).Subsequently, the resist was developed in AR 600-546 (Allresist) for 60 s, stopped in AR 600-60 (Allresist) for 30 s and immersed in propan-2-ol for 30 s.Using electron beam evaporation at a pressure of 5x10 -7 mbar, a chromium (Cr) adhesion layer of 3 nm followed by 30 nm of gold (Au) has been deposited on the developed sample.Lift-off in N-Ethyl-2-pyrrolidon (Allresist) at 80°C was performed for at least 8 hours, to remove the gold-covered and non-exposed CSAR resist and reveal the fabricated structure.

Experimental Set-up
In our experiments, CEP-stable ultrashort laser pulses were split into two arms of the Mach-Zehnder interferometer (

Section II. Finite element simulations of a single bowtie
Numerical simulations were performed using the commercial software COMSOL Multiphysics based on a finite element method.A single bowtie structure is implemented by two identically but opposing isosceles triangles with a base of 250 nm and a height of 300 nm supported by a substrate.The opposing tips and edges of the bowties are modelled with filets (radius of 5 nm).
The height of the bowtie is 30 nm and a junction size of 10 nm is exemplarily selected.The dielectric function of gold was taken from Johnson and Christy 1 and the refractive index of the fused silica substrate was approximated with 1.5 in the spectral range of interest.A refined mesh size of at least 1 nm was used in a volume (50 nm x 50 nm x 50 nm) centered in the bowtie junction to map the fine features of the junction.Perfectly matched layers were placed around the simulation domain to completely absorb the waves leaving the domain.The spectral response and the 2D electrical field distributions of a single bowtie structure are numerically calculated using full field formulation and background field conditions.
Figure S2 shows the extinction cross section of a single bowtie with the abovementioned dimensions.The insets depict the electrical field distributions (taken at the half height of the bowtie) normalized to the background electrical field at the respective wavelengths.The polarization of the incident electrical field is parallel to the bowtie axis.Based on the field distributions we identify the peak at 1420 nm as the first order plasmonic mode and the peaks at 770 nm and 810 nm as a higher order excitation originating from the hybridization of the two opposing isosceles triangles.The double peak feature results from plasmonic excitations in the electrical connections and is not present for bowties without electrical connections (not shown).The shoulder at ~1150 nm has the same origin.

Section III. Homodyne beating detection of the photo-assisted tunnelling current in Nanodevices
An ultrashort CEP stable laser pulse produced in a mode-locked oscillator entails an underlying frequency comb, whose teeth are separated by the repetition rate (fr ~ 80 MHz) of the oscillator.
The frequency comb spans over the entire spectral bandwidth of the laser pulses, i.e. from 650 nm to 1050 nm, with the central wavelength of the spectrum being at ~ 810 nm.The electric field of this laser pulse in the frequency domain can be expressed as where fn = nfr , is the n th multiple of the repetition rate of the laser pulses.() n ff   is the Dirac delta function describing the position of an individual tooth of the frequency comb, which are separated from each other by fr and () n f  describes the spectral weight of each individual comb line in the laser spectrum.The central (or the carrier) frequency (f1) of our laser pulses is ~ 0.37  10 15 Hz (for a central wavelength of λ ~ 810 nm), implying that n ~ 1 r f f is approximately 4.6  10 6 .This frequency comb is emitted in every shot from the oscillator at its repetition rate.
On interaction with the radio frequency (RF) wave in the acousto-optic-frequency-shifter (AOFS) being driven at the frequency of fr + f0, the ultrashort laser pulses undergo diffraction (Fig. 1d, main-text).The frequency comb of the 1 st order diffracted laser pulse ('pulse-1' in Fig. 1e) out of the AOFS is upshifted in the frequency by fr + f0 with respect to the zeroth order laser pulse ('pulse-2' in Fig. 1e).The electric field of the first-order diffracted laser pulse from the AOFS can be expressed as; .
) ( ( 1) Since the order of the frequency comb lines n is much greater than 1 (n ~ 4.6 10 6 ), implying (n+1) ~ n, the above equation simplifies to Thus, the offset between the carrier frequencies of the 1 st and 0 th order diffracted laser pulses is only ~ f0.The excited collective electron oscillations in the junction will produce a local electric field, which can be expressed as a convolution of the incident laser field Ei(t) and the optical response function (R(t)) of the nanoantennas; , where the subscripts, i = 1, 2 denote the two different laser pulses (pulse-1 and pulse-2, in Fig. 1e).The net electric field generated by the combination of the two pulses (0 th and 1 st order diffracted beams, E1 and E2) with a delay τ between them at the nanoantenna junction, in the time domain can be written as: .
where f1 is the carrier frequency of the 0 th order diffracted beam (pulse-2 in Fig. 1e).The total polarization response (linear as well as nonlinear) induced in the nanoantenna junction owing to its interaction with the ultrashort pulses can be expressed as; (  , (2)  and (3)  are linear, second and third order optical susceptibilities, respectively.
Photo-assisted tunnelling current (I1 T ) generated in the nanoantenna junction due to the linear polarization response (one-photon absorption) of the nanoantennas will be proportional to the square of the net first order polarization response induced by the two laser pulses.
22 11 ( ) ( ) Here, most of the components of the tunnelling current due to the linear polarization response come at very high frequencies, such as 2f1 and 2(f1 + f0), which is mixed with the tunnelling current signal at 0 Hz in the nanoantenna junction, thus cannot be measured by lock-in detection owing to the limited bandwidth of the high gain current amplifier.The cross-terms in the above equation, arising due to interference of the plasmon oscillations induced by the two frequencyshifted laser pulses in the nanoantenna junction contribute to the generation of the photocurrent, which oscillates at the small offset frequency of f0 (< 1 kHz) in the nanoantenna junction, which can be measured in the experiments.
In the above analysis we have only considered the linear phase terms of the laser pulses, which is the group delay i.e. f1t and (f1+f0)t, where f1 = nfr.If we consider all the phase terms of the laser pulses, the electron tunnelling current can be expressed as: ) , where ϕ1 and ϕ2 are the complete temporal phases of the two laser pulses.
The phase terms for the two laser pulses can be expanded by Taylor's series with all linear and nonlinear phase terms The dispersion of the two laser pulses, i.e.GDD and higher order phases are identical in our experiment, 1 '' Thus, the polarization term at a fixed delay of τ between the two pulses can be expressed as; Therefore, measuring the linear polarization induced tunneling current arising due to one-photon excitations in the nanoantenna junction as a function of the delay between the two pulses at their carrier offset frequency (f0) enables complete temporal characterization of the laser pulses.
At τ = 0 fs delay between the two pulses (pulse-1 and pulse-2), the above equation imitates the photo-assisted tunnelling current generated by the polarization response induced by a single laser pulse coming at the repetition rate of the small offset frequency of f0 at the nanoantenna junction, The electric field strengths of the pulse-1 and pulse-2 in our experiments are identical, , hence simplifying the above equation to; In the case of a higher order nonlinear interaction of the laser pulses with the nanoantenna junction, the photo-assisted tunnelling current produced in the nanoantenna junction will be proportional to the multiple power of the corresponding terms in the total polarization response.
For example, the photo-assisted tunnelling current due to a coherent two-photon absorption or the 2 nd order nonlinear polarization response in the nanoantenna junction will be; Similarly, the third order nonlinear response would generate a photo-assisted tunnelling current as given by; Measurement of the time-resolved photo-assisted tunnelling current as a function of the delay between pulse-1 and pulse-2 at the lock-in frequency of 2f0 enables sampling of the 2 nd order polarization response (or the 2 nd order nonlinear electron oscillations) of the nanoantennas to the ultrashort laser pulses; . Likewise, time-resolved measurement at 3f0 frequency in the lock-in detection enables sampling of the 3 rd order nonlinear polarization response; . The oscillation period of the second and third-order nonlinear electron oscillations will be one-half (~1.4 fs) and one-third (~0.9 fs) of the local plasmon oscillations (~2.7 fs), respectively.In the current experiments using laser power up to 220 pJ, the contributions of third-order nonlinear are usually weak.
In the intensity scaling experiment shown in Fig. 3e (main-text), measurement of photoassisted tunnelling current at the lock-in frequency of f0 at the zero delay between pulse- However, all the individual terms in the above equation arising due to different orders of the optical response follow completely different scaling laws with respect to the increasing field strengths of the incident laser pulses.Scaling of the photo-assisted tunnelling current in Fig. 3e for the lock-in frequency of f0 shows a switching from the slope of one to the slope of two, indicating the presence of only linear response at the lower intensity of the laser pulses and presence of both linear as well as second order responses in the nanoantenna junction at higher intensity of the laser pulses.Third-order response, which would contribute to a slope of three in Fig. 3e are not present for the intensities of the laser pulses used in this experiment.The intensity of the laser pulses was intentionally kept below ~220 pJ in order to avoid irreversible physical damage of the nanoantennas, which occurs at higher intensity of the laser pulses.
Nevertheless, at a higher intensity (~ 300 pJ) of the laser pulses, third-order nonlinear response can be measured as shown in Fig. 3b (main text).
In the measurement shown in Fig. 3e (main-text), photo-assisted tunnelling current at the lockin frequency of 2f0 can only arise due to the presence of the second-order response in the nanoantenna junction, as it shows a purely quadratic scaling in the experiment (Fig. 3e, maintext).
Measurement of the amplitude of the photocurrent at the 2f0 frequency enables a direct access to the contribution of second-order nonlinear response in the photocurrent signal measured at the lock-in frequency of f0 in Eqn. ( 12).At lower incident intensities of the laser pulse, i.e.
below 100 pJ in Fig. 3e (main-text), only linear response can be excited in the junction of the nanoantennas, as the signal measured at the 2f0 frequency is below the noise level and the slope of the scaling curve is one.Only when the signal at 2f0 frequency start to emerge, i.e. above 100 pJ, the second order nonlinear response are generated in the nanojunction and this is when the slope of the scaling curve in the f0 signal gradually changes its value from one to two.
Therefore, by recording the homodyne beating signal at f0 and its harmonic frequencies (2f0 and higher), we demonstrate a direct identification of contributions of 1 st and 2 nd order lightinduced polarizations (electron oscillations) induced in the nanoantenna junction in the measured photocurrents in the nanodevice.Transport of electrons across the nanoantenna junction is primarily determined by the probability of electron tunneling between the apexes of the two antennas (Fig. 4a), which in turn is significantly influenced by the occupation of high-lying electronic sates above the Fermi level of Au nanoantennas ensuing absorption of photons from the laser pulse.We consider here a one-dimensional potential barrier model formed between the two apexes of the nanoantennas and the vacuum tunneling gap as schematically shown in Fig. 4a of the main-text.In the perturbative regime of light-matter interaction, with Keldysh parameter 2 γ > 1, the photoninduced excitation of the electrons above the Fermi level can be modeled by an effective timeaveraged Fermi population distribution function, feff.In the case of non-perturbative lightmatter interaction, where γ < 1, the potential barrier formed between the apexes of the nanoantennas can be significantly modified, this would happen at much higher local field strengths of the laser pulses at the nanoantenna junction, > 10 V/nm.
Briefly, by utilizing the Simmons tunneling model 3,4 , the tunneling probability of electrons across the potential barrier (UB) can be expressed as; 0 ( , ) ( ) ( , , ) where d is the tunnelling gap between the apexes of the nanoantennas, ( , , )

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Coherent collective oscillations of electrons excited in metallic nanostructures (localized surface plasmons) can confine incident light to atomic scales and enable strong light-matter interactions, which depend nonlinearly on the local field.Direct sampling of such collective electron oscillations in real-time is crucial to performing petahertz scale optical modulation, control, and readout in a quantum nanodevice.Here, we demonstrate real-time tracking of collective electron oscillations in an Au bowtie nanoantenna, by recording photo-assisted tunnelling currents generated by such oscillations in this quantum nanodevice.The collective electron oscillations show a noninstantaneous response to the driving laser fields with a decay time of nearly 10 femtoseconds.The temporal evolution of nonlinear electron oscillations resulting from the coherent nonlinear optical response of the nanodevice were also traced in real-time.The contributions of linear and nonlinear electron oscillations in the generated tunnelling currents in the nanodevice were precisely determined.A coherent control of electron oscillations in the nanodevice is illustrated directly in the time domain.Functioning in ambient conditions, the excitation, coherent control, and read-out of coherent electron oscillations pave the way toward on-chip light-wave electronics in quantum nanodevices.
photon processes open new prospects for understanding strong light-matter interaction in solids directly in the time domain as well as pave the way towards on-chip light-wave electronics at petahertz switching and read-out frequency49,50  .

Figure
Figure 1| a, Nanodevice and optical homodyne beating technique for tracing coherent oscillations of electrons.a, Photograph of the nanodevice.The nanodevice consists of a series of seven connected, identically designed, Au bowtie nanoantennas fabricated on top of a fused silica substrate.b, A scanning electron microscope (SEM) image of seven bowties.. c, Numerically calculated plasmonic response (left panel) of a single Au bowtie (junction size of 10 nm) showing the 2 nd order resonance as deduced from the spatial near field distribution (right panel) at 770 nm.The local field-enhancement factor (E/E0) is denoted by the colour code in the colour bar.In the simulations, the electrical field is polarized along the long bowtie axis.d, Laser pulses with a very small offset frequency, f0, in their carrier frequencies are generated by selecting the zeroth-order beam E1(nfr) 'pulse-1' and first-order diffracted beam E2(nfr+f0) 'pulse-2' of laser pulses traversing through an acousto-optic frequency shifter (AOFS).e, Schematic illustration of the optical homodyne beating technique: zeroth and first order diffracted laser pulses from the AOFS are combined and focused onto the nanodevice.Photocurrent generated by the laser pulses in the nanodevice is measured by lock-in detection at the offset frequency of f0.

Figure
Figure 2| Real-time tracking of coherent collective electron oscillations.a, Variation of the laserinduced photocurrent as a function of the delay between pulse-1 and pulse-2 laser pulses of slightly different carrier frequencies (Fig. 1e) in a biased nanodevice, with the bias in the nanoantenna junction being 2.5 V. b, Comparison of the experimental and calculated plasmonic response in the nanoantenna junction.The dashed-blue and dashed-red curves represent the phases of experimentally measured and theoretically simulated plasmonic oscillations, respectively.The dotted-black curve shows the spectrum of the incident laser pulses on the nanoantenna junction.c, Comparison of the electric field of ~ 7 fs long driving laser pulse (dashed black curve) with the theoretically calculated plasmonic field (solid blue curve).Black double arrow indicates a delayed response of the plasmonic oscillations with respect to the electric field of the driving laser pulse.

Figure 3|
Figure 3| Quantifying single and multi-photon light-matter interactions.a, b Variation of the laserinduced photocurrent as a function of the delay between pulse-1 and pulse-2 laser pulses measured at the lock-in frequency of 2f0 and 3f0, respectively.The bias in the nanoantenna junction is 3.0 V. c, Spectral response of the time-resolved electron oscillations in a (red-curve).d, Spectral response of the timeresolved electron oscillations in b (green-curve).e, Measured variation of the photo-assisted tunnelling current as a function of increasing field-strength of the incident laser pulses (top x-axis) on the nanodevices.Violet and red-points show the variation of the photocurrent signal measured with the lock-in detection frequency of f0 and 2f0, respectively.Measurements were performed at the zero-delay between pulse-1 and pulse-2 (Fig.1e).Peak-field strength refers to the maximum of the net electric field produced by the two pulses.Black-points show the variation of the total photo-assisted tunnelling current generated in the nanodevice.Dashed green and orange curves indicate a slope of one (linear) and two (quadratic) in the dual logarithmic plot.

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Figure 4| Photo-assisted electron tunnelling in the nanoantenna junction.a, Black points: Variation of the photocurrent signal measured at the lock-in frequency of f0 as a function of the increasing bias in the nanodevice.The pulse energy of the laser pulses was set at ~ 100 pJ and the delay between the pulses (pulse-1 and pulse-2) was set to zero.Black and red-curves show the calculated electron tunnelling probability, considering only single-photon excitation as a function of the increasing bias in the nanoantenna junction, where the junction gap is 1.2 and 3 nm, respectively.b, Bottom-panel: Schematic of the energy-level alignment in the biased nanodevice.Fermi level (EF) of the Au nanoantenna on the left side (Apex 1, top-panel) is upshifted with respect to the Fermi level of the nanoantenna on the right side (Apex 2, top-panel).Electron oscillations above the Fermi level stimulated by one-, two-or three-photon absorption, can lead to photo-assisted electron tunnelling to the other side of the junction, as also schematically illustrated in the top-panel.

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Figure 5| Coherent control of plasmon oscillations.a, A series of time-resolved plasmonic oscillations measured as a function of the CEP of pulse-1 in the nanodevice.The CEP of pulse-2 was kept fixed.Electron oscillations were sampled at the lock-in frequency of f0.b, Measured temporal variation of electron oscillations from the measurement in a for four different CEPs of pulse-1.The CEPs of the pulse-1 are annotated on top of each curve.The traces are shifted vertically for clarity.
Fig. S1) by a beam splitter (BS).Laser pulses in one arm of the interferometer were loosely focused onto an acousto-optic-frequency-shifter (AOFS) by a biconvex lens of a focal length of ~ 25 cm.A transverse radio frequency wave of frequency fr + f0 (~ 80 MHz + 700 Hz) runs through the AOFS with the average power of the incident laser pulses being ~300 mW.The fused silica AOFS is driven by an arbitrary waveform synthesizer and the power of the radio frequency (RF) wave is amplified by an RF amplifier up to ~ 2W as schematically shown in Fig. S1.The 0 th order diffracted beam out of the AOFS is blocked, whereas the 1 st order frequency upshifted laser beam (see also section II) is combined with the laser pulses from the other arm of the interferometer.The two combined laser pulses (pulse-1 and pulse-2 in Fig. 1e, main-text) are then focused by an off-axis parabolic mirror (OAPM) of focal length ~ 2.5 cm to the bowtie nanoantenna device mounted on a precision 3D stage.Markers in the nanodevice and a high zoom objective (10 ) placed behind the nanodevice enable the precise positioning of the nanoantenna junction in the laser focal spot (~ 10 μm diameter).The photo-assisted tunneling current induced by the ultrashort laser pulses in the nanoantenna junction is amplified by a high gain ( 9 10  V/A) current amplifier (Femto, DLPCA-200) and measured with a lock-in amplifier.

Fig. S2 |
Fig. S2 | Bottom Panel: Simulated extinction cross section of a single bowtie as a function of the wavelength with dimensions given in the text.Top Panels: Distributions of the enhanced near fields at different wavelengths.
Fig. S3 | Comparison of the spectra of the experimentally measured and calculated plasmonic response of the nanoantenna junction.The blue curve shows the spectrum of the plasmonic response of the nanoantennas as measured in the experiments.The red curve shows the spectral shape of the plasmonic response as evaluated from the finite element simulations

Fig. S4 |
Fig. S4 | Comparison of the experimentally measured plasmon oscillations (the blue curve) with the electric field of the driving laser pulse (the dashed black curve) and the calculated plasmonic field (the red curve).The plasmon oscillations were measured by recording the variation of the laser-induced photocurrent as a function of the delay between pulse-1 and electrons, which is a function of the energy of the electron (E), tunneling gap (d) and the applied bias in the nanoantenna junction (UB).feff (E) can be modelled as a parameterized sum over different Fermi population distribution functions of amplitudes j  , energy intervals Ej and energy widths multiples of the central photon energy of the laser pulse and its bandwidth;