Abstract
Quantum coherent evolution, interference between multiple distinct paths^{1,2,3,4} and phasecontrolled sequential interactions are the basis for powerful multidimensional optical^{5} and nuclear magnetic resonance^{3} spectroscopies, including Ramsey’s method of separated fields^{6}. Recent developments in the quantum state preparation of free electrons^{7} suggest a transfer of such concepts to ultrafast electron imaging and spectroscopy. Here, we demonstrate the sequential coherent manipulation of freeelectron superposition states in an ultrashort electron pulse, using nanostructures featuring two spatially separated nearfields with polarization anisotropy. The incident light polarization controls the relative phase of these nearfields, yielding constructive and destructive quantum interference of the subsequent interactions. Future implementations of such electron–light interferometers may provide access to optically phaseresolved electronic dynamics and dephasing mechanisms with attosecond precision.
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Main
A central objective of attosecond science is the optical control over electron motion in and near atoms, molecules and solids, leading to the generation of attosecond light pulses or the study of static and dynamic properties of bound electronic wavefunctions^{8,9,10,11}. One of the most elementary forms of optical control is the dressing of freeelectron states in a periodic field^{12,13}, which is observed, for example, in twocolour ionization^{14,15}, free–free transitions near atoms^{12,16}, and in photoemission from surfaces^{17,18,19,20}. Similarly, beams of free electrons can be manipulated by the interaction with optical nearfields^{7,21,22,23,24}. In this process, field localization at nanostructures facilitates the exchange of energy and momentum between free electrons and light. In the past few years, inelastic electron–light scattering^{22,23,25} found application in socalled photoninduced nearfield electron microscopy^{7,21,26,27}, the characterization of ultrashort electron pulses^{23,24}, or in work towards optically driven electron accelerators^{28,29}. Very recently, the quantum coherence of such interactions was demonstrated by observing multilevel Rabioscillations in the electron populations of the comb of photon sidebands^{7,22}. Access to these quantum features, gained by nanoscopic electron sources of high spatial coherence^{30,31,32}, opens up a wide range of possibilities in coherent manipulations, control schemes and interferometry with freeelectron states.
Here, we present a first implementation of quantum coherent sequential interactions with freeelectron pulses. In particular, we employ a nanostructure that facilitates phasecontrolled double interactions, leading to a selectable enhancement or cancellation of the quantum phase modulation in the final electron wavefunction. Figure 1a illustrates the basic principle of our approach: traversal of the first nearfield induces photon sidebands (labelled 2 in Fig. 1a) to the initially narrow electron kinetic energy spectrum (labelled 1), which correspond to a sinusoidal phase modulation of the free electron wavefunction. Following free propagation, the electrons coherently interact with a second nearfield and, in analogy to Ramsey’s method^{6}, the final electronic state sensitively depends on the relative phase between the two acting fields. In particular, a further broadening (labelled 4) or a recompression (labelled 3) of the momentum distribution can be achieved.
For a single interaction of a free, quasimonoenergetic electron state with an optical nearfield, the resulting final state is composed of a superposition of momentum sidebands associated with energy changes by ±N photon energies^{22,23}, populated with amplitudes A_{N} according to
where J_{N} are the Nthorder Bessel functions. The dimensionless coupling parameter g describes the efficiency of momentum exchange with the electron and scales linearly with the longitudinal vector component of the optical nearfield amplitude, that is, the electric field component parallel to the electron trajectory (denoted F_{z} in Fig. 1c; see also Methods ‘Sinusoidal phase modulation’). In the spatial representation of the free electron state, this Besseltype distribution of sideband amplitudes is manifest in a sinusoidal modulation of the phase of the wavefunction in the form^{23}
where ψ_{in} and ψ_{fin} are the initial and final state wavefunctions, respectively, ω is the optical frequency, v is the electron velocity and z is the spatial coordinate along the electron trajectory.
In the present experiment, schematically depicted in Fig. 1a, c, we demonstrate that two spatially separated optical nearfields may cause an overall interaction of strength g_{tot}, which is describable as the coherent sum of the individual, generally complexvalued interactions g_{1} and g_{2},
where ϕ_{0} is a constant phase offset that depends on the spatial separation of the interaction regions (see Methods ‘Sinusoidal phase modulation’ and ‘Coordinate system and geometric phase offset’). In terms of the spatial wavefunction, this then corresponds to an overall enhancement or cancellation of the subsequent interactioninduced phase modulations (equation (2)).
The desired control over g_{tot} requires the ability to separately address the two nearfields in a phaselocked manner. We achieve this by tailoring the nanostructure geometry, employing the strong polarization anisotropy of a pair of perpendicular plates (Fig. 1b). This approach allows us to control the nearfield strengths and their relative phase by selecting the polarization state of the overall excitation. In the following, we describe the experimental implementation of this principle.
A narrow beam of ultrashort electron pulses passes the optically excited nanostructure in close vicinity (Fig. 1c). The final electronic state resulting from inelastic electron–light scattering is analysed by electron spectroscopy for a systematic variation of the incident light polarization. The polarization state is described by the Jones vector J, which we set in the standard fashion^{33} by the combination of a half and quarterwave plate at rotation angles θ and ξ, respectively. The Jones vector for sample excitation is then given by the product of the initial (in our case, diagonal) polarization state and wave plate Jones matrices M, scaled by the field strength F = 0.08 V nm^{−1}: .
In a first set of measurements, the nearfield responses of the two nanoscopic plates to the incident polarization state are independently characterized. To this end, the electron beam is placed close to each of the edges, and distant from the corner (red, blue circles in Fig. 2f), such that the electrons traverse only one of the two nearfield regions in each case. Figure 2a, b displays electron spectra for a continuous variation of polarization states (achieved by wave plate rotation), including polarizations parallel and perpendicular to the plates. The widths of these spectra directly reflect the respective coupling constants g_{1,2}, as the highest populated sideband is given by 2 g (ref. 7). It is evident that both edges exhibit strong nearfields only for excitation conditions with polarization perpendicular to the respective edge orientation. This behaviour can be regarded as a linear analyser response, in which each edge projects the incident polarization state onto a quasipolarizability α_{1,2}, yielding scalar coupling constants g_{1,2} = α_{1,2} ⋅ J. By the design of the structure, the vectors α_{1,2} are linearly independent, in fact nearly orthogonal, which allows for separate amplitude and phase control.
To demonstrate a modulation of the total coupling constant g_{tot} by mere manipulation of the relative phase of the interactions g_{1} and g_{2}, we vary the incident polarization state in such a way as to keep the projections onto the vertical (∼g_{1}) and horizontal (∼g_{2}) axes fixed. This is the case for all elliptical polarization states with main axes rotated by 45° with respect to the edges, including ±45° linear as well as left and righthand circular polarizations. Figure 2c, d displays the nearly constant coupling strengths at the individual edges resulting from this pure phase variation. Placing the beam at the corner, however, such that it sequentially interacts with both nearfields, we find a strong change in the spectral width of the final electronic state after a variation through the same set of polarization states (Fig. 2e). This conclusively demonstrates the quantum coherence and thus reversibility of the two subsequent interactions. Specifically, a strong recompression of the spectrum is achieved near θ = 39°. In the spatial wavefunction picture, this corresponds to a cancellation of the initially imprinted phase modulation by the second interaction. The effect of sequential coherent interactions can also be illustrated by spatial maps, in which the total coupling constant is displayed as a function of beam position near the nanostructure (Fig. 1d). The individual edges exhibit largely homogeneous coupling constants decaying over a distance of about 100 nm away from the edge (orange regions). In addition, for a destructive relative phase of the individual interactions, a substantially reduced total coupling constant is evident near the corner, at which the electrons traverse both nearfields (green dashed circle).
To identify the individual nearfield responses, we map the interaction strength for arbitrary incident polarization states by a systematic variation of both wave plate angles. Figure 3a displays the measured coupling constants g_{1}, g_{2} and g_{tot}, with higherresolved lineouts in Fig. 3c, d (symbols). For the individual edges (left and middle in Fig. 3a, red and blue symbols in Fig. 3c, d), we obtain quasipolarizabilities α_{1,2} = α_{1,2}n_{1,2} with the normalized projection vectors and , close to the design aim of and , and amplitude prefactors of α_{1} = 35 (V nm^{−1})^{−1} and α_{2} = 44 (V nm^{−1})^{−1}. The arbitrary overall phase of both vectors was chosen to yield real values for the respective dominant element. While the vectors n_{1,2} are universal and spatially independent for each of the edges, the specific prefactor sensitively depends on the particular distance from the respective surface. Employing amplitudes α_{1} = 52 (V nm^{−1})^{−1}, α_{2} = 29 (V nm^{−1})^{−1} and a constant phase offset ϕ_{0} = 1.30, the entire set of measurements near the corner of the structure is successfully described by a summation {g}_{\text{tot}}=({\alpha}_{1}+{\text{e}}^{i{\phi}_{0}}\times {\alpha}_{2})\u2027J, again clearly demonstrating the phasecontrolled quantum coherent interaction with both nearfields. Minor deviations, for example in the incomplete spectral recompression near the minima of g_{tot}, are attributed to a spatial average over nearfield strengths across the electron beam (see Methods ‘Determination of coupling constant and spatial averaging’). This leads to small residual sideband populations and highlights the importance of carrying out such experiments with lowemittance electron beams, as performed here, using nanotip sources. Dispersive reshaping of the wavefunction, on the other hand, can be excluded for the given spatial separation of the interaction planes (see Methods ‘Influence of dispersion’).
A comment should be made about the invoked phase offset ϕ_{0}. The precise polarization state, at which maximum recompression occurs, is governed by the phase relation between the optical farfield and the respective nearfields, and the phase lag arising from the electron and light propagation between the two interaction planes. Although these phases are physically distinct, in practice, they can be combined in the single phase offset ϕ_{0}, which is sufficient to account for all experiments. For the present measurements, we identify this phase with a precision that corresponds to a timing uncertainty of a few attoseconds. This implies a sensitivity of the scheme to phase or timing changes to the freeelectron wavefunction of this very same magnitude, rendering the presented interferometer an ideal tool to study excitationinduced phase shifts in new forms of electron holography employing the longitudinal degree of freedom. Utilizing this approach to imprint phase information onto the electron wavefunction could be translated to attosecond temporal resolution by, for example, energyresolved electron diffraction.
Whereas the present work considers the longitudinal momentum, the transverse momentum component can also be accessed in coherent control experiments, for example, by multiple Kapitza–Dirac interactions^{34} or diffraction from surface plasmon waves^{35}. Similarly, coupling to both transverse momentum components will allow for the optical preparation of freeelectron angular momentum states in chiral nearfields^{36}. More generally, the absence of efficient decoherence mechanisms in vacuum renders freeelectron wave packets an ideal system for coherent control schemes, which can be extended to multicolour approaches and additional interaction stages. Future experiments may utilize this type of ‘electron–light interferometer’ by inserting optically excited materials in the gap for precision measurements of electronic dephasing with subcycle resolution. Various further applications include phaseresolved nearfield imaging, possible quantum computation schemes using free electrons, or the tailored structuring of electron densities in accelerator beamlines with attosecond accuracy.
Methods
The experiments were performed in a recently developed ultrafast transmission electron microscope, featuring a nanoscale photoemitter as a pulsed electron source for electron pulses with high spatial coherence. Specifically, ultrashort electron pulses are generated by localized photoemission from a ZrO/W tip emitter, accelerated to a kinetic energy of 120 keV and focused tightly (with a beam divergence of 5.3 mrad) in close vicinity to a nanostructure. Electron spot diameters down to 3 nm and pulse durations as short as 300 fs were achieved. A scanning electron micrograph of the nanostructure design is shown in Fig. 1b. The two plates with a distance of 5 μm were milled by a focused ion beam from a single, annealed gold wire (30 μm diameter). The experimental scenario is sketched in Fig. 1c: a pump laser beam (800 nm wavelength, dispersively stretched to a pulse duration of 3.4 ps, 250 kHz repetition rate, 23 mW average power) passes a half and a quarterwave plate for polarization control and is focused onto the sample to a spot diameter of about 50 μm (fullwidth at halfmaximum). The electron kinetic energy spectra are recorded with an electron energy loss spectrometer.
Sinusoidal phase modulation.
To obtain the electron wavefunction ψ(z, t) after interaction with the optical nearfields, we apply the scattering (Smatrix) approach in the interaction picture (see also ref. 7). The final wavefunction is given by ψ(z, ∞)〉 = S ψ(z, −∞)〉 with the timeordered unitary operator
and the interaction Hamiltonian
where v is the electron velocity and e is the electron charge. The vector potential A(z, t) for the two nearfields separated by the distance L is given by
ϕ denotes the phase lag of the second nearfield induced by the optical path length difference of the driving laser field (corresponding to d_{l} in Supplementary Fig. 2). For the wavefunction after interaction we obtain
In the second step, we introduced the coupling constant g = e/2ℏω∫ _{−∞}^{∞}F(z)exp(−iΔkz)dz, as in ref. 23. It is proportional to the spatial Fourier component of the longitudinal vector component of the nearfield F(z) along the electron trajectory at the spatial frequency Δk = ω/v, which corresponds to the momentum change of an electron at velocity v gaining or losing an energy ℏω. Equation (7) evidences that the interaction of the free electrons with the two optical nearfields is describable as a single sinusoidal phase modulation of the electron wavefunction and that the two consecutive interactions coherently add up in the way stated in equation (3) in the main text.
Influence of dispersion.
Between the two interaction regions, the electron wavefunction propagates in free space. The momentumdependent propagation operator is given by
where γ is the Lorentz factor and Δp is the momentum change due to the interaction with the optical nearfield, with Δp given by Nℏω/v for the Nthorder sideband. During free propagation, the sideband orders acquire different phases, which leads to a dispersive reshaping of the electron wavefunction and, at a certain propagation distance, to a temporal focusing into a train of attosecond pulses^{7}. In the present study, the propagation distance is much shorter than the distance to the temporal focus (typically millimetre scale). Specifically, the experimental parameters employed here (coupling constants g ≈ 5, propagation distance L = 6 μm, v = 0.6c, γ = 1.24) yield very small sidebanddependent phase shifts on the order of N^{2} ⋅ 0.17 mrad, such that dispersive effects are negligible.
Coordinate system and geometric phase offset.
The angles θ and ξ define the orientation of the fast axes of the wave plates relative to the horizontal x axis of the coordinate system, which is indicated by black arrows in Supplementary Fig. 1a. The wave plate setting θ = ξ = −45°, for example, yields linear laser polarization at −45° to the x axis.
In the following, we discuss the influence of the sample and beam geometry on the constant phase offset ϕ_{0}. The difference in electron group and laser phase velocity leads to a phase lag, which can be calculated as follows: for a given plate distance d, the electron and laser path lengths are d_{e} = d/cosα and d_{l} = dcosβ/cosα, respectively, where α = 37° is the sample tilt angle and β = 55° is the angle between laser and electron beam. The path length difference corresponds to a timing difference of
with v = 0.6c. A small variation of α by about 2.4° shifts Δt by a quarter laser period, that is, ϕ_{0} by 90°. Sample tilting thus presents a convenient way to externally control ϕ_{0}. Note that the data displayed in Figs 2e and 3c in the main text were recorded at two slightly different sample tilts, resulting in a relative phase shift of Δϕ = 81.5°.
Determination of coupling constant and spatial averaging.
In principle, the coupling constants can be inferred from the cutoff energy of the electron energy spectra, which is given by 2 g ℏω (ref. 7). For a more precise determination, we extracted coupling constants from a fit of Bessel amplitudes to the data, according to equation (1). Due to the finite electron beam size, a small spatial average over different coupling constants needs to be taken into account, for which we adopt a Gaussian distribution of the electron intensity in the beam. At the gold edges, the nearfield strength can be regarded as homogeneous in directions parallel to an edge, and exponentially decaying along the perpendicular direction (see Supplementary Fig. 2g). In this case, the probability distribution of coupling constants is given as
where g_{0} is the expectation value of the coupling constant, l is the decay length of the nearfield strength and Σ is the electron beam width (standard deviation). For the analysis, we consider a constant ratio l/Σ for each nearfield.
When averaging is taken into account, the experimental data are well reproduced. A comparison of Supplementary Figs 2c, e illustrates that spatial averaging only weakly affects the visibility of quantum coherent features in the electron energy spectra (see ref. 7). The spectra recorded at the upper edge show stronger averaging compared with the lower edge, since the electron focus is not perfectly centred between the two edges (small displacement Δz). For the data set shown here, we obtain l/Σ_{U} ≈ 5 and l/Σ_{L} ≈ 10. Together with the nearfield decay length of l ≈ 90 nm (determined from the raster scan in Fig. 1d), we find Σ_{U} = 18 nm and Σ_{L} = 9 nm, in accordance with the electron focal spot diameter of 8 nm used in the experiment.
Data availability.
The data that support the plots within this paper and other findings of this study are available from the corresponding authors on request.
References
Feynman, R. P. Spacetime approach to nonrelativistic quantum mechanics. Rev. Mod. Phys. 20, 367–387 (1948).
Hasselbach, F. Progress in electron and ioninterferometry. Rep. Prog. Phys. 73, 016101 (2010).
Spiess, H. W. Multidimensional SolidState NMR and Polymers (Academic Press, 1994).
Bordé, Ch. J. Atomic interferometry with internal state labelling. Phys. Lett. A 140, 10–12 (1989).
Mukamel, S. Principles of Nonlinear Optical Spectroscopy Vol. 6 (Oxford Series on Optical Sciences, Oxford Univ. Press, 1999).
Ramsey, N. F. Experiments with separated oscillatory fields and hydrogen masers. Rev. Mod. Phys. 62, 541–552 (1990).
Feist, A. et al. Quantum coherent optical phase modulation in an ultrafast transmission electron microscope. Nature 521, 200–203 (2015).
Krausz, F. & Ivanov, M. Attosecond physics. Rev. Mod. Phys. 81, 163–234 (2009).
Itatani, J. et al. Tomographic imaging of molecular orbitals. Nature 432, 867–871 (2004).
Haessler, S. et al. Attosecond imaging of molecular electronic wavepackets. Nature Phys. 6, 200–206 (2010).
Xie, X. et al. Attosecond probe of valenceelectron wave packets by subcycle sculpted laser fields. Phys. Rev. Lett. 108, 193004 (2012).
Weingartshofer, A., Holmes, J., Caudle, G., Clarke, E. & Krüger, H. Direct observation of multiphoton processes in laserinduced freefree transitions. Phys. Rev. Lett. 39, 269–270 (1977).
Agostini, P., Fabre, F., Mainfray, G., Petite, G. & Rahman, N. K. Freefree transitions following sixphoton ionization of xenon atoms. Phys. Rev. Lett. 42, 1127–1130 (1979).
Radcliffe, P. et al. Atomic photoionization in combined intense XUV freeelectron and infrared laser fields. New J. Phys. 14, 043008 (2012).
Meyer, M. et al. Angleresolved electron spectroscopy of laserassisted Auger decay induced by a fewfemtosecond xray pulse. Phys. Rev. Lett. 108, 063007 (2012).
Morimoto, Y., Kanya, R. & Yamanouchi, K. Lightdressing effect in laserassisted elastic electron scattering by Xe. Phys. Rev. Lett. 115, 123201 (2015).
Ogawa, S., Nagano, H., Petek, H. & Heberle, A. P. Optical dephasing in Cu(111) measured by interferometric twophoton timeresolved photoemission. Phys. Rev. Lett. 78, 1339–1342 (1997).
Petek, H. et al. Optical phase control of coherent electron dynamics in metals. Phys. Rev. Lett. 79, 4649–4652 (1997).
Saathoff, G., MiajaAvila, L., Aeschlimann, M., Murnane, M. M. & Kapteyn, H. C. Laserassisted photoemission from surfaces. Phys. Rev. A 77, 022903 (2008).
Mahmood, F. et al. Selective scattering between Floquet–Bloch and Volkov states in a topological insulator. Nature Phys. 12, 306–310 (2016).
Barwick, B., Flannigan, D. J. & Zewail, A. H. Photoninduced nearfield electron microscopy. Nature 462, 902–906 (2009).
García de Abajo, F. J., AsenjoGarcia, A. & Kociak, M. Multiphoton absorption and emission by interaction of swift electrons with evanescent light fields. Nano Lett. 10, 1859–1863 (2010).
Park, S. T., Lin, M. & Zewail, A. H. Photoninduced nearfield electron microscopy (PINEM): theoretical and experimental. New J. Phys. 12, 123028 (2010).
Kirchner, F. O., Gliserin, A., Krausz, F. & Baum, P. Laser streaking of free electrons at 25 keV. Nature Photon. 8, 52–57 (2014).
García de Abajo, F. J. & Kociak, M. Electron energygain spectroscopy. New J. Phys. 10, 073035 (2008).
Yurtsever, A., van der Veen, R. M. & Zewail, A. H. Subparticle ultrafast spectrum imaging in 4D electron microscopy. Science 335, 59–64 (2012).
Piazza, L. et al. Simultaneous observation of the quantization and the interference pattern of a plasmonic nearfield. Nature Commun. 6, 6407 (2015).
England, R. J. et al. Dielectric laser accelerators. Rev. Mod. Phys. 86, 1337–1389 (2014).
Kozak, M. et al. Optical gating and streaking of freeelectrons with attosecond precision. Preprint at http://arXiv.org/abs/1512.04394v1 (2015).
Gulde, M. et al. Ultrafast lowenergy electron diffraction in transmission resolves polymer/graphene superstructure dynamics. Science 345, 200–204 (2014).
Ehberger, D. et al. Highly coherent electron beam from a lasertriggered tungsten needle tip. Phys. Rev. Lett. 114, 227601 (2015).
Müller, M., Paarmann, A. & Ernstorfer, R. Femtosecond electrons probing currents and atomic structure in nanomaterials. Nature Commun. 5, 5292 (2014).
Saleh, B. E. A. & Teich, M. C. Fundamentals of Photonics Vol. 2, 203–209 (John Wiley & Sons, 2007).
Batelaan, H. Illuminating the Kapitza–Dirac effect with electron matter optics. Rev. Mod. Phys. 79, 929–941 (2007).
García de Abajo, F. J., Barwick, B. & Carbone, F. Electron diffraction by plasmon waves. Preprint at http://arXiv.org/abs/1603.07551v1 (2016).
AsenjoGarcia, A. & Garcia de Abajo, F. J. Dichroism in the interaction between vortex electron beams, plasmons, and molecules. Phys. Rev. Lett. 113, 066102 (2014).
Acknowledgements
We gratefully acknowledge funding by the Deutsche Forschungsgemeinschaft (DFGSPP1840 ‘Quantum Dynamics in Tailored Intense Fields’, and DFGSFB1073 ‘Atomic Scale Control of Energy Conversion’, project A05). We thank S. V. Yalunin for useful discussions, and M. Sivis for help in sample preparation.
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K.E.E. prepared the nanostructure, conducted the experiments with contributions from A.F., and analysed the data. The manuscript was written by K.E.E. and C.R., with contributions from S.S. C.R. and S.S. conceived and directed the study. All authors discussed the results and the interpretation.
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Echternkamp, K., Feist, A., Schäfer, S. et al. Ramseytype phase control of freeelectron beams. Nature Phys 12, 1000–1004 (2016). https://doi.org/10.1038/nphys3844
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DOI: https://doi.org/10.1038/nphys3844
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