Abstract

Electrons in atoms and molecules are versatile physical systems allowing a vast range of light–matter interactions. Spontaneous emission, which appears in a wide variety of applications, depends crucially on the bound electron energy levels. The discrete nature of these electron energy levels and the ionization threshold constrain the energy scale of all light–matter interactions involving bound electrons. To bypass these constraints, we take ideas from optical and electronic beam shaping and propose creating new electron states as superpositions of extended states above the ionization threshold. We show that such superpositions enable the control of spontaneous emission with tunable spectra in the eV–keV range. We find that the specific shaping lengthens the diffraction and radiative lifetimes of the wavepackets in exchange for increasing their spatial spreads. Our approach could have applications toward developing novel kinds of light emitters at hard-to-access spectral ranges.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (1)

G. M. Vanacore, G. Berruto, I. Madan, E. Pomarico, P. Biagioni, R. J. Lamb, D. McGrouther, O. Reinhardt, I. Kaminer, B. Barwick, V. Grillo, E. Karimi, F. J. García de Abajo, and F. Carbone, “Ultrafast generation and control of an electron vortex beam via chiral plasmonic near fields,” Nat. Mater. 18, 573–579 (2019).
[Crossref]

2018 (7)

C. Heide, T. Higuchi, H. B. Weber, and P. Hommelhoff, “Coherent electron trajectory control in graphene,” Phys. Rev. Lett. 121, 207401 (2018).
[Crossref]

W. Cai, O. Reinhardt, I. Kaminer, and J. G. D. Abajo, “Efficient orbital angular momentum transfer between plasmons and free electrons,” Phys. Rev. B 98, 045424 (2018).
[Crossref]

G. M. Vanacore, I. Madan, G. Berruto, K. Wang, E. Pomarico, R. J. Lamb, D. McGrouther, I. Kaminer, B. Barwick, F. J. García de Abajo, and F. Carbone, “Attosecond coherent control of free-electron wave functions using semi-infinite light fields,” Nat. Commun. 9, 2694 (2018).
[Crossref]

Y. Morimoto and P. Baum, “Diffraction and microscopy with attosecond electron pulse trains,” Nat. Phys. 14, 252–256 (2018).
[Crossref]

Q.-C. Ning, U. Saalmann, and J. M. Rost, “Electron dynamics driven by light-pulse derivatives,” Phys. Rev. Lett. 120, 033203 (2018).
[Crossref]

A. Patsyk, M. A. Bandres, R. Bekenstein, and M. Segev, “Observation of accelerating wave packets in curved space,” Phys. Rev. X 8, 011001 (2018).
[Crossref]

D. Popmintchev, B. R. Galloway, M. Chen, F. Dollar, C. A. Mancuso, A. Hankla, L. Miaja-Avila, G. O’Neil, J. M. Shaw, G. Fan, S. Ališauskas, G. Andriukaitis, O. D. Balčiunas, T. Mücke, A. Pugzlys, A. Baltuška, H. C. Kapteyn, T. Popmintchev, and M. M. Murnane, “Near- and extended-edge x-ray-absorption fine-structure spectroscopy using ultrafast coherent high-order harmonic supercontinua,” Phys. Rev. Lett. 120, 093002 (2018).
[Crossref]

2017 (6)

J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond x-ray pulses reach the carbon K-edge,” Nat. Commun. 8, 186 (2017).
[Crossref]

J. Palmer and S. D. Hogan, “Experimental demonstration of a Rydberg-atom beam splitter,” Phys. Rev. A 95, 053413 (2017).
[Crossref]

M. Mutzafi, I. Kaminer, G. Harari, and M. Segev, “Non-diffracting multi-electron vortex beams balancing their electron-electron interactions,” Nat. Commun. 8, 650 (2017).
[Crossref]

P. Baum, “Quantum dynamics of attosecond electron pulse compression,” J. Appl. Phys. 122, 223105 (2017).
[Crossref]

K. E. Priebe, C. Rathje, S. V. Yalunin, T. Hohage, A. Feist, S. Schäfer, and C. Ropers, “Attosecond electron pulse trains and quantum state reconstruction in ultrafast transmission electron microscopy,” Nat. Photonics 11, 793–797 (2017).
[Crossref]

J. L. Wong and I. Kaminer, “Abruptly focusing and defocusing needles of light and closed-form electromagnetic wavepackets,” ACS Photon. 4, 1131–1137 (2017).
[Crossref]

2016 (10)

A. Ryabov and P. Baum, “Electron microscopy of electromagnetic waveforms,” Science 353, 374–377 (2016).
[Crossref]

R. Penciu, K. Makris, and N. Efremidis, “Nonparaxial abruptly autofocusing beams,” Opt. Lett. 41, 1042–1045 (2016).
[Crossref]

D. W. Schumacher, J. H. Hoogenraad, D. Pinkos, and P. H. Bucksbaum, “Programmable cesium Rydberg wave packets,” Phys. Rev. A 6, 4719–4726 (2016).
[Crossref]

R. E. Goetz, A. Karamatskou, R. Santra, and C. P. Koch, “Quantum optimal control of photoelectron spectra and angular distributions,” Phys. Rev. A 93, 013413 (2016).
[Crossref]

R. E. Goetz, M. Merkel, A. Karamatskou, R. Santra, and C. P. Koch, “Maximizing hole coherence in ultrafast photoionization of argon with an optimization by sequential parametrization update,” Phys. Rev. A 94, 023420 (2016).
[Crossref]

K. E. Echternkamp, A. Feist, S. Schäfer, and C. Ropers, “Ramsey-type phase control of free-electron beams,” Nat. Phys. 12, 1000–1004 (2016).
[Crossref]

G. J. Stein, P. D. Keathley, P. Krogen, H. Liang, J. P. Siqueira, C.-L. Chang, C.-J. Lai, K.-H. Hong, G. M. Laurent, and F. X. Kärtner, “Water-window soft x-ray high-harmonic generation up to the nitrogen k-edge driven by a kHz, 2.1  μm OPCPA source,” J. Phys. B 49, 155601 (2016).
[Crossref]

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1, 331–335 (2016).
[Crossref]

N. Rivera, C. W. Hsu, B. Zhen, H. Buljan, J. D. Joannopoulos, and M. Soljačić, “Controlling directionality and dimensionality of radiation by perturbing separable bound states in the continuum,” Sci. Rep. 6, 33394 (2016).
[Crossref]

K. Makris, D. Papazoglou, and S. Tzortzakis, “Invariant superoscillatory electromagnetic fields in 3D-space,” J. Opt. 19, 014003 (2016).
[Crossref]

2015 (3)

I. Kaminer, J. Nemirovsky, M. Rechtsman, R. Bekenstein, and M. Segev, “Self-accelerating dirac particles and prolonging the lifetime of relativistic fermions,” Nat. Phys. 11, 261–267 (2015).
[Crossref]

J. Harris, V. Grillo, E. Mafakheri, G. C. Gazzadi, S. Frabboni, R. W. Boyd, and E. Karimi, “Structured quantum waves,” Nat. Phys. 11, 629–634 (2015).
[Crossref]

A. Feist, K. E. Echternkamp, J. Schauss, S. V. Yalunin, S. Schäfer, and C. Ropers, “Quantum coherent optical phase modulation in an ultrafast transmission electron microscope,” Nat. Phys. 521, 200–203 (2015).
[Crossref]

2014 (7)

V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R. Dennis, and R. W. Boyd, “Generation of nondiffracting electron Bessel beams,” Phys. Rev. X 4, 011013 (2014).
[Crossref]

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-preserving accelerating electromagnetic wave packets in curved space,” Phys. Rev. X 4, 011038 (2014).
[Crossref]

N. Ishii, K. Kaneshima, K. Kitano, T. Kanai, S. Watanabe, and J. Itatani, “Carrier-envelope phase-dependent high harmonic generation in the water window using few-cycle infrared pulses,” Nat. Commun. 5, 3331 (2014).
[Crossref]

R. Shiloh, Y. Lereah, Y. Lilach, and A. Arie, “Sculpturing the electron wave function using nanoscale phase masks,” Ultramicroscopy 144, 26–31 (2014).
[Crossref]

C. Ott, A. Kaldun, L. Argenti, P. Raith, K. Meyer, M. Laux, Y. Zhang, A. Blättermann, S. Hagstotz, T. Ding, R. Heck, J. Madronero, F. Martin, and T. Pfeifer, “Reconstruction and control of a time-dependent two-electron wave packet,” Nature 516, 374–378 (2014).
[Crossref]

R. Schley, I. Kaminer, E. Greenfield, R. Bekenstein, Y. Lumer, and M. Segev, “Loss-proof self-accelerating beams and their use in non-paraxial manipulation of particles’ trajectories,” Nat. Commun. 5, 5189 (2014).
[Crossref]

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113, 257401 (2014).
[Crossref]

2013 (4)

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Supplementary Material (2)

NameDescription
» Code 1       Code for our experiments.
» Supplement 1       Supplemental document

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Figures (4)

Fig. 1.
Fig. 1. Shaping of electron states in the continuum of energy levels of the hydrogen atom creates localized and quasi-shape-invariant high-energy wavepackets, unlimited in energy. The possibility of decay from the continuum to bound states enables a photon emission with energy higher than the ionization threshold. Color, r2|Ψ|2/max{r2|Ψ|2}.
Fig. 2.
Fig. 2. By shaping the (Whittaker) electron wavepacket, we can prolong the lifetime to a desired duration in exchange for increasing the spatial spread. (a) As we decrease the energy spread ΔE, the probability density r2|Ψ|2 spreads out farther in space according to (6). (b) The benefit of making the energy spread small, however, is that the lifetime grows according to (7). Here, Δt=19.14ns and the colors match the Whittaker wavepackets in (a). (c) Envelope of the wavepacket at three points in time: A, B, C [also marked in (b)].
Fig. 3.
Fig. 3. Dynamics of the Whittaker wavepacket. (a) The wavepacket begins at t=0, with a set of nodes marked by dots. The dotted lines trace the nodes, and the cross signals a vanished node. Time frames and overlap points linked in color. (b) A larger range in space that includes more oscillations (only the envelope is plotted) and has a longer lifetime Δt. (c) The geometry of the Whittaker wavepacket is roughly the same as that of the free particle with Bessel functions as modes. Baseline ΔE0/E0 set in (a).
Fig. 4.
Fig. 4. Radiative decay of the Whittaker wavepackets to bound states. (a) The decay probability reaches a saturation line for infrared energy E=1eV. The average rate approximates a slope. (b) The profiles of the decay rates (small panels) for the soft-x-ray energy E=217.6eV are monotonously decreasing in comparison to the profiles in (a). Baseline ΔE0/E0 set in (a). Note that (b) may have quantitative corrections due to beyond-dipole corrections arising from the short wavelength of the electromagnetic field, since for our spontaneous emission calculations, we use the dipole approximation.

Tables (1)

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Table 1. Spread-Lifetime Trade-Off of Whittaker Wavepacketsa

Equations (12)

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wκ(x,0)=4iκ2eiκxπcsch(π/2κ)01e2iκxs(s1s)i2κds,
ΨE,ΔE(r,t)=Ne(κμ)2/2σ2wκ(x,t)dκ,
Δr=var(envelope(ΨE,ΔE(r,0))),
Δt=var(O(t))
O(t)=|0Ψ*(r,0)Ψ(r,t)r2dr|2.
Δr2.471a0(ΔE)/eV.
Δt0.136eV·fs(ΔE)E.
Γ˜=P(2(Δt))2(Δt)
dPfi(k,λ)=Vd3k(2π)3|Sfi(k,λ)|2
Sfi=f|Tei0tH^intdt|i
H^int[ψ]=iemed3xψ*A^(x,t)·ψ,
Sfi(k,λ)=eme2ϵ0ωkV0tdteiωktε^kλ·d3xψf*(x,t)eik·x·ψi(x,t)

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