Abstract

Probing optical excitations with nanometer resolution is important for understanding their dynamics and interactions down to the atomic scale. Electron microscopes currently offer the unparalleled ability of rendering spatially resolved electron spectra with combined meV and sub-nm resolution, while the use of ultrafast optical pulses enables fs temporal resolution and exposure of the electrons to ultraintense confined optical fields. Here, we theoretically investigate fundamental aspects of the interaction of fast electrons with localized optical modes that are made possible by these advances. We use a quantum optics description of the optical field to predict that the resulting electron spectra strongly depend on the statistics of the sample excitations (bosonic or fermionic) and their population (Fock, coherent, or thermal), whose autocorrelation functions are directly retrieved from the ratios of electron gain intensities. We further explore feasible experimental scenarios to probe the quantum characteristics of the sampled excitations and their populations. In particular, we present realistic simulations for electron beams interacting with optical cavities infiltrated with optically pumped quantum emitters, which we show to undergo a varied temporal evolution in the cavity mode statistics that causes radical modifications in the transmitted electron spectra depending on pump-electron delay.

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

Full Article  |  PDF Article

Corrections

17 December 2019: A typographical correction was made to the final paragraph of Section 2.


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References

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2019 (4)

O. Kfir, “Entanglements of electrons and cavity photons in the strong-coupling regime,” Phys. Rev. Lett. 123, 103602 (2019).
[Crossref]

S. Franke, S. Hughes, M. K. Dezfouli, P. T. Kristensen, K. Busch, A. Knorr, and M. Richter, “Quantization of quasinormal modes for open cavities and plasmonic cavity quantum electrodynamics,” Phys. Rev. Lett. 122, 213901 (2019).
[Crossref]

P. Das, J. D. Blazit, M. Tencé, L. F. Zagonel, Y. Auad, Y. H. Lee, X. Y. Ling, A. Losquin, O. S. C. Colliex, F. J. G. de Abajo, and M. Kociak, “Stimulated electron energy loss and gain in an electron microscope without a pulsed electron gun,” Ultramicroscopy 203, 44–51 (2019).
[Crossref]

E. J. C. Dias and F. J. G. de Abajo, “Fundamental limits to the coupling between light and 2D polaritons,” ACS Nano 13, 5184–5197 (2019).
[Crossref]

2018 (4)

E. Pomarico, I. Madan, G. Berruto, G. M. Vanacore, K. Wang, I. Kaminer, F. J. G. de Abajo, and F. Carbone, “meV resolution in laser-assisted energy-filtered transmission electron microscopy,” ACS Photon. 5, 759–764 (2018).
[Crossref]

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

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

Y. Morimoto and P. Baum, “Attosecond control of electron beams at dielectric and absorbing membranes,” Phys. Rev. A 97, 033815 (2018).
[Crossref]

2017 (5)

A. Feist, N. Bach, T. D. N. R. da Silva, M. Mäller, K. E. Priebe, T. Domräse, J. G. Gatzmann, S. Rost, J. Schauss, S. Strauch, R. Bormann, M. Sivis, S. Schäfer, and C. Ropers, “Ultrafast transmission electron microscopy using a laser-driven field emitter: femtosecond resolution with a high coherence electron beam,” Ultramicroscopy 176, 63–73 (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]

G. Guzzinati, A. Beche, H. Lourenco-Martins, J. Martin, M. Kociak, and J. Verbeeck, “Probing the symmetry of the potential of localized surface plasmon resonances with phase-shaped electron beams,” Nat. Commun. 8, 14999 (2017).
[Crossref]

M. J. Lagos, A. Trügler, U. Hohenester, and P. E. Batson, “Mapping vibrational surface and bulk modes in a single nanocube,” Nature 543, 529–532 (2017).
[Crossref]

M. Kozák, J. McNeur, K. J. Leedle, H. Deng, N. Schönenberger, A. Ruehl, I. Hartl, J. S. Harris, R. L. Byer, and P. Hommelhoff, “Optical gating and streaking of free electrons with sub-optical cycle precision,” Nat. Commun. 8, 14342 (2017).
[Crossref]

2016 (5)

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

T. T. A. Lummen, R. J. Lamb, G. Berruto, T. LaGrange, L. D. Negro, F. J. G. de Abajo, D. McGrouther, B. Barwick, and F. Carbone, “Imaging and controlling plasmonic interference fields at buried interfaces,” Nat. Commun. 7, 13156 (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]

R. Bourrellier, S. Meuret, A. Tararan, O. Stéphan, M. Kociak, L. H. G. Tizei, and A. Zobelli, “Bright UV single photon emission at point defects in h-BN,” Nano Lett. 16, 4317–4321 (2016).
[Crossref]

F. J. G. de Abajo, B. Barwick, and F. Carbone, “Electron diffraction by plasmon waves,” Phys. Rev. B 94, 041404 (2016).
[Crossref]

2015 (5)

S. Meuret, L. H. G. Tizei, T. Cazimajou, R. Bourrellier, H. C. Chang, F. Treussart, and M. Kociak, “Photon bunching in cathodoluminescence,” Phys. Rev. Lett. 114, 197401 (2015).
[Crossref]

A. T. A. Hörl and U. Hohenester, “Full three-dimensional reconstruction of the dyadic green tensor from electron energy loss spectroscopy of plasmonic nanoparticles,” ACS Photon. 2, 1429–1435 (2015).
[Crossref]

L. Piazza, T. T. A. Lummen, E. Quiñonez, Y. Murooka, B. Reed, B. Barwick, and F. Carbone, “Simultaneous observation of the quantization and the interference pattern of a plasmonic near-field,” Nat. Commun. 6, 6407 (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,” Nature 521, 200–203 (2015).
[Crossref]

A. Reiserer and G. Rempe, “Cavity-based quantum networks with single atoms and optical photons,” Rev. Mod. Phys. 87, 1379–1418 (2015).
[Crossref]

2014 (4)

R. Barbosa-Silva, A. F. Silva, A. M. Brito-Silva, and C. B. de Araújo, “Bichromatic random laser from a powder of rhodamine-doped sub-micrometer silica particles,” J. Appl. Phys. 115, 043515 (2014).
[Crossref]

O. L. Krivanek, T. C. Lovejoy, N. Dellby, T. Aoki, R. W. Carpenter, P. Rez, E. Soignard, J. Zhu, P. E. Batson, M. J. Lagos, R. F. Egerton, and P. A. Crozier, “Vibrational spectroscopy in the electron microscope,” Nature 514, 209–214 (2014).
[Crossref]

M. Kociak and O. Stephan, “Mapping plasmons at the nanometer scale in an electron microscope,” Chem. Soc. Rev. 43, 3865–3883 (2014).
[Crossref]

F. O. Kirchner, A. Gliserin, F. Krausz, and P. Baum, “Laser streaking of free electrons at 25  keV,” Nat. Photonics 8, 52–57 (2014).
[Crossref]

2013 (4)

L. H. G. Tizei and M. Kociak, “Spatially resolved quantum nano-optics of single photons using an electron microscope,” Phys. Rev. Lett. 110, 153604 (2013).
[Crossref]

D. Rossouw and G. A. Botton, “Plasmonic response of bent silver nanowires for nanophotonic subwavelength waveguiding,” Phys. Rev. Lett. 110, 066801 (2013).
[Crossref]

I. Gierz, J. C. Petersen, M. Mitrano, C. Cacho, I. C. E. Turcu, E. Springate, A. Stöhr, A. Köhler, U. Starke, and A. Cavalleri, “Snapshots of non-equilibrium Dirac carrier distributions in graphene,” Nat. Mater. 12, 1119–1124 (2013).
[Crossref]

F. J. G. de Abajo, “Multiple excitation of confined graphene plasmons by single free electrons,” ACS Nano 7, 11409–11419 (2013).
[Crossref]

2012 (2)

S. T. Park and A. H. Zewail, “Relativistic effects in photon-induced near field electron microscopy,” J. Phys. Chem. A 116, 11128–11133 (2012).
[Crossref]

G. Zhu, G. Radtke, and G. A. Botton, “Bonding and structure of a reconstructed (001) surface of SrTiO3 from TEM,” Nature 490, 384–387 (2012).
[Crossref]

2010 (3)

F. J. G. de Abajo, “Optical excitations in electron microscopy,” Rev. Mod. Phys. 82, 209–275 (2010).
[Crossref]

F. J. G. de Abajo, A. A. Garcia, and M. Kociak, “Multiphoton absorption and emission by interaction of swift electrons with evanescent light fields,” Nano Lett. 10, 1859–1863 (2010).
[Crossref]

S. T. Park, M. Lin, and A. H. Zewail, “Photon-induced near-field electron microscopy (PINEM): theoretical and experimental,” New J. Phys. 12, 123028 (2010).
[Crossref]

2009 (2)

2008 (3)

J. K. Hyun, M. Couillard, P. Rajendran, C. M. Liddell, and D. A. Muller, “Measuring far-ultraviolet whispering gallery modes with high energy electrons,” Appl. Phys. Lett. 93, 243106 (2008).
[Crossref]

F. J. G. de Abajo and M. Kociak, “Probing the photonic local density of states with electron energy loss spectroscopy,” Phys. Rev. Lett. 100, 106804 (2008).
[Crossref]

D. A. Muller, L. Fitting Kourkoutis, M. Murfitt, J. H. Song, H. Y. Hwang, J. Silcox, N. Dellby, and O. L. Krivanek, “Atomic-scale chemical imaging of composition and bonding by aberration-corrected microscopy,” Science 319, 1073–1076 (2008).
[Crossref]

2007 (1)

K. A. Mkhoyan, T. Babinec, S. E. Maccagnano, E. J. Kirkland, and J. Silcox, “Separation of bulk and surface-losses in low-loss eels measurements in stem,” Ultramicroscopy 107, 345–355 (2007).
[Crossref]

2003 (1)

A. Howie, “Valence excitations in electron microscopy: resolved and unresolved issues,” Micron 34, 121–125 (2003).
[Crossref]

1991 (1)

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
[Crossref]

1977 (1)

J. H. Eberly, B. Shore, Z. Białynicka-Birula, and I. Białynicki-Birula, “Coherent dynamics of n-level atoms and molecules. I. Numerical experiments,” Phys. Rev. A 16, 2038–2047 (1977).
[Crossref]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

1970 (1)

A. A. Lucas, E. Kartheuser, and R. G. Badro, “Electron-phonon interaction in dielectric films. application to electron energy loss and gain spectra,” Phys. Rev. B 2, 2488–2499 (1970).
[Crossref]

1965 (1)

P. Carruthers and M. M. Nieto, “Coherent states and the forced quantum oscillator,” Am. J. Phys. 33, 537–544 (1965).
[Crossref]

1963 (1)

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[Crossref]

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 37–38 (1946).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Abrikosov, A. A.

A. A. Abrikosov, L. P. Gor’kov, and I. Y. Dzyaloshinskii, Quantum Field Theoretical Methods in Statistical Physics (Pergamon, 1965).

Adiv, Y.

R. Dahan, S. Nehemia, M. Shentcis, O. Reinhardt, Y. Adiv, K. Wang, O. Beer, Y. Kurman, X. Shi, M. H. Lynch, and I. Kaminer, “Observation of the stimulated quantum Cherenkov effect,” arXiv:1909.00757.

Aoki, T.

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Phys. Rev. Lett. (6)

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Supplementary Material (1)

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Figures (3)

Fig. 1.
Fig. 1. Coupling regimes in the interaction of a beam electron with an optical mode. Weak and strong coupling corresponds to the regions roughly separated by the contour $ \bar n{|{\beta _0}|^2} \sim 1 $ (white line), where $ \bar n $ is the average mode population and $ {\beta _0} $ is the single-mode interaction coefficient [Eq. (5)]. The density plots show the ratio of integrated gains and losses in the electron spectra for (a) Fock, (b) coherent, and (c) thermal populations.
Fig. 2.
Fig. 2. Dependence on boson population distribution in the interaction with an electron beam. (a)–(c) Distribution of the probability $ {p_n} $ for the occupation of each state $ |n\rangle $ in the three types of population statistics considered at the moment of electron interaction: (a) Fock, (b) coherent, and (c) thermal, with average values $ {\bar n} = 2 $, 10, and 50. (d)–(o) Electron spectra after interaction with a mode having the initial populations of (a)–(c) as a function of the electron-mode coupling parameter $ \sqrt {\bar n} |{\beta _0}| $. The energy loss $ \Delta E $ is normalized to the boson energy $ \hbar {\omega _0} $, and a peak Lorentzian broadening of $ 0.1 {\omega _0} $ is introduced for clarity.
Fig. 3.
Fig. 3. Interaction with an optical cavity coupled to pumped quantum emitters. (a) We consider a bosonic optical cavity (e.g., a Mie resonator) sustaining a single mode (frequency $ {\omega _0} $, inelastic decay rate $ \kappa $) and infiltrated with $ N $ three-level QEs. Optical pumping prepares the emitters in their upper energy state at time $ t = 0 $, from which they decay to an intermediate level by resonant coupling to the cavity mode at a rate $ g $. (b) Temporal evolution of the average cavity mode population $ \bar n $ and (c) second-order autocorrelation function at zero delay $ {\text{g}^{(2)}} $ for $ N = 2 $, 10, and 50 with $ \kappa = 0 $ (solid curves) and $ \kappa /Ng = 0.1 $ (dashed curves). (d)–(f) Evolution of the populations $ {p_n} $ and (g)–(i) the electron spectra as a function of the pump-electron delay time for $ N = 50 $ and different cavity decay rates: $ \kappa /Ng = 0 $ in (d) and (g); 0.01 in (e) and (h); and 0.1 in (f) and (i). We assume an electron-mode coupling $ |{\beta _0}| = 0.7 $.

Equations (46)

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H ^ 0 = ω 0 a ^ a ^ + E 0 v ( i + k 0 ) , H ^ 1 = ( e v / c ) A ^ ,
A ^ = ( i c / ω 0 ) [ E 0 ( r ) a ^ E 0 ( r ) a ^ ] ,
| ψ ( r , t ) = ψ 0 ( r , t ) = n = 0 e i ω 0 [ ( z / v t ) n t ] f n ( z ) | n ,
d f n d z = n u f + 1 n 1 n + 1 u f 1 n + 1 ,
P = n = max { 0 , } | f n ( ) | 2
f n ( ) = p n + e i χ ( n + ) ! n ! e | β 0 | 2 / 2 ( β 0 ) × n ( | β 0 | 2 ) n n ! ( + n ) ! ( n n ) ! ,
β 0 = e ω 0 d z E 0 z e i ω 0 z / v
f n ( ) = p n + e i arg { β 0 } J ( 2 n + | β 0 | ) ,
P = { J 2 ( 2 | β | ) , Fock, coherent e 2 | β | 2 I ( 2 | β | 2 ) , thermal ,
[ m e c 2 β + c α ( p + e c A ) e φ ] Ψ = i Ψ t ,
β = [ I 0 0 I ] , α = [ 0 σ σ 0 ]
Ψ = V 1 / 2 k ψ k e i k r i E k t / Ψ k ,
Ψ k = A k [ s ^ B k ( k σ ) s ^ ]
( m e c 2 β + c α p ) Ψ k = E k Ψ k ,
[ E 0 ( 2 c 2 / E 0 ) k 0 ( i + k 0 ) + e α A e φ ] Ψ = i Ψ t .
Ψ ψ ( r , t ) Ψ k 0 ,
[ E 0 ( 2 c 2 / E 0 ) k 0 ( i + k 0 ) + ( e c / E 0 ) k 0 A e φ ] ψ ( r , t ) = i ψ ( r , t ) t ,
[ E 0 v ( i + k 0 ) + ( e v / c ) A e φ ] ψ ( r , t ) = i ψ ( r , t ) t .
A ^ = j ( i c / ω j ) [ E j ( r ) a ^ j E j ( r ) a ^ j ] ,
( H ^ 0 + H ^ 1 ) | ψ ( r , t ) = i | ψ ( r , t ) t
H ^ 0 = j ω j a ^ j a ^ j + E 0 v ( i + k 0 ) ,
H ^ 1 = ( e v / c ) A ^ ,
i d α n d t = n g e i ω 0 t α n 1 + n + 1 g e i ω 0 t α n + 1 ,
| ψ i ( t ) = n α n ( t ) | n .
| ψ i ( t ) = S ^ ( t , t 0 ) | ψ i ( t 0 ) ,
S ^ = e i χ e β 0 a ^ β 0 a ^
β 0 ( t , t 0 ) = i t 0 t d t g ( t ) e i ω 0 t
χ ( t , t 0 ) = 1 t 0 t d t Re { β 0 ( t , t 0 ) g ( t ) e i ω 0 t } .
n | S ^ | n 0 = e i χ n 0 ! n ! e | β 0 | 2 / 2 ( β 0 ) n 0 n × n ( | β 0 | 2 ) n n ! ( n 0 n + n ) ! ( n n ) ! ,
d f 1 n + 1 , 1 d z = n + 1 u , d f 1 n 1 , 1 d z = n u , d f 2 n + 2 , 2 d z = n + 2 u f 1 n + 1 , 1 , d f 0 n , 2 d z = n u f 1 n 1 , 1 n + 1 u f 1 n + 1 , 1 , d f 2 n 2 , 2 d z = n 1 u f 1 n 1 , 1 , ,
P 1 = ( 1 + n ¯ ) | β 0 | 2 ,
P 1 = n ¯ | β 0 | 2 ,
d f n , d z = n + 1 u f 1 n + 1 , 1 .
f n , ( ) = ( 1 ) n ( n 1 ) ( n + 1 ) d z 1 z 1 d z 2 z 1 d z u ( z 1 ) u ( z 2 ) u ( z ) = 1 ! n ( n 1 ) ( n + 1 ) ( β 0 ) ,
P P 1 = g ( ) ( ! ) 2 ,
P n = 0 p n J 2 ( 2 n | β 0 | ) .
P 0 x d x e x 2 J 2 ( 2 x | β | ) = e 2 | β | 2 I ( 2 | β | 2 ) ,
d ρ ^ d t = i [ ρ ^ , H ^ ] + γ 2 ( 2 σ ^ ρ ^ σ ^ σ ^ σ ^ ρ ^ ρ ^ σ ^ σ ^ ) ,
n ¯ ˙ = ( 2 / ) Im { ρ 10 g e i ω 0 t } κ n ¯ , ρ ˙ 10 = ( i / ) ( 1 2 n ¯ ) g e i ω 0 t κ ρ 10 / 2
n ¯ = 1 2 1 1 + I s / I ,
P 1 = p 0 | β 0 | 2 = ( 1 n ¯ ) | β 0 | 2 , P 1 = p 1 | β 0 | 2 = n ¯ | β 0 | 2 ,
ξ ( t ) = E 0 p [ e i ( ω 0 ω ) t ( ω 0 ω i κ / 2 ) + e i ( ω 0 + ω ) t ( ω 0 + ω i κ / 2 ) ] .
E 0 = [ k 0 2 p + ( p ) ] e i k 0 r r ,
β 0 = 2 e ω 0 v 2 γ [ i p x K 1 ( ζ ) + p z γ K 0 ( ζ ) ] ,
P 1 isotropic = | β 0 | 2 = | 2 e ω 0 p / v 2 γ | 2 f ( ω 0 b / v γ ) ,
d p n m d t = g [ n ( m + 1 ) p n 1 m + 1 ( n + 1 ) m p n m ] + κ [ ( n + 1 ) p n + 1 m n p n m ]

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