Free-Electron–Bound-Electron Resonant Interaction
Avraham Gover
1,*
and Amnon Yariv
2
1
Department of Electrical Engineering Physical Electronics, Tel Aviv University, Ramat Aviv 69978, Israel
2
California Institute of Technology (Caltech), Pasadena, California 91125, USA
(Received 25 September 2019; revised manuscript received 2 December 2019;
accepted 13 January 2020; published 13 February 2020)
Here we present a new paradigm of free-electron–bound-electron resonant interaction. This concept is
based on a recent demonstration of the optical frequency modulation of the free-electron quantum electron
wave function (QEW) by an ultrafast laser beam. We assert that pulses of such QEWs correlated in their
modulation phase, interact resonantly with two-level systems, inducing resonant quantum transitions when
the transition energy Δ E ¼ ℏω
21
matches a harmonic of the modulation frequency ω
21
¼ nω
b
. Employing
this scheme for resonant cathodoluminescence and resonant EELS combines the atomic level spatial
resolution of electron microscopy with the high spectral resolution of lasers.
DOI: 10.1103/PhysRevLett.124.064801
The reality of the quantum electron wave function and its
interpretation have been a matter of debate since the
inception of quantum theory [1,2]. Recent developments
in ultrafast electron microscopy, and particularly photon-
induced near-field electron microscopy (PINEM) [3–10]
demonstrated the possibility of modulating the energy
spectrum of single quantum electron wave packets
(QEW) at discrete energy sidebands ΔE
n
¼ nℏω
b
by
interaction with a laser beam of frequency ω
b
. The
interaction is made possible by a multiphoton emission
or absorption process in the near field of a nanostructure
[9,11], a foil [8,10], or a laser beat (pondermotive potential)
[12,13]. It was also shown that due to the nonlinear energy
dispersion of electrons in free space drift, the discrete
energy modulation of the QEW turns into a tight bunching
density modulation at attosecond short levels, correspond-
ing to high spectral harmonics contents ω
n
¼ nω
b
in the
expectation value of the QEW density hjΨðr; tÞj
2
i. The
physical reality of this sculpting of the QEW in the time and
space (propagation coordinate, z) dimensions can be
demonstrated in the interaction of such modulated
QEWs with radiation [14,15]. Such bunching-phase-sensi-
tive resonant stimulated radiative interactions (acceleration
or deceleration) of QEW have been demonstrated recently
experimentally with a second laser beam, phase-locked to
the bunching frequency or its harmonic [10,12,16,17].
Here we propose a new concept of free-electron–bound-
electron resonant interaction (FEBERI) based on the idea
that optical frequency density modulated QEWs can
interact resonantly with quantum electron transitions in
matter at harmonics of its modulation frequency. Such
interaction is shown schematically in Figs. 1(a) and 1(b) for
the simple case of interaction with a single two-level
system (2-LS) of a bound electron, e.g., in an atom,
quantum-dot structure, defect center in crystal, etc. The
interaction would lead to resonant transitions between the
quantum levels 1 and 2 and corresponding energy loss or
gain in the free electron energy. The resonant interaction
can be monitored by measurement of the electron energy
loss or gain spectrum (EELS, EEGS) [18], or by measuring
the fluorescence due to excitation of the bound electron to
the upper level 2 and its radiative relaxation to the lower
level 1 or possible other levels. In this sense, the effect will
be a resonant cathodoluminescence (RCL) effect, showing
enhanced CL [19] emission of the sample when the
harmonic frequency of the interacting QEW nω
b
matches
the transition energy nℏω
b
≅ ΔE ¼ ℏω
1;2
. Such a scheme
can have a major impact on electron microscopy and
material spectroscopy, combining the atomic level spatial
resolution of electron microscope with the high spectral
resolution of the laser. Such resolution can be instrumental
also in quantum computing, addressing Q-bits based on
2-LS defect centers in crystals [20,21]. Furthermore, with
intense localized pumping of atomic or nanometric 2-LS
systems and microresonators, one may even consider
development of microscopic single atom lasers [22].
(a)(b)(c)
FIG. 1. A density-modulated quantum electron wave packet
(a) passing near a two-level system (2LS) target (b), and exciting
transitions and luminescence. (c) Enhanced excitation by a train
of modulation-phase correlated QEWs.
PHYSICAL REVIEW LETTERS 124, 064801 (2020)
Editors' Suggestion Featured in Physics
0031-9007=20=124(6)=064801(6) 064801-1 © 2020 American Physical Society
A possible way of analyzing the proposed interaction
scheme is by solving the Schrödinger equation for the
scattering of an incident electron quantum wave packet by
the electrons of an atomic system. This can be done
numerically using a time dependent density functional
theory (TDDFT) formulation based on the time dependent
Kohn-Sham (TDKS) equations [23] (See Supplemental
Material [24] SM-A). Taking an analytical approximation
approach, the problem can be presented in terms of dipole
coupling between the free QEW and the bound electron
through their near-field induced electric fields. In this Letter
we neglect the effect of the 2-LS dipole moment on the
QEWs (neglecting their interaction quantum recoil), and
assume that the bound electron transition is governed by the
Schrödinger equation
iℏ
∂
∂t
Ψ
b
ðr;tÞ¼½H
0
þ V
wp
ðrÞΨ
b
ðr;tÞ; ð1Þ
where V
wp
¼ −eEðtÞ · r, and EðtÞ is the field induced by
the modulated QEW at the 2-LS location. For a single
QEW, modulated at frequency ω
b
and tightly bunched [9],
we model its density as
n
e
ðr;tÞ¼f
e⊥
ðr
⊥
Þf
et
ðt − t
0
− z=vÞf
mod
ðt − z=v
0
Þ; ð2Þ
where f
e⊥
ðr
⊥
Þ is a narrow normalized transverse distribu-
tion, f
et
ðtÞ¼exp ð−t
2
=2σ
2
et
Þ=
ffiffiffiffiffiffi
2π
p
σ
et
, σ
et
is the quantum
wave packet duration, f
mod
ðtÞ¼
P
∞
n¼−∞
B
n
e
−inω
b
t
,
and B
n
are the coefficients of the harmonics. Maximal
tight density bunching is attained after drift time t
D;max
¼
T
b
=ð2Δp
m
=p
0
Þ past the laser modulation point, where
T
b
¼ 2π=ω
b
, and Δp
m
=p
0
is the momentum modulation
amplitude of the electron relative to the average momentum
[9,15]. Substantially high amplitude (B
n
> 0.3) harmonics
are attainable up to the 20th harmonic [15] and beyond [25]
(see Supplemental Material [24] SM-B).
The self-fields of the charge-modulated QEW are found
from Maxwell equations similarly to the semiclassical
calculation of EELS in electron microscopy [18]. For the
assumption of a transverse distribution f
e⊥
ðr
⊥
Þ narrow
relative to the bunching wavelengths and an impact
parameter r
⊥
longer than the QEW width, but within the
near field of the modulated QEW [see Figs. 1(a) and 1(b)]:
Eðr;tÞ¼
e
2πϵ
0
f
et
ðt − t
0
− z=vÞ
v
2
γ
ϵ
ϵ
r
X
n
ω
n
B
n
g
n
ðrÞe
iω
n
½t−ðz=vÞ
;
ð3Þ
g
n
ðr
⊥
Þ¼
1
γ
ϵ
K
0
ω
n
r
⊥
vγ
ϵ
ˆ
z − K
1
ω
n
r
⊥
vγ
ϵ
ˆ
r
⊥
; ð4Þ
with ω
n
¼ nω
b
, γ
ϵ
¼ 1=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 − ϵ
r
β
2
p
, ϵ
r
¼ ϵ=ϵ
0
β ¼ v=c
r ¼ðr
⊥
;zÞ, r
⊥
¼ðx; yÞ and K
m
are modified Bessel
functions. We then need to solve a simple 2-LS equation
for the bound electron wave function Ψ
b
ðr;tÞ¼
a
1
ðtÞφ
1
ðrÞe
−iE
1
t=ℏ
þ a
2
ðtÞφ
2
ðrÞe
−iE
2
t=ℏ
, where φ
1
ðrÞ,
φ
2
ðrÞ are the eigenvalues of the noninteracting 2-LS.
This is a standard problem of coherent light interaction
with quantum levels in matter [26,27], except that the field
here is not a laser field, but the near field of the modulated
QEW. It leads to the coupled-modes equations (see
Supplemental Material [24] SM-C):
a
0
1
ðtÞ¼
i
ℏ
V
∗
21
ðtÞe
iω
21
t
a
2
ðtÞ; ð5aÞ
a
0
2
ðtÞ¼
i
ℏ
V
21
ðtÞe
−iω
21
t
a
1
ðtÞ; ð5bÞ
where ω
21
¼ðE
2
− E
1
Þ=ℏ, V
ij
¼
R
φ
i
ðrÞV
wp
ðr;tÞφ
j
ðrÞd
3
r,
and it is assumed that V
11
¼ 0 V
22
¼ 0 V
21
¼ V
∗
12
.
We use the solution of these equations in two cases of
interest: first, for a π-pulse half period Rabi oscillation case,
in which the interaction Hamiltonian VðtÞ and interaction
time are large enough to produce complete transition of the
bound electron from the ground to top level. Second, for the
case of weak coupling, where we can only calculate the
probability of exciting the upper level in one interac-
tion event.
In the first case, we assume for simplicity a constant
amplitude harmonic field EðtÞ¼E
0
cosðω
0
t − φ
0
Þ¼
1
2
E
0
e
iðω
0
t−φ
0
Þ
þ c:c: and present the solution for the case
of exact resonance, ω
0
¼ ω
1;2
¼ ΔE=ℏ ¼ðE
2
− E
1
Þ=ℏ,
starting from the ground state a
1
ð0Þ¼1, a
2
ð0Þ¼0 (see
solution for the general case in Refs. [26,27]). The
occupation probabilities of the states are
P
2
ðtÞ¼ja
2
ðtÞj
2
¼ sin
2
ðΩ
R
t=2Þ;
P
1
ðtÞ¼ja
1
ðtÞj
2
¼ cos
2
ðΩ
R
t=2Þ; ð6Þ
where
Ω
R
¼ 2jV
21
j=ℏ ¼ μ
21
· E
0
=ℏ; ð7Þ
μ
21
¼ −er
21
is the dipole moment of the 2-LS transition.
We now extend this model to the case of inducing a
coherent 2-LS transition with a pulse of modulated QEWs
[Eq. (2)]. Such an ensemble of QEWs may excite tran-
sitions in the 2-LS incoherently and randomly, with
probability proportional to N
e
. However, when the modu-
lated envelopes of the QEWs are phase correlated, as shown
in Fig. 1(c), the Rabi oscillation process will build up
coherently throughout the entire pulse. The correlated
optically modulated QEWs pulse density can be presented
as a sum of individual QEW densities (2) arriving each at
random time t
0j
(j ¼ 1 to N
e
) at the 2-LS location
n
e
ðr;tÞ¼f
e⊥
ðr
⊥
Þf
mod
t −
z
v
X
N
e
1
f
et
t − t
0j
−
z
v
: ð8Þ
This scenario [Fig. 1(c)], is similar to the case of
superradiance of a bunched electron beam. In this case
PHYSICAL REVIEW LETTERS 124, 064801 (2020)
064801-2
the bunching of the electrons is modeled in terms of point
particles [28] or quantum wave packets [15], and the
bunched beam radiates at all harmonics nω
b
of the
bunching frequency in proportion to N
2
e
, similarly to
Dicke’s atomic superradiance [29,30].
In the present case, the nth harmonic component of the
modulation-correlated electrons interacts coherently with a
2-LS. Consequently, when the modulation phases of the
different QEWs are correlated [see Fig. 1(c)] (for example,
if they were all a priori bunched by the same coherent laser
beam), then the near fields of all wave packets, irrespective
of their arrival times t
oj
, add up coherently, but the sum of
the QEW density distribution in Eq. (8) is replaced by the
temporal distribution of the electron pulse (SM-D).
Assuming for simplicity that the pulse envelope is uniform,
of duration T
p
, shorter than the relaxation or the
decoherence time of the 2-LS, we replace f
e
ðtÞ with
f
p
ðtÞ¼1=T
p
in Eq. (3), and the relevant field component
in Eq. (7) for the nth harmonic is
E
0
¼
1
2πϵ
0
eω
n
v
2
γ
ϵ
ϵ
r
g
n
ðr
⊥
Þ
B
n
N
e
T
p
: ð9Þ
Interestingly enough the quantum features disappear in
this case and the pulse duration T
p
takes the role of the
quantum wave packet size σ
et
. It is noteworthy that the
expressions for the density bunching amplitudes B
n
have
been derived independently for a pulse of laser-modulated
particles beam in a point-particles model in connection to
harmonic superradiance in FEL [28]. Here, following
Ref. [9] we used a quantum wave packet model for the
bunching, because the wave packet size σ
ez
¼ v
0
σ
et
is
usually longer than an optical wavelength in a high quality
TEM [14,31]. The consistency of the quantum and classical
analyses of the bunching process in the case of multiple
electrons is satisfying. Of course, the transition process in
the 2-LS is by itself a quantum effect in any case.
Substituting Eq. (9) in Eq. (7), one can calculate, given
the dipole moment μ
12
and the electron number N
e
in the
pulse, the condition for complete population inversion:
Ω
R
T
p
¼
2αω
n
cβ
2
γ
ϵ
ϵ
r
μ
21
· g
n
ðr
⊥
ÞB
n
N
e;π
¼ π; ð10Þ
with α ¼ð1=4πε
0
Þe
2
=ℏc ¼ 1=137 the fine-structure con-
stant. Note that the pulse duration T
p
cancelled out of
condition (10), and it is insignificant, as long as it is shorter
than the relaxation time of the upper level t
r
. If this
condition can be satisfied, it should be possible to obtain
an efficient resonant CL (RCL) with a finite pulse of N
e;π
modulated QEWs interacting with a single 2-LS atom or a
quantum dot or in bulk.
Another possible scenario is when Ω
R
T
p
≪ π (weak
coupling). In this case, from Eq. (6),
P
2
ðT
p
Þ ≈
Ω
R
T
p
2
2
¼
αω
n
cβ
2
γ
ϵ
ϵ
r
μ
21
· g
n
ðr
⊥
ÞB
n
N
e
2
:
ð11Þ
Noteworthy is the quadratic dependence on N
e
of the
transition probability of coherent resonant FEBERI and
RCL as opposed to the linear dependence in conven-
tional CL.
The coherent buildup of Rabi oscillation by a pulse of
phase-correlated QEWs is demonstrated in Fig. 2(a),
showing simulation (see Supplemental Material [24]
SM-E) of population buildup of level 2 due to interaction
with a pulse of N
e
correlated QEWs having the same phase
but arriving at random time t
0j
. For comparison, simulation
parameters are normalized to accumulate the same
Rabi phase Φ
R
ðtÞ¼
R
t
0
μ
12
¯
E
0
ðt
0
Þdt
0
=ℏ, on the average.
FIG. 2. Compar ison of upper level population probability buildups by the field excitation of a pulse of modulated QEWs.
(a) Excitation of Rabi oscillation with correlated wave packets: Green curve, QEWs arrive at equal time spacing; blue, at random time
t
0j
; black, a continuous modulated QEW. (b) Same, with uncorrelated modulated QEWs (φ
0j
random); black is the average of events.
PHYSICAL REVIEW LETTERS 124, 064801 (2020)
064801-3
Therefore, their time is normalized to an effective Rabi
frequency:
¯
Ω
R
¼ Ω
R
ðN
e
T
e
=T
p
Þ. Figure 2(b) shows the
buildup of the occupation probability of level 2 in the case
of uncorrelated QEWs, namely, their phases φ
0j
¼ ω
0
t
0j
are random. Evidently the population growth P
2
is slower,
and does not arrive to full occupation. The black curve is an
average over several simulation events, confirming the
initial linear buildup of the upper level population as a
function of N
e
in the absence of relaxation.
We now go back to present a general solution of Eq. (5B)
in the limit of weak coupling, and apply it to compare the
cases of a single finite size QEW, modulated QEWs, and
the conventional point-particle limit. Whether a single
QEW interaction with matter can be observed and mea-
sured is a fundamental physics question that has been
considered in connection to interaction of QEW with light
[15,32,33]; here we consider it in the case of interaction
with matter.
The first-order solution of Eq. (5B) for a general finite
time field pulse EðtÞ with perturbation Hamiltonian
V
wp
¼ μ
21
· E, under the assumptions a
1
ðtÞ ≈ a
1
ð0Þ¼1,
ja
2
ðtÞj ≪ 1,is
a
2
¼ −
i
ℏ
μ
21
·
Z
∞
−∞
EðtÞe
iω
21
t
dt ¼ −
i
ℏ
μ
21
· E
⌣
ðω
21
Þ; ð12Þ
where E
⌣
ðωÞ¼F fEðtÞg.
We first apply this expression for the case of an
unmodulated QEW—Eq. (3) with B
n
¼ δ
n;0
. The proba-
bility of exciting level 2 by N
e
uncorrelated QEWs is
P
WP
2
¼ N
e
ja
2
j
2
¼ N
e
P
par
2
jF
e
ðω
21
Þj
2
; ð13Þ
P
Part
2
¼
αω
21
cβ
2
γ
ϵ
ϵ
r
μ
21
· g
n
ðr
⊥
Þ
2
; ð14Þ
where for a Gaussian wave packet envelope f
et
ðtÞ¼
ð
ffiffiffiffiffiffi
2π
p
σ
et
Þ
−1
expð−t
2
=2σ
2
et
Þ, one has F
e
ðωÞ¼F ff
et
ðtÞg ¼
e
−ω
2
σ
2
et
=2
. In the limit σ
et
→ 0 of a point particle
P
WP
2
→ P
Par
2
, as expected. On the other hand, for a long
QEW ω
21
≫ 1=σ
et
the excitation probability decays. This
is evidently a quantum effect, not predictable by a point-
particle model of the electron. It is consistent with previous
conjectures of decay of radiative interaction of a QEW
in the limit ωσ
et
¼ 2πσ
ez
=βλ ≫ 1 with σ
ez
¼ vσ
et
,
λ ¼ 2πc=ω, that are predicted in a semiclassical interaction
model [14,15], but not verified in a QEW analysis of
spontaneous emission by a single QEW [33].
In the interesting case of a modulated QEW, we insert in
Eq. (13) the expression of the Fourier transform of Eq. (3)
of a harmonic frequency n, such that ω
n
≈ ω
21
.ForN
e
uncorrelated modulated QEWs this results in
P
M−WP
2
¼ N
e
ja
2
j
2
¼ N
e
P
par
2
jB
n
j
2
jF
e
ðω
21
Þj
2
¼ N
e
P
par
2
jB
n
j
2
e
−ðω
21
−ω
n
Þ
2
σ
2
et
: ð15Þ
For very tight bunching it is possible to get high
harmonic amplitudes up to about 20th harmonic [9] and
higher [25], and attain resonant FEBERI transitions at
frequencies ω
n
¼ nω
b
≫ ω
b
beyond the cutoff frequency
of an unmodulated finite QEW 1=σ
et
. Comparison of
Eqs. (15) to (13) (with N
e
¼ 1) reveals the special
characteristics of a modulated QEW of finite size.
The significant enhancement of FEBERI transition with
a pulse of modulation correlated QEWs, as compared to
conventional (point-particle) interaction, is evident when
one compares Eq. (11) to Eqs. (14), (15) (for the case
N
e;π
≫ N
e
≫ 1). In this case, the probability of excitation
of the upper level by N
e
modulated QEWs can be written as
P
2
ðT
p
Þ¼B
2
n
N
2
e
P
par
2
, namely, there is an enhancement
factor B
2
n
N
e
relative to the excitation probability with N
e
uncorrelated point-particle interaction events. This
enhancement may amount to many orders of magnitude,
and it would be also the enhancement ratio of resonant
cathodoluminescence in a two-level system, whether the
excited electrons relax radiatively to the ground level or to
other quantum levels.
We can now estimate the viability of our new concepts of
FEBERI and RCL, referring to real parameters of a material
target of interest, such as NV defect centers in diamond. We
first check how many phase-correlated modulated QEWs
would be required to produce a full π-phase Rabi transition
and satisfy Eq. (10). In diamond NV centers there is a 2-LS
quantum transition of ΔE ¼ 1.945 eV [21], and it thus can
be excited resonantly by the second harmonic
field component of a pulse of QEWs, modulated by an
infrared laser of λ
b
¼ 1.27 μm. With a rough estimate
½g
n
ðr
⊥
ÞB
n
=β
2
γ
ε
ε
r
≈ 1, one obtains N
π
¼ 2.2 × 10
4
, which
corresponds to 3.7 × 10
−15
Coulomb. This may be exces-
sive charge for the femtosecond laser driven photoemission
techniques used in PINEM [9,34], and one may be
concerned about energy spread and loss of modulation
coherence due to Coulomb interaction scattering in the
electron pulse [35]. However, since the relaxation time of
the 2-LS can be quite long (t
r
¼ 13.5 × 10
−9
sec for a
diamond NV center [21]), one may resort to distributing the
charge over longer electron pulses or employing high
rep-rate mode locked laser techniques [36], in order to
mitigate the Coulomb scattering problem. Alternatively, if
one operates in the weak coupling regime with a smaller
number of correlated modulated QEWs (say, N
e
¼
10
2
⟪N
π
), one can still attain an enhancement factor of
RCL by a factor N
e
relative to the conventional CL from
the same number of electrons.
The new concepts of FEBERI and RCL with single
QEWs and with an ensemble of correlated QEWs, were
presented here in the framework of a simplified semi-
classical model. They should lead to more elaborate
theoretical formulation and experimental studies in both
fundamental and applied physics research. The reality of
the size and shape of a single electron wave function in its
PHYSICAL REVIEW LETTERS 124, 064801 (2020)
064801-4
interaction with radiation has raised new interest in the old
question of particle-wave duality [14,31–33,37]. The CL
process provides an alternative way for probing the reality
of the single electron wave function through interaction
with a well defined 2-LS quantum transition. In this Letter
we proposed that the interaction of a single electron with an
atomic system in matter can depend, and thus be controlled,
by the modulation of its quantum electron wave function by
a coherent laser beam and by the history of its transport to
the interaction point. In the case of interaction with single
QEWs, this picture may be contrasted by an argument of
collapse of the wave function to a point particle at the single
interaction event [38]. In the case of an ensemble of
identical correlated modulated QEWs, the wavepacket
modulation resonant enhancement effect is well justified
[39] and consistent with classical point-particle bunching
analysis.
These ideas are expected to lead to a new way for
studying fundamental questions of quantum theory. On the
application side, the combination of these concepts with the
atomic scale spatial resolution of electron microscopes can
lead to development of a new kind of electron microscopy
and spectroscopy on the level of single atom resolution.
Besides diagnosis on the basis of the emitted radiation
(RCL), the imprint of the resonant interaction on the
free electrons spectrum can be revealed in a TEM instru-
ment also through EELS measurement [resonant EELS
(REELS)]. This, as well as other applications such as
addressing individual 2-LS targets as Q bits (for example,
diamond NV centers in quantum computer schemes [20]),
resonant CL in bulk (rather than isolated atoms), quantum
dots, etc, resonant directed superradiant emission in a
grating configuration [40], and possibly localized lasing
in microcavity lasers [22], are promising directions of
further development of the new concepts presented here.
This Letter presented only the theoretical principles of a
new interaction scheme. Their realization in the laboratory
may require dedicated development of electron microscopy
technology for improving the quality of the modulated
QEW, controlling the electron emission from the cathode,
mitigating the deleterious effect of Coulomb scattering
[35], and improving the efficiency of QEW modulation by
laser beam.
This research was supported in part by a grant from the
Israel Science Foundation (ISF) and the German-Israeli
Project Cooperation (DIP). We thank Tom Shalev and Bin
Zhang for providing simulation results.
*
Corresponding author.
gover@eng.tau.ac.il
[1] E. Schrödinger, An undulatory theory of the mechanics of
atoms and molecules, Phys. Rev. 28, 1049 (1926).
[2] M. Z. Born, Zur Quantenmechanik der Stoßvorgänge,
Z. Phys. 37, 863 (1926).
[3] B. Barwick, D. J. Flannigan, and A. H. Zewail, Photon-
induced near-field electron microscopy, Nature (London)
462, 902 (2009).
[4] F. J. García de Abajo, A. Asenjo-Garcia, and M. Kociak,
Multiphoton absorption and emission by interaction of swift
electrons with evanescent light fields, Nano Lett. 10, 1859
(2010).
[5] S. T. Park, M. Lin, and A. H. Zewail, Photon-induced near-
field electron microscopy (PINEM): Theoretical and ex-
perimental, New J. Phys. 12, 123028 (2010).
[6] S. T. Park and A. H. Zewail, Relativistic effects in photon-
induced near field electron microscopy, J. Phys. Chem. A
116, 11128 (2012).
[7] K. E. Echternkamp, A. Feist, S. Schäfer, and C. Ropers,
Ramsey-type phase control of free-electron beams, Nat.
Phys. 12, 1000 (2016).
[8] 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).
[9] A. Feist, K. E. Echternkamp, J. Schauss, S. V. Yalunin, S.
Schfer, and C. Ropers, Quantum coherent optical phase
modulation in an ultrafast transmission electron microscope,
Nature (London)521, 200 (2015).
[10] 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 (2017) .
[11] L. U. C. A. Piazza, T. T. A. Lummen, E. Quinonez, Y.
Murooka, B. W. Reed, B. Barwi ck, and F. Carbone, Simul-
taneous observation of the quantization and the interference
pattern of a plasmonic near-field, Nat. Commun. 6, 6407
(2015).
[12] M. Kozák, N. Schönenberger, and P. Hommelhoff, Ponder-
omotive Generation and Detection of Attosecond Free-
Electron Pulse Trains, Phys. Rev. Lett. 120, 103203 (2018).
[13] M. Kozák, T. Eckstein, N. Schönenberger, and P. Hommelh-
off, Inelastic ponderomotive scattering of electrons at a
high-intensity optical travelling wave in vacuum, Nat. Phys.
14, 121 (2018).
[14] A. Gover and Y. Pan, Dimension-dependent stimulated
radiative interaction of a single electron quantum wave-
packet, Phys. Lett. A 382, 1550 (2018).
[15] Y. Pan and A. Gover, Spontaneous and stimulated
radiative emission of modulated free-electron quantum
wavepackets—
semiclassical analysis, J. Phys. Commun.
2, 115026 (2018).
[16] D. S. Black, U. Niedermayer, Y. Miao, Z. Zhao, O.
Solgaard, R. L. Byer, and K. J. Leedle, Net Acceleration
and Direct Measurement of Attosecond Electron Pulses in a
Silicon Dielectric Laser Accelerator, Phys. Rev. Lett. 123,
264802 (2019).
[17] N. Schönenberger, A. Mittelbach, P. Yousefi, U. Nieder-
mayer, and P. Hommelhoff, Generation and Characteriza-
tion of Attosecond Micro-Bunched Electron Pulse Trains
via Dielectric Laser Acceleration, Phys. Rev. Lett. 123,
264803 (2019).
PHYSICAL REVIEW LETTERS 124, 064801 (2020)
064801-5
