of 6
Quantum Wave-Particle Duality in Free-Electron
Bound-Electron Interaction
Bin Zhang ,
1
Du Ran ,
1,2
,*
Reuven Ianconescu ,
1,3
Aharon Friedman ,
4
Jacob Scheuer ,
1
Amnon Yariv,
5
and Avraham Gover
1
,
1
Department of Electrical Engineering Physical Electronics, Center for Laser-Matter Interaction (LMI),
Tel Aviv University, Ramat Aviv 69978, Israel
2
School of Electronic Information Engineering, Yangtze Normal University, Chongqing 408100, China
3
Shenkar College of Engineering and Design, 12, Anna Frank Street, Ramat Gan 5252626, Israel
4
Ariel University, Ariel 40700, West Bank
5
California Institute of Technology, Pasadena, California 91125, USA
(Received 31 December 2020; accepted 8 April 2021; published 17 June 2021)
We present a comprehensive relativistic quantum-mechanical theory for interaction of a free electron
with a bound electron in a model, where the free electron is represented as a finite-size quantum electron
wave packet (QEW) and the bound electron is modeled by a quantum two-level system (TLS). The analysis
reveals the wave-particle duality nature of the QEW, delineating the point-particle-like and wavelike
interaction regimes and manifesting the physical reality of the wave function dimensions when interacting
with matter. This QEW size dependence may be used for interrogation and coherent control of
superposition states in a TLS and for enhancement of cathodoluminescence and electron energy-loss
spectroscopy in electron microscopy.
DOI:
10.1103/PhysRevLett.126.244801
The interpretation of the quantum electron wave function
Ψ
ð
r
;t
Þ
has been a matter of debate since the inception of
quantum theory
[1,2]
. The accepted Born interpretation is
that the expectation value
hj
Ψ
ð
r
;t
Þj
2
i
represents the prob-
ability density of finding the electron at point
r
and time
t
.
The reality of the quantum electron wave packet (QEW)
and the measurability of its dimensions, as well as the
transition from the quantum wave function presentation
to the classical point-particle theory (the wave-particle
duality) were considered previously in the context of
electron interactions with light
[3
10]
. It was shown that
the transition of the QEW radiative interaction from the
wavelike regime, exhibiting characteristic multisideband
photon-induced near-field electron microscopy (PINEM)
energy spectrum
[11
21]
to the classical point-particle-like
acceleration-deceleration regime
[22]
, takes place when
[3
6,23]
Γ
¼
ωσ
et
¼
2
πσ
ez
=
βλ
<
1
, where
β
¼
v
0
=c
is the
centroid velocity of the QEW. Namely, the transition takes
place when the wave packet duration
σ
et
or its length
σ
ez
are
short relative to the optical radiation period
2
π
=
ω
or
wavelength
λ
, respectively.
Recent technological advances enable the shaping of
single QEWs in the transverse and longitudinal dimensions
[24
26]
. Furthermore, it has been demonstrated that the
QEW density expectation value can be modulated at
optical frequencies by interaction with a laser beam
[17
21]
, and that this modulation is detectable by inter-
action with another synchronous laser beam, attesting to the
reality of the QEW periodic sculpting (modulation) in the
context of a stimulated radiative interaction
[18,19,27,28]
.
The reality of QEW modulation features has also been
asserted in the case of multiple modulation-correlated
electron wave packets, where coherent superradiant emis-
sion
[29]
, proportional to the number of electrons squared
N
2
, is expected
[6]
, as in the classical case of a prebunched
particle beam
[30]
.
In analogy with the interaction of a QEW with radiation,
the reality of the QEW shape and its modulation features
were also claimed to be manifested in an interaction with
matter in a newly proposed effect of free-electron
bound-
electron resonant interaction (FEBERI)
[31]
. Based on a
simple semiclassical model it was asserted in Ref.
[31]
that
a QEW, passing in the vicinity of a two-level system (TLS)
target (e.g., an atom, quantum dot, crystal color center, or
trapped ion), would induce QEW size- and shape-depen-
dent transitions in the TLS. Specifically, it was suggested
that an ensemble of optical frequency modulated QEWs
would excite resonantly TLS transitions if their frequency
of modulation (produced by a laser of frequency
ω
b
in a
PINEM setup
[17]
) is a subharmonic of the TLS transition
frequency:
n
ω
b
¼
ω
2
;
1
¼
E
2
;
1
=
, where
E
2
;
1
¼
E
2
E
1
is
the energy gap of the TLS quantum levels. Further, it was
argued that, if all QEWs are modulation correlated (by the
same modulating laser), the transition rate would be
enhanced in proportion to
N
2
in analogy with the super-
radiance effect. The analogy is quite straightforward
when the QEWs are in the near-point-particle limit
Γ
¼
ω
2
;
1
σ
et
<
1
, and injected as a periodic spatial and temporal
train of pulses, a process recently termed
quantum
klystron
[32]
and closely related to the effect of
pulsed
PHYSICAL REVIEW LETTERS
126,
244801 (2021)
0031-9007
=
21
=
126(24)
=
244801(6)
244801-1
© 2021 American Physical Society
beam scattering
[33,34]
. The extension to the case of
optical frequency density modulated QEWs and multiple
modulation-correlated QEWs was hypothesized in
Refs.
[31,35]
on the basis of semiclassical theory and a
Born
s probability interpretation of the wave function
envelope modulation.
The semiclassical analysis of FEBERI and its proposed
dependence on the QEW dimension were contested and
caused a debate
[31,36
38]
. It also triggered numerous
recent publications on the subject
[32,35,39
42]
. In this
Letter we substantiate the reality of the QEW shaping in
FEBERI by presenting a complete quantum-mechanical
and relativistically valid formulation, demonstrating the
wave packet size dependence of the FEBERI in the case of
a single QEW, as well as the quantum wave-to-particle
transition in this interaction. Control over the dimensions of
single QEWs, and eventually over multiple modulation-
correlated QEWs, may enable implementation of the
FEBERI effect in new applications in electron microscopy
and atomic-scale probing of quantum excitations in matter,
and in diagnostics and coherent control of quantum bits
(qubits) and quantum emitters.
Model.
Our comprehensive quantum-mechanical theory
model is based on the configuration depicted in Fig.
1
,
comprising a thin free-electron QEW, propagating in the
vicinity ofa TLSthat is modeledasa hydrogenlike atom.For
simplicity we assume that the interaction of the free and
bound electrons is Coulombic. We start with a Schrödinger
equation for the combined wave function of the free and
bound electrons
i
Ψ
ð
r
;
r
0
;t
Þ
t
¼ð
H
0
þ
H
I
Þ
Ψ
ð
r
;
r
0
;t
Þ
;
ð
1
Þ
where
H
0
¼
H
0
F
þ
H
0
B
is the kinetic Hamiltonian of the
free and bound electrons and
H
I
is the interaction
Hamiltonian. To apply this to relativistic electrons, we
use the
relativistic
Schrödinger equation Hamiltonian
for a free electron of energy
E
0
¼
γ
mc
2
and momentum
p
0
¼
γ
m
v
0
:
H
0
F
ð
r
Þ¼
E
0
þ
v
0
·
ð
i
p
0
Þþð
1
=
2
γ
3
m
Þ
ð
i
p
0
Þ
2
. This Hamiltonian was derived in Ref.
[3]
and the Supplemental Material of Ref.
[4]
by a second order
iterative approximation of the Klein-Gordon equation, and
consequently does not include spin effects. It was also
derived recently directly from the Dirac equation
[43]
without the quadratic term that is needed to account for
electron wave packet chirp and size expansion in free
drift. It corresponds to a second order expansion of
the relativistic energy dispersion of a free electron:
E
p
¼
E
0
þ
v
0
·
ð
p
p
0
Þþð
1
=
2
γ
3
m
Þð
p
p
0
Þ
2
.
The eigenfunction solutions of the bound-electron
Hamiltonian are modeled as the solutions of a TLS:
H
0
B
ψ
j
ð
r
0
;t
Þ¼
E
j
ψ
j
ð
r
0
;t
Þ
(
j
¼
1
, 2), where
ψ
j
ð
r
0
;t
Þ¼
φ
j
ð
r
0
Þ
e
iE
j
t=
. In this case the wave function of the bound
electron is
Ψ
B
ð
r
0
;t
Þ¼
P
2
j
¼
1
C
j
ψ
j
ð
r
0
;t
Þ
, where the co-
efficients satisfy
P
2
j
¼
1
j
C
j
j
2
¼
1
. The wave function
solution of the free-electron Hamiltonian in zero order
(no interaction) is taken to be a wave packet:
Ψ
ð
0
Þ
F
ð
z; t
Þ¼
Z
dp
ffiffiffiffiffiffiffiffi
2
π
p
c
ð
0
Þ
p
e
iE
p
t=
e
ipz=
:
ð
2
Þ
We assume that the wave packet momentum distribution
before interaction is a Gaussian:
c
ð
0
Þ
p
¼
1
ð
2
πσ
2
p
0
Þ
1
=
4
exp

ð
p
p
0
Þ
2
4
σ
2
p
0
iE
p
z
0
v
0

;
ð
3
Þ
where
z
0
is the initial location of the QEW and
σ
p
0
is the
QEW momentum spread. For simplicity we assume here
that the QEW reaches the interaction point
z
¼
0
at time
t
0
at its longitudinal waist, so that the axial coordinate spread
of the QEW is
σ
z
0
¼
=
2
σ
p
0
(the expansion of the QEW
during the short interaction time is negligible
[4]
).
In this simplified model, the spin is neglected. As shown
in Sec. S.1 of the Supplemental Material
[44]
, the spin-spin
interaction potential is estimated to be more than 4 orders of
magnitude smaller than the Coulomb dipole interaction in
the relevant regime. Furthermore, we assume that the free
and bound electrons do not overlap spatially. In this case,
there is no exchange energy, and we can avoid the intricate
second quantization of many-body interaction theory
[49]
.
Considering only the Coulomb interaction, we neglect the
retardation effect, assuming that the electron transit time
through the interaction region is much shorter than the
transition frequency period
[50]
, and the interaction
Hamiltonian is (see Sec. S.2 of the Supplemental
Material
[44]
)
H
I
ð
r
;
r
0
Þ¼
e
2
4
πε
0
γ
r
0
·
ð
ˆ
e
z
z
þ
ˆ
e
r
r
0
Þ
ð
γ
2
z
2
þ
r
2
0
Þ
3
=
2
;
ð
4
Þ
where
r
,
r
0
are the coordinates of the free and bound
electrons, respectively. Here we used a dipole interaction
Hamiltonian, under the assumption that the free electron
passing by the modeled neutral hydrogen atom (
r
0
r
0
),
has no interaction with it other than the dipolar interaction
with its electronic quantum states
[42]
. To keep the analysis
valid in the relativistic regime, we used Feynman
s
FIG. 1. Two-level system (TLS) quantum interaction model of
quantum electron wave packet interaction with a bound electron.
PHYSICAL REVIEW LETTERS
126,
244801 (2021)
244801-2
expression
[51]
for the Coulomb field induced by the free
electron on the atom site (Sec. S.2 of the Supplemental
Material
[44]
). In the interaction process, the expa-
nsion coefficients of the QEW [Eq.
(3)
] are enta-
ngled with the coefficients of the bound electron
C
j
,
and the combined wave function is
Ψ
ð
r
0
;
r
;t
Þ¼
P
2
j
¼
1
R
p
dpc
jp
ð
t
Þ
φ
j
ð
r
0
Þ
e
i
ð
E
j
þ
E
p
Þ
t=
e
ipz=
.
Projection onto momentum space.
After substituting
the expansion of
Ψ
ð
r
0
;
r
;t
Þ
into the Schrödinger equation
(1)
and canceling out the no-interaction terms, projection into a
single momentum and TLS state results in (Sec. S.2 of the
Supplemental Material
[44]
)
_
c
i;p
0
ð
t
Þ¼
1
i
Z
dp
h
p
0
;i
j
H
I
j
p; j
i
c
j;p
0
ð
t
Þ
e
i
̃
Et=
;
ð
5
Þ
which describes the dynamic evolution of the entangled
free electron and bound electron in momentum space,
where
̃
E
¼
E
p
E
p
0
E
i;j
, with
E
i;j
¼
E
i
E
j
. The
matrix element is calculated for the specific interaction
Hamiltonian
(4)
for longitudinally and transversely aligned
dipoles (Sec. S.5 of the Supplemental Material
[44]
):
H
I;i;j;
k
ð
p
Þ¼
i
ð
e
μ
i;j
p=
ε
0
γ
2
Þ
K
0
ð
pr
=
γ
Þ
;
H
I;i;j;
ð
p
Þ¼
ð
e
μ
i;j
p=
ε
0
γ
Þ
K
1
ð
pr
=
γ
Þ
, where
μ
2
;
1
¼
e
r
2
;
1
¼
e
R
φ

2
ð
r
0
Þ
r
0
φ
1
ð
r
0
Þ
d
3
r
0
is a dipole transition element of
the bound electron,
K
0
and
K
1
are the zeroth and first order
modified Bessel functions of the second kind, respectively.
The QEW and the TLS are disentangled before the
interaction:
c
j;p
ð
t
0
Þ¼
C
ð
0
Þ
j
ð
t
0
Þ
c
ð
0
Þ
p
. In a first order iterative
approximation (Sec. S.3 of the Supplemental Material
[44]
), the differential equation can be solved under
the approximation
c
j;p
ð
t
Þ
C
ð
0
Þ
j
ð
t
0
Þ
c
ð
0
Þ
p
(stationary TLS
during the interaction where
t
0
is the arrival time of
the QEW centroid to the TLS at
z
¼
0
, and
t
0
and
t
þ
0
are the starting and ending interaction times):
c
i;p
0
ð
t
þ
0
Þ¼
C
ð
0
Þ
i
ð
t
0
Þ
c
ð
0
Þ
p
0
þ
Δ
c
i;p
0
, where
Δ
c
i;p
0
¼ð
2
π
=iv
0
Þ
H
I;i;j
ð
p
rec
Þ
C
ð
0
Þ
j
ð
t
0
Þ
c
ð
0
Þ
p
0
p
rec
and
p
rec
¼
p
0
p
¼
E
i;j
=v
0
. The transi-
tion probability of the TLS after interaction is obtained by
integrating the squared modulus of the combined coeffi-
cient
c
i;p
0
over
p
0
:
P
i
ð
t
þ
0
Þ¼
Z
−∞
j
c
i;p
0
ð
t
þ
0
Þj
2
dp
0
¼
P
ð
0
Þ
i
þ
Δ
P
ð
1
Þ
i
þ
Δ
P
ð
2
Þ
i
;
ð
6
Þ
where
P
ð
0
Þ
i
¼j
C
i
ð
t
0
Þj
2
is the initial occupation probability
of level
i
and the incremental probabilities are
Δ
P
ð
1
Þ
i
¼
2
Re

C
ð
0
Þ
i
ð
t
0
Þ
Z
−∞
dp
0
c
ð
0
Þ
p
0
ð
t
0
Þ
Δ
c
i;p
0

¼
4
π
e
Γ
2
=
2
sin
ζ
v
0
j
H
I;i;j
ð
p
rec
Þj
×
j
C
ð
0
Þ
i
ð
t
0
Þ
C
ð
0
Þ
j
ð
t
0
Þj
ð
7
Þ
and
Δ
P
ð
2
Þ
i
ð
t
þ
0
Þ¼
4
π
2
v
2
0
j
H
I;i;j
ð
p
rec
Þj
2
×
j
C
ð
0
Þ
j
ð
t
0
Þj
2
:
ð
8
Þ
In the case of excitation of a TLS from the ground state
C
ð
0
Þ
2
ð
t
0
Þ¼
0
;C
ð
0
Þ
1
ð
t
0
Þ¼
1
, the excitation probability of the
TLS [Eq.
(8)
] is given by
P
2
ð
t
þ
0
Þ¼
4
π
2
j
H
I;i;j
ð
p
rec
Þj
2
=v
2
0
.
Evidently the excitation probability in this case is inde-
pendent of the QEW shape or dimensions
[36]
(see the
discussion in Sec. S.4 of the Supplemental Material
[44]
).
However, if the TLS is initially in a superposition state,
terms
(7)
and
(8)
are both present, but term
Δ
P
ð
1
Þ
i
[Eq.
(7)
]
may be dominant. In the derivation of Eq.
(7)
, we expressed
C
ð
0
Þ
i
ð
t
0
Þ
C
ð
0
Þ
j
ð
t
0
Þ¼j
C
ð
0
Þ
i
ð
t
0
Þj
×
j
C
ð
0
Þ
j
ð
t
0
Þj
e
i
φ
in terms of
the dipole moment excitation amplitude and the phase of
the superposition state (azimuth angle on the Bloch sphere)
[44]
. We defined the phase of the QEW arrival time relative
to the TLS dipole moment excitation phase,
ζ
¼
ω
i;j
t
0
φ
,
and the interaction decay coefficient of the finite size QEW:
Γ
¼
ω
i;j
σ
et
. This makes the incremental probability
Δ
P
ð
1
Þ
i
dependent on the wave packet dimensions for a given
predetermined superposition state of the TLS, and it
vanishes for a long wave packet. Instructively, Eq.
(7)
and the decay coefficient
Γ
are analogous to the decay of
QEW acceleration in the transition from the classical point-
particle to the quantum (PINEM) regime in a stimulated
radiative interaction of a finite-size QEW
[4,23]
.
An alternative analytic iterative solution of the
Schrödinger equation for the FEBERI effect leads to
general expressions for the transition probability in terms
of Born
s probability distribution interpretation of the QEW
hj
Ψ
F
ð
r
;t
Þj
2
i
(Sec. S.4 of the Supplemental Material
[44]
),
which is conducive to further generalization to the cases of
modulated multiple particle QEWs
[35]
. In the case of a
single QEW, this approximation results in the same
approximate expressions
(7)
and
(8)
in its range of validity
of short QEW. To check the validity of the analytical
approximations and extend the solution to any size of the
QEW and a general TLS initial state, we have developed
two kinds of numerical computation codes for solving the
FEBERI problem in a rigorous quantum-mechanical
model, starting from the momentum projection differential
equation
(5)
, or directly from the Schrödinger equation
(1)
(see Sec. S.7 of the Supplemental Material
[44]
). The
computation results shown in Figs.
2
4
were performed for
a model of a Gaussian QEW, showing the claimed
dependence of the interaction on its size
σ
et
and delineating
the transition from the quantum-wave-like limit
(
σ
et
>T
2
;
1
¼
2
π
=
ω
2
;
1
) to the point-particle-like limit
(
σ
et
<T
2
;
1
) of the QEW. The parameters used in the
examples are typical of electron microscope PINEM-
type experiments
[17]
:
E
0
¼
200
keV,
r
0
¼
2
.
4
nm,
E
i;j
¼
2
eV, and
μ
i;j
¼
5
D. To focus on the parameter
PHYSICAL REVIEW LETTERS
126,
244801 (2021)
244801-3
scaling with the wave packet size, in all cases the transit time
parameter is short relative to the QEW:
t
r
¼
r
0
=
γβ
c
σ
et
.
Figure
2(a)
depicts the time dependence of the transition
probability of the TLS, starting from the ground state,
and the corresponding energy decrement of the free
electron, demonstrating maintenance of energy conserva-
tion
E
F
ð
t
Þ
E
F;
in
þ
ω
2
;
1
P
2
¼
0
throughout the process.
The transition buildup time depends on the QEW size
σ
et
,
but the postinteraction occupation probability of the upper
level and the free-electron energy decrement are found to
be independent of
σ
et
, in good agreement with the
computed analytical expression [Eq.
(8)
for
j
C
ð
0
Þ
1
j
2
¼
1
],
which is marked with a dot in the figure.
Figure
2(b)
depicts the same time dependence of the
transition probability of a TLS and the corresponding
energy increment of the free electron, calculated for
different values of
σ
et
but starting from a superposition
state, taken to be
ðj
1
i
j
2
=
ffiffiffi
2
p
(on the equator of the
Bloch sphere). Unlike in the ground state case, the
incremental probability for transition to the upper level
strongly depends on the wave packet size in the case of a
superposition state and decreases for a longer QEW, which
is consistent with Eq.
(7)
.
The strong wave-packet-size-dependent exponential
decay exp
ð
Γ
2
=
2
Þ
predicted by the analytical approximate
expression
(7)
is clearly demonstrated in Fig.
3
for a general
superposition initial state on the equator of the Bloch
sphere. The dependence on the incidence matching phase
ζ
¼
ω
2
;
1
t
0
φ
and on the QEW size
σ
et
is in agreement
with the analytical approximate expression
(7)
. Note the
large (3 orders of magnitude) enhancement of the maxi-
mum incremental transition probability in the limit of
short QEW. This is explained with a comparison of
Δ
P
ð
1
Þ
2
[Eq.
(7)
] and
Δ
P
ð
2
Þ
2
[Eq.
(8)
] that results in a relation,
½
Δ
P
ð
1
Þ
2

max
¼
ffiffiffiffiffiffiffiffiffiffiffiffi
Δ
P
ð
2
Þ
2
q
. The total transition increment is
always
Δ
P
2
¼
Δ
P
ð
1
Þ
2
þ
Δ
P
ð
2
Þ
2
; namely, the incremental
transition probability shown in Fig.
3
never decays to
zero, but the contribution of
Δ
P
ð
2
Þ
2
is negligible.
Figures
3(b)
and
3(c)
display the sinusoidal dependence
on the incidence phase and the exponential decay with the
QEW size, respectively, again in excellent agreement of the
analytical and numerical results.
Finally, we present the computation of the postinter-
action spectral energy distribution increment of the free
electron
Δ
ρ
ð
E
Þ¼ð
1
=v
0
Þ
P
2
i
¼
1
½j
c
i;p
ð
t
þ
0
Þj
2
j
c
i;p
ð
t
0
Þj
2

for the case of a long QEW. In this case, the incidence
phase dependence of the short QEW is lost, but the
measurable EELS spectrum depicts a dependence on the
latitudes of the TLS state on the Bloch sphere. In Fig.
4
, for
σ
et
=T
2
;
1
¼
1
, the QEW interaction with a TLS at ground
state
j
1
i
(south pole) exhibits only energy loss (TLS
excitation to upper level). At any other latitude, a
PINEM-like two-sideband spectrum shifted by the recoil
energy

E
2
;
1
relative to the initial energy is exhibited.
FIG. 2. Temporal evolution of excitation probability
P
2
and the
free-electron energy decrement
Δ
E
F
for different
σ
et
. (a) TLS
initially in the ground state. (b) TLS initially in an equatorial
superposition state.
FIG. 3. (a) Numerically computed
P
2
as a function of
σ
et
and
ζ
with the TLS initially in a superposition state. (b) Numerical
(solid lines) and analytical (dashed lines) transition probability
P
2
as a function of
ζ
for
σ
et
=T
2
;
1
¼
0
.
1
and 0.3. (c) Numerical (solid
lines) and analytical (dashed lines)
P
2
as a function of
σ
et
for
different relative incidence phases.
FIG. 4. EELS spectrum of long size QEWs (
σ
et
=T
2
;
1
¼
1
)
exhibiting a negative energy recoil (
E
2
;
1
) shift single sideband
for the initial TLS ground state
j
1
i
, symmetric

E
2
;
1
sidebands
for the equatorial TLS states
ðj
1
e
i
φ
j
2
=
ffiffiffi
2
p
, and asymmetric
net positive energy gain sidebands for initial states on the 60°
latitude angle of the Bloch sphere
ðj
1
ffiffiffi
3
p
e
i
φ
j
2
=
2
.
PHYSICAL REVIEW LETTERS
126,
244801 (2021)
244801-4
The sidebands are
symmetric
on the equator, and exhibit
asymmetric
net positive energy gain spectrum for the initial
TLS state in the northern hemisphere, and vice versa in the
southern hemisphere.
Discussion and conclusion.
We presented a compre-
hensive quantum-mechanical, relativistically valid analysis
of the FEBERI effect in a model in which the bound
electron is represented by a TLS and the free electron is
represented as a QEW. The Coulombic interaction of the
two-body system was solved in terms of the relativisticly
extended Schrödinger equation. Since we started from a
wave packet model for the free electron, the analysis
applies to the two limits of interaction, wavelike and
point-particle-like cases, manifesting the wave-particle
duality nature of quantum mechanics and the transition
from quantum to classical point-particle presentation. This
observation is similar to the analogous cases of QEW
interactions with light: stimulated radiative interaction and
superradiance
[4
6,23]
. The results show that the electron
wave packet dimension is a physically observable param-
eter that can be measured in lab experiments by interaction
with matter or light. This dependence is consistent with
Born
s interpretation of the quantum wave function and
conducive to an extension of the process to multiple
modulated QEWs, as suggested in Refs.
[31,35]
, that
would enable an enhanced resonant cathodoluminescence
effect with modulation-correlated QEWs.
This Letter focused on the analysis of FEBERI with a
single finite-size QEW (an extension to multiple entangled
QEWs in a rigorous quantum model will be presented
elsewhere). We found that the transition probability to an
upper or lower quantum level of the TLS depends on its
initial excitation level, and in the case of excitation from a
superposition state it can be many orders of magnitude
larger than the excitation from the ground state if the QEW
is in the short (near point-particle) limit (typically sub
ft
sec), and its arrival phase is in phase with the preset dipole
oscillation phase of the TLS superposition state. Moreover,
the initial quantum state of the TLS on the Bloch sphere
impresses a specific signature on the EELS spectrum of the
interacting electrons that depends on their size: in the short-
QEW limit it displays sinusoidal phase dependent accel-
eration or deceleration energy shift, depending on the
azimuth angle of the TLS state, and in the long QEW
limit it displays a symmetric or asymmetric two-sideband
spectrum that depends on the latitude angle on the northern
or southern hemisphere of the TLS state on the Bloch
sphere. Thus, such controlled interactions can provide
diagnoses of both the QEW shape and the TLS state. In
the latter case, this observation may lead to potential
applications in new electron microscopy atomic-scale
probing of quantum excitations in matter, and it may be
particularly useful as an important application of diagnosis
and coherent control of qubits and quantum emitters
[42,52]
.
Practical implementation of these concepts requires
substantial technological development and elaboration of
electron microscopy techniques. Shaping the wave packet
length in the range of an optical period requires develop-
ment of filtering and wave packet compression techniques
[26,48,53
56]
. Preinteraction setting up of a single TLS
target at a distinct superposition state, and measuring
enough EELS data within a time period shorter than the
relaxation time of the TLS, is an experimental challenge.
However, there are possibilities for enhancing the meas-
urement signal with multiple modulation-correlated QEWs
[31,35,53]
and with multiple TLS targets, including
enhanced superradiant cathodoluminescence
[57]
. Many
of the technical difficulties would be mitigated in scenarios
of low frequency (microwave, terahertz) TLS transition
frequencies, as in Ref.
[32]
.
We acknowledge the support of ISF (Israel Science
Foundation) Grant No. 00010001000 and the PBC program
of the Israel Council of Higher Education.
*
Corresponding author.
randu11111@163.com
Corresponding author.
gover@eng.tau.ac.il
[1] E. Schrödinger,
Phys. Rev.
28
, 1049 (1926)
.
[2] M. Born,
Z. Phys.
38
, 803 (1926)
.
[3] A. Friedman, A. Gover, G. Kurizki, S. Ruschin, and A.
Yariv,
Rev. Mod. Phys.
60
, 471 (1988)
.
[4] A. Gover and Y. Pan,
Phys. Lett. A
382
, 1550 (2018)
.
[5] Y. Pan, B. Zhang, and A. Gover,
Phys. Rev. Lett.
122
,
183204 (2019)
.
[6] Y. Pan and A. Gover,
J. Phys. Commun.
2
, 115026 (2018)
.
[7] H. Fares,
Phys. Lett. A
384
, 126883 (2020)
.
[8] J. P. Corson and J. Peatross,
Phys. Rev. A
84
, 053832
(2011)
.
[9] I. Kaminer, M. Mutzafi, A. Levy, G. Harari, H. Herzig
Sheinfux, S. Skirlo, J. Nemirovsky, J. D. Joannopoulos, M.
Segev, and M. Solja
č
i
ć
,
Phys. Rev. X
6
, 011006 (2016)
.
[10] R. Remez, A. Karnieli, S. Trajtenberg-Mills, N. Shapira, I.
Kaminer, Y. Lereah, and A. Arie,
Phys. Rev. Lett.
123
,
060401 (2019)
.
[11] B. Barwick, D. J. Flannigan, and A. H. Zewail,
Nature
(London)
462
, 902 (2009)
.
[12] F. J. García de Abajo, A. Asenjo-Garcia, and M. Kociak,
Nano Lett.
10
, 1859 (2010)
.
[13] S. T. Park, M. M. Lin, and A. H. Zewail,
New J. Phys.
12
,
123028 (2010)
.
[14] S. T. Park and A. H. Zewail,
J. Phys. Chem. A
116
, 11128
(2012)
.
[15] K. E. Echternkamp, A. Feist, S. Schäfer, and C. Ropers,
Nat.
Phys.
12
, 1000 (2016)
.
[16] 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,
Nat.
Commun.
9
, 2694 (2018)
.
PHYSICAL REVIEW LETTERS
126,
244801 (2021)
244801-5
[17] A. Feist, K. E. Echternkamp, J. Schauss, S. V. Yalunin, S.
Schaefer, and C. Ropers,
Nature (London)
521
, 200 (2015)
.
[18] K. E. Priebe, C. Rathje, S. V. Yalunin, T. Hohage, A. Feist,
S. Schäfer, and C. Ropers,
Nat. Photonics
11
, 793 (2017)
.
[19] L. Piazza, T. T. A. Lummen, E. Quiñonez, Y. Murooka, B.
W. Reed, B. Barwick, and F. Carbone,
Nat. Commun.
6
,
6407 (2015)
.
[20] M. Kozák, N. Schönenberger, and P. Hommelhoff,
Phys.
Rev. Lett.
120
, 103203 (2018)
.
[21] M. Kozák, T. Eckstein, N. Schönenberger, and P.
Hommelhoff,
Nat. Phys.
14
, 121 (2018)
.
[22] C. A. Brau,
Modern Problems in Classical Electrodynamics
(Oxford University Press, Oxford, 2004).
[23] Y. Pan and A. Gover, New J. Phys. (2021),
https://doi.org/
10.1088/1367-2630/abd35c.
[24] J. Verbeeck, H. Tian, and P. Schattschneider,
Nature
(London)
467
, 301 (2010)
.
[25] N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A.
Arie,
Nature (London)
494
, 331 (2013)
.
[26] P. Baum,
J. Appl. Phys.
122
, 223105 (2017)
.
[27] D. S. Black, U. Niedermayer, Y. Miao, Z. Zhao, O.
Solgaard, R. L. Byer, and K. J. Leedle,
Phys. Rev. Lett.
123
, 264802 (2019)
.
[28] N. Schönenberger, A. Mittelbach, P. Yousefi, J. McNeur, U.
Niedermayer, and P. Hommelhoff,
Phys. Rev. Lett.
123
,
264803 (2019)
.
[29] R. H. Dicke,
Phys. Rev.
93
, 99 (1954)
.
[30] A. Gover, R. Ianconescu, A. Friedman, C. Emma, N. Sudar,
P. Musumeci, and C. Pellegrini,
Rev. Mod. Phys.
91
,
035003 (2019)
.
[31] A. Gover and A. Yariv,
Phys. Rev. Lett.
124
, 064801 (2020)
.
[32] D. Rätzel, D. Hartley, O. Schwartz, and P. Haslinger,
arXiv:2004.10168
.
[33] F. Robicheaux and L. D. Noordam,
Phys. Rev. Lett.
84
,
3735 (2000)
.
[34] M. S. Pindzola, M. Witthoeft, and F. Robicheaux,
J. Phys. B
33
, L839 (2000)
.
[35] A. Gover, B. Zhang, D. Ran, R. Ianconescu, A. Friedman, J.
Scheuer, and A. Yariv,
arXiv:2010.15756
.
[36] F. J. G. de Abajo,
Phys. Rev. Lett.
126
, 019501 (2021)
;
126
,
069902(E) (2021)
.
[37] A. Gover and A. Yariv,
Phys. Rev. Lett.
126
, 019502 (2021)
.
[38] F. J. G. de Abajo and V. Di Giulio,
arXiv:2010.13510
.
[39] Z. Zhao, X.-Q. Sun, and S. Fan,
arXiv:2010.11396
[Phys.
Rev. Lett. (to be published)].
[40] F. J. García de Abajo and V. Di Giulio,
ACS Photonics
8
,
945 (2021)
.
[41] O. Kfir, V. Di Giulio, F. J. García de Abajo, and C. Ropers,
arXiv:2010.14948
.
[42] R. Ruimy, A. Gorlach, C. Mechel, N. Rivera, and I.
Kaminer,
arXiv:2011.00348
[Phys. Rev. Lett. (to be
published)].
[43] V. Di Giulio, M. Kociak, and F. J. García de Abajo,
Optica
6
,
1524 (2019)
.
[44] See Supplemental Material at
http://link.aps.org/
supplemental/10.1103/PhysRevLett.126.244801
, which in-
cludes Refs. [45
48], for detailed and elaborate derivations.
[45] A. D. McLachlan,
Mol. Phys.
6
, 441 (1963)
.
[46] Ilya Kuprov,
http://spindynamics.org/documents/sd_m1_
lecture_06.pdf
.
[47] P. Feynman,
https://www.feynmanlectures.caltech.edu/
II_26.html
.
[48] H. Bateman,
Tables of Integral Transforms
(McGraw-Hill,
New York, 1954).
[49] R. J. Bartlett,
Annu. Rev. Phys. Chem.
32
, 359 (1981)
.
[50] F. J. García de Abajo,
Rev. Mod. Phys.
82
, 209 (2010)
.
[51] R. P. Feynman, R. B. Leighton, and M. Sands,
The Feynman
Lectures on Physics, Mainly Electromagnetism and matter
(Addison-Wesley, Reading, MA, 1964), Vol. 2.
[52] Z. C. Shi, Y. H. Chen, W. Qin, Y. Xia, X. X. Yi, S. B. Zheng,
and F. Nori,
arXiv:2011.12473
.
[53] V. Di Giulio and F. J. G. de Abajo,
Optica
7
, 1820
(2020)
.
[54] Y. Morimoto and P. Baum,
Phys. Rev. Lett.
125
, 193202
(2020)
.
[55] U. Niedermayer, T. Egenolf, O. Boine-Frankenheim, and P.
Hommelhoff,
Phys. Rev. Lett.
121
, 214801 (2018)
.
[56] O. Reinhardt and I. Kaminer,
ACS Photonics,
7
, 2859
(2020)
.
[57] A. Halperin, A. Gover, and A. Yariv,
Phys. Rev. A
50
, 3316
(1994)
.
PHYSICAL REVIEW LETTERS
126,
244801 (2021)
244801-6