Supplementary Material: 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, Tel Av
iv University,
Ramat Aviv 69978, Israel
2
School of Electronic Information Engineering, Yangtze Normal Un
iversity,
Chongqing 408100, China
3
Shenkar College of Engineering and Design 12, Anna Frank St., Ramat
Gan, Israel
4
Ariel University, Ariel 40700, Israel
5
California Institute of Technology, Pasadena, California 91125, US
A
Contents
S.1 Spin-Spin interaction
2
S.2 Derivation of Eq. (5) in the main text
3
S.3 Iterative solution in momentum space
6
S.4 Probabilistic Model for FEBERI
8
S.5 Coulomb Interaction Matrix element
11
S.6 Calculate
I
=
R
c
(0)
∗
p
c
(0)
p
−
p
rec
dp
12
S.7 Numerical computation solution
13
1
S.1 Spin-Spin interaction
In this section, we compare the spin-spin interaction Hamiltonian to t
he Coulomb interaction Hamil-
tonian. Suppose
γ
1
and
γ
2
are gyromagnetic ratios of two particles with spin quanta
S
1
and
S
2
.
The potential energy
H
m
of the spin-spin interaction is then given by [
1
,
2
]
H
m
=
μ
0
γ
1
γ
2
~
2
4
π
|
R
|
3
[3(
S
1
·
ˆ
r
)(
S
2
·
ˆ
r
)
−
S
1
·
S
2
]
,
(S1)
where
ˆ
r
is a unit vector in the direction of the line joining the two spins and
|
R
|
is the distance
between the two spins. For an isolated electron, the gyromagnetic
ratio for the self-spinning electron
is
γ
e
=
g
e
μ
B
~
=
e
2
m
e
g
e
≃
e
m
e
,
(S2)
where
μ
B
is Bohr magneton and
g
e
≃
2. By substituting
μ
0
= 1
/ǫ
0
c
2
and
γ
i
=
γ
e
(
i
= 1
,
2) into
equation (S1), we get
H
m
=
1
4
πǫ
0
e
2
~
2
c
2
m
2
e
|
R
|
3
[3(
S
1
·
ˆ
r
)(
S
2
·
ˆ
r
)
−
S
1
·
S
2
]
.
(S3)
On the other hand, the Coulomb interaction Hamiltonian of a free elec
tron with the dipole moment
of a bound electron is
H
coul
=
e
2
4
πǫ
0
r
2
,
1
r
⊥
0
|
R
|
3
,
(S4)
where
r
2
,
1
=
μ
2
,
1
/e
with
μ
2
,
1
being the electric dipole moment of (2
,
1) levels transition, and
r
⊥
0
is the impact parameter of the free electron relative to the model n
ucleus (see Fig. 1 in the main
text).
The ratio of the spin-spin interaction Hamiltonian
H
m
to the Coulomb interaction Hamiltonian
H
coul
within the interaction range
|
R
| ≃
r
⊥
0
, assuming [3(
S
1
·
ˆ
r
)(
S
2
·
ˆ
r
)
−
S
1
·
S
2
] =
O
(1), is:
H
m
H
coul
≃
~
2
m
2
e
c
2
1
r
2
,
1
r
⊥
0
=
λ
2
c
r
2
,
1
r
⊥
0
,
(S5)
where
λ
c
=
~
/m
e
c
= 2
.
43
×
10
−
12
m
is the Compton wavelength and we estimate
r
2
,
1
to be of the
order of the Bohr radius (
r
B
),
r
2
,
1
≃
r
B
= 5
×
10
−
11
m
and
r
⊥
0
= 2
.
4
nm
. For this example,
H
m
H
coul
= 5
×
10
−
5
.
(S6)
Therefore, the relative strength of electron spin-spin interactio
n compared to the Coulomb dipole
interaction is negligibly small.
2
S.2 Derivation of Eq. (5) in the main text
In this section, we present the detailed derivation of the integro-d
ifferential equation (5) in the main
text. We start from the Schr ̈odinger equation for the joint wave
function of the free and bound
electrons:
i
~
∂
Ψ(
r
,
r
′
,t
)
∂t
= (
H
0
+
H
I
)Ψ(
r
,
r
′
,t
)
,
(S7)
where
H
0
=
H
0
F
+
H
0
B
is the kinetic Hamiltonian of the free electron and bound system (Hyd
rogen-
like atom modeled as a TLS). We apply the dipole approximation for the b
ound system. This implies
that the bypassing free electron has no interaction with the neutr
al charge atom, except for resonant
dipole interaction with its quantum states [
3
]. Then the interaction Hamiltonian between the dipole
moment and the electric field generated by the incident free electro
n is:
H
I
=
−
e
r
′
·
E
. In order
to keep the analysis valid in the relativistic regime, we use Feynman’s ex
pression for the electric
field [
4
]. For an electron moving with relativistic velocity
v
0
on a straight line along the
z
axis, the
electric field at a position (
x,y,z
) is calculated by
E
=
−
∇
φ
−
∂
A
∂t
, where the potentials are
φ
=
γ
4
πǫ
0
e
[
x
2
+
y
2
+
γ
2
(
z
−
v
0
t
)
2
]
1
/
2
,
A
z
=
γ
4
πǫ
0
ev
0
[
x
2
+
y
2
+
γ
2
(
z
−
v
0
t
)
2
]
1
/
2
,
A
x
=
A
y
= 0
,
(S8)
with
γ
= 1
/
p
1
−
v
2
0
. The
x
and
y
components of the electric field are
E
x
=
−
∂φ
∂x
=
eγ
4
πǫ
0
x
[
x
2
+
y
2
+
γ
2
(
z
−
v
0
t
)
2
]
3
/
2
,
E
y
=
−
∂φ
∂y
=
eγ
4
πǫ
0
y
[
x
2
+
y
2
+
γ
2
(
z
−
v
0
t
)
2
]
3
/
2
.
(S9)
Differentiating
φ
with respect to
z
and differentiating
A
z
with respect to
t
in Eq. (S8), the
z
-
component of the electric field is
E
z
=
−
∂φ
∂z
−
∂A
z
∂t
=
γ
4
πǫ
0
eγ
2
(
z
−
v
0
t
)
[
x
2
+
y
2
+
γ
2
(
z
−
v
0
t
)
2
]
3
/
2
+
γ
4
πǫ
0
ev
2
0
γ
2
(
z
−
v
0
t
)
[
x
2
+
y
2
+
γ
2
(
z
−
v
0
t
)
2
]
3
/
2
=
γ
4
πǫ
0
e
(
z
−
v
0
t
)
[
x
2
+
y
2
+
γ
2
(
z
−
v
0
t
)
2
]
3
/
2
.
(S10)
Then the electric field can thus be represented by
E
=
E
x
ˆ
e
x
+
E
y
ˆ
e
y
+
E
z
ˆ
e
z
=
eγ
4
πǫ
0
x
ˆ
e
x
+
y
ˆ
e
y
+ (
z
−
v
0
t
)
ˆ
e
z
[
x
2
+
y
2
+
γ
2
(
z
−
v
0
t
)
2
]
3
/
2
.
(S11)
In our model (see Fig. 1 of main text), the position of the bound sys
tem is (
x,y,z
) = (
−
r
⊥
0
,
0
,
0)
and the coordinate of the
z
-component of free electron is
v
0
t
. Thus the electric field at the position
of the TLS is
E
=
−
eγ
4
πǫ
0
r
⊥
0
ˆ
e
r
+
z
ˆ
e
z
[
r
2
⊥
0
+
γ
2
z
2
]
3
/
2
.
(S12)
Because we applied the dipole approximation for the bound system, t
he interaction Hamiltonian
between the incident electron and the bound system is
H
I
=
e
2
γ
4
πǫ
0
r
′
·
(
z
ˆ
e
z
+
r
⊥
0
ˆ
e
r
)
[
r
2
⊥
0
+
γ
2
z
2
]
3
/
2
,
(S13)
which we use in Eq. (4) in the main text.
3
The expansion coefficients of the QEW are entangled with the coefficie
nts of the bound electron
and the combined wavefunction is:
Ψ(
r
′
,
r
,t
) =
2
X
j
=1
Z
p
dpc
j,p
(
t
)
φ
j
(
r
′
)
e
−
iE
j
t/
~
e
iE
p
t/
~
e
ipz/
~
.
(S14)
After substituting this expansion into Schr ̈odinger Eq. (
S7
), we obtain
i
~
∂
Ψ(
r
,
r
′
,t
)
∂t
=
i
~
2
X
j
=1
Z
p
dp
̇
c
j,p
(
t
)
−
E
j
+
E
p
~
c
j,p
(
t
)
φ
j
(
r
′
)
e
−
iE
j
t/
~
e
iE
p
t/
~
e
ipz/
~
= (
H
0
B
+
H
0
F
+
H
I
)
2
X
j
=1
Z
p
dpc
j,p
(
t
)
φ
j
(
r
′
)
e
−
iE
j
t/
~
e
iE
p
t/
~
e
ipz/
~
.
(S15)
When cancelling out the no-interaction terms, we are left with:
i
~
2
X
j
=1
Z
p
dp
̇
c
j,p
(
t
)
e
iE
j
t
~
e
i
(
E
p
t
−
pz
)
~
=
H
I
(
r
,
r
′
)
2
X
j
=1
Z
p
dpc
j,p
(
t
)
φ
j
(
r
′
)
e
iE
j
t
~
e
i
(
E
p
t
−
pz
)
~
.
(S16)
We multiply by
φ
∗
i
(
r
′
) and integrate over space, and by using the ortho-normality relat
ion
R
φ
i
(
r
′
)
φ
∗
j
(
r
′
)
d
3
r
′
=
δ
i,j
:
i
~
Z
p
dp
̇
c
j,p
(
t
)
e
−
iE
i
t/
~
e
iE
p
t/
~
e
ipz/
~
=
e
−
iE
i
t/
~
Z
p
dpc
i,p
(
t
)
h
i
|
H
I
(
r
,
r
′
)
|
i
i
e
iE
p
t/
~
e
ipz/
~
+
e
−
iE
i
t/
~
Z
p
dpc
j
6
=
i,p
(
t
)
h
i
|
H
I
(
r
,
r
′
)
|
j
i
e
iE
p
t/
~
e
ipz/
~
,
(S17)
where
M
i,j
(
r
) =
h
i
|
H
I
(
r
,
r
′
)
|
j
i ≡
R
d
3
r
′
φ
∗
i
(
r
′
)
H
I
(
r
,
r
′
)
φ
j
(
r
′
). For simplicity, we redefine the self-
interaction terms, so that
h
i
|
H
I
(
r
,
r
′
)
|
i
i
= 0, then Eq. (
S17
) simplify to:
i
~
Z
p
dp
̇
c
j,p
(
t
)
e
−
iE
i
t/
~
e
−
iE
p
t/
~
e
ipz/
~
=
e
−
iE
i
t/
~
Z
p
dpc
j
6
=
i,p
(
t
)
M
i,j
(
r
)
e
iE
p
t/
~
e
ipz/
~
.
(S18)
This is an integro-differential equation that needs to be solved as a f
unction of time for the initial
condition
c
j,p
(
t
−
0
) =
C
(0)
j
(
t
−
0
)
c
(0)
p
. If
|
r
′
| ≪ |
r
−
r
′
| ≈
(
r
2
⊥
0
+
γ
2
z
2
)
1
/
2
, then the integration over
r
′
can be carried out independently of
r
:
M
i,j
(
r
⊥
0
,r
) =
Z
φ
∗
i
(
r
′
)
H
I
(
r
′
,
r
)
φ
j
(
r
′
)
d
3
r
′
.
(S19)
For the interaction
H
I
=
e
2
4
πǫ
0
γ
r
′
·
(
ˆ
e
z
z
+
ˆ
e
r
r
⊥
0
)
(
γ
2
z
2
+
r
2
⊥
0
)
3
/
2
,
M
i,j
=
e
2
4
πǫ
0
γ
r
i,j
·
(
ˆ
e
z
z
+
ˆ
e
r
r
⊥
0
)
γ
2
z
2
+
r
2
⊥
0
.
(S20)
Here, we define the dipole transition matrix element as
μ
2
,
1
=
−
e
r
2
,
1
≡ −
e
Z
φ
∗
2
(
r
′
)
r
′
φ
1
(
r
′
)
d
3
r
.
(S21)
We project the integro-differential Eq. (
S18
) onto momentum space by multiplying with
e
−
ip
′
z/
~
and integrating over z. With
R
e
i
(
p
′
−
p
)
z/
~
dz
= 2
π
~
δ
(
p
′
−
p
), we get:
2
πi
~
2
Z
p
dp
̇
c
i,p
e
−
iE
p
t/
~
δ
(
p
′
−
p
) =
e
−
i
(
E
j
−
E
i
)
t
~
Z
p
dpc
j,p
(
t
)
Z
∞
−∞
dzM
i,j
(
r
)
e
i
(
p
−
p
′
)
z
e
−
iE
p
t/
~
,
(S22)
4
then,
̇
c
i,p
′
=
1
2
πi
~
2
Z
dp
̃
M
i,j
(
p
′
−
p
)
c
j,p
(
t
)
e
−
i
(
E
p
−
E
p
′
−
E
i,j
)
t/
~
,
(S23)
where
E
i,j
=
E
i
−
E
j
, and
̃
M
i,j
(
p
′
−
p
) =
R
∞
−∞
dzM
i,j
(
r
)
e
i
(
p
−
p
′
)
z/h
. Equation (5) in the main text
is obtained when we present this function in terms of the interaction
matrix element in momentum
space:
H
I,i,j
(
p
′
−
p
) =
h
p
′
,i
|
H
I
(
r
,
r
′
)
|
p,j
i
=
h
p
′
|
M
i,j
(
r
)
|
p
i
=
1
2
π
~
̃
M
i,j
(
p
′
−
p
)
.
(S24)
In
S.5
we present the explicit expressions of
̃
M
i,j
(
p
) for the longitudinal and transverse components
of the dipole moment for the specific Coulomb interaction Hamiltonian (
S20
).
5
S.3 Iterative solution in momentum space
The integro-differential equation (
S23
) describes the dynamic evolution of the entangled free electron
and bound electron in momentum space. Before the start interact
ion time, the free and bound
electrons are disentangled:
c
j,p
(
t
) =
C
(0)
j
(
t
)
c
(0)
p
.
(S25)
Equation (
S23
) can be solved by an iterative process, in which we assume to first or
der that the free
and bound electrons are not evolving in time during an effective intera
ction time
t
int
:
c
j,p
(
t
) =
C
(0)
j
(
t
0
)
c
(0)
p
,
(S26)
where
t
0
is the arrival time of the centroid of the QEW to
z
= 0 – the position of the TLS, and
t
−
0
,
t
+
0
are the start and end times of the QEW interaction with the TLS, ass
uming nearly uniform
wavepacket distribution (long QEW).
Substituting this time independent amplitude in the RHS of
Eq. (
S23
), it is possible to integrate over time, and get a factor:
Z
t
+
0
t
−
0
dte
−
i
(
E
p
−
E
p
′
−
E
i,j
)
t
0
/
~
t
+
0
−
t
−
0
~
sinc[(
E
p
−
E
p
′
−
E
i,j
)(
t
+
0
−
t
−
0
)
/
2
~
]
=
e
−
i
(
E
p
−
E
p
′
−
E
i,j
)
t
0
/
~
2
t
int
~
sinc[(
E
p
−
E
p
′
−
E
i,j
)
t
int
/
~
]
→
2
π
~
δ
(
E
p
−
E
p
′
−
E
i,j
)
,
(S27)
where the last limit is taken for an infinite interaction time
(this would be valid also for other
models of QEW distribution, such as a long gaussian).
This dictates a conservation of energy
transfer condition:
E
p
′
−
E
p
=
−
E
i,j
. This condition determines the recoil momentum of the QEW
during the interaction:
p
rec
=
p
′
−
p
=
−
E
i,j
/v
0
. The momentum recoil
p
rec
is defined here so
that if the transition is from the lower level
j
= 1 to the upper level
i
= 2, the momentum recoil
is negative, and vice versa. Here we used only the first order expan
sion of the dispersion equation
E
p
=
ǫ
0
+
v
0
·
(
p
−
p
0
) +
1
2
γ
3
m
(
p
−
p
0
)
2
(the dispersive second order term would introduce a small
interaction quantum recoil correction differentiating up and down t
ransitions that can be neglected
in the present context). Thus, integration of Eq. (
S23
) using Eqs. (
S25
) and (
S26
) results in:
c
i,p
′
(
t
+
0
) =
C
(0)
i
(
t
−
0
)
c
(0)
p
′
+ ∆
c
i,p
′
,
(S28)
where ∆
c
i,p
′
=
1
i
~
v
0
̃
M
i,j
(
p
rec
)
C
(0)
j
(
t
0
)
c
(0)
p
′
−
p
rec
.
The transition probability is:
P
i
(
t
+
0
) =
Z
p
′
|
c
i,p
′
(
t
+
0
)
|
2
dp
′
=
Z
p
′
|
C
i
(
t
0
)
c
(0)
p
′
(
t
0
) + ∆
c
i,p
′
(
t
+
0
)
|
2
dp
′
=
P
(0)
i
+ ∆
P
(1)
i
+ ∆
P
(2)
i
,
(S29)
where
P
(0)
i
=
|
C
i
(
t
0
)
|
2
R
|
c
(0)
p
′
|
2
(
t
0
)
dp
′
=
|
C
i
(
t
0
)
|
2
is the initial occupation probability of level
i
, and
the incremental probabilities are:
∆
P
(1)
i
= 2
Re
C
(0)
∗
i
(
t
0
)
Z
∞
−∞
dp
′
c
(0)
∗
p
′
(
t
0
)∆
c
i,p
′
,
(S30)
∆
P
(2)
i
=
Z
∞
−∞
dp
′
|
∆
c
i,p
′
|
2
.
(S31)
Substituting ∆
c
i,p
′
in Eq. (
S30
), and integrating over momentum, using
R
dp
|
c
(0)
p
−
p
rec
|
2
= 1, one
obtains:
∆
P
(2)
i
(
t
+
0
) =
Z
dp
′
|
c
(0)
i,p
′
|
2
=
1
~
2
v
2
0
|
̃
M
i,j
(
p
rec
)
|
2
|
C
(0)
j
(
t
0
)
|
2
=
4
π
2
v
2
0
|
C
(0)
j
(
t
0
)
H
I,i,j
(
p
rec
)
|
2
.
(S32)
6
In the case of excitation of a TLS from ground state:
C
(0)
2
(
t
−
0
) = 0,
C
(0)
1
(
t
−
0
) = 1, the excitation
probability of the TLS is given by:
P
2
(
t
+
0
) = ∆
P
(2)
2
=
Z
dp
′
|
c
(0)
2
,p
′
|
2
=
1
~
2
v
2
0
|
̃
M
2
,
1
(
p
rec
)
|
2
=
4
π
2
v
2
0
|
C
(0)
1
(
t
0
)
H
I,
2
,
1
(
p
rec
)
|
2
.
(S33)
Evidently the excitation probability in this case is independent of the Q
EW shape or dimensions.
However, considering the case where the TLS is in a superposition st
ate at the interaction time,
the first order incremental probability term Eq. (
S30
) may be dominant. Substituting ∆
c
i,p
′
in Eq.
(
S30
) and integrating over momentum results in:
∆
P
(1)
i
=
2
~
v
0
Re
"
C
(0)
∗
i
(
t
0
)
C
(0)
j
(
t
0
)
̃
M
i,j
(
p
rec
)
i
I
(
p
rec
)
#
=
4
π
v
0
Re
C
(0)
∗
i
(
t
0
)
C
(0)
j
(
t
0
)
H
I,i,j
(
p
rec
)
i
I
(
p
rec
)
.
(S34)
For a Gaussian distribution of the QEW, the integral
I
(
p
rec
) =
R
c
(0)
∗
p
c
(0)
∗
p
−
p
rec
dp
is evaluated in
S.6
:
I
(
p
rec
) =
e
−
1
2
(
p
rec
/
2
σ
p
0
)
2
e
−
iω
i,j
t
0
.
(S35)
Substituting
p
rec
=
~
ω
1
,
2
/v
,
σ
z
0
=
~
/
2
σ
p
0
,
σ
t
0
=
σ
z
0
/v
, one gets:
∆
P
(1)
i
=
2
i
~
v
0
Re
[
C
(0)
∗
i
(
t
0
)
C
(0)
j
(
t
0
)
e
−
iω
i,j
t
0
(
̃
M
i,j
(
p
rec
)
/i
)]
e
−
Γ
2
/
2
=
4
πe
−
Γ
2
/
2
sin
ζ
v
0
|
H
I,i,j
(
p
rec
)
C
(0)
∗
i
(
t
0
)
C
(0)
j
(
t
0
)
|
,
(S36)
where we substituted
C
(0)
∗
i
(
t
0
)
C
(0)
j
(
t
0
=
|
C
(0)
∗
i
(
t
0
)
C
(0)
j
(
t
0
|
e
iφ
, expressed in terms of the dipole mo-
ment excitation amplitude and phase in the TLS Bloch sphere present
ation, defined
ζ
=
ω
i,j
t
0
−
φ
:
the phase of the QEW arrival time relative to the TLS dipole moment ex
citation phase, and
Γ =
p
rec
2
σ
p
0
=
~
ω
2
,
1
2
vσ
p
0
=
ω
2
,
1
σ
t
0
= 2
π
σ
z
0
βλ
2
,
1
.
(S37)
This means that the incremental probability is dependent on the wav
epacket dimensions if the
phase of the superposition state of the TLS is pre-determined, an
d it vanishes for a long wavepacket.
Instructively, Eq. (
S36
) and the decay constant (
S37
) are analogous to the decay of point-particle
electron acceleration and transition from classical point-particle t
o quantum regime PINEM inter-
action in stimulated radiative interaction of a finite size QEW [
7
].
We note that the assumption that
c
j,p
(
t
) is constant during the interaction time
t
int
, as assumed
in Eq. (
S26
), may have partial validity, and the transition of the sinc function t
o delta function in
(
S27
) is questionable when the interaction time
t
int
is short:
E
2
,
1
t
int
<
~
/
2
,
(S38)
We assume that
t
int
is longer than both the interaction transit time
t
r
=
r
⊥
0
/γv
0
and the wavepacket
duration
σ
et
=
σ
z
0
/v
0
:
t
r
,σ
et
< t
int
<
~
/E
2
,
1
= 1
/ω
2
,
1
,
(S39)
Note that the limit
σ
et
<
1
/ω
2
,
1
is exactly the near-point-particle limit Γ =
ωσ
et
= 2
πσ
ez
/βλ
,
discussed in the introduction. Also, instructively, when
ω
2
,
1
σ
et
<
1
/
2, then
σ
ep
> E
2
,
1
/v
0
, namely,
the momentum spread is larger than the recoil
p
rec
=
p
′
−
p
=
−
E
i,j
/v
0
.
In the next section we present an alternative approximate solution
of the Schr ̈odinger equation
that contrary to the approximate iteration in the momentum proje
ction approach takes into con-
sideration also the dynamics of the TLS transition, and may be suitab
le for the short interaction
regime Eq. (
S39
).
7
S.4 Probabilistic Model for FEBERI
The zero-order iteration of the momentum projection source equ
ation (
S23
) with the zero-order
approximation of stationary TLS state (
S26
) helps describing the modification of the QEW distri-
bution due to the FEBERI interaction, and infers to the dynamics of
the TLS only through the
conservation of energy condition. In the near-point-particle QEW
limits (
S39
) this approximation
does not describe the TLS dynamics. Rather than continuing this ite
rative approach in the momen-
tum space, and solving (
S23
) without taking the delta function limit in (
S27
), we present in this
section an alternative approach, going back to the source equatio
n (
S23
) and solving it directly with
a similar first order iteration approximation by substituting on the RH
S of (
S23
):
c
j,p
(
t
)
≃
C
(0)
j
(
t
)
c
(0)
p
,
(S40)
which is the same as in the previous assumption Eq. (
S26
), but here allowing development in time
of the TLS. After multiplying Eq. (
S18
) by the complex conjugate of the free electron wavefunction
and integrating over space:
R
d
3
r
Ψ
(0)
∗
F
(
r
,t
), one obtains:
i
2
π
Z
p
dp
′
̇
c
i,p
(
t
)
c
(0)
∗
p
e
i
(
E
p
′
−
E
p
)
t
/
~
Z
z
dze
i
(
p
−
p
′
)
z/
~
=
C
(0)
j
(
t
)
e
i
(
E
i
−
E
j
)
t
/
~
Z
d
3
rM
i,j
(
r
)
ψ
(0)
F
(
r
,t
)
2
,
(S41)
with
R
z
dze
i
(
p
−
p
′
)
z/
~
= 2
π
~
δ
(
p
′
−
p
):
2
πi
~
Z
p
dp
′
̇
c
i,p
(
t
)
c
(0)
∗
p
=
C
(0)
j
(
t
)
e
i
(
E
i
−
E
j
)
t
/
~
Z
d
3
rM
i,j
(
r
)
ψ
(0)
F
(
r
,t
)
2
,
(S42)
This presentation is reminiscent of point-particle interaction with th
e Born quantum wavefunction
probability
|
Ψ
(0)
F
(
r
,t
)
|
2
of electron arrival time
t
at the TLS location
z
= 0.
It should be stressed that
|
Ψ
(0)
F
(
r
,t
)
|
2
is not well determined for a single electron. We assume
that it is possible to solve Eq. (
S42
) with substitution of its expectation value
h|
Ψ
(0)
F
(
r
,t
)
|
2
i
, and
the solution will then represent the result of interaction with an ens
emble of identical QEWs.
The probability distribution of a single electron QEW of narrow width is:
Ψ
(0)
F
(
r
,t
)
2
=
δ
(
r
⊥
)
f
ez
(
z
−
v
0
(
t
−
t
0
)) =
δ
(
r
⊥
)
f
et
(
t
−
t
0
−
z/v
0
)
/v
0
,
(S43)
where
f
et
is normalized over time. Then:
i
~
Z
p
dp
′
̇
c
i,p
(
t
)
c
(0)
∗
p
=
C
(0)
j
(
t
)
e
iω
i,j
t
f
(
t
−
t
0
)
,
(S44)
f
(
t
−
t
0
) =
1
v
0
Z
dzM
i,j
(
z
)
f
et
(
t
−
t
0
−
z/v
)
.
(S45)
where
M
i,j
is given in Eq. (
S20
).
With approximation Eq. (
S40
), assuming negligible change in the QEW around the interaction
time
t
0
, we can turn Eq. (
S44
) into coupled differential equations for the TLS (
i,j
= 1
,
2):
̇
C
i
(
t
) =
1
i
~
C
j
(
t
)
e
−
iω
i,j
t
f
(
t
−
t
0
)
.
(S46)
After integration, one obtains
C
i