Initial Time CDR and CLS of a Slab (Friedberg et al., 1973)
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368
Jamal T. Manassah
(supposing the probability to be small), and the total excitation energy of the
X
2
sample is
jbðaÞj . The CDR is then
a
0X !
1
!
b *aị b aị
B a
C
d
C,
(12)
ln jbaịj2 ẳ À2ReB
X
@
! 2 A
dt
b ðaÞ
where “∘” is being used to denote vector scalar (dot) product.
At the same time, one can extract a frequency shift by supposing that not the
!
magnitude of b ðaÞ but also its phase is changing. Ideally, supposing that all the
atoms are subject to the same time-dependent factor exp((ÀΓ CDR + iΔΩCLS)t)
! 2
!
!
!
entering into b ðaÞ. Then, one has b *aị b aị ẳ b aị exp2 CDR tị.
One makes the following identifications:
2Re t ẳ 0ịị ẳ Initial CDR,
Im t ẳ 0ịị ẳ Initial CLS,
where
(13a)
(13b)
!
X!
b *aị b_ aị
ẳ
a
X
!
j b aịj2
:
(13c)
a
In the continuum approximation Eq. (13c) can be written
N2 k30
0ị ẳ
!
!0 ! !0
r
r e
ℏ d3 r d3 r e
ð
X À! Á À! Á
À! ! Á À!Á * À!Á
!
ρ r 0 Gi, j r À r 0 e
ρ r bj r :
bi r 0 e
Â d3 r d3 r 0
i, j
(14)
Using expression (11) for the 1D Kernel and assuming that the initial
state is a phase resonant state (i.e., b(z, 0) ¼ exp(ik0z)), we obtain for the
CDR and CLS:
!
CDR C
sin 2 2u0 ị
,
(15)
ẳ
2u0 +
2
4
2u0
CLS ẳ
!
C sin 4u0 ị 4u0
:
4u0
4
(16)
Quantum Electrodynamics of Two-Level Atoms in 1D Configurations
2.3
369
Eigenfunctions and Eigenvalues of a Slab (Friedberg
and Manassah, 2008c,d,e)
The computation of the CDR and CLS at t ¼ 0, discussed in the previous section, consisted essentially of computing an integral whose integrand is a product of factors representing the spatial dependence of
the initial state of the system, the normalized number density profile,
and the Lienard–Wiechert Green’s function. To obtain the time development of the system, an attractive option will be the use of spectral
decomposition.
We note, however, that in this instance, the Green’s function operator is
non-Hermitian. A study of the “orthogonality,” normalization, and completeness of the eigenfunctions of the Lienard–Wiechert operator in 1D will
be the focus of the rest of this subsection.
2.3.1 Functional Form of the Eigenfunctions
o,e
The eigenvalues λo,e
s and eigenfunctions φs (Z) associated with the 1D
kernel are obtained by solving the integral equation
Ck0
so, e so, e zị ẳ
2
z0
dz0 expðik0 jz À z0 jÞφso, e ðz0 Þ:
(17)
Àz0
This integral equation admits two families of solutions, where the superscript refers, respectively, to (odd, even) parity in space, which are given,
respectively, by
À
os Z ị ẳ sin vos Z ,
(18)
e
e
s Z ị ẳ cos vs Z ,
(19)
where Z ẳ z/z0. The complex wavevectors (vos , ves ) are solutions of the transcendental equations
À Á u0
(20)
cot v0s ¼ i o ,
vs
À Á
u0
(21)
tan ves ¼ Ài e ,
vs
where u0 ¼ k0z0 and s is the index of the solution. We plot in Figs. 4 and 5
the values of the odd and even complex wavevectors as a function of the
index s for a slab having u0 ¼ 10.25π.
À Á
We find that for odd modes ðs À 12Þπ < Re vos < sπ, while for even mode
À Á
ðs À 1Þπ < Re ves < ðs À 12Þπ.
370
Jamal T. Manassah
(a)
Re(vo)
60
50
40
30
20
10
o
o
o
o
o
o
o
o
o
o
5
10
5
10
o
o
o
o
o
o
15
o
o
o
o
20
(b)
Im(vo)
o
−0.5
−1.0
o
o
o
o
o
15
o
o
o
o
−1.5
o
o
20
s
s
o
o o
o o
o
o o
Fig. 4. (a) The real and (b) imaginary parts of the wavevectors for the odd modes are
plotted as a function of the index s. u0 ¼ k0z0 ¼ 10.25π.
Associated with the above eigenfunctions are the temporal eigenvalues
λos , e ¼ i
Cu20
À Á2 :
u20 À vos , e
(22)
The normalized eigenvalues are defined as Λs ¼ λs/C. We plot in Figs. 6
and 7 the locus in the complex plane of the odd and even eigenmodes. We
note that for the considered value of u0 ¼ 10.25π in these figures,
max(Re(Λ)) would belong to the even mode whose Re(ves ) % u0, while
for u0 ¼ 10.75π (not shown), max(Re(Λ)) would belong to the odd mode
whose Re(vos ) % u0. This is a general feature for any value of u0, having,
À
Á
À
Á
respectively, the values m + 14 π and m + 34 π. For values of u0 having
the values mπ or ðm + 12Þπ, two modes, one odd and one even have nearly
the max(Re(Λ)). This is, as will be pointed out in the remarks at the end
of this section, the source of asymmetry observed in the numerical simulation of superradiant emission for slabs with these particular thicknesses.
371
Quantum Electrodynamics of Two-Level Atoms in 1D Configurations
(a)
Re(v e)
60
50
40
30
20
10
e
e
e
e
e
e
e
e
e
5
e
e
e
e
e
10
e
e
e
e
e
e
15
20
15
20
(b)
Im(ve)
e
e
e
−0.5
5
e
e
−1.0
10
e
e
e
e
e
−1.5
e
s
s
e e
e e
e
e
e
e
e
Fig. 5. (a) The real and (b) imaginary parts of the wavevectors for the even modes are
plotted as a function of the index s. u0 ¼ k0z0 ¼ 10.25π.
Im(Lo)
4
o
o
2 oo
o
oo
-2
-4
oo
oo
1
2
3
4
5
6
o
Re(L )
o
o
o
Fig. 6. The locus in the complex plane of the eigenvalues of the odd modes.
u0 ¼ k0z0 ¼ 10.25π.
(This asymmetry is found numerically to be restricted to values of u0 in very
narrow windows centered at the values m or m + 12ị.)
For u0 ẳ m Ỉ 14 π, the value of vdom corresponding to the dominant
mode is given approximately by
vdom ¼ u0 À iΔ,
(23a)
372
Jamal T. Manassah
Im(Le)
e
4
2 ee
e
ee
ee
ee
-2 e
-4
2
4
6
8
e Re(Le)
e
e
Fig. 7. The locus in the complex plane of the eigenvalues of the even modes.
u0 ¼ k0z0 ẳ 10.25.
where is the solution of the equation
exp2ị ﬃ 2u0 ,
(23b)
1
Δ ﬃ W ð4u0 Þ,
2
(23c)
giving
where W(x) is the ProductLog function. The corresponding dominant
eigenvalue is given by
dom
u0 exp2ị
ẳ
:
2
4
(23d)
It is worth noting at this point that Λdom, which determines the duration
of the linear regime in superradiant emission (the time by which the system
dumps
P o oute most of its energy), is not the same as the sum of all eigenmodes
s(Λs + Λs ), i.e., the trace of the kernel of Eq. (10); this latter quantity can be
computed easily in the position space representation, namely:
XÀ
s
Λos
+ Λes
Á
k0
¼ lim0
2 z!z
zð0
expðik0 jz À z0 jịdz0 ẳ k0 z0 ẳ u0 :
(24)
z0
Having found the eigenfunctions and eigenvalues of this non-Hermitian
operator, the next steps are to find the pseudo-orthogonality relations and
prove Parseval’s identity for these basis functions. Once these results are
established, techniques similar to those of standard Fourier expansion for
finding a system’s temporal development can then be used: i.e., one expands
an arbitrary initial spatial distribution of the polarization in these
373
Quantum Electrodynamics of Two-Level Atoms in 1D Configurations
eigenfunctions and then obtains the time development of the system by tagging to each of the eigenfunctions in the initial distribution decomposition a
factor of exp(Àiλst) with each of the terms of the expansion, where λs is the
eigenvalue of the respective mode.
2.3.2 Pseudo-Orthogonality Relations
As the Lienard–Weichert kernel is non-Hermitian, its eigenfunctions do not
obey the usual orthogonality relations of Hermitian operators (familiar from
quantum mechanics). Instead, the eigenfunctions obey the following
pseudo-orthogonal relations.
2.3.2.1
Odd Eigenfunctions
For s 6¼ s0
"
À o Á À oÁ
À o Á À o Á#
o
o
À
Á
sin
v
À
2v
2v
cos
v
cos
vs sin vs0
0
0
s
s
s
s
sin vos Z sin vos0 Z dZ ¼
À Á2 À o Á2
vos À vs0
À1
À Á À ÁÀ
À Á
À ÁÁ
2
¼ hÀ Á2 À Á2 i sin vos sin vos0 vos0 cot vos0 À vos cot vos
vos À vos0
À Á À Á
2
¼ hÀ Á2 À Á2 i sin vos sin vos0 iu0 iu0 ị ẳ 0,
vos vos0
1
(25a)
where we used Eq. (20) going from the second line to the third line.
For
À Á
À Á
À o Á À o Á
sin 2vos
cos 2 vos
¼ 1 À o À oÁ
sin vs Z sin vs Z dZ ¼ 1 À
s¼s :
2vos
vs cot vs
À1
À Á
cos 2 vos
¼1À
¼ N os :
iu0
(25b)
0
ð1
Equations (25a) and (25b) combine into the general expression:
ð1
À1
À Á À
Á
sin vos Z sin vos0 Z dZ ¼ N os δs, s0
(25c)
374
Jamal T. Manassah
2.3.2.1
Even Eigenfunctions
0
For s 6¼ s
ð1
cos
À1
À
vos Z
Á
cos
À
vos0 Z
Á
"
À Á À Á
À Á À Á#
2ves cos ves0 sin vos À 2ves0 cos ves sin ves0
dZ ¼
À Á2 À e Á2
ves À vs0
À Á À ÁÀ
À Á
À ÁÁ
2
¼ hÀ Á2 À Á2 i cos ves cos ves0 ves tan ves À ves0 tan ves0
ves À ves0
À Á À Á
2
¼ hÀ Á2 À Á2 i cos ves cos ves0 iu0 + iu0 ị ẳ 0,
ves ves0
(26a)
where we used Eq. (21) going from the second line to the third line.
For
À Á
À e Á À e Á
sin 2ves
s¼s :
cos vs Z cos vs Z dZ ¼ 1 +
2ves
À1 À Á
À
Á
sin 2 ve
sin 2 ves
¼ N es :
¼ 1 + e À seÁ ¼ 1 À
iu0
vs tan vs
0
ð1
(26b)
Equations (26a) and (26b) combine into the general expression:
ð1
À Á À
Á
cos ves Z cos ves0 Z dZ ¼ N es δs, s0 :
(26c)
À1
We note that neither of the two normalizations ever vanishes. (For Nos
to vanish, we must have cos(2vos ) ¼ À 1 + 2iu0, while for Nes to vanish, we
must have cos(2ves) ¼ 1 À 2iu0.)
We plot in Figs. 8 and 9 the values of the normalization constants
Nos and Nes for the system described in Figs. 4 and 5.
2.3.3 Parseval's Identity
If we take any of the elements of the complete basis functions {ψ m ¼ sin
(mπZ)} for odd functions over the interval À1 Z 1, we shall prove that
ð1
1¼
2
dZjψ m Z ịj ẳ
1
1
X
A2 sị
,
N os
sẳ1
(27)
375
Quantum Electrodynamics of Two-Level Atoms in 1D Configurations
(a)
Re(No)
1.2
o o
o o o o o o o o o
o o o o o o o o o
1.0
0.8
0.6
0.4
0.2
5
10
15
20
s
(b)
Im(N o)
0.15
o
0.10
0.05
o o
o o o o o
5
o
o
10
−0.05
−0.10
o
o
o
o
o o 20
15
o o o
s
o
Fig. 8. (a) The real and (b) imaginary parts of the normalization constants for the odd
modes are plotted as a function of the index s. u0 ¼ k0z0 ¼ 10.25π.
where Nos has been defined in Eq. (25b) and
1
Asị ẳ
1
2y
dZ m Z ịsin vos Z ẳ 2
sin vos ,
o
2
vs À y
(28)
where y ¼ mπ.
Defining the functions g(v) and p(v) as
gvị ẳ vcot vị iu0 ,
2y
pvị ẳ 2
,
v y2
then
Asị ẳ p vos sin vos :
(29)
(30)
(31)
376
Jamal T. Manassah
(a)
Re(N e)
e
1.2
1.0
e e e e e e e e e
e
e
e e e e e e e e
0.8
0.6
0.4
0.2
5
10
(b)
Im(N e)
0.15
20
s
e
0.10
0.05
15
e e
e e e e e
e
e
e
5
10
−0.05
e
−0.10
e
e
15 e e e e 20
e
s
e
Fig. 9. (a) The real and (b) imaginary parts of the normalization for the even modes are
plotted as a function of the index s. u0 ẳ k0z0 ẳ 10.25.
Now let
ỵ
1
vp2 vị
Jẳ
dv
,
2i
gvị
(32)
C
where the contour encircles all poles with jvj jvmaxj. In the limit that
jvmaxj ! 1, J ¼ 0. But J is also the sum of residues of all poles within
the contour.
The poles are at vos and Àvos for all s, and at y and Ày. The residue at Àvos is
the same as the one at vos , and likewise for y and Ày. Therefore,
X
(33)
Res: at vos :
0 ¼ lim J ¼ 2 Res: at y +
jvmax !1j
At y, we have p(y) infinite but v/g(y) ¼ 0. Consequently, one of the infinite factors is cancelled and we have only a simple pole whose residue is 1.
At vos , the residue is vos p2(vos )/g0 (vos ) where
Quantum Electrodynamics of Two-Level Atoms in 1D Configurations
g0 vị ẳ
dg
v
ẳ cot vị 2 ,
dv
sin vị
377
(34)
so that
Á
g0 vos ¼ À
vos
À Á N os :
sin 2 vos
(35)
It follows that the residue is À ANðosÞ.
! s
1
X
A2 ðsÞ
QED.
Hence, 0 ¼ 2 1 À
N os
s¼1
2
Similar steps can be followed to establish Parseval’s identity for the even
eigenfunctions.
2.4
Differential Form of the Field Equation (Friedberg
and Manassah, 2008c)
Starting with Eq. (10), we found its stationary solution. For these solutions,
b_ðz, t Þ has the same spatial distribution as b(z, t). The time evolution of b(z, t)
has the form exp(Ài(ω À ω0)t), where ω is complex with negative imaginary
part. Thus, we make the replacement b_ ! Àiðω À ω0 Þb. Giving for the stationary form of the integral equation
i
Ck0
bzị ẳ
0 + L + iγ T Þ 2
zð0
dz0 expðik0 jz À z0 jÞbðz0 Þ:
(36)
Àz0
In this section, we derive the differential form of this equation. This will
allow a more familiar treatment of the problem when considering mixed
multiple layer problems.
We define, for all z from À1 to 1, a field (z) by
z0
zị ẳ
dz0 expik0 jz z0 jịbz0 ị:
(37)
z0
Evidently,
d2
2
+ k0 zị ẳ 0 forjzj ! z0 ,
dz2
(38)
378
Jamal T. Manassah
d2
2
+ k0 zị ẳ 2ik0 bzị for z0
dz2
z
z0 :
(39)
Now combining Eq. (36) with Eq. (39), we have in the interval
z0 z z0:
2
d
Ck20
2
z
ị
ẳ
+
k
zị,
(40a)
0
dz2
0 + L Þ + iγ T
or
d2
2
+
k
ε
ð
ω
Þ
ΦðzÞ ¼ 0,
0
dz2
(40b)
where
εðωÞ ¼ 1 À
C
:
ðω À ω0 + ωL Þ + iγ T
(40c)
In addition, from Eq. (37), both Φ and dΦ
dz should be continuous at
z ¼ Æ z0. Solving Eqs. (40a)–(40c) with these boundary conditions gives
us again the same eigenvalue conditions previously derived (Eqs. 18–22).
This formulation of the problem allows us to use the powerful transfer
matrix method to solve multislab structures (see Appendix).
2.5
Inverted System in the Superradiant Linear Regime
(Friedberg and Manassah, 2008e)
The first step in generalizing our treatment to solve the superradiance problem in the linear regime is to reconcile the eigenmode analysis with the
traditional Maxwell–Bloch equations approach.
The Bloch equations for the response of a system of two-level atom to an
external resonant field are given by
@χ
ne
ψ
¼ ẵinL + T + i ,
@t
2
@n
e *ị + 1 1 nị,
ẳ i*e
@t
(41)
(42)
where is the complex polarization with exp(Àiω0t) factored out and normalized so that jχ j ¼ 12 at maximum; n is the fraction of atoms in the ground
state minus the fraction in the excited state, and the other parameters in the
e is the Rabi frequency associated with
equations are as previously defined. ψ