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Prog. Theor. Exp. Phys. 2017, 013A03 (17 pages)
DOI: 10.1093/ptep/ptw177
Nonlinear Schrödinger equation and semiclassical
description of the light–atom interaction
Sergey A. Rashkovskiy∗
Institute for Problems in Mechanics of the Russian Academy of Sciences,
Vernadskogo Ave., 101/1, Moscow, 119526, Russia
Tomsk State University, 36 Lenina Avenue, Tomsk, 634050, Russia
∗
E-mail: [email protected]
Received August 19, 2016; Revised October 25, 2016; Accepted November 8, 2016; Published January 21, 2017
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In this paper, we consider the nonlinear Schrödinger equation describing the light–atom interaction while taking into account damping due to spontaneous emission. We demonstrate that
the optical Bloch equations with damping due to spontaneous emission can be derived immediately from the nonlinear Schrödinger equation without any phenomenology and quantization
of radiation. The steady-state solutions of the nonlinear optical Bloch equations are obtained
and compared with those obtained for linear optical equations. In this approximation, the light
scattering by an atom is considered.
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Subject Index
1.
A34, A60, A64
Introduction
The atom–field interaction is one of the most fundamental problems of quantum optics.
It is believed that a complete description of the light–atom interaction is possible only within the
framework of quantum electrodynamics (QED), when both the states of an atom and the radiation
itself are quantized. There have been attempts to build a so-called semiclassical theory in which
only the states of an atom are quantized, while the radiation is considered to be a classical Maxwell
field [1–6]. Despite the success of this approach, there are many optical phenomena that cannot
be described by semiclassical theory. It is believed that an accurate description of these phenomena requires a full quantum mechanical treatment of both the atom and the field. In particular,
this statement is related to a description of the light–atom interaction with damping due to spontaneous emission. In quantum optics, such an interaction is described by the optical Bloch equation
in which the linear term describing the damping due to spontaneous emission is introduced phenomenologically. A justification of this term and the calculation of the damping rate are made only
in QED.
In Refs. [7–11], an attempt has been made to extend the semiclassical theory and construct a completely classical theory, which is similar to classical field theory [12]. Thus, in Refs. [7–9], it was
shown that the discrete events (e.g., clicks of a detector, emergence of the spots on a photographic
plate) that are observed in some experiments with light (especially in the double-slit experiments)
can be explained within classical electrodynamics without quantization of the radiation. Similarly,
considering the electron waves as a classical continuous field, similar to the classical electromagnetic field, one can consistently explain the “wave–particle duality of electrons” in the double-slit
© The Author(s) 2017. Published by Oxford University Press on behalf of the Physical Society of Japan.
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experiments [10]. In this case, the Dirac equation and its specific cases (Klein–Gordon, Pauli, and
Schrödinger) can be considered to be the field equations, which are similar to Maxwell’s equations
for classical electromagnetic fields. This viewpoint enables the description, in the framework of
classical field theory, of the many observed phenomena that involve electrons and are considered to
be paradoxical from the standpoint of classical mechanics.
From a formal perspective, the hydrogen atom can be considered to be a classical open volume
resonator in which an electrically charged continuous electron wave is held by the electrostatic
field of the nucleus. This resonator is described by the Schrödinger equation just as the classical
electromagnetic dielectric resonator is described by Maxwell’s equations.
In the hydrogen atom, as in any volume resonator, there are eigenmodes that correspond to a discrete
spectrum of eigenfrequencies, which are the eigenvalues of the field equation (e.g., Schrödinger and
Dirac). As usual, the standing electron waves correspond to the eigenmodes. If only one of the
eigenmodes is excited in the atom as the volume resonator, then such a state of the atom is a pure
state. If several (two or more) eigenmodes are simultaneously excited in the atom, then such a state
is a mixed state [11].
Using this viewpoint, it was shown [11] that all of the basic optical properties of the hydrogen
atom have a simple and clear explanation in the framework of classical electrodynamics without
quantization of radiation. In particular, it was shown that the atom can be in a pure state indefinitely.
This arrangement means that the atom has a discrete set of stationary states, which correspond to all
possible pure states, but only the pure state that corresponds to the lowest eigenfrequency is stable.
Precisely this state is the ground state of the atom. The remaining pure states are unstable, although
they are the stationary states. Any mixed state of an atom in which several eigenmodes are excited
simultaneously is nonstationary, and according to classical electrodynamics, the atom that is in that
state continuously emits electromagnetic waves of the discrete spectrum, which is interpreted as a
spontaneous emission.
Thus, a fully classical description of spontaneous emission was given in Ref. [11], and all of its
basic properties that are traditionally described within the framework of quantum electrodynamics
were obtained. It was shown that the “jump-like quantum transitions between the discrete energy
levels of the atom” do not exist, and the spontaneous emission of an atom occurs not in the form of
discrete quanta but continuously.
As is well known, the linear wave equation, e.g., the Schrödinger equation, cannot explain the
spontaneous emission and the changes that occur in the atom in the process of spontaneous emission
(so-called “quantum transitions”). To explain spontaneous transitions, quantum mechanics must
be extended to quantum electrodynamics, which introduces such an object as a QED vacuum, the
fluctuations of which are considered to be the cause of the “quantum transitions.”
It was shown in [11] that the Schrödinger equation, which describes the electron wave as a classical
field, is sufficient for a description of the spontaneous emission of a hydrogen atom. However, it
should be complemented by a term that accounts for the inverse action of self-electromagnetic
radiation on the electron wave. In the framework of classical electrodynamics [12], it was shown
that the electron wave is described in the hydrogen atom by a nonlinear equation [11],
∂ψ
2
e2
∂3
2e2
i
=−
ψ − ψ − 3 ψr 3 r |ψ|2 dr,
(1)
∂t
2me
r
3c
∂t
where the last term on the right-hand side describes the inverse action of the self-electromagnetic
radiation on the electron wave and is responsible for the degeneration of any mixed state of the
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hydrogen atom. This term “provides” a degeneration of the mixed state of the hydrogen atom to a
pure state, which corresponds to the lower excited eigenmodes of an atom. As shown in Ref. [11], this
term has a fully classical meaning and fits into the semiclassical concept developed in Refs. [7–11].
In this paper, using Eq. (1), the light–atom interactions are considered from the standpoint of
classical field theory. We will show that the optical Bloch equations with damping due to spontaneous
emission and with the correct damping rate can be directly derived from the nonlinear equation (1)
without any phenomenology and without quantization of radiation.
2.
Optical Bloch equations with damping due to spontaneous emission
Let us consider a hydrogen atom that is in a light field and account for its spontaneous emission.
Under the action of the light field, the excitation of some eigenmodes of the electron wave occurs
in the hydrogen atom at the expense of others. This phenomenon will be manifested in a stimulated
redistribution of the electric charge of the electron wave in the atom between its eigenmodes that
occurs under the action of the nonstationary electromagnetic field [11]. As a result, the atom, as
an open volume resonator, will go into a mixed state in which, in full compliance with classical
electrodynamics, electric dipole radiation will occur [11], which is customarily called a spontaneous
emission. As shown in Ref. [11], the emission will be accompanied by a spontaneous “crossflow”
of the electric charge of the electron wave from an eigenmode that has a greater frequency to
an eigenmode that has a lower frequency. Under certain conditions, between the spontaneous and
stimulated redistribution of the electric charge of an electron wave within the atom, a detailed
equilibrium can be established, in which the amount of electric charge that crossflows from an
eigenmode k into an eigenmode n per unit time will be equal to the amount of electric charge that
crossflows from eigenmode n into eigenmode k per unit time. Clearly, such equilibrium will be
forced and will depend on the intensity of the light field. This finding is easily observed when the
light field is instantaneously removed. As shown in Ref. [11], the atom, which was previously in a
mixed state, will spontaneously emit electromagnetic waves, and this spontaneous emission will be
accompanied by a spontaneous redistribution of the electric charge of the electron wave between the
excited eigenmodes of the atom. As a result, over time, the whole electric charge of the electron wave
crossflows into a lower eigenmode (i.e., with a lower frequency) of the initially excited eigenmodes,
and the atom enters into a pure state that can be retained indefinitely because spontaneous emission
is absent in this state.
Let us consider this process in more detail using the example of the hydrogen atom.
If an atom is in an external electromagnetic field, then Eq. (1) should be supplemented by terms
that take into account the external action. As a result, one obtains the equation
2
e
2e2 ∂ 3 ∂ψ
1
e
i
(2)
=
∇ + A −e
+ ϕ − 3 r 3 r |ψ|2 dr ψ,
∂t
2me i
c
r
3c ∂t
where ϕ(tr) and A(tr) are the scalar and vector potentials of an external electromagnetic field.
We consider a linearly polarized electromagnetic wave with wavelength λ, which is considerably
greater than the characteristic size of the hydrogen atom:
λ r |ψ|2 dr.
(3)
In this case, one can consider the electric field of the electromagnetic wave in the vicinity of the
hydrogen atom to be homogeneous but nonstationary: E (t, r) = E0 cos ωt, where E0 and ω are
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constants. For such a field, one can select the gauge [12], at which
A = 0, ϕ = −rE0 cos ωt.
(4)
In this case, Eq. (2) takes the form
e2
2e2
∂3
∂ψ
2
=−
ψ − ψ + ψerE0 cos ωt − 3 ψr 3
i
∂t
2me
r
3c
∂t
r |ψ|2 dr,
As usual, the solution of Eq. (5) can be found in the form
cn (t)un (r) exp (−iωn t),
ψ(tr) =
(5)
(6)
n
where the constants ωn and the functions un (r) are the eigenvalues and eigenfunctions of the linear
Schrödinger equation:
ωn un = −
2
e2
un − un .
2me
r
The functions un (r) form the orthonormal system:
un (r)uk∗ (r)dr = δnk .
(8)
Substituting Eq. (6) into Eq. (5) while accounting for Eqs. (7) and (8), we obtain
...
dcn
2 ck (dnk E0 ) exp (iωnk t) cos ωt − 3
ck (dnk d ) exp (iωnk t),
=−
i
dt
3c
k
(7)
(9)
k
where
ωnk = ωn − ωk ,
d = −e r |ψ|2 dr
(10)
(11)
is the electric dipole moment of the electron wave in the hydrogen atom:
∗
= −e run∗ (r)uk (r)dr.
dnk = dkn
(12)
Considering expressions (6), (11) and (12), we can write
ck cn∗ dnk exp (iωnk t).
d=
(13)
n
k
Let us consider the so-called “two-level atom”, i.e., the case in which only two eigenmodes k and n
of an atom are excited simultaneously.
For definiteness, one assumes that ωn > ωk , and correspondingly,
ωnk > 0.
(14)
In this case, Eqs. (9) and (11) take the form
i
dcn
= −[ck (dnk E0 ) exp (iωnk t) + cn (dnn E0 )] cos ωt
dt
... ...
2
− 3 [ck dnk d exp (iωnk t) + cn (dnn d )],
3c
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(15)
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S. A. Rashkovskiy
∗
dck
= −[cn dnk
E0 exp (−iωnk t) + ck (dkk E0 )] cos ωt
dt
∗ ... ...
2
− 3 [cn dnk
d exp (−iωnk t) + ck (dkk d )],
3c
∗
exp (−iωnk t).
d = |cn |2 dnn + |ck |2 dkk + ck cn∗ dnk exp (iωnk t) + cn ck∗ dnk
(16)
(17)
In general, differentiating vector d with respect to time, we must account for the fact that the
parameters cn are functions of time. As shown below, the parameters cn are changed with time
much more slowly than the oscillating factor exp (iωnk t). This arrangement means that there is the
condition
|ċn | ωnk |cn |.
(18)
Considering Eqs. (17) and (18), we obtain approximately
...
3
∗ ∗
∗
d = iωnk [cn ck dnk exp (−iωnk t) − ck cn dnk exp (iωnk t)].
(19)
Substituting expression (19) into Eqs. (15) and (16), we obtain
dcn
= − [ck (dnk E0 ) exp (iωnk t) + cn (dnn E0 )] cos ωt
dt
2
3
∗
cn |ck |2 |dnk |2 + cn cn ck∗ dnn dnk
exp (−iωnk t)
− 3 iωnk
3c
2
3
[ck ck cn∗ (dnk )2 exp (2iωnk t) + |cn |2 ck (dnn dnk ) exp (iωnk t)],
(20)
+ 3 iωnk
3c
∗
dck
= − cn dnk
E0 exp (−iωnk t) + ck (dkk E0 ) cos ωt
i
dt
2
3
∗ 2
∗
exp (−iωnk t)
) exp (−2iωnk t) + |ck |2 cn dkk dnk
cn cn ck∗ (dnk
− 3 iωnk
3c
2
3
[ck |cn |2 |dnk |2 + ck ck cn∗ (dkk dnk exp (iωnk t) )],
(21)
+ 3 iωnk
3c
∗ , (d )2 = (d d )
where |dnk |2 = dnk dnk
nk
nk nk
Equations (20) and (21) contain rapidly oscillating terms with frequencies of ω, ωnk , and 2ωnk ,
which in view of (18) can be removed by averaging over the fast oscillations. As a result, we obtain
the equations
i
2ω3
dcn
= −ck (dnk E0 ) exp (iωnk t) cos ωt − i nk
cn |ck |2 |dnk |2 ,
dt
3c3
∗
2ω3
dck
i
= −cn dnk
E0 exp (−iωnk t) cos ωt + i nk
ck |cn |2 |dnk |2 ,
dt
3c3
i
(22)
(23)
Equations (22) and (23) describe the Rabi oscillations with damping due to spontaneous emission.
The first term on the right-hand side of Eq. (22) describes the excitation of mode n due to the impact
of the incident electromagnetic wave on mode k. In quantum mechanics, it is traditionally interpreted
to be an induced transition from a lower energy level k to a higher energy level n due to “absorption
of the photon.” The first term on the right-hand side of Eq. (23) describes the excitation of mode
k due to the impact of the incident electromagnetic wave on mode n. In quantum mechanics, it is
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traditionally interpreted as an induced transition from a higher energy level n to a lower energy level
k due to the “emission of a photon.” The last term on the right-hand side of Eqs. (22) and (23) is
traditionally interpreted as a spontaneous transition from a higher level n to a lower level k, which
is accompanied by an “emission of a photon” with a frequency of ωnk .
From the viewpoint of classical field theory considered here, the first terms on the right-hand
sides of Eqs. (22) and (23) describe an induced (by the action of the light field) redistribution of the
electric charge of the continuous electron wave between the excited modes n and k, while the second
terms describe a spontaneous redistribution of the electric charge between these two modes, which
is accompanied by a spontaneous emission. In our semiclassical analysis, there are neither photons
nor electrons; there are no “jump-like transitions” between the atom’s discrete energy levels, which,
incidentally, are also absent, while the terms of Eqs. (22) and (23) describe the interaction of two
classical wave fields: the electromagnetic wave and the electron wave.
Let us introduce the notations
ρnn = |cn |2 , ρkk = |ck |2 , ρnk = cn ck∗ , ρkn = ck cn∗ .
(24)
ρnn + ρkk = 1,
(25)
∗
.
ρnk = ρkn
(26)
Obviously,
Condition (25) according to Ref. [11] expresses the law of conservation of electric charge, which
means that in the process under consideration, the electric charge of the electron wave is simply
redistributed between modes n and k.
Using Eqs. (22) and (23) for the parameters in Eq. (24), we obtain the equation
dρnn
dρkk
=−
= i[ρkn bnk exp (iωnk t) − ρnk b∗nk exp (−iωnk t)] cos ωt − 2γnk ρnn ρkk ,
dt
dt
∗
dρkn
dρnk
=
= (ρkk − ρnn )[ibnk exp (iωnk t) cos ωt − γnk ρnk ],
dt
dt
(27)
(28)
where
3
2ωnk
|dnk |2 ,
3c3
1
= (dnk E0 ).
γnk =
bnk = b∗kn
(29)
(30)
Equations (27) and (28) describe the interaction of electromagnetic waves with a hydrogen atom
taking into account a spontaneous emission. They are, in fact, the optical Bloch equations with
damping due to spontaneous emission [13]. However, there are fundamental differences in Eqs.
(27) and (28) from the conventional (linear) optical Bloch equations [13]. Thus, in quantum optics,
the optical Bloch equations that account for damping due to spontaneous emission are not derived
strictly but are postulated by the addition of the corresponding linear damping terms in the linear wave
equations. This arrangement is because the damping in a spontaneous emission cannot be obtained
from the linear Schrödinger equation and requires a “second quantization.” In our approach, Eqs. (27)
and (28) are strictly derived and are a direct and natural consequence of the nonlinear Schrödinger
equation (5), which accounts for the spontaneous emission. Equations (27) and (28) are nonlinear
in contrast to conventional (linear) optical Bloch equations because the last terms on the right-hand
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side of Eqs. (27) and (28) nonlinearly depend on ρkk and ρnn . They become almost linear if the
excitation of the upper mode n is weak, and we have approximately ρkk ≈ ρkk − ρnn ≈ 1. In this
case, Eqs. (27) and (28) turn into the conventional (linear) optical Bloch equations [13]. Moreover,
the damping rate γnk cannot be obtained within the framework of the linear Schrödinger equation
and is introduced into the linear optical Bloch equations phenomenologically [13]; its value (29) is
derived only in the framework of quantum electrodynamics [14]. In the theory under consideration,
the damping rate (29) is a direct and natural consequence of the nonlinear Schrödinger equation (5).
3.
Solution of the nonlinear optical Bloch equations
Let us consider a conventional approach for such systems in the case in which |ωnk − ω| ωnk
(rotating wave approximation) [13]. Substituting cos ωt = 12 [exp (iωt) + exp (−iωt)] and discarding
(by averaging) the rapidly oscillating terms, one transforms Eqs. (22) and (23) into the form
dcn
1
= − bnk ck exp (it) − iγnk cn |ck |2 ,
dt
2
1
dck
= − b∗nk cn exp (−it) + iγnk ck |cn |2 ,
i
dt
2
i
(31)
(32)
where
= ωnk − ω.
(33)
Accordingly, equations (27) and (28) in this case take the form
dρnn
dρkk
1
=−
= i [ρkn bnk exp (it) − ρnk b∗nk exp (−it)] − 2γnk ρnn ρkk ,
dt
dt
2
∗
dρkn
1
dρnk
=
= (ρkk − ρnn ) [i bnk exp (it) − γnk ρnk ].
dt
dt
2
(34)
(35)
Let us consider the stationary solution of Eqs. (34) and (35), which corresponds to ρnn = const and
ρkk = const, while the parameters ρnk and ρkn will be the oscillating functions. The solution of Eqs.
(34) and (35) can be found in the form
ρnk = a exp (it),
(36)
where a is a constant.
Substituting expression (36) into Eqs. (34) and (35), we obtain the equations
i
1 ∗
a bnk − ab∗nk − 2γnk ρnn ρkk = 0,
2
1
ia = (ρkk − ρnn ) (i bnk − γnk a).
2
(37)
(38)
Hence, using Eq. (25), we obtain
1
bnk (1 − 2ρnn )
,
2 [γnk (1 − 2ρnn ) + i]
(39)
(1 − 2ρnn )2
− 4ρnn (1 − ρnn ) = 0.
2 (1 − 2ρ )2 + 2
γnk
nn
(40)
a=i
|bnk |2
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In particular,
|a|2 = ρnn (1 − ρnn ).
Equation (40) can be rewritten in the form
2
|bnk |2 /γnk
2
(/γnk ) =
− 1 (1 − 2ρnn )2 .
4ρnn (1 − ρnn )
(41)
(42)
The solution of equation (42) exists only if its right-hand side is positive. This arrangement is possible
only if
2
|bnk |2 /γnk
> 4ρnn (1 − ρnn ).
(43)
One can rewrite this condition in the form
2
ρnn
− ρnn +
1
2
|bnk |2 /γnk
> 0,
4
(44)
which is satisfied for any ρnn if
2
|bnk |2 /γnk
1.
(45)
2
|bnk |2 /γnk
< 1,
(46)
At the same time, if
then condition (44) is satisfied only if
1
2
2
1 − 1 − |bnk | /γnk
2
(47)
1
2
2
1 + 1 − |bnk | /γnk ρnn 1.
2
(48)
0 ρnn
or if
The solutions of Eq. (42) for different values of the parameter |bnk | /γnk are shown in Fig. 1. Equation
(42) has two solutions for the same (see Fig. 1). Their sum is equal to one, which means that one root
can be considered to be ρnn , while the other is considered to be ρkk . Theoretically, each of these roots
can correspond to ρnn . Recall that condition (14) was accepted. Therefore, due to the spontaneous
emission, mode n always loses an electric charge, while mode k receives it. This arrangement means
that only the smaller root of Eq. (42) should correspond to mode n, while the larger root corresponds
to mode k. Such a solution will always be stable: there are no small perturbations that could violate
this condition because the system will always return to it. In contrast, the second solution, in which
ρnn corresponds to a larger root of Eq. (42), will be unstable, and any small perturbations lead to the
system spontaneously returning to the first stable state due to crossflow of the electric charge of the
electron wave from the upper excited mode to the lower mode. Thus, in the stable state, the smaller
of the two roots of Eq. (42) always corresponds to ρnn , while the larger root corresponds to ρkk .
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Fig. 1. Dependencies of ρnn (/γnk ) and ρkk (/γnk ) for different values of the parameter |bnk | /γnk . The lower
branch corresponds to ρnn , and the upper branch corresponds to ρkk .
Then, for the case in Eq. (46),
maximum value of ρnn , which can be achieved at resonance
the 2 . The corresponding minimum value of ρ
( = 0), will be (ρnn )max = 12 1 − 1 − |bnk |2 /γnk
kk
2 . It should be noted that under condition (46), the
will be (ρkk )min = 12 1 + 1 − |bnk |2 /γnk
solution of Eq. (42) has a discontinuity at = 0. In fact, at the point = 0 (i.e., at resonance),
this equation has
solution ρnn = ρkk = 0.5, but in the limit → 0, it has a solution
a formal
1
2 . The resonant solution (at = 0) can be realized in an
lim ρnn = 2 1 ± 1 − |bnk |2 /γnk
→0
experiment only if the condition = 0 is provided exactly. Any deviation from this condition will
lead to the “destruction” of the resonant solutions. Because the exact condition = 0 cannot be
realized in an experiment due to the presence of the fluctuations in any system, it is clear that the
resonance solution has no practical interest.
Let us compare the solution of the exact nonlinear equations (34) and (35) with the solution of the
linear optical Bloch equations that are considered in quantum optics [13]. The linear optical Bloch
equations can be obtained as a linear approximation of Eqs. (34) and (35), which correspond to the
case of weak excitation of the upper mode n, i.e., when ρnn 1. In our notation, the stationary
solution of the linear optical Bloch equations with damping due to spontaneous emission has the
following form [13]:
ρnn =
1
4
|bnk |2
2 + 1 |b |2
2 + γnk
2 nk
.
(49)
Figure 2 shows a comparison of the stationary solutions (42) and (49).
In contrast to the nonlinear equations (34) and (35), the linear optical Bloch equations have a
unique stationary solution ρnn , which corresponds to the smaller (stable) of two stationary solutions
of the nonlinear equations (34) and (35).
Figure 2 shows that the lower (stable) of the two stationary solutions of nonlinear optical Bloch
equations (34) and (35) tends to the stationary solution (49) of the linear optical Bloch equations
only for small values of the parameter |bnk | /γnk < 0.4, as well as for asymptotically large values
|ωnk − ω| /γnk 1. Both correspond to the relatively weak actions of the optical field on the atom.
Thus, the linear optical Bloch equation with damping due to spontaneous emission can describe only
2 << 1. In the description of strong
the impact of a weak optical field on the atom when |bnk |2 /γnk
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Fig. 2. Comparison of the dependencies ρnn (/γnk ) for different values of parameter |bnk | /γnk , which corresponds to stationary solutions of the nonlinear optical Bloch equations (34) and (35) (solid black lines) and
linear optical Bloch equations (dashed red lines).
impacts, especially near the resonance frequency, it is necessary to use the nonlinear optical Bloch
equations (34) and (35).
2 1, at resonance ( = 0), a simultaneous
In particular, in a strong light field when |bnk |2 /γnk
strong excitation of both modes at which ρnn (0) = ρkk (0) = 12 occurs. In this case, a saturation occurs
2.
when the value ρnn (0) = ρkk (0) does not depend on the intensity of exposure to the atom |bnk |2 /γnk
2 < 1, it will always be ρ
At the same time, at the weak excitation when |bnk |2 /γnk
nn < ρkk , even
under resonance conditions.
2ω2
2
Let us estimate the ratio γnk /ωnk = 3cnk3 |dnk |2 while accounting for |dnk | ∼ eaB and aB = me
2,
ωnk ∼
me4
. As
3
a result, we obtain
γnk /ωnk ∼ α 3 1,
(50)
2
where α = e c is the fine-structure constant.
It follows from Figs. 1 and 2 that when |bnk | /γnk ∼ 1, and even more so when |bnk | /γnk < 1,
the width of the frequency range on which a noticeable change in the parameters cn occurs is
of the order of ∼γnk , and therefore, ωnk . Using expressions (36) and (39), we obtain
|ċn | ∼ |cn | |bnk | /γnk . As a result, we can conclude that in this case, |ċn | ωnk |cn |. At the same
time, when |bnk | /γnk 1, the width of the frequency range γnk , and in this case, it could be
|ċn | ∼ ωnk |cn | or even |ċn | > ωnk |cn |. Thus, we conclude that condition (18) can be violated only
for very strong electromagnetic waves, for which |bnk | /γnk 1. Such an electromagnetic wave can
cause a so-called tunnel ionization of the atom, which is not considered here.
Note that the parameters cn and ck that correspond to the stationary solution of Eqs. (34) and (35)
can be found by direct solution of Eqs. (31) and (32) (see Appendix).
4.
Einstein A and B coefficients
Let us consider the solution of Eqs. (34) and (35) near the stationary solution. As before, we assume
that the parameters ρnk and ρkn are the fast oscillating functions, while the parameters ρnn and
ρkk change slowly compared with the parameters ρnk and ρkn . In this case, when solving Eq. (35),
the parameters ρnn and ρkk may be considered to be constant, and the solution of Eq. (35) will
coincide with the above-obtained solution (36), (39). Substituting this solution into Eq. (34), one
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obtains
dρnn
dρkk
1
γnk (ρkk − ρnn )2
− 2γnk ρnn ρkk .
=−
= |bnk |2 2
dt
dt
2
γnk (ρkk − ρnn )2 + 2
(51)
This equation describes the slow change in the amplitudes ρnn and ρkk of excitation of the modes
n and k of the two-level atom under the action of a classical monochromatic electromagnetic wave
when the orientation of the component dnk of the atom dipole moment with respect to vector E0 is
considered to be fixed and predetermined.
In experiments, in most cases we address an ensemble of atoms for which the mutual orientation
of the vectors dnk and E0 is random. In this case, it is necessary to average the first term of Eq. (51)
over all possible mutual orientations of the vectors dnk and E0 .
Considering random orientations of the electric dipole moments dnk with respect to the vectors E0
and their statistical independence, one can write
∗
∗
(dnk
E0 )(dnk E0 )
= dnk,i dnk,j
E0i E0j ,
(52)
where . . .
denotes averaging over all mutual orientations of the vectors dnk and E0 , and i and j are
the vector indices; assume summation over repeated vector indices.
Because all orientations of the vector dnk are equally probable, one obtains
dnk,i dnk,j =
1
|dnk |2 δij
3
(53)
If the radiation field is isotropic, then obviously
1
|E0 (ω)|2 δij .
3
(54)
1
|dnk |2 |E0 (ω)|2 .
3 2
(55)
E0i E0j =
Thus, in this case,
|bnk |2 =
If the electromagnetic wave is not monochromatic, then it must be characterized by the spectral
energy density Uω (ω) instead of the amplitude E0 :
|E0 (ω)|2 =8πUω (ω) dω.
(56)
In this case, taking into account the expressions (55), (56), and (33), Eq. (51) can be written in the
form
dρnn
dρkk
4π
=−
= 2 |dnk |2
dt
dt
3
∞
0
γnk (ρkk − ρnn )2
2 (ρ − ρ )2 + (ω − ω)2
γnk
nn
kk
nk
Uω (ω) dω − 2γnk ρnn ρkk . (57)
Taking into account the assessment (50), one concludes that the function
(ρkk −ρnn )2 γnk
2
(ω−ωnk )2 +(ρkk −ρnn )2 γnk
has
a bell-shaped form in the vicinity of the frequency ωnk with bandwidth ω ∼ γnk ωnk .
Let us consider the continuous spectrum of the radiation field, which satisfies the condition
dUω (ωnk ) Uω (ωnk ).
γnk (58)
dω
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In this case taking into account the expressions (44) and (45), one obtains
∞
0
(ρkk − ρnn )2 γnk
U (ω) dω
2 ω
(ω − ωnk )2 + (ρkk − ρnn )2 γnk
∞
≈Uω (ωnk )
0
(ρkk − ρnn )2 γnk
dω
2
(ω − ωnk )2 + (ρkk − ρnn )2 γnk
∞
≈Uω (ωnk )
−∞
(ρkk − ρnn )2 γnk
dw=πUω (ωnk ) (ρkk − ρnn ).
2
w2 + (ρkk − ρnn )2 γnk
(59)
Substituting expression (59) into Eq. (57), one obtains
dρnn
dρkk
4π 2 |dnk |2
=−
=
Uω (ωnk ) (ρkk − ρnn ) − 2γnk ρnn ρkk .
dt
dt
32
(60)
More detailed derivation and analysis of this equation and its application to the description of the
thermal radiation will be considered in the next report.
Here we consider a probabilistic interpretation of this equation. For this purpose, we rewrite Eq.
(60) in the form
dρnn
dρkk
=−
= Bkn Uω (ωnk ) ρkk − Bkn Uω (ωnk ) ρnn − Ank ρnn ρkk ,
dt
dt
(61)
where
Bkn = Bnk =
4π 2 |dnk |2
,
3 2
Ank = 2γnk .
(62)
(63)
Using expression (29), one can find the connection between the coefficients (62) and (63):
Bkn = Bnk =
π 2 c3
Ank .
3
ωnk
(64)
Note that this connection is universal because it does not depend on the electric dipole moments
|dnk |2 and is determined only by the frequency of spontaneous emission ωnk . Therefore, although
the relation (64) was obtained for the hydrogen atom, one can expect that it will be valid for other
atoms having different frequencies of spontaneous emission ωnk .
It should be noted that up to this point there was no need to use a probabilistic interpretation of
the parameters ρnn and ρkk ; they described the distribution of the electric charge of the continuous
electron wave between excited modes n and k of the atom. At the same time, it is easy to see that Eq.
(61), formally, has the form of the kinetic equation describing the transitions of some fictitious system
between two discrete states n and k; in this case, taking condition (25) into account, the parameters
ρnn and ρkk can be interpreted as the probabilities of finding the system in the corresponding discrete
states. Then, the first and second terms on the right-hand side of Eq. (61) can be interpreted as
induced transitions k → n (due to absorption of radiation) and n → k (due to stimulated emission),
respectively, and the third term as a spontaneous transition n → k (due to spontaneous emission).
The probabilities of the corresponding transitions are as follows:
ind
wkn
= Bkn Uω (ωnk ) ,
ind
= Bnk Uω (ωnk ) ,
wnk
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sp
wnk = Ank ρkk .
(65)
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S. A. Rashkovskiy
In this case, the parameters Bkn , Bnk , and Ank in Eqs. (62) and (63) should be interpreted as the
Einstein coefficients. Taking expression (63) into account, the “probabilities” of the corresponding
“transitions” are connected by the relations
ind
ind
wkn
= wnk
=
π 2 c3 1 sp
wnk Uω (ωnk ).
3 ρ
ωnk
kk
(66)
These expressions are different from those derived in quantum electrodynamics [14] only by the
additional factor ρ1kk .
In the case when the excitation of the upper eigenmode n is weak, i.e., when ρnn ρkk , one can
take ρkk ≈ 1, and then relation (66) becomes the well-known result of quantum electrodynamics
[14]. In this case, the correct expressions (62) and (63) for the Einstein A and B coefficients are
obtained.
However, the above analysis shows that the probabilistic interpretation of these equations is not
only unnecessary but also erroneous because it is based only on the outer analogy. We see that the
so-called Einstein coefficients for spontaneous emission can be obtained in a natural way within the
framework of classical field theory without any quantization of radiation.
5.
Light scattering by an atom
Using the results of the previous section, it is easy to calculate the secondary radiation (induced and
spontaneous) that is created by an atom that is in the field of a classical electromagnetic wave. This
radiation will be perceived as the scattering of the incident electromagnetic wave.
The intensity of the electric dipole radiation according to classical electrodynamics is defined by the
expression [12]
I=
2 2
d̈ ,
3c3
(67)
where d is the electric dipole moment of the electron wave in a hydrogen atom; the bar denotes
averaging over time.
For a two-level atom, as discussed in the previous section, accounting for expressions (17), (18),
and (24) implies that
2
∗
d̈ = −ωnk
[ρkn dnk exp (iωnk t) + ρnk dnk
exp (−iωnk t)].
(68)
If the atom is in a stationary forced excited state, then the parameter ρnk is determined by the
expressions (36) and (39), and it oscillates at the frequency given in Eq. (33).
In this case,
2
∗
d̈ = −ωnk
[a∗ dnk exp (iωt) + adnk
exp (−iωt)],
(69)
and for the intensity of the scattered radiation, we obtain
I=
4
4ωnk
|a|2 |dnk |2 ,
3c3
(70)
or when accounting for expressions (29) and (41), we obtain
I = 2γnk ρnn (1 − ρnn )ωnk .
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S. A. Rashkovskiy
According to the expression in Eq. (69), in the approximation under consideration, we are contending
with Rayleigh scattering. However, if we account for the rapidly oscillating terms in Eqs. (27) and
(28), the non-Rayleigh components in the scattering spectrum will be detected, but their intensity
will be negligible.
Let us calculate the scattering cross-section
dσ =
dI
S
,
(72)
where [12]
dI =
d̈ 2
sin2 θdo
4π c3
(73)
is the amount of energy that is emitted (scattered) by the atom per unit time per unit solid angle do;
θ is the angle between the vector d̈ and the direction of the scattering; and
c
|E|2
S=
(74)
4π
is the energy flux density of the incident electromagnetic wave.
In our case,
S=
c
|E0 |2 .
8π
(75)
Then, accounting for expressions (29) and (41), we obtain
dσ = 6γnk
ωnk
ρnn (1 − ρnn ) sin2 θdo.
c |E0 |2
(76)
Using expressions (29) and (40), we can also write
dσ =
2
92 c2 γnk
2
4ωnk
|bnk |2
(1 − 2ρnn )2
sin2 θdo.
2 (1 − 2ρ )2 + 2 ]
|dnk |2 |E0 |2 [γnk
nn
(77)
The scattering pattern will be determined by the parameter bnk .
In that case, when the vector dnk is real valued, based on definition (30), we can write
|bnk |2 =
1
|dnk |2 |E0 |2 cos2 ϑ
2
(78)
where ϑ is the angle between the vectors dnk and E0 .
Then, we obtain
dσ =
2
9c2 γnk
2
4ωnk
(1 − 2ρnn )2
sin2 θ cos2 ϑdo.
2
2
2
[γnk (1 − 2ρnn ) + ]
(79)
In particular, at the resonance frequency ( = 0),
dσ =
9c2
sin2 θ cos2 ϑdo.
2
4ωnk
(80)
In other cases ( = 0), in the calculation of the right-hand side of expression (79), it is necessary
to account for the fact that parameter ρnn , which is a solution of Eq. (42), will depend on the
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2 (see Fig. 1). Accounting for expressions (78) and (29), we
nondimensional parameter |bnk |2 /γnk
obtain
2
|bnk |
2
2
/γnk
9c6 |E0 |
=
cos2 ϑ.
6 |d |2
4ωnk
nk
(81)
At a low intensity of the incident electromagnetic wave, when
2
9c6 |E0 |
1,
6 |d |2
4ωnk
nk
(82)
we obtain ρnn 1, and then,
dσ ≈
2
γnk
9c2
sin2 θ cos2 ϑdo.
2 (γ 2 + 2 )
4ωnk
nk
(83)
(r)
(i)
(r)
In cases when the vector dnk is complex valued, we can write dnk = dnk + idnk , where dnk
(i)
In this case, it is possible to introduce the angle χ, such that
and dnk are real-valued vectors.
(i) (r) dnk / |dnk | = cos χ and dnk / |dnk | = sin χ. Then, the relation (30) can be written as bnk =
b∗kn = 1 |dnk | |E0 | (cos χ cos ϑr + i sin χ cos ϑi ), where ϑr and ϑi are the angles between the vector
(r)
(i)
E0 and the vectors dnk and dnk , respectively. Hence, we obtain
|bnk |2 =
1
|dnk |2 |E0 |2 (cos2 χ cos2 ϑr + sin2 χ cos2 ϑi ).
2
(84)
Then, using expression (77), we obtain
dσ =
2
9c2 γnk
2
4ωnk
(1 − 2ρnn )2
sin2 θ(cos2 χ cos2 ϑr + sin2 χ cos2 ϑi )do.
2 (1 − 2ρ )2 + 2 ]
[γnk
nn
(85)
In particular, at the resonance frequency ( = 0),
dσ =
9c2
sin2 θ(cos2 χ cos2 ϑr + sin2 χ cos2 ϑi )do.
2
4ωnk
(86)
In other cases ( = 0), in the calculation of the right-hand side of expression (85), it is necessary
to account for the fact that the parameter ρnn , which is a solution of Eq. (42), will depend on the
2 (see Fig. 1). Using the expressions (84) and (29), we obtain
nondimensional parameter |bnk |2 /γnk
in this case
2
|bnk |2 /γnk
=
9c6 |E0 |2
(cos2 χ cos2 ϑr + sin2 χ cos2 ϑi ).
2
6
4ωnk |dnk |
(87)
At a low intensity of the incident electromagnetic wave, when condition (82) is satisfied and ρnn 1,
we obtain
dσ ≈
2
γnk
9c2
sin2 θ(cos2 χ cos2 ϑr + sin2 χ cos2 ϑi )do.
2 [γ 2 + 2 ]
4ωnk
nk
(88)
Thus, the Rayleigh scattering of an electromagnetic wave by a hydrogen atom is fully described
within the framework of classical field theory without any quantization.
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S. A. Rashkovskiy
Concluding remarks
We demonstrated that the light–atom interaction can be fully described within the framework of
classical field theory without the use of quantum electrodynamics. In particular, we determined that
the optical Bloch equations with damping due to spontaneous emission and with correct damping
rate can be directly derived from the nonlinear Schrödinger equation (2) without quantization of
radiation. We see that the optical Bloch equations are nonlinear and their solutions essentially differ
from the solutions of the corresponding linear optical Bloch equations. This nonlinearity can play
an essential role in light–atom interaction and should be taken into account in all calculations.
In this paper, in the calculations of the light–atom interaction, we did not account for a property
of an electron wave: the spin. To describe the light–atom interaction taking into account the spin, it
is necessary to use the Dirac equation—or, in the Schrödinger long-wave approximation [10], the
Pauli equation, which should be supplemented by the terms that account for the inverse action of the
self-radiation field on the electron wave. This issue will be considered in subsequent papers.
Acknowledgements
Funding was provided by Tomsk State University competitiveness improvement program.
Appendix
We seek the stationary solution of Eqs. (31) and (32) in the form
cn = an exp (in t) , ck = ak exp (ik t),
(A.1)
where n , k an and ak are constants. For the stationary solution, the parameters n , k are real
valued.
Obviously
ρnn = |an |2 , ρkk = |ak |2 ,
(A.2)
where ρnn and ρkk are the stationary solutions of Eqs. (34) and (35) which were obtained above.
According to expression (36)
a = an a∗k , =n − k .
(A.3)
Substituting expressions (A.1) into Eqs. (31) and (32) and taking into account the expressions (A.2)
and (A.3), one obtains the equations
1
(n − iγnk ρkk ) an − bnk ak = 0,
2
(A.4)
1
− b∗nk an + (n − + iγnk ρnn ) ak = 0.
2
(A.5)
These equations have nonzero solutions only if
(n − iγnk ρkk ) (n − + iγnk ρnn ) −
1
|bnk |2 = 0.
4
(A.6)
Separating the real and imaginary parts of Eq. (A.6) one obtains
2
n (n − ) + γnk
ρnn ρkk −
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1
|bnk |2 = 0,
4
(A.7)
PTEP 2017, 013A03
S. A. Rashkovskiy
ρnn n − (n − ) ρkk = 0.
(A.8)
From Eq. (A.8) we obtain
n = ρkk
.
ρkk − ρnn
(A.9)
Taking into account the expressions (A.3) one obtains
k = ρnn
.
ρkk − ρnn
(A.10)
Substituting expression (A.9) into Eq. (A.7) we obtain the expression (40).
Substituting expression (A.9) into Eq. (A.4) we obtain the expression (39) which, when taking into
account expressions (A.2) and (A.3) can be rewritten as follows:
ak a = an ρkk .
(A.11)
This expression connects the amplitudes ak and an . Believing
an =
√
ρ nn exp (iθn ),
where θn is a phase, then according to (A.11) one obtains
√
ρkk ρ nn
exp (iθn ).
ak =
a
(A.12)
(A.13)
Thus, the expressions (A.1), (A.9), (A.10), (A.12) and (A.13), along with considering the stationary
solution of Eqs. (34) and (35) obtained in Sect 3, completely define the functions cn (t) and ck (t) up
to arbitrary constant phase θn
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