#### Spectral boundary conditions and solitonic solutions in a classical Sellmeier dielectric

Eur. Phys. J. C
Spectral boundary conditions and solitonic solutions in a classical Sellmeier dielectric
F. Belgiorno 1 2 4
S. L. Cacciatori 0 4
A. Viganò 3
0 Department of Science and High Technology, Università dell'Insubria , Via Valleggio 11, 22100 Como , Italy
1 INdAM-GNFM , Rome , Italy
2 Dipartimento di Matematica, Politecnico di Milano , Piazza Leonardo 32, 20133 Milan , Italy
3 Dipartimento di Fisica, Universitá degli Studi di Milano , Via Celoria 16, 20133 Milan , Italy
4 INFN sezione di Milano , Via Celoria 16, 20133 Milan , Italy
Electromagnetic field interactions in a dielectric medium represent a longstanding field of investigation, both at the classical level and at the quantum one. We propose a 1 + 1 dimensional toy-model which consists of an half-line filling dielectric medium, with the aim to set up a simplified situation where technicalities related to gauge invariance and, as a consequence, physics of constrained systems are avoided, and still interesting features appear. In particular, we simulate the electromagnetic field and the polarization field by means of two coupled scalar fields φ , ψ , respectively, in a Hopfield-like model. We find that, in order to obtain a physically meaningful behavior for the model, one has to introduce spectral boundary conditions depending on the particle spectrum one is dealing with. This is the first interesting achievement of our analysis. The second relevant achievement is that, by introducing a nonlinear contribution in the polarization field ψ , with the aim of mimicking a third order nonlinearity in a nonlinear dielectric, we obtain solitonic solutions in the Hopfield model framework, whose classical behavior is analyzed too.
1 Introduction
In the framework of electromagnetic field interactions in
a dielectric medium, both at the classical level and at the
quantum one, a very rich phenomenology appears,
involving several phenomena, from the standard dispersion law to
Hawking-like pair creation. We have developed in our
previous studies an analysis of the Hopfield model, which has
be made relativistically covariant, and suitably extended in
order to keep into account in a semi-phenomenological way
the possibility that e.g. the dielectric susceptibility (and/or the
resonance frequency) depends on spacetime variables, with
the aim of simulating the standard Kerr effect in a nonlinear
dielectric [
1, 2
]. Quantization has been taken into account in
[
1, 3
]. The analysis with scalar models has been developed
in [
2, 4
], with the aim of gain knowledge of the basic physics
at hand without all tricky technicalities which are associated
with gauge invariance. A further step towards a more
complete analysis is contained in [5], where a full four
dimensional electromagnetic field in a nonlinear dielectric medium
and the analogue Hawking effect have been investigated.
As a further contribution to our investigation of the
Hopfield model, we extend our analysis by considering a
dielectric medium which does not fill all the space as in our previous
work. We propose a 1 + 1 dimensional toy-model which
consists of an half-line filling dielectric medium, with the aim
to set up a simplified situation where technicalities related
to gauge invariance and, as a consequence, physics of
constrained systems are avoided, and still interesting features
appear. In particular, we simulate the electromagnetic field
and the polarization field by means of two coupled scalar
fields which are indicated as φ , ψ respectively, in a model
which is inherited by the Hopfield model. The interface
between the vacuum region and the dielectric one is
represented by z = 0, and the dielectric medium fills the region
z ≥ 0. The electromagnetic field φ is involved with both the
vacuum region and the dielectric one, whereas the
polarization ψ is different from zero only in the dielectric region. By
analyzing the particle spectrum of the model we find that,
in order to obtain a physically meaningful behavior, one has
to introduce boundary conditions depending on the particle
spectrum one is dealing with. Indeed, for the electromagnetic
field one finds that smooth solutions with continuous φ , ∂z φ
at the interface z = 0 does not correspond to a complete
scattering basis, due to the presence of a spectral gap (i.e. a gap
in the particle spectrum) associated with the presence of the
dielectric medium. Note that, for simplicity, we are
purposefully dealing with a transparent dielectric medium
(absorption would require further efforts at the quantum level). For
particles in the spectral gap, we have to impose Dirichlet
boundary conditions at the interface, meaning a complete
reflection for the associated electromagnetic modes. This is
the first interesting achievement of our analysis.
We also introduce a nonlinear contribution in the
polarization field ψ , with the aim of mimicking a third order
nonlinearity in a nonlinear dielectric medium. We obtain exact
solutions, which correspond to propagating solitons, and study
their energy propagation both in a global sense (spatial
integrals) and in a local one (Poynting vectors). This is the second
achievement of our analysis.
It is worth mentioning that there exists a huge literature
concerning electromagnetic field in the presence of a
dielectric medium filling an half-space, mainly in a framework
where phenomenological refractive index appears. We limit
ourselves to quoting a classical textbook [
6
], for classical
scattering of light, and the seminal study [
7
], concerning the
effects of spatial dispersion. See also [
8
] on energy
propagation. As to quantization of the 1 + 1 dimensional system,
we refer to [
9, 10
], where a phenomenological approach to
the electromagnetic field in inhomogeneous and dispersive
media is assumed.
2 The half-line filling model
We will consider a 1 + 1 dimensional problem, where a
straight line, parametrized by the coordinate z, is filled by
a dielectric medium for z ≥ 0. The dielectric is described by
a field ψ , and interacts with a “scalar” electromagnetic field
φ. The system is described by the action
S[φ , ψ ] =
R
+
dt
z≥0
1
R 2 ∂μφ∂ μφ dz
21 ψ˙ 2 − ω202 ψ 2 − gφψ˙
where the dot indicates time derivative.
We require for ψ to be smooth in z ≥ 0, to vanish
elsewhere, but we do not add continuity conditions in z = 0. For
φ there is the requirement of smoothness in z ≥ 0, and in
z < 0, and to be of class C 1(R). Our aim is to show that such
boundary conditions are not sufficient in order to get a “good
problem”.
The equations of motion are thus
φ = 0,
ψ = 0, for
z < 0,
φ + gψ˙ = 0,
2
ψ¨ + ω0ψ − gφ˙ = 0, for
z ≥ 0, (2.3)
(2.2)
(2.1)
where
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
with the condition
φ (0+) = φ (0−),
φ (0+) = φ (0−),
where the prime indicates the spatial derivative.
We further require for the energy to be finite, which is
equivalent to the condition
R
[φ˙ 2 + φ 2 + ψ˙ 2 + ω02ψ 2]dz < ∞.
Physically, any initial condition compatible with (2.5) should
be possible.
2.1 General solution and plane wave bases
We can take the Fourier transform of the fields in order to get
the general solution. If we define φ = φ< + φ≥, where
φ<(t , z) = φ (t , z)χ(−∞,0)(z),
φ≥(t , z) = φ (t , z)χ[0,∞)(z),
and χ is the characteristic function, and similar for ψ , then
we get
dk
R 4π ω (k)
c(k)e−iω(k)t + ikz +c(k)∗eiω(k)t−ikz ,
dk
R 4π ωa (k)n(ωa (k))
ba (k)e−iωa (k)t+ikz + ba (k)∗eiωa (k)t−ikz ,
dk ωa2(k) − k2
R 4π ωa (k)n(ωa (k)) i gωa (k)
ba (k)e−iωa (k)t+ikz − ba (k)∗eiωa (k)t−ikz ,
φ<(t , z) =
φ≥(t , z) =
ψ<(t , z) = 0,
ψ≥(t , z) =
a=±
×
a=±
×
ω (k) = |k|,
1
ω±(k) = 2
2 2
n(ω) = 1 + (ω02g−ωω0 2)2 .
k02 = k2
2 2
+ k02g−k0ω02 ,
g2 + (ω0 + |k|)2 ± 21
g2 + (ω0 − |k|)2,
In particular, ω± are the positive branches corresponding to
the dispersion relation
Fig. 1 The Sellmeier
dispersion relation in the lab.
The shaded region evidences the
gap
where k0 is the time component of the two-momentum.
The coefficients c, b± are related by imposing the
boundary conditions for φ on z = 0. Equivalently, we can look for
a basis of plane wave solutions, say, a scattering basis. After
some tedious algebra we get the positive frequency “basis”
(φk , ψk )t defined by
ei|k|t φk (t, z)
= θ (k) eikz
+ θ (−k)
+ θ (k)
× χ[0,∞)(z),
2q
k + q
2k
k + q
ei|k|t ψk (t, z) =
k − q e−ikz
+ k + q
eikz χ(−∞,0)(z)
eiqz + θ (−k) eiqz
k2 − q2
i g|k|
+ θ (−k) eiqz
θ (k)
2k eiqz
k + q
q − k e−iqz
+ k + q
q − k e−iqz
+ k + q
(2.16)
χ[0,∞)(z),
(2.17)
(2.18)
where
q = q(k) = k
2
k2 − g2 − ω0
2
k2 − ω0
is such that ω(k) = ωa (q(k)).
Notice that q(k), and then the scattering basis, is defined
only for |k| < ω0 or |k| > ω02 + g2 ≡ ω¯ . This leads to the
problem being ill defined, which we will now investigate.
3 The spectral boundary conditions
In order to understand the ill definiteness of the problem let
us first discuss the simple origin of the trouble. The point is
where
β( p) =
p + k+ − p − k−
p + k−
p − k+
−1
.
that the modes with ω0 ≤ |k| ≤ ω¯ correspond to the gap in
the dispersion relations in the medium (see the Fig. 1).
Thus, for such modes the relation ω(k) = ωa (q(k))
cannot be satisfied. For these, ba = 0 and the Neuman condition
on z = 0 implies that c(k) = 0 for the modes in the gap.
From the physical point of view this means that modes with
k-vector in the gap cannot propagate in z < 0, which sounds
absurd! As to say that a φ-laser with frequency centered, say,
at ω = (ω0 +ω¯ )/2 cannot work because somewhere far away
(no matter how much) is present a dielectric with a gap in the
spectrum.
From the mathematical point of view, this corresponds to
incompleteness of the scattering basis, because it does not
allow for describing all possible finite energy initial states,
since initial states living in vacuum with modes in the gap
are complementary to the scattering basis, as we will now
argue.
3.1 Incompleteness of the scattering basis
In some sense we can say that the scattering basis is complete
in the right side, inside the matter. Indeed, if we define k± =
ω±( p) p/| p|, for z > 0 we can write
eipzθ(z) = θ(z)θ(− p)β( p)
p + k+ φk+ (0, z) − p − k−
p + k− φk− (0, z)
p − k+
+ θ(z)θ( p)β( p)
2 p 2 p
× p − k+ φk+ (0, z) − p − k− φk− (0, z)
+ φ−k+ (0, z) − φ−k− (0, z)
=: θ(z)φ¯ p(z),
(3.1)
(3.2)
φ¯ p(z) = β( p)
p −2pk+ eik+z + p −2pk− eik−z .
In a similar way we can reproduce the combination
reproducing e−i pz in z > 0. This does not provide an equivalent set
of functions since there remain further possible combination,
which are
This way, the generators ei pz are realized for z > 0. Notice
that for z < 0
where (., .) stands for the standard product in L2. Then we
obtain
×
×
p
k+
p
−
p
k+ φk+ − φk− − φ−k− − φ−k+ = 2θ (−z)
sin(k+z) − i cos(k−z) ,
φk+ − k
φk− + φ−k− + φ−k+ = 2θ (−z)
p
cos(k+z) − i k
−
sin(k−z) ,
which do not provide a complete set of solutions since exactly
the modes with vector k in the gap are absent. We have looked
at the field φ only, since for ψ we can add arbitrary k modes
with frequency ω0 so that there are no problems of
completeness.
Thus we see that in order to have a complete set of
solutions we should add those modes which vanish in z > 0 and
have the spectral parameter k in the gap ω0 ≤ |k| ≤ ω¯ .
The completion of the basis is obtained by adding the gap
modes
φg,k (z) = c(k) sin(kz)θ (−z).
Now we can construct the projection operators in order to
specify the spectral boundary conditions.
3.2 Inner product, Hamiltonian and boundary conditions
Let us define
χ> = χ[0,∞)(z),
χ< = χ(−∞,0)(z).
We also define the symplectic matrix
:=
02×2 −i I2×2
i I2×2 02×2
,
the multicomponent field
⎛ φ ⎞
ψ
:= ⎜⎜ πφ ⎟⎠⎟ ,
⎝
πψ
and also the inner product
<
1, 2 > = ( 1,
2),
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
<
×
i
1, 2 > = − 2
dz
φ1∗←→∂0φ2 + (ψ1∗←→∂0ψ2 + g(ψ1∗φ2 − φ1∗ψ2))χ> .
In order to better understand the problem of completeness and
of the boundary conditions, we introduce the Hamiltonian
operator H such that the equations of motion are written in
a Hamiltonian form:
⎛ φ ⎞ ⎛ πφ ⎞
∂0 ⎜⎜ πφψχ> ⎟⎟ = ⎜⎜⎝ ∂z2φ −(πgψπψ+χg>φ−)χg>2φχ> ⎟⎠⎟ ⇐⇒ ∂0 = H ψ,
⎝ πψ χ> ⎠ −ω02ψχ>
where
⎡ 0
gχ>
H := ⎢⎢⎣ ∂z2 − g2χ>
0
is formally selfadjoint with respect to the inner product <
., . > defined above. In order to verify this, we must integrate
by part the derivative contributions appearing in the operator
Hˆ . So doing, we discover that in z = 0 some boundary
terms appear, which are a priori possible hindrances to the
hermiticity of the operator itself. These boundary terms are
of the form
φ (0+)∂z φ (0+) − φ (0−)∂z φ (0−);
we can get rid of them in three ways: (a) we can impose the
continuity of the field and of its derivative at z = 0, as in
the case of a standard problem of scattering in the presence
of a step-like potential barrier; (b) we can impose Dirichlet
boundary conditions at z = 0; (c) we can apply Neumann
boundary conditions at z = 0. We point out that our function
space, endowed with the aforementioned inner product, is a
Krein space (negative norm states, which amount to
antiparticles in a quantum field theory framework, appear).
Selfadjointness means in this case that
Hc := ( Hˆ )†
coincides with Hˆ . We note that the spectrum of the
Hamiltonian operator coincides with the one-particle frequencies
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
(3.16)
ω, ωα which were discussed in the previous section. Note
also that such frequencies are conserved, as separation of
variables easily shows.
In order to judge about the selfadjointness problem, we
could proceed as follows: let us consider eigenstates =
exp(−i ωt ) f (z), with the spatial part f (z) smooth with
compact support. This requirement is such that boundary terms
immediately disappear, as in the case (a) above, but there
remains a problem. Indeed, such a choice of functional space
implies that the fields and their partial derivatives with respect
to z are continuous in z = 0, and this requirement eliminates
the boundary terms. Still, there is an unsatisfactory property
from a physical point of view, i.e. the electric field would
vanish for all the frequencies belonging to the mass gap,
which is not a physically acceptable property for what was
discussed previously. Then we must provide a further
specification for the physical domain of Hˆ . If we require that
Dirichlet boundary conditions are satisfied at z = 0 for all
the frequencies in the mass gap of our dispersion law, then
we obtain a satisfactory behavior for our operator.
As it is a selfadjoint operator, its eigenfunctions are
orthogonal for ω = ω . We can define a projector Pgap which
is such that the boundary conditions become
Bφ := Pgapφ|z=0 = 0.
(3.17)
With these boundary conditions one takes into account
properly the fact that, in the mass gap, z = 0 becomes a sort of
infinite barrier which expels the field from inside the
dielectric medium because the Sellmeier relation displays a
massgap region in the spectrum. This unusual feature is due to
the transparency of the medium, which simplifies greatly the
quantum version of the model but does not allow one to obtain
a completely satisfactory model. Notice that continuity of the
field and of its z-derivative is not a boundary condition, but
takes into account the finiteness of the barrier for all
frequencies outside the mass gap (the electric field can live both
inside and outside). Our analysis for the Hopfield model is
in agreement with the short discussion appearing in [
7
] and
concerning the exciton behavior in the case of absence of
spatial dispersion (see Sect. 2 therein).
It is also remarkable that a spectral boundary condition of
the type
Bs φ := Pgapφ|z=0 ⊕ ∂z (1 − Pgap)φ|z=0 = 0,
(3.18)
would produce unphysical results, as it would imply no
transmission in the scattering basis for the field φ.
4 Solitonic solution
In this section, we introduce a nonlinear term in the
polarization field ψ and look for solitonic solutions of the field
equations. We recall that solitons in Kerr dielectric media
are usually derived in the framework of the so-called
nonlinear Schroedinger equation (NLS). See e.g. [
11
] for NLS, and
[
12
] for solitonic solutions in fiber optics. See also [
13
] for
further discussion.
We look for a static solution in the dielectric, in the
comoving frame; we rewrite the Hamiltonian action (2.1) in a
covariant form, and add a self interaction ψ 4 term simulating the
Kerr effect, i.e. generating a dielectric perturbation
moving with substantially constant velocity in the bulk dielectric
medium:
1
R 2 ∂μφ∂μφ dz
S[φ, ψ ] =
R
dt
+
(vμ∂μψ )2 − ω202 ψ 2 − gφvμ∂μψ − 4λ ψ 4 dz
!
Note that the static solution exists if and only if b > 0, so
v > ω0/g.
All of this is true in the comoving frame; in order to obtain
solutions in the lab frame, we boost our functions by z →
γ (z − V t ), so
a
ψ (t, z) = cosh(bγ (z − V t )) ,
φ (t, z) =
arctan tanh
γ (z − V t )
Note that (2.1) is obtained by means of vμ = (1, 0) and
λ = 0.
We set v = γ V for the velocity of the dielectric
perturbation on the z-axis, and imposing the staticity φ (t, z) ≡ φ (z),
ψ (t, z) ≡ ψ (z), we obtain
a
ψ (z) = cosh(bz) ,
φ (z) =
where
a :=
1
b := v
This is our solution in the dielectric (z > 0). It is useful to
provide also for z > 0
φ (t, z) = agvγ
1
cosh(bγ (z − V t ))
Now, we want to glue this solution with that in the vacuum;
since the dielectric is moving, the field in vacuum is
timedependent, so the general solution to φ = 0 is
φ (t, z) = α(z − t ) + β(z + t ),
while ψ = 0 in z < 0.
We have to impose the gluing conditions at z = 0 for
every time t : we want the continuity of the field φ and of its
normal derivative ∂nφ (which is equivalent, in the lab frame,
to ∂z φ).
We obtain
aγ gv(1 + V )
α (z − t ) = 2 cosh(bγ V (z − t )) ,
aγ gv(1 − V )
β (z + t ) = 2 cosh(bγ V (z + t )) .
We can interpret our solution in this way: we have a
progressive wave α(z − t ) that hits upon the dielectric. After
the collision, we will have a reflected wave β(z + t ) and a
transmitted wave (4.7), plus the polarization field (4.6).
In order to confirm our physical interpretation, we are
going to calculate the total energy of the system.
4.1 Solitonic energy
The Hamiltonian of the theory (in lab frame) is
H = 21 φ˙ 2 + φ
2
+ ψ˙ 2 + ω02ψ 2
+ 4λ! ψ 4;
we split our calculus in vacuum and dielectric parts: we start
with the vacuum part.
In vacuum, the Hamiltonian is reduced to (ψ = 0)
HV = 21 φ˙ 2 + φ
2
=
a2γ 2g2v2
4
(1 + V )2 (1 − V )2
× cosh2(bγ V (z − t )) + cosh2(bγ V (z + t )) ;
(4.13)
therefore, the total energy in z < 0 is
0
−∞
EV =
dz HV =
a2γ g2v2
2bV
(1+V 2)−2V tanh(bγ V t ) .
(4.14)
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
We note immediately that EV is strictly positive, and that is
a decreasing function of t , as we expected.
The Hamiltonian in the dielectric is equivalent to (4.12),
and explicitly
(4.15)
E D is an increasing function, and it is positive too.
This calculus confirms our physical picture: we have an
energy flux from z → −∞ to z → +∞, because energy
in vacuum decreases in time, while the dielectric is
progressively filled and its energy increases.
It is worthwhile noting that energy in vacuum does not
converge to zero, because there exists a reflected wave β(z +
t ). At t → +∞, both EV and E D become constant, and this
situation corresponds to a “fulfilled” system.
Now, we obtain the total amount of energy: since
action (4.1) is invariant under temporal translation, total
energy is a Noether charge, so we expect that it is
timeindependent. Indeed
Etot = EV + E D =
a2γ g2v2
2bV
(1 + V )2;
(4.17)
as expected it is time-independent, and Etot > 0.
It is interesting to observe that our solitonic solution fulfills
four properties which are associated as ‘definitory
properties’ to solitons in [
14
]: (1) finite total energy; (2) finite,
non-singular and localized energy density, where
localization means that at any time t the region where H ≥ δ, for
any δ ∈ (0, maxz H), is bounded; (3) the solitonic solution is
non-singular; (4) the solitonic solution is non-dissipative (in
the sense that maxz H does not vanishes as t → ∞). There
is also a fifth (and last) property, i.e. classical stability in the
sense of Lyapunov, which is shown to be implemented in the
following section.
4.2 Transmission and reflection coefficients
The naive expectation for the process at hand is the following:
one would expect that the scattering of the solitonic wave
starts from the left, z 0, at very early times, t → −∞,
with a progressive wave moving towards the interface z = 0,
and that at very far times in the future, t → +∞, one gets two
separate packets, the first counter-propagating in the vacuum
region z 0 (reflected wave) and the second one progressing
in the dielectric medium z 0 with velocity V (transmitted
wave). We show that this is the case, and we provide the
reflection and the transmission coefficients.
Usually, one could approach the problem by using the
component Jz of the current density. In the present case, this
is not possible because there is no charge displacement in the
medium and then Jz = 0. We can then approach the problem
by recalling that the canonical stress-energy tensor can be
calculated, and in particular the component T0z represents
the flux of energy through a surface orthogonal to the
zdirection. To be more specific, we calculate T0z for the only
field which propagates in our system: the ‘electromagnetic
field’ φ.1 We get
T μν
∂ L
= ∂ ∂μφ
∂ ν φ − ημν L ,
and, in particular, we are interested in
T 0z = (∂ 0φ)(∂ z φ) =: Sz ,
where Sz is the equivalent of the Poynting vector for the
electromagnetic field. Then we get
(4.18)
(4.19)
(4.20)
(4.21)
(4.22)
(4.23)
Sz = −φ˙ φ .
Sz = (α )2 − (β )2.
Let us consider the vacuum region z < 0:
In the dielectric region z ≥ 0 we obtain
Sz = V (φD )2,
where we indicate with φD the field φ in the dielectric region.
Let us consider the field at t 0 and for z 0. In order
to get a field which is appreciably different from zero, we
should impose t ∼ z, in such a way to obtain the peak value
for α , whereas β ∼ 0 and φD = 0. In such a situation, we
have
α (z − t ) ∼
aγ gv(1 + V )
2
This situation represents the initial pulse which moves
towards z = 0 from the left. It is also useful to note that
for t 0 and z > 0 one has φD (z + V |t |) ∼ 0.
1 Indeed, there is no real propagation of the polarization field ψ , which
is present only in the z ≥ 0 region and is in some sense pathological:
it represents fixed dipoles oscillating around a fixed position in space,
with a given frequency ω0, and exists just in the dielectric medium.
Then it cannot be involved in energy transport. It can be noted that its
contribution to the stress-energy tensor would be non-symmetric, even
if it is just a scalar field. A Belinfante–Rosenfeld procedure could be
taken into account, or even an Abraham-like tensor could be set up for
the polarization part. Still, a scattering picture would be meaningless,
as, by definition, ψ is just present for z ≥ 0.
Let us now consider t 0, i.e. the final state. In the region
z 0, we get a field appreciably different from zero only
for t ∼ −z, i.e. we get a reflected packet with
β (z + t ) ∼
aγ gv(1 − V )
2
In the dielectric region, we obtain a non-vanishing
contribution only for t ∼ z/ V , which correspond to the peak of
the transmitted packet. Summarizing, we have the Poynting
vector for the initial packet, for the reflected one and for the
transmitted one, respectively:
It is interesting to point out that the Poynting vector Sz is
continuous at the surface z = 0, as would be expected for
the Poynting vector in the full electromagnetic case:
(α )2(t ) − (β )2(t ) = V (φD )2(t ).
Indeed, one obtains
a2γ 2g2v2(1 + V )2
4 cosh2(bγ V t )
−
a2γ 2g2v2(1 − V )2
4 cosh2(bγ V t )
a2γ 2g2v2 V
= cosh2(bγ V t )
(4.32)
.
Sz (t
Sz (t
R : =
T : =
Stability of solitons is a nontrivial problem, addressing which
one has to face in dimension greater than two with a strong
no-go theorem due to Hobart and Derrick. A subtle
distinction between absolute stability and stability in the sense of
Lyapunov has to be taken into account, as discussed e.g. in
[
15, 16
]. We consider for simplicity the case of an infinite
dielectric, but extensions to our previous framework are
possible (see below). In our analysis, we follow the ideas
contained in [16], and we are able to infer the stability of our
soliton solution.
The starting point consists in writing the Hamiltonian
operator as a function of the field momenta and the fields
themselves:
H = H (πφ , πψ , φ , ψ ).
Then one has to consider an expansion of H up to the second
order in the field and momenta variations δπφ , δπψ , δφ , δψ
around the soliton solution, taking into account that the first
order contribution vanishes (as the soliton solution is ‘on
shell’, i.e. satisfies the Hamiltonian equations of motion). By
defining
δ P =
δ Q =
δπφ
δπψ
δφ
δψ
,
,
and indicating by δ P t , δ Qt the transposed vectors, we get
1
H (πφ , πψ , φ , ψ ) = H0 + 2 δ P t T δ P
1
+ 2 δ Qt V δ Q + δ P G δ Q + · · · ,
where H0 is the zeroth-order contribution associated with the
solitonic solutions, and where higher order contributions are
neglected. For simplicity, we work in the lab frame.
Explicitly, we get
(5.1)
(5.2)
(5.3)
(5.5)
(5.6)
(5.7)
(5.8)
T =
G =
and
V =
where ψ0 corresponds to the solitonic solution. Then one
finds a second order operator,
where Gt stays for the transposed matrix. It can be noted that,
having G = 0, in our model appear gyroscopic-like
contributions, which require that stability in the Lyapunov sense
is shifted from the requirement of minimality of the energy
e−iωt X (z)
det(K − ω
) = 0
K X = ω
where X (z) is the suitable vector function depending only
on z, is stable in the sense of Lyapunov if the equation
admits only real solutions ω ∈ R. Note that the symplectic
eigenvector X satisfies
In the present case, by using with some ingenuity the rule
det A = det A11 det( A22 − A21 A1−11 A12),
which holds for a block square matrix A with equal square
blocks A11, A12, A21, A22,
functional, to more general conditions [
16
]. According to the
analysis in [
16
], by introducing the symplectic matrix (3.8),
a stationary solution
(5.9)
(5.10)
(5.11)
(5.12)
(5.13)
= 0.
(5.14)
(5.15)
(5.16)
(5.17)
A =
A11 A12
A21 A22
,
(5.4)
one obtains
det(K − ω
) = det
−∂z2 − ω2
i ωg
−i ωg
ω02 − ω2 + λ2 ψ02
We can observe that momentarily neglecting the
contribution associated with ψ0, one in essence obtains the
dispersion relation in the dielectric medium, as is easy to realize
by means of a Fourier analysis of the above operator. We
know that ω is real when ω2 is outside the forbidden interval
(ω02, ω02 + g2), which corresponds to the well-known mass
gap in the Sellmeier dispersion relation delimiting the reality
of the refractive index in optics. We have also to keep into
account that the ψ0 contribution, for positive λ, is a positive
and bounded operator which represents a small perturbation
with respect to the (unbounded) operator ω2
0 −ω2. Such a
perturbation is substantially not able to perturb in any sensible
way the reality of ω, in the sense that if we define
s2 := ω02 + λ2 sup(ψ02),
ω2 < ω02,
ω2 > s2 + g2,
then stability in the above sense is ensured as far as
which correspond to the stability conditions for the case at
hand.
In the case of semi-infinite dielectric medium, one finds
that the Hamiltonian H is split into two expressions, one for
the vacuum region and the other for the dielectric region.
As a consequence, the operator K above is split into two
expressions too:
K =
where K> is formally the same as in the infinite dielectric
medium discussed above, and K< is a free field contribution
associated only with πφ , φ (as the polarization field
contribution vanishes). The latter contribution does not affect the
stability properties discussed above.
6 Conclusions
In the framework of a 1 + 1 dimensional toy-model which
consists of an half-line filling dielectric medium,
simulating the electromagnetic field and the polarization field by
means of two coupled scalar fields φ, ψ , respectively, in a
Hopfield-like model, we have achieved substantially two
relevant results. The first one is that boundary conditions
associated with the spectrum of quasi-particles (polaritons) are
necessary in order to match with meaningful physics. Indeed,
in order to avoid an unphysical behavior in the case of the
‘mass gap’, one is forced to impose reflection at z = 0 for the
electromagnetic field φω when ω falls in the mass gap. This is
a somewhat unexpected condition, which can be related to the
requirement of transparency of the dielectric medium. The
second result is that, in a nonlinear model, solitonic solutions
exist, whose behavior has been studied. All relevant
properties of solitons are shown to be fulfilled. Indeed, we have
considered the energy behavior both in a global sense, by
studying how the energy changes with time in the different
parts of our setting, and in a local sense, by means of the
analysis, based on T0z , which represents the flux of energy
through a surface orthogonal to the z-direction. We have also
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