ˇCeské Vysoké Ucen´ı Technické Teze k disertacn´ı práci

České Vysoké Učenı́ Technické
Teze k disertačnı́ práci
České vysoké učenı́ technické v Praze
Fakulta jaderná a fyzikálně inženýrská
Katedra fyziky
Jan Smotlacha
Numerická řešenı́ modelů v kvantové mechanice
Doktorský studijnı́ program: Aplikace přı́rodnı́ch věd
Studijnı́ obor: matematická fyzika
Teze disertace k zı́skánı́ akademického titulu ”doktor”, ve zkratce ”Ph.D.”
Praha, Únor 2012
Disertačnı́ práce byla vypracována v kombinované formě doktorského studia na katedře fyziky
Fakulty jaderné a fyzikálně inženýrské ČVUT v Praze.
Uchazeč: Ing. Jan Smotlacha
katedra fyziky
Fakulta jaderná a fyzikálně inřenýrská ČVUT
Břehová 7, Praha 1
Školitel: Doc. Ing. Goce Chadzitaskos, CSc.
katedra fyziky
Fakulta jaderná a fyzikálně inřenýrská ČVUT
Břehová 7, Praha 1
Oponenti: Prof. Anatol Odzijewicz
Wydzial Fizyki, Uniwersytet w Bialymstoku, Poland
Prof. Vladimir Andrejevič Osipov
Laboratorija těoretičeskoj fiziki im. N. N. Bogoljubova, OIJAI, Dubna, Russia
Teze byly rozeslány dne:..............................................
Obhajoba disertace se koná dne .............................................. v .............. hod. před komisı́ pro obhajobu disertačnı́ práce ve studijnı́m oboru matematická fyzika v zasedacı́ mı́stnosti č. .......... Fakulty
jaderné a fyzikálně inženýrské ČVUT v Praze.
S disertacı́ je možno se seznámit na děkanátě Fakulty jaderné a fyzikálně inženýrské ČVUT v Praze,
na oddělenı́ pro vědeckou a výzkumnou činnost, Břehová 7, Praha 1.
............................................
Prof. Ing. Miloslav Havlı́ček, DrSc.
předseda komise pro obhajobu disertačnı́ práce
ve studijnı́m oboru
matematická fyzika
Fakulta jaderná a fyzikálně inženýrská ČVUT, Břehová 7, Praha 1
Abstract
Thesis is divided into two parts, each of them deals with an application of quantum theory on
different problems.
First part is based on publications:
J. Smotlacha, R. Pincak, M. Pudlak, Electronic Structure of Disclinated Graphene in a Uniform
Magnetic Field, Eur. Phys. J. B 84, 255 (2011) [1],
R. Pincak, J. Smotlacha, M. Pudlak, Electronic properties of disclinated nanostructured cylinder,
prepaired to be published in European Journal of Physics [2],
R. Pincak, M. Pudlak, J. Smotlacha, chapter with title Electronic properties of single and double wall
carbon nanotubes for the book ”Carbon Nanotubes: Synthesis, Properties and Applications”, Publisher Nova Sciences, USA, Edited by Prof. Ajay Kumar Mishra, accepted for publication January
2012, in print [3].
In this review, we will speak about the contents of the first two publications. There the electronic properties of the chosen kinds of carbon nanostructures are described: the disclinated graphene
with the hyperboloidal structure and the nanocylinder with and without a defect. The electronic
properties are characterized by the local density of states (LDoS). For the calculation of LDoS, we
use the theoretical background coming from the effective mass theory and tight-binding approximation [4, 5]. The use of incorporated gauge fields on the disclinated surface is described in [6]. The
investigated problem of the electronic structure in close vicinity of the disclination took part in [7],
where the conical geometry was used. But as it was shown in [8], the hyperboloidal geometry is more
appropriate.
Second part is based on publication:
J. Smotlacha, G. Chadzitaskos, C. Daskaloyannis, Spectra of Photon Down Conversion, Proceedings
of the XXVIII. Workshop on Geometric Methods in Physics, Bialowieza (2009) (see [1] in Part II).
The spectra of down conversion and the construction of coherent states for this process are presented here. Down conversion is considered to be the simplest nonlinear model producing entangled
photons ([2] in Part II). In the beginning, the Hamiltonian of the investigated deformed parafermionic
oscillator is introduced. Then, the corresponding algebra and its Fock-like representation is derived
with the help of the operators commuting with the Hamiltonian and satisfying given commutation
relations. We calculate the energy spectrum using the method of orthogonal polynomials discussed
in [3, 4]-Part II and used in [5]-Part II. Then the coherent states are constructed. Needs for using the
para-Grassmann variables emerges. The method of construction of coherent states of su(2) deformed
Lie algebra is used.
Obsah
I
Electronic Properties of Carbon Nanostructures
1
1 Introduction
1
2 Electronic Structure of Disclinated Graphene
2.1 Heptagonal defects . . . . . . . . . . . . . . .
2.2 Pentagonal defects . . . . . . . . . . . . . . .
2.3 Local density of states . . . . . . . . . . . . .
in a
. . .
. . .
. . .
Uniform Magnetic
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
Field
. . . . . . .
. . . . . . .
. . . . . . .
3
3
5
6
3 Electronic Properties of Disclinated Nanostructured Cylinder
3.1 Defect-free cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Perturbed cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
8
9
4 Conclusion
11
References
11
II
13
Quantum Optics: Spectrum and Coherent States
1 Three Boson Interaction Process
1.1 Introduction . . . . . . . . . . . . . . .
1.2 Deformed Parafermionic Algebra of the
1.3 Calculation of the eigenvalues . . . . .
1.4 Coherent states . . . . . . . . . . . . .
. . . .
Model
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13
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16
References
20
A List of author’s papers
22
B List of author’s conference contributions
22
Část I
Electronic Properties of Carbon
Nanostructures
1
Introduction
The carbon nanostructures play a key role in constructing nanoscale devices like quantum wires,
nonlinear electronic elements, transistors, molecular memory devices or electron field emitters. Their
molecules are variously-shaped geometrical forms its surface is composed of disclinated hexagonal
carbon lattice. The term ”disclinated lattice”means that in the lattice, which was initially constituted by carbon atoms forming the hexagonal structure, are places where the carbon atoms form
other kinds of polygons than hexagons. We call these places topological defects. The main structure
of this kind is graphene - the completely hexagonal carbon plane lattice from which all other kinds
of nanostructures are derived.
The topological defects are most often presented by the pentagons and by the heptagons. In the
closest vicinity of these defects, the positive resp. the negative curvature arises for the case of the
pentagons or the heptagons, respectively. Generally, for the n-sided polygon, n < 6 corresponds to
the positive curvature and n > 6 corresponds to the negative curvature.
To calculate the (LDoS), knowledge of the solution of the corresponding Dirac equation is necessary. This solution is represented by the wave-function and to find it, we have to know the geometry
of the molecular surface. As discovered in [8], the most suitable geometry for the description of the
close vicinity of the defects is the hyperboloidal geometry. Very often, for a given geometry, the
number of possible defects is limited.
The solutions for spherical, conical and 2-fold-hyperboloidal cases were found in [9, 7, 8]. In this
paper, we use the presented model for calculation of the electronic properties of the structures with
the geometry of the 1-fold hyperboloid and of a defect-free and a perturbed nanocylinder including
2 heptagons at the opposite sides of the surface. For a higher number of defects, next investigation
of the possible geometry would have to be performed.
In the Dirac equation, the disclination is represented by a SO(2) gauge vortex and the corresponding metrics follows from the elasticity properties of the graphene membrane. To enhance the
interval of energies, different methods are used: a self-consistent perturbation scheme is used for the
1
case of the vicinity of the 1-defects and an analytical solution is found for a first order expansion
of the metric coefficients for the case of the perturbed nanocylinder. Furthermore, the Landau states are investigated and a discussion on the influence of the Zeeman effect is included for the first case.
To research the electronic properties, we have to solve the Dirac equation in (2+1) dimensions.
It means the we solve the Dirac equation on a 2D surface which is immersed into the third dimension. The metric gµν of the 2D surface comes from following parametrisation, with the help of
two parameters ξ, ϕ:
→
−
(ξ, ϕ) → R = (x(ξ, ϕ), y(ξ, ϕ), z(ξ, ϕ)),
(1.1)
where
0 < ξ < ∞,
0 ≤ ϕ < 2π.
(1.2)
The Dirac equation has the form
iσ α eµα [∂µ + Ωµ − iaµ − aW
µ − iAµ ]ψ = Eψ,
where ∂µ means the partial derivation according to the µ parameter, i.e. ∂µ =
(1.3)
∂
.
∂xµ
In this equation, besides the energy E, the particular constituents have the following sense:
σ α , α = 1, 2, have the same purpose as the Dirac γ matrices for the 4-component Dirac equation.
The symbol eµα , µ = ξ, ϕ, denotes the zweibein. It must satisfy [10]
∂µ eαν − Γρµν eαρ + (ωµ )αβ eβν = 0,
(1.4)
where Γρµν is the Levi-Civita connection and ωµ is called the spin connection. This term is present in
(1.3) by
1
Ωµ = ωµαβ [σα , σβ ]
(1.5)
8
which is the spin connection in the spinor representation. The gauge fields aW
µ , aµ stand to ensure
the wave function ψ be a smooth function for the cases of rotationally symmetric surface without
and with defects. The component Aµ denotes the magnetic field.
Because of the rotational symmetry used in this paper, the only nonzero components of the gauge
fields Ωµ , aW
µ , aµ and Aµ are the ϕ components.
The wave function ψ is composed of two parts:
ψA
,
(1.6)
ψ=
ψB
each corresponding to different sublattices of the curved graphene sheet. This means that the graphene plane can be divided into two sublattices of carbon atoms. The sublattices are labeled by A
−→
and B and they are shifted by the vector BA. Every atom of the graphene plane has three nearest
2
Obrázek 1: The decomposition of the graphene plane into sublattices A and B
neighbours, each of them being involved in the opposite sublattice. So, a concrete sublattice consists
of next-neighbouring atoms (see Fig. 1).
To solve the Dirac equation, we write the wave function in the form
1
u(ξ)eiϕj
ψA
,
j = 0, ±1, ...
= √
4 g
ψB
v(ξ)eiϕ(j+1)
ϕϕ
(1.7)
and substituting (1.7) into (1.3) we obtain
e
∂ξ u
j
−√
u = Ev,
√
gξξ
gϕϕ
where
and gξξ , gϕϕ are the metric coefficients.
2
2.1
e
∂ξ v
j
−√
−√
v = Eu,
gξξ
gϕϕ
e
j = j + 1/2 − aϕ
(1.8)
(1.9)
Electronic Structure of Disclinated Graphene in a Uniform Magnetic Field
Heptagonal defects
3
In the case of negative curvature and associated heptagonal defects, the parametrisation (1.1) for
the case of the hyperboloid is:
(ξ, ϕ) → (a cosh ξ cos ϕ, a cosh ξ sin ϕ, c sinh ξ),
(2.1)
where a and c are some dimensionless parameters. The corresponding diagonal components of the
metric are:
gξξ = a2 sinh2 ξ + c2 cosh2 ξ,
gϕϕ = a2 cosh2 ξ
(2.2)
The form of (1.8) will be:
q
√
2
e
e
∂ξ u − (j − Φ cosh ξ) tanh2 ξ + ηu = E gξξ v,
where:
(2.3)
q
e cosh2 ξ) tanh2 ξ + ηv = E √gξξ u,
−∂ξ v − (e
j−Φ
(2.4)
e = Φ/α,
Φ
η = c2 /a2 .
→
−
If the magnetic field is chosen in such a way that A = B(y, −x, 0)/2, then:
e
j = (j + 1/2 − aϕ )/α,
Aϕ = −Φ cosh2 ξ,
Aξ = 0,
where:
1
Φ = a2 Φ0 B,
2
The geometric units are used, i.e. e = ~ = c = 1.
(2.5)
Φ0 =
(2.6)
e
.
~c
(2.7)
The parameter η ≪ 1 is a dimensionless parameter which describes the elasticity properties of the
initial graphene plane. Due to these properties, the defects can be interpreted as small perturbations
√
in the graphene plane. In the case of finite elasticity, we can use an approximation η ∼ νǫ, where
ǫ ≤ 0.1 [8]. In this way, the elasticity is described by a small parameter ǫ. Its value is usually taken
between 0.01 and 0.1. If we perform some necessary corrections to the gauge field ωµ , then as we
take ǫ → 0 we obtain the metric of the cone. For a closer description of the form of the particular
variables, see [1].
Let us now suppose E = 0. This energy corresponds to the so-called zero-energy mode which is
appropriate for the electron states at the Fermi level. Then, the solution of (2.3),(2.4) is:
!
ej
e
e
Φ
Φ△(ξ)
sinh ξ
cosh ξ
ke
j− η2k
,
(2.8)
exp −
u0 (ξ) = C(△(ξ) + k sinh ξ)
△(ξ) + sinh ξ
2
v0 (ξ) = C ′ (△(ξ) + k sinh ξ)
e
Φ
−ke
j+ η2k
cosh ξ
△(ξ) + sinh ξ
4
−ej
exp
e
Φ△(ξ)
sinh ξ
2
!
,
(2.9)
where:
p
k = 1 + η,
q
△(ξ) = k 2 cosh2 ξ − 1,
(2.10)
and C, C ′ are the normalisation constants.
For nonzero values of E a perturbation scheme is developed which enables to find the solution
numerically. The solution has the form [11]:
u(ξ) = u0 (ξ)U(ξ),
v(ξ) = v0 (ξ)V(ξ),
(2.11)
where U(ξ), V(ξ) form the perturbed part. Then, for a given ξ0 , the local density of states is defined
as:
LDoS(E) = u2 (E, ξ0 ) + v 2 (E, ξ0 ).
(2.12)
To evaluate the local density of states, we have to calculate the normalisation constants C,C ′ . They
differ for different values of E. For unnormalised solutions u′ (ξ), v ′(ξ) of (2.3),(2.4) and each value of
energy:
ξZmax
(u′(ξ)2 + v ′ (ξ)2 )dξ.
(2.13)
1/C 2 = 1/C ′2 =
0
Since for ξmax = ∞ the integral diverges, some finite value of ξmax in some interval which is of
particular interest is needed. In this work, we take ξmax = 2.5 and ξmax = 2. It follows from the
parametrisation (2.1) that for the given values of ξ, the corresponding distance is r = a cosh ξ, which
means that for a = 1 Å we have rmax = 6.13 Å, or rmax = 3.76 Å. These values are of the same
order as the size of the Brillouin zone which is formed by the single hexagons. Each atom in the
hexagon lies at a distance 1.42 Å from its nearest neighbours [4, 12]. This is the main principle of the
tight-binding approximation [13] in which we only account for the influence of the nearest neighbours.
2.2
Pentagonal defects
The case of positive curvature is described in more detail in [8]. The parametrisation is changed
into:
(ξ, ϕ) → (a sinh ξ cos ϕ, a sinh ξ sin ϕ, c cosh ξ),
(2.14)
and the diagonal components of the metric are:
gξξ = a2 cosh2 ξ + c2 sinh2 ξ,
gϕϕ = a2 sinh2 ξ.
(2.15)
The form of (1.8) is:
q
√
2
e
e
∂ξ u − (j − Φ sinh ξ) coth2 ξ + ηu = E gξξ v,
5
(2.16)
q
√
2
e
e
−∂ξ v − (j − Φ sinh ξ) coth2 ξ + ηv = E gξξ u.
(2.17)
In the case E = 0, the corresponding solution is:
u0 (ξ) = C(△(ξ) + k cosh ξ)
v0 (ξ) = C ′ (△(ξ) + k cosh ξ)
where:
e
Φ
ke
j+ η2k
e
Φ
−ke
j− η2k
p
k = 1 + η,
sinh ξ
△(ξ) + cosh ξ
sinh ξ
△(ξ) + cosh ξ
△(ξ) =
The magnetic field has the form
Aϕ = −Φ sinh2 ξ,
q
ej
−ej
e
Φ△(ξ)
cosh ξ
exp −
2
exp
e
Φ△(ξ)
cosh ξ
2
!
!
,
,
(2.18)
(2.19)
k 2 sinh2 ξ + 1.
(2.20)
Aξ = 0,
(2.21)
where Φ is defined as in (2.7).
To calculate the solution for nonzero values of E and the local density of states we use the same
procedure as presented for the heptagonal defects.
2.3
Local density of states
In Figs. 2 and 3, the LDoS as a function of energy E and the parameter ξ is presented for
hyperboloidal surfaces with the defects formed by 1 polygon. In all of these figures, we set j = 0 in
(2.5) and ǫ = 0.01 in the expression for η. We can see the evidence that for increasing B or ξmax ,
the LDoS is decreasing and the decrease is faster for the pentagonal defects. If we took larger ξmax ,
the LDoS would go to zero with the exception of a small number of energies for which we would
obtain plane waves. The larger values of ξ are, however, unphysical because of the limited interval of
validity of the tight-binding approximation.
3
Electronic Properties of Disclinated Nanostructured Cylinder
6
Obrázek 2: LDoS as a function of E ∈ (−0.5, 0.5) and ξ ∈ (0, 2.5) for 1-heptagon defects (left part)
and 1-pentagon defects (right part) for B = 0; ǫ = 0.01.
Obrázek 3: LDoS as a function of E ∈ (−0.5, 0.5) and ξ ∈ (0, 2.5) for 1-heptagon defects (left part)
and 1-pentagon defects (right part) for B = 0.5; ǫ = 0.01.
7
Obrázek 4: Nanocylinder with a small perturbation
3.1
Defect-free cylinder
The defect-free cylinder can be described with the help of the parametrization
→
−
R (z, ϕ) = (a cos ϕ, a sin ϕ, z),
(3.1)
where a is the radius. Compared to (1.1), here we use z instead of ξ. But the procedure of calculation
will be similar.
The solution has the form
u(z) = C1 exp(αz) + C2 exp(−αz),
C1
v(z) =
E
where
e
j
α−
a
!
C2
exp(αz) −
E
α = α(E) =
s
e
j
α+
a
!
(3.2)
exp(−αz),
e
j2
− E2
a2
(3.3)
(3.4)
and C1 (E), C2 (E) are calculated from the normalization condition
zZmax
(|u(E, z)|2 + |v(E, z)|2 )dzdϕ = 4π
−zmax
zZmax
0
(|u(E, z)|2 + |v(E, z)|2 )dz = 1,
(3.5)
where zmax is the length of the cylinder. In Fig. 5 LDoS is plotted as a function of 2 variables, E and z.
8
Obrázek 5: LDoS as a function of E ∈ (−1, 1) and z ∈ (0, 100) for defect-free cylinder
3.2
Perturbed cylinder
In the case of a small perturbation (see Fig. 4), we have
p
p
→
−
R (z, ϕ) = a 1 + △z 2 cos ϕ, a 1 + △z 2 sin ϕ, z ,
(3.6)
where △ is a positive real parameter, △ << 1. For △ = 0, we get the defect-free cylinder discussed
in previous chapter.
Then we solve the system of equations
where
p
∂z u
1 + a2 △2 z 2
−
e
j
u
p
= Ev,
a 1 + △z 2
−p
∂z v
1 + a2 △2 z 2
−
e
j
v
p
= Eu,
a 1 + △z 2
(3.7)
N
1
e
(3.8)
j = j + 1/2 + (2m + n) − ,
3
4
with (n, m) being so-called chiral vector, i.e. we consider the perturbation to be caused by N heptagonal defects. For small △ and neglecting the second order of △, it can be simplified as
e
e
1 2
1 2
j
j
1 − △z u = Ev,
1 − △z v = Eu.
−∂z v −
(3.9)
∂z u −
a
2
a
2
9
Obrázek 6: LDoS as a function of E ∈ (−1, 1) and z ∈ (0, 100) for perturbed cylinder with △ = 0.05
(left) and △ = 0.1 (right)
The solution is
u(z) = C△1 Dν1 (ξ(z)) + C△2 Dν2 (iξ(z)),
C△1
v(z) =
E
where
!
e
jDν1 (ξ(z))
1 2 2
C△2
∂z Dν1 (ξ(z)) −
(1 − △ z ) +
a
2
E
√
a2 △ − 4a2 E 2 + 4ia △e
j + 4e
j2
ν1 = i
,
√
8a △e
j
(3.10)
!
e
jDν2 (iξ(z))
1 2 2
∂z Dν2 (iξ(z)) −
(1 − △ z ) ,
a
2
(3.11)
√
j + 4e
j2
a2 △ − 4a2 E 2 − 4ia △e
ν2 = −i
,
√
8a △e
j
s


r
e
a
2j 
+
z ,
ξ(z) = (−△)1/4 
a
2e
j
(3.12)
(3.13)
Dν (ξ) being the parabolic cylinder function [14]. Again, the normalization constants will be calculated
form 3.5.
In Fig. 6, a comparison of the 3D plots of LDoS is made for different values of △. The number
of defects N = 2 and the next values, m = 20, n = 20, a = 14, zmax = 100 and j = 0. On the whole,
we can conclude from the comparison that LDoS is slightly increasing for higher △. Although this
is only the first order approximation, the real solution of (3.7) behaves in a similar manner.
10
4
Conclusion
We have studied the electronic structure of two distinct kinds of nanostructures: the disclinated
graphene in the vicinity of heptagonal and pentagonal defects and the nanostructured cylinder.
In the first case, hyperboloidal parametrisations (2.1),(2.14) were assumed after rejection of the
conical metric. The defects are involved in (2.5) with the help of the parameter ǫ, which comes from
the elasticity properties of the graphene. Next, we incorporated a uniform magnetic field (2.6),(2.21)
that can significantly influence the LDoS. These were calculated from the solution of the Dirac
equation, which we obtained numerically with the help of the extension of an analytical solution for
zero-energy modes (2.8),(2.9),(2.18),(2.19). In the presented figures, the behaviour of the LDoS is
compared for both kinds of defects. For very small values of ǫ, this behaviour is similar for both kinds
of defects, but it is more spread for heptagonal defects.
For the nanostructured cylinder, we compare the results for the defect-free and the perturbed
cylinder. For the defect-free cylinder, we get a simple exponential solution presented in Fig. 5, for
the perturbed cylinder we get the solution presented in Fig. 6 for different values of pertubation. We
see that the from small values of the perturbation, the LDoS is influenced significantly.
Reference
[1] J. Smotlacha, R. Pincak, M. Pudlak, Eur. Phys. J. B 84, 255 (2011).
[2] R. Pincak, J. Smotlacha, M. Pudlak, Electronic properties of disclinated nanostructured cylinder,
prepaired to be published.
[3] R. Pincak, M. Pudlak, J. Smotlacha, chapter with title Electronic properties of single and double
wall carbon nanotubes for the book ”Carbon Nanotubes: Synthesis, Properties and Applications”,
Publisher Nova Sciences, USA, Edited by Prof. Ajay Kumar Mishra, accepted for publication
January 2012, in print.
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[6] E. A. Kochetov, V. A. Osipov, J. Phys. A: Math. Gen. 32, 1961, (1999).
[7] P. E. Lammert, V. H. Crespi, Phys. Rev. B 69, 035406 (2004).
[8] E.A. Kochetov, V.A. Osipov and R. Pincak, J. Phys.: Condens. Matter 22, 395502 (2010).
[9] R. Pincak, M. Pudlak, chapter in Progress in Fullerene Research, ed. F. Columbus, Nova Science
Publishers, New York (2007).
[10] E. A. Kochetov, V. A. Osipov, JETP Letters 91, 110 (2010).
11
[11] D. V. Kolesnikov, V. A. Osipov: JETP Letters 79, 532 (2004).
[12] J. C. Slonczewski, P. R. Weiss, Phys. Rev. 109, 272 (1958).
[13] J. Gonzalez, F. Guinea, M. A. H. Vozmediano, Nucl. Phys. B 406, 771 (1993).
[14] M. Abramowitz, I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and
mathematical tables, Dover Publications, New York (1964).
12
Část II
Quantum Optics: Spectrum and Coherent
States
There is a lot of different kinds of processes in quantum optics. Some of the methods of finding
the eigensystems are usually connected with the conserved quantities which help us to simplify the
solution of the problem. The higher is the number of these quantities, the more complicated is the
task.
In this work, we present the numerical solutions of spectrum of the Hamiltonian for the three
boson interaction process (namely, down conversion and second-harmonic generation). We use the
method of orthogonal polynomials [5] and demonstrate that this method can be used in nonlinear
optical processes. After reduction we use the Turbiner polynomials approach. The eigenvalues of
Hamiltonian for low number of photons are calculated and the approximation formula is found out.
Finally, we find the corresponding coherent states. For this purpose, we apply the methods of constructions of su(2) and deformed su(2) coherent states.
1
1.1
Three Boson Interaction Process
Introduction
During the recent years, the down conversion process and the second harmonic generation process has been intensively investigated, because of development of experimental techniques. It is the
simplest nonlinear model in which entangled photons are produced (see for example [2]). In this
process a photon of frequency ω2 is converted into two photons of frequency ω1 , or vice versa in
second harmonic generation process. In order to constitute coherent states, we first investigate the
Hamiltonian which describes this effect. We use the techniques developed in [3, 4], which is strictly
related to the theory of orthogonal polynomials, and we make the reduction of the Hamiltonian. The
Hamiltonian of down conversion is
a2 , b
a†2
b
a1 , b
a†1 , b
b =H
b0 + H
b I = ωb
H
a†1b
a1 + 2ωb
a†2b
a2 + (b
a†1 )2b
a2 + b
a21b
a†2 ,
(1.1)
denote the creation and annihilation operators of the appropriate bosons. The
where
b
b
b I . It can be reduced
both parts H0 and HI commute and because of this fact we investigate only H
13
to the form
2
b = (b
b+ A
b† ,
H
a1 )2 b
a†2 + b
a†1 b
a2 = A
(1.2)
of class of Hamiltonians discussed in [6] – the quasi-exactly solvable models. The integral of motion
gives the maximal number of photons of low frequency that take part in the process, and it divides
Hilbert space into the invariant Hilbert subspaces. The eigenvalues are calculated with the help
of MATHEMATICA and the approximate formula for eigenvalues and eigenstates of low number
of photons – up to 900 – we present. Selected graphs of dependence on eigenvalues are presented.
Having eigenvalues, it is straightforward to express the eigenvectors via the values of polynomials
at the roots of higher polynomial [7]. The eigenstates of reduced operator, which describe the down
conversion are coherent states of the model - it corresponds to the coherent states of finite open chain
[8].
In next section we start with the description of algebra of down conversion process, in section
3 the calculation of eigenvalues and approximation formula is found out. The coherent states are
described in section 4. Instead of the approach in [9, 10], we use the [11, 12] coherent states for down
conversion with para-Grassmann variables as in [13].
1.2
Deformed Parafermionic Algebra of the Model
The nonlinear two boson operators from the Hamiltonian (1.2)
satisfy the relations
2
†
b
A = b
a†1 b
a2
b = (b
A
a1 )2 b
a†2 ,
b† A
b = (n1 − 1)n1 (n2 + 1),
A
bA
b† = (n1 + 1)(n1 + 2)n2
A
(1.3)
where n1 = b
a†1b
a1 and n2 = b
a†2b
a2 . Now, let us define operator
c = n1 + 2n2 .
M
b [M
c, H]
b = 0. Its physical meaning is a number of photons of frequency
It is an integral of motion of H:
ω1 that can take part in the process of second harmonic generation, and it is proportional to the
c, m is the number of photons of
intensity of light in the pulse. For even m - an eigenvalue of M
2
frequency ω2 , proportional to the intensity of initial pulse for down conversion.
From this follows that n1 is odd (even) just in the case when m is odd (even). We choose the
even case. The solution for the odd case would be analogous.
14
c
b = 1 n1 . Starting from the above definitions of the operators M
We next define the operator N
2
†
b we can formulate the algebra generated by the operators N,
b A,
b A
b as a deformed oscillator
and N
algebra [11]
i
h
i
h
†
†
b
b
b
b
b
b
N, A = −A
N, A = A ,
(1.4)
b† A
b = Φ(M
c, N),
b
bA
b† = Φ(M
c, N
b + 1)
A
A
This deformed oscillator is characterized by the structure function
Φ(m, n) = 2n(2n − 1)(
m
− n + 1)
2
(1.5)
There is a unitary Fock-like representation of this algebra. The vector space-carrier of the representation possess the basis
|m, ni , m, n ∈ N0
We do a substitution for m:
c |m, ni = m |m, ni , N
b |m, ni = n |m, ni
M
p
b† |m, ni = Φ(m, n + 1) |m, n + 1i ,
A
p
b |m, ni = Φ(m, n) |m, n − 1i
A
m = 2p,
where p = 0, 1, 2, . . .
Finally, the structure function (1.5) is written as
1
Φ(p, n) = 4n n −
(p − n + 1)
2
(1.6)
(1.7)
(1.8)
(the index m in (1.5) is replaced by the index p).
The above structure function satisfies the relation
Φ(p, p + 1) = 0,
therefore it is the characteristic structure function of a deformed parafermionic oscillator, which was
studied by Quesne [14]. This parafermionic oscillator is a finite dimensional deformed oscillator [11],
which has a faithful finite dimensional representation of dimension p + 1. The vector space which
carries the representation has the Fock basis
|p, ni ,
where n = 0, 1, 2, . . . , p
This basis is the initial Fock basis (1.6), where the index m is replaced by the index p. Obviously in
b and A
b† satisfy the relations:
this basis the operators A
p+1
p+1
b
b†
A
|p, ni = 0,
A
|p, ni = 0.
15
1.3
Calculation of the eigenvalues
Now we calculate the energy spectrum of the investigated process. For a given eigenvalue E and
a given value of p, we denote the corresponding eigenvector |E, pi. It should satisfy
b |E, pi = E |E, pi .
H
(1.9)
With the help of the Fock basis, the eigenvector can be expressed as
p
1 X cn (p, E)
p
|p, ni ,
|E, pi = √
N n=0 Φ(p, n)!
where
Φ(p, 0)! = 1,
Φ(p, n)! =
n
Y
(1.10)
Φ(p, k)
k=1
and N is the normalization constant.
The coefficients cn (p, E) are polynomials of the energy E. They are a kind of Hermite-like polynomials, because they satisfy these recurrence relations (we fix the form of c0 (p, E)):
c0 (p, E) = 1,
c1 (p, E) = E,
cn+1 (p, E) = Ecn (p, E) − 4n(n − 1/2)(p + 1 − n)cn−1 (p, E),
(1.11)
n = 2, ..., p − 1.
Furthermore,
Ecp (p, E) − 4p(p − 1/2)cp−1(p, E) = 0
(1.12)
which (after comparison with the third line in(1.11) ) can be formally defined as cp+1 (p, E). Then
[p]
the energy eigenvalues Ek can be calculated as the roots of the polynomials
[p]
cp+1(p, E)(Ek ) = 0,
k = 0, 1, . . . , p.
(1.13)
The eigenvalues are symmetrically placed around the origin, i. e. they take values with sign +
and - (see Figure 1). The approximation formulas for the highest eigenvalues for different values of
p are presented in Figures 2 and 3.
1.4
Coherent states
Coherent states will be made using the standard coherent states of harmonic oscillator. For the
construction of coherent states inside the invariant subspaces we will use para-Grassmann variables
as in [8]. The dimension p of invariant subspaces is done by the intensity of input light in a nonlinear medium. The para-Grassmann variables z are defined on p-dimensional complex space C. The
16
Obrázek 1: The eigenvalues of the Hamiltonian of the 3-boson interaction for p=50
20 000
Λmax (p)
15 000
10 000
5000
p
0
100
200
300
400
500
Obrázek 2: The highest eigenvalues: approximation formula λmax = apb with a = 1.53 and b = 1.50
17
Λmin H pL
14
12
10
8
6
4
2
p
0
100
200
300
400
500
Obrázek 3: The lowest eigenvalues: approximation formula λmin = apb with a = 1.035 and b = 0.386
ring is converted into a Hilbert space Mp by providing it with an inner product. Then the coherent
states map K is defined as a map from Mp into the tensor product of space Mp and an abstract
(p + 1)-dimensional Hilbert space Hp .
During the construction of the coherent states we will see that it holds for the eigenvalue z which
is at the same time one of the parameters labeling the coherent state,
z p+1 = 0.
(1.14)
This can’t hold for any complex z except 0. So, we suppose that z ∈ Cp+1,p+1 and denote z = z a
matrix with zeros everywhere except the elements placed one position right from the diagonal, i.e.


0 z1
0 ··· 0
 0
0 z2 · · · 0 



(1.15)
z=
 ··· ··· ··· ··· ··· ,

 0
0
0 · · · zp
0
0
0 ··· 0
which is the most general nontrivial mathematical object with the required property (1.14), and we
call it the para-Grassmann variable.
bA
b† are defined
For the down conversion, the annihilation and creation operators A,
p
p
b ni = Φ(p, n)|p, n − 1i, A
b† |p, ni = Φ(p, n + 1)|p, n + 1i, A|p,
b 0i = 0,
A|p,
18
(1.16)
where the set {|p, ii} is the Fock base of the (p+1)-dimensional Hilbert space Hp . Defining the
coherent states |K(p, z)i of down conversion as the eigenstates of the operator  on Hp for given p,
we get
b
A|K(p,
z)i = z|K(p, z)i.
(1.17)
We suppose the form of the eigenstate
|K(p, z)i =
p
1 X
kn z n |p, ni
N n=0
(1.18)
with N being the normalization constant. The domain of the eigenvalue z is derived below. After
the substitution into (1.17),
p
X
n=0
=
p−1
X
b ni =
kn z A|p,
n
kn+1 z
n=0
Now we see that
n+1
p
X
kn z n
n=0
p
Φ(p, n)|p, n − 1i =
p
X
p
kn z n |p, ni.
Φ(p, n + 1)|p, ni = z
(1.19)
n=0
kn+1
p
Φ(p, n + 1) = kn
(1.20)
for n = 0, ..., p − 1 and (1.14) holds, which is the reason for using the para-Grassmann variable and
denotion z = z, where z is defined in (1.15). Now, we put k0 = 1 and solve (1.20) for different p. For
arbitrary z, the solution will be of the form
|K(p, z)i =
The coefficients kn are
1
k0 z0 |p, 0i + k1 z1 |p, 1i + ... + kp zp |p, pi .
N
kn = p
1
Φ(p, n)!
,
n = 0, . . . , p.
(1.21)
(1.22)
In Table 1, we see the coefficients k0 , ..., kp for some of the low values of p.
In accordance with the construction of su(2) coherent states introduced [12, 15], the coherent
states labeled by two parameters, one inside the invariant Hilbert space and the other one labeling
the superposition of spaces, can be constructed.
Now, we take into consideration the superposition of invariant Hilbert subspaces, i.e. taking part
of the coherent state from the arbitrary subspace of the Hilbert space on the input of the process.
We denote H the Hilbert space
H = H1 ⊕ H2 ⊕ . . . ,
(1.23)
19
p
1
2
3
4
k0
1
1
1
1
k1
√1
2
1
2
√1
6
1
√
2 2
Tabulka 1: Coefficients of the series (1.21)
k2
k3
k4
1
√
4 3
1
1
√
12
12 30
1√
12 2
1
√
24 30
√1
96 105
where Hp , p = 1, 2, . . . are (p + 1)-dimensional invariant subspaces. Then, for every p we denote
(supposing z1 = z2 = . . . = zp = z) the p × p matrix




0
1
0 ··· 0
0
z
0 ··· 0
 0
 0
0
1 ··· 0 
0
z ··· 0 




 = z  · · · · · · · · · · · · · · ·  = zIp ,
(1.24)
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
zp = 




 0
 0
0
0 ··· 1 
0
0 ··· z 
0
0
0 ··· 0
0
0
0 ··· 0
and we define the coherent states formally
∞
1 2 X αp
√ |K(p, zIp )i.
|α, zi = exp(− |α| )
2
p!
p=1
Substituting (1.18), we finally have the formula
p
∞
1 2 X α p X z n Ip n
1
√
p
exp(− |α| )
|p, ni.
|α, zi =
N
2
p! n=0 Φ(p, n)!
p=1
where α is a complex number.
Reference
[1] J. Smotlacha, G. Chadzitaskos, C. Daskaloyannis, Spectra of Photon Down Conversion, Proceedings of the XXVIII. Workshop on Geometric Methods in Physics, Bialowieza (2009).
[2] L. Mista , R.Filip ”Non perturbative Treatment of Nonlinear Optical Systems”Fortschr. Phys.
49, (2001) 1047–1052
[3] A. Odzijewicz, M. Horowski, A.Tereszkiewicz ”Integrable multi-boson systems and orthogonal
polynomials”J. Phys. A: Math. Gen. 34, (2001) 4353.
[4] M. Horowski, A.Odzijewicz and A.Tereszkiewicz, ”Some integrable system in nonlinear quantum
optics”, J.Math.Phys. 44 (2003), 480.
20
[5] T. Brougham, G. Chadzitaskos, I. Jex: J. Mod. Opt. 56, 1588-1597 (2010).
[6] A. Turbiner, Sov Phys., J.E.T.P. 67, (1988) 230.
[7] H.S. Wilf, Mathematics for physical science, Dover,New York, (1962).
[8] G. Chadzitaskos, Coherent States on Open Finite Chains. Proceedings of XXI Workshop on
Geometrical Methods in Physics, Melville, New York (2007) 240.
[9] A. Luis, J. Perina, SU(2) coherent states in parametric down conversion, Phys. Rev. A 53, 1886
(1996).
[10] X. Wang, B. Sanders, S. Pan, Entangled coherent states for systems with SU(2) and SU(1,1)
symmetries, J. Phys. A: Math. Gen. 33, 7451, (2000).
[11] Daskaloyannis C., Generalized Deformed Oscillator and Nonlinear Algebras. J. Phys.
A:Mathematical and General 24 , L789 (1991).
[12] J. P. Gazeau, J. Klauder, Coherent states for systems with discrete and continuous spectrum,
J. Phys. A: Math. Gen. 32 (1999) 123.
[13] M. Chaichian and A. P. Demichev, Czech J. Phys. 46(1997), p.155.
[14] Quesne C., Generalized Deformed Parafermions, Nonlinear Deformations of so(3) and Exactly
Solvable Potentials, Phys. Lett. A193, 248 (1994).
[15] M. Sadiq, A Inomata, Coherent states for polynomial su(2) algebra, J. Phys. A: Math. Theor.
40 (2007) 11105.
21
A
List of author’s papers
• J. Smotlacha, R. Pincak, M. Pudlak, Eur. Phys. J. B 84, 255 (2011).
• R. Pincak, J. Smotlacha, M. Pudlak, Electronic properties of disclinated nanostructured cylinder, prepaired to be published.
• R. Pincak, M. Pudlak, J. Smotlacha, chapter with title Electronic properties of single and
double wall carbon nanotubes for the book ”Carbon Nanotubes: Synthesis, Properties and
Applications”, Publisher Nova Sciences, USA, Edited by Prof. Ajay Kumar Mishra, accepted
for publication January 2012, in print.
B
List of author’s conference contributions
• XXVIII. Workshop on Geometric Methods in Physics, Bialowieza 2009: Spectra of Photon Down
Conversion.
• XXXI. Winter School Geometry and Physics in January, Srni, 2011: Four boson interaction
process.
22
Czech summary
Disertačnı́ práce se zabývá řešenı́m různých modelů týkajı́cı́ch se kvantové mechaniky. Je rozdělena
do dvou částı́. Prvnı́ část se týká studia elektronických vlastnostı́ nanočástic.
V úvodu se seznámı́me se strukturou nanočástic, jejich využitı́m a způsoby, pomocı́ kterých je
dosaženo zakřivenı́ původnı́ struktury - šestiúhelnı́kové uhlı́kové mřı́že zvané grafén. Těmi jsou přidánı́
resp. vyřı́znutı́ výsečı́ svı́rajı́cı́mi úhel 60◦ , resp. jeho násobek. Výsledkem je vznik n-úhelnı́ku v mı́stě
zakřivenı́, kde n < 6 pro kladné zakřivenı́ (s kladnou hodnotou křivosti), resp. n > 6 pro záporné
zakřivenı́.
Následujı́cı́ teoretická část je nejdřı́ve věnována zavedenı́ základnı́ch pojmů a vztahů. Po objasněnı́
vztahu mezi elementárnı́ buňkou grafénu a Brillouinovou zónou v přı́slušném reciprokém prostoru
je uveden Blochův teorém, kterým poskytuje důležité informace o vlastnostech vlnové funkce, která
je řešenı́m Hamiltoniánu pro elektrony v krystalu. Dále jsou uvedeny základnı́ metody pro výzkum
daného problému: je to předevšı́m ”tight-binding”aproximace spočı́vajı́cı́ v započtenı́ vlivu pouze
nejbližšı́ch atomů uhlı́ku šestiúhelnı́kové mřı́že a teorie efektivnı́ hmoty, pro kterou s využitı́m dané
aproximace zapı́šeme Hamiltonián. Pomocı́ variačnı́ch principů spočteme z této rovnice matici odpovı́dajı́cı́ přechodům mezi jednotlivými stavy přı́slušnými atomům z různých podmřı́žı́. Vlastnı́
čı́sla této matice odpovı́dajı́ vlastnı́m energiı́m původnı́ho Hamiltoniánu. Jako přı́klad je tu uvedeno
→
−
energetické spektrum grafénu pro různou orientaci vektoru k reciprokého prostoru.
Po zavedenı́ dalšı́ch členů v původnı́m Hamiltoniánu je dokázána korespondence mezi tı́mto Hamiltoniánem a Diracovou rovnicı́. Jejı́ struktuře je pak věnována dalšı́ kapitola, v nı́ž jsou do Hamiltoniánu přidána kalibračnı́ pole souvisejı́cı́ s geometrickou strukturou molekul, invariancı́ vlnové
funkce vůči konkrétnı́m typům transformacı́ a magnetickým polem. Nakonec je uvedena soustava
rovnic, jejı́ž řešenı́ vede k výpočtu hledané veličiny - lokálnı́ hustoty stavů.
V dalšı́m se setkáváme s přı́slušnými výpočty pro jednotlivé zkoumané přı́pady: okolı́ defektu
pro záporné, resp. kladné zakřivenı́ pro přı́pad předpokládané hyperboloidálnı́ geometrie a válec bez
defektu, resp. s dvěma protilehlými defekty. V prvnı́m přı́padě nejprve spočteme řešenı́ pro nulové
módy a pro řešenı́ přı́padu nenulových energiı́ využijeme poruchové schéma. V druhém přı́padě
spočteme pro všechny energie přı́slušná analytická řešenı́, v přı́padě s defekty ovšem využı́váme
zjednodušený tvar rovnice.
Zvláštnı́ kapitolu věnujeme výpočtu energetického spektra dvojitých nanotrubek různých typů.
Rozdı́l oproti předchozı́mu spočı́vá zvláště v nutnosti započı́tat vliv vzájemné interakce mezi vnějšı́
a vnitřnı́ nanotrubkou. Vliv geometrie na energetické spektrum je zahrnut dodatečně.
Výpočty dále vedou na řešenı́ složitějšı́ch přı́padů s vı́ce defekty a grafénovými vrstvami, se
započtenı́m dalšı́ch vlivů.
V druhé části se zabýváme řešenı́m energetického spektra a koherentnı́ch stavů deformovaného
parafermionového oscilátoru. Nalezenı́m veličin komutujı́cı́ch s Hamiltoniánem, resp. splňujı́cı́ch dané
relace, zkonstruujeme přı́slušnou algebru, jejı́ž možnou reprezentacı́ je Fockova báze přı́slušného Hilbertova prostoru. Pomocı́ strukturnı́ funkce odvodı́me pro výpočet vlastnı́ch hodnot přı́slušný rekurentnı́ vztah, který vede na ortogonálnı́ polynomy. V grafech jsou vynesena vlastnı́ čı́sla pro konkrétnı́
hodnotu zachovávajı́cı́ se veličiny a také funkce aproximujı́cı́ nejvyššı́, resp. nejnižšı́ vlastnı́ čı́sla pro
jednotlivé hodnoty této veličiny.
Dále je uveden výpočet a konstrukce koherentnı́ch stavů, při které je nutno zavést tzv. paraGrassmannovu proměnnou. Vlastnı́ hodnotou je pak čtvercová matice splňujı́cı́ danou vlastnost.
Způsob konstrukce koherentnı́ch stavů je stejný jako v přı́padě su(2) deformované Lieovy algebry.
Výpočtem energetického spektra se dále zabýváme v přı́padě 4-bosonového procesu. Nejdřı́ve
odvodı́me obecný tvar matice přı́slušné soustavy pro dané hodnoty veličin komutujı́cı́ch s Hamiltoniánem a poté vypočteme vlastnı́ čı́sla. Z jejich grafické závislosti na integrálech pohybu odvodı́me
přibližný tvar aproximativnı́ formule.