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Saint Petersburg National Research University
of Informational Technologies, Mechanics, and Optics
MATHEMATICAL CHALLENGE
OF QUANTUM TRANSPORT
IN NANOSYSTEMS
International Conference
Saint Petersburg, March 12 – 15, 2013
Book of Abstracts
Saint Petersburg
2013
1
Mathematical Challenge of Quantum Transport in Nanosystems: Book of
Abstracts of the International Conference (Saint Petersburg, March 12 – 15, 2013). –
Saint Petersburg: University ITMO, 2013. – 24 p.
Organized by the Saint Petersburg National Research University of Informational
Technologies, Mechanics, and Optics.
 Saint Petersburg National Research University of
Informational Technologies, Mechanics, and
Optics, 2013
2
PROGRAM COMMITTEE
Vladimir N. Vasilyev (University ITMO, Russia) – Chairman
Igor Yu. Popov (University ITMO, Russia) – Vice-Chairman
Sergey N. Naboko (St. Petersburg State University, Russia)
Boris S. Pavlov (Massey University, New Zealand)
George P. Miroshnichenko (University ITMO, Russia)
LOCAL ORGANIZING COMMITTEE
Leonid M. Studenikin – Chairman
Igor Yu. Popov – Vice-Chairman
Igor S. Lobanov
Ekaterina S. Trifanova
Irina V. Blinova
Alexander I. Trifanov
Maxim A. Skryabin
Anton I. Popov
INVITED SPEAKERS
Pavel Exner (Prague, Czech Republic)
Horia Cornean (Aalborg, Denmark)
Hagen Neidhardt (Berlin, Germany)
Valentin Zagrebnov (Marseille, France)
Konstantin Pankrashkin (Paris, France)
Ashok Chatterjee (Hyderabad, India)
Mikhail Belishev (St. Petersburg, Russia)
Jaroslav Dittrich (Rez, Czech Republic)
Paul Tackley (Zurich, Switzerland)
THE MAIN TOPICS OF THE CONFERENCE:
Spectral theory
Scattering
Quantum transport
Quantum communications and computations
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Conference Program
Young Researchers Symposium (YRS)
March 12, 2013
930 – 1030 Registration
1030 – 1040 Opening
1040 – 1055 Mikkel Brynildsen (Aalborg, Denmark)
On spectral gaps in graphene in a weak constant magnetic field
55
10
10 – 11 Victor Mikhailov (St. Petersburg, Russia)
Inverse problems for the wave equation and two-velocity system on a tree
10
25
11 – 11 Alina Anikevich (St. Petersburg, Russia)
On the spectrum of Y-type chain of weakly coupled conglobated
resonators
25
40
11 – 11 Oleg Sokolov (St. Petersburg, Russia)
Hamiltonian with delta-potentials and with infinite number of eigenvalues
40
55
11 – 11 Dmitrii Eremin (Saransk, Russia)
Point spectrum for bent sphere chains in a magnetic field
55
10
11 – 12 Anton Boitsev (St. Petersburg, Russia)
Boundary triplets approach for sum of tensor products of operators
1210 – 1240 Coffee
1240 – 1255 Konstantin Pravdin (St. Petersburg, Russia)
Model of the point sources electromagnetic field interaction with
metamaterials
55
10
12 – 13 Alexander Trifanov (St. Petersburg, Russia)
Kraus representation of conditional evolution operators
10
25
13 – 13 Natalia Konobeeva (Volgograd, Russia)
Tunneling current in carbon nanotubes with deep impurities
25
40
13 – 13 Anastasia Pak (Volgograd, Russia)
Electrophysical properties of carbon nanotubes. The method of linear
augmented cylindrical wave
40
55
13 – 13 Pavel Smirnov (St. Petersburg, Russia)
The spectrum of the chain quantum graph with Y-branching
55
10
13 – 14 Ivan Melikhov (St. Petersburg, Russia)
Hartree-Fock approximation for interacting particles in deformed
nanolayer
1410 – 1440 Coffee
1440 – 1455 Olga Rodygina (St. Petersburg, Russia)
Soliton induced flow in nanotube
4
1455 – 1510 Dmitrii Vavulin (St. Petersburg, Russia)
Multiplexing of classical and quantum channels in optical communication
systems
10
25
15 – 15 Artur Gleim (St. Petersburg, Russia)
Sidebands-based quantum cryptography system: theory and practice
25
40
15 – 15 Alena Ivanova (St. Petersburg, Russia)
Investigation of quantum random number generation based on space-time
division of photons
40
55
15 – 15 Anton Popov (St. Petersburg, Russia)
Benchmark solutions for Stokes flow algorithm testing
1555 – 1600 Closing
Pierre Duclos Workshop
March 13, 2013
930 – 1030 Registration
1030 – 1040 Opening
1040 – 1140 Pavel Exner (Rez near Prague, Czech Republic)
Control of vertex coupling in quantum graphs
1140 – 1210 Coffee
1210 – 1310 Horia Cornean (Aalborg, Denmark)
Non-equilibrium steady-states for interacting open systems
1310 – 1430 Lunch
1430 – 1505 Jaroslav Dittrich (Rez, Czech Republic)
Scattering through a straight quantum waveguide with combined
boundary conditions
05
30
15 – 15 Radu Purice (Bucharest, Romania)
The Peierls - Onsager substitution in the gauge covariant magnetic
pseudodifferential calculus
30
55
15 – 15 Aurel Gabris (Prague, Czech Republic)
Simulations using optical quantum walk implementations
1555 – 1625 Coffee
1625 – 1650 Jan Smotlacha (Dubna, Russia; Czech Republic)
Electronic structure of disordered graphene and Green's function approach
50
15
16 – 17 Natalie Firsova (St. Petersburg, Russia)
Asymptotic behavior of scattering data and conductivity for small Fermi
energy in monolayer graphene
5
1715 – 1740 Yuriy Firsov (St. Petersburg, Russia)
Synthetic electric fields influence on monolayer graphene nonlinear
electromagnetic response
40
00
17 – 18 Igor Lobanov, Ekaterina Trifanova (St. Petersburg, Russia)
Large quantum graph simulation for mesoscopic system modeling:
current–voltage characteristic of graphene nanoribbon
March 14, 2013
1000 – 1100 Hagen Neidhardt (Berlin, Germany)
Boundary triplets and tunnel junction formula with applications
1100 – 1130 Coffee
1130 – 1230 Konstantin Pankrashkin (Orsay, France)
Gaps opening and split band edges for quantum waveguides with periodic
perturbations
30
10
12 – 13 Paul Tackley (Zurich, Switzerland)
To be announced
1310 – 1430 Lunch
1430 – 1530 Mikhail Belishev (St. Petersburg, Russia)
To be announced
30
55
15 – 15 Yuriy Belyaev (Syktyvkar, Russia)
Method of symmetric polynomials in the computations of scattering
matrix
1555 – 1625 Coffee
1625 – 1650 Eugene Kanzieper (Holon, Israel)
Statistics of reflection eigenvalues in chaotic cavities with non-ideal leads
50
15
16 – 17 Yusup Eshkabilov (Tashkent, Uzbekistan)
Efimov's effect for partial integral operators of Fredholm type
March 15, 2013
1000 – 1100 Valentin Zagrebnov (Marseille, France)
A non-equilibrium system with repeated interactions
1100 – 1130 Coffee
1130 – 1155 Alexei Vagov (Bayreuth, Germany)
Multiple-band superconductors: extended Ginzburg-Landau formalism by
a systematic expansion in small deviation from the critical temperature
6
1155 – 1220 Evgueni Karpov (Brussels, Belgium)
Gaussian classical capacity of Gaussian quantum channels
20
45
12 – 12 Victor Zalipaev (St. Petersburg, Russia)
Tunneling through a smooth potential barrier and localised states in
graphene monolayer with mass gap
45
10
12 – 13 Tigran Vartanyan (St. Petersburg, Russia)
Charge transport through the planar array of metal nanoparticles
1310 – 1430 Lunch
1430 – 1455 Alexander Vasilchenko (Krasnodar, Russia)
Phase diagram of the electron transition in the spin-polarized state in onedimensional quantum dot
55
20
14 – 15 Maarif Jafarov (Baku, Azerbaijan)
Modeling of neqatron phenomenon in in nanoscale films A2B6
1520 – 1545 Irina Blinova (St. Petersburg, Russia)
Model of non-axisymmetric flow in nanotube
20
45
15 – 15 Sergey Chivilikhin (St. Petersburg, Russia)
Stokes flows in nanoscale regions with free boundary
1545 – 1600 Closing
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Abstracts
On the spectrum of Y-type chain of weakly coupled
conglobated resonators
Alina Anikevich
Saint Petersburg National Research University of Informational Technologies,
Mechanics, and Optics, Saint Petersburg, Russia
E-mail: [email protected]
Spectral properties of the system are strongly associated with its geometry. The
spectral problem for the Y-bent chain of weakly coupled conglobated resonators is
investigated. Y-bent system can be described as the central ball linking three chains
consisted of balls of the same radius. There is a delta-coupling condition with
parameter α in every contact point. Specifically, it is supposed that for each branch
there is an axis passing through all sphere’s coupling points and these three axes lie in
the same plane. And it also supposed that centers of balls that are the closest to the
central ball are the vertices of an equilateral triangle. The transfer-matrix approach
and the theory of operator extensions are employed to solve the spectral problem for
this system. The band structure of the continuous spectrum is described. It is also
shown that such system for a certain value of parameter α has at most one negative
eigenvalue.
Small-angle scattering of electrons in undoped structures InAs/AlSb
under red light emitting diode illumination
Marina Afanasova
Ryazan State University named by S.A.Esenin, Ryazan, Russia
E-mail: [email protected]
M.M. Afanasova, M.V. Khavronina investigation shows the transverse
magnetoresistance dependence on magnetic field. It was found, that under red LED
illumination concentration of electrons is changed in the limits from (3.8-6.1)1015 m2.
Estimation of quantum relaxation time is done. It is shown, that in the magnetic field
B=0-8 Tesla it is about (8.1-9.2)10-14 c. Weak non-monotonic dependence of
quantum time of relaxation on concentration and external dependence indicates the
identical dominating mechanism of electrons scattering. Theoretical calculation
shows, that dominating mechanism is electron scattering on roughness of the
heterointerfaces.
8
Method of symmetric polynomials in the computations
of scattering matrix
Yuriy Belyaev
Syktyvkar State University, Syktyvkar, Russia
E-mail: [email protected]
Every square matrix is characterized by its symmetric polynomials. The method for
calculating any analytic matrix function by means of symmetric polynomials is
presented. This symmetric polynomials method is demonstrated on the example of
matrix exponential. The scattering matrix of n-th order is calculated in this way. The
scattering matrix of electrons in a layered periodic structure is analyzed.
Model of non-axisymmetric flow in nanotube
Irina Blinova
St, Petersburg National Research University of Information Technologies, Mechanics
and Optics, Saint Petersburg, Russia
E-mail: [email protected]
Stokes approximation for the fluid flow in nanotube is considered. The flow is
induced by arbitrary oriented rotlet (point-like source of vorticity) inside the
nanotube. The system of 3D Stokes and continuity equations is reduced to a system
for two scalar functions. The solution of the problem is obtained.
Boundary triplets approach for sum of tensor products of operators
Anton Boitsev
Saint Petersburg National Research University of Informational Technologies,
Mechanics, and Optics, Saint Petersburg, Russia
E-mail: [email protected]
A model of two particles quantum system with the Hamiltonian having a form of a
sum of tensor products is considered. Boundary triplet is constructed for the case
when one operator assumed to be self-adjoint. The application of the suggested
technique to a system of 1D particles is described.
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On spectral gaps in graphene in a weak constant magnetic field
Mikkel Brynildsen
Aalborg University, Aalborg, Denmark
E-mail: [email protected]
We introduce a discrete tight-binding model for graphene, a monolayer of carbon
atoms in a honeycomb lattice. This model, due to P. R. Wallace (1947), features a
two-band dispersion relation with conical singularities at two points in the Brillouin
zone, the so-called “Dirac cones” and “Dirac points”. Motivated by the Wallacemodel, we consider a simplified situation, where the discrete zero-field Hamiltonian
has a dispersion relation with a single Dirac-cone. We show that gaps open in the
spectrum of the associated magnetic Hamiltonian, when a weak orthogonal constant
magnetic field is incorporated by means of the “Peierls-substitution”. This is joint
work (in progress) with Horia Cornean and Ira Herbst.
Stokes flows in nanoscale regions with free boundary
Sergey Chivilikhin
St, Petersburg National Research University of Information Technologies, Mechanics
and Optics, Saint Petersburg, Russia
E-mail: [email protected]
Fluid flows in nanoscale regions with small Reynolds numbers are described with
using of Stokes approximation. In 2D case the pressure can be presented as an
expansion in a complete system of harmonic functions. Using the pressure
distribution, we calculate the velocity and deformation of free boundary. The specific
feature of nanoflow is slip boundary conditions on the solid walls. This approach
gives us the possibility to calculate the flows with free boundary in nanoscale regions
with complicate geometry, for example the filling and emptying of nanotubes
systems and the flows along superhydrophobic surfaces.
Non-equilibrium steady-states for interacting open systems
Horia Cornean
Aalborg University, Aalborg, Denmark
E-mail: [email protected]
We give sufficient conditions for the existence of a steady-state transport regime for
interacting mesoscopic systems coupled to reservoirs (semi-infinite leads). The
partitioning and partition-free scenarios are treated on an equal footing. Moreover,
our time-dependent scattering approach proves the independence of the steady-state
quantities from the initial state of the sample, and that the stationary current vanishes
10
when the bias is zero. This is joint work with V. Moldoveanu (Bucharest) and C.-A.
Pillet (Toulon).
Scattering through a straight quantum waveguide
with combined boundary conditions
Jaroslav Dittrich
Nuclear Physics Institute ASCR, Rez, Czech Republic
E-mail: [email protected]
Scattering through a straight two-dimensional quantum waveguide with a very simple
combination of the Dirichlet and Neumann boundary conditions is studied for the
energies between the two lowest transverse modes. The existence of matching
conditions solution at the interface of waveguide parts with different boundary
conditions are proved. The use of stationary scattering theory is justified showing its
relation to the wave packets motion. The matching conditions are solved numerically
and the reflection and transmission coefficients are calculated.
Point spectrum for bent sphere chains in a magnetic field
Dmitrii Eremin
Mordovian State University, Saransk, Russia
E-mail: [email protected]
Infinite bent chain on nanospheres connected by wires is considered. Mathematical
model based on the theory of self-adjoint extensions of symmetric operators is
constructed. The positions of the eigenvalues for different values of the system
parameters (the length of the connecting wires, the bent angle, and the intensity of
magnetic field) are found.
Efimov’s effect for partial integral operators of Fredholm type
Yusup Eshkabilov
National Universty of Uzbekistan, Tashkent, Uzbekistan
E-mail: [email protected]
A sufficient condition for existence Efimov’s effect for self-adjoint partial integral
operators of Fredholm type is obtained.
11
Control of vertex coupling in quantum graphs
Pavel Exner
Czech Academy of Sciences, Rez near Prague, Czech Republic
E-mail: [email protected]
It is a longstanding problem how to understand the coupling in vertices of a quantum
graph using approximations, either by a family of appropriate “fat graphs” or by
operators on the graph itself. In particular, within an approximation by Neumann
Laplacians on a tube network the squeezing limit yields only the free (or Kirchhoff)
boundary conditions. In this talk I will describe two recent ideas. The first comes
from a common work with Olaf Post: it will be shown that adding families of suitably
scaled potentials to those Laplacians one can get spectrally nontrivial vertex
couplings, including those with wave functions discontinuous at the vertices. Next I
will describe another result obtained together with Taksu Cheon and Ondrej Turek on
approximations by Schröedinger operators on graphs which shows a way how the
graph problem can be solved in full generality. Combining the two techniques, one
can approximate any coupling using families of scaled Schrödinger operators on
Neumann networks. The technique also suggests a way how a transport through such
junctions can be controlled.
Asymptotic behavior of scattering data and conductivity
for small Fermi energy in monolayer graphene
Natalie Firsova
Institute for Problems of Mechanical Engineering, the Russian Academy of Sciences,
Saint Petersburg, Russia
E-mail: [email protected]
Electron scattering problem in the monolayer graphene with short range impurities is
considered. The main novel element in the suggested model is the band asymmetry of
the defect potential in the 2+1-dimensionel Dirac equation. This asymmetry appears
naturally if the defect violates the symmetry between sublattices. Our goal in the
present paper is to take into account a local band asymmetry violation arising due to
the defect presence. We analyze the effect of the electron scattering on the electronic
transport parameters in the monolayer graphene. The explicit exact formulae obtained
for S-matrix for δ-shell and annular well potential cases allowed us to analyze the
asymptotic behavior of such scattering data as scattering phases, transport cross
section, the transport relaxation time and then conductivity for small values of Fermi
energies. The obtained results are in good agreement with experimental data which
shows that the considered model is reasonable.
12
Synthetic electric fields influence on monolayer grapheme
nonlinear electromagnetic response
Yuriy Firsov
A.F.Ioffe Physical-Technical Institute, the Rusian Academy of Sciences,
Saint Petersburg, Russia
E-mail: [email protected]
More than 70 years ago Peierls and Landau have shown that absolutely plane 2Dcrystall cannot exist due to the thermal fluctuations which inevitably will destroy 2D
long range order at any temperature. Indeed, all the experimentally investigated
monolayer graphene membranes were not plane but had inherent out-of-plane
deformations (ripples, bubbles etc.). These corrugations generate so called
pseudomagnetic fields (gauge fields). In the presence of external time-dependant
fields so called synthetic (pseudo electric) fields should be in addition generated
which should lead to observable consequences. For instance we recently showed that
these fields lead to a new early not known loss mechanism in graphene
nanoresonators which was responsible for significantly increase of dissipation. The
experimental facts confirming our prediction were found. This effect we studied in
megahertz frequency range. In the present communication we consider the influence
of the alternating synthetic electric fields generated by time-dependent corrugations
motions stipulated by optical radiation. To take this effect into account we essentially
generalize the known Mikhailov’s photonic equations with self-consistent field effect
taken into account but written for an absolutely plane monolayer graphene membrane
in terahertz frequency range. It allows us to give more correct description of
nonlinear graphene electromagnetic response in terahertz frequency range. Both
harmonic and pulse excitation is considered.
Simulations using optical quantum walk implementations
Aurel Gabris
Czech Technical University in Prague, Prague, Czech Republic
E-mail: [email protected]
Efficient simulation of quantum systems has been a central question in quantum
mechanics from early times. In the present talk we present a table-top-sized
experimental implementation of quantum walks using single photons. The
implementation is capable of simulating various physical effects such as Andersontype localization, and two particle interactions within the quantum walk framework.
13
Sidebands-based quantum cryptography system:
theory and practice
Artur Gleim
Saint Petersburg National Research University of Informational Technologies,
Mechanics, and Optics, Saint Petersburg, Russia
E-mail: [email protected]
Quantum cryptography is a branch of modern physics aiming at creating
cryptographic devices that allow performing unconditionally secure distribution of a
binary secret key between several users. In this work we are presenting a new type of
such system. The main difference that determines its advantages lies in the method of
photon generation. Unlike any other known implementation, it uses neither laser
radiation attenuated to critical level (coherent states) nor single photon sources (e.g.
based on spontaneous parametric scattering). Single photons appear on spectral
sidebands as a result of phase modulation of a high-intensity carrier wavelength. This
method has several advantages: the setup is one-directional, does not include
interferometers that require precise justification and simplifies phase control.
Therefore, it allows one to create a device capable of distributing secret key to the
distances up to 200 km with bitrates up to 1 Mbit/s with QBER not exceeding the
maximum level allowed by theory. In our report we present theoretical analysis of the
sidebands-based method along with the latest experimental data received on system
prototype.
Investigation of quantum random number generation
based on space-time division of photons
Alena Ivanova
Saint Petersburg National Research University of Informational Technologies,
Mechanics, and Optics, Saint Petersburg, Russia
E-mail: [email protected]
One of the simplest optical quantum random number generators is based on
separation of photons, emitted by a laser, with a beam splitter. We read values from
two detectors at the beam splitter output. Encoding of random bits is performed as
follows: if a non-zero number of photons comes to one of the detectors, and the other
detector does not read any photons, such a condition is considered a binary value 1.
The opposite case is considered to be a binary value 0. Situations when both sensors
detect photons, or both do not detect photons, are ignored. In this paper we
theoretically study statistics of random numbers, depending on a parameter that
characterizes symmetry of the beam splitter. Degree of deviation of the obtained
distribution from the uniform random distribution is investigated on a basis of series
of statistical tests.
14
Modeling of negatron phenomenon in nanoscale films A2B6
Maarif Jafarov
Baku State University, Baku, Azerbaijan
E-mail: [email protected]
Results of studies of some negatron phenomena in films are presented. Also an
analysis of physical processes leading to appearance of a portion with negative
differential resistance in VCC (voltage-current characteristics) and a portion with
negative photocapacitance in various samples as a function of composition (anion and
cation replacement) regimes of deposition and thermal processing is given.
Investigation of processes like negative differential resistance (NDR) with formation
of electric inhomogeneity and anomalous temperature dependence of dark current
leading to nonlinearity of VCC allows us to develop models of phenomena and
theories that have found application in solving of some practical problems in
negatronics. As a result of electronic-molecular interaction between surfaces of
studied films and atmosphere the centers with absorbing origin are formed, that were
separated from majority carriers of current by surface potential barriers.
Gaussian classical capacity of Gaussian quantum channels
Evgueni Karpov
Université Libre de Bruxelles, Brussels, Belgium
E-mail: [email protected]
Classical capacity of quantum channels is the tight upper bound for the (classical)
information transmission rate when quantum nature of information carries is taken
into account. This is an extension to the quantum world of the foundational notion of
the capacity of information channels introduced by Shannon. Bosonic Gaussian
channels provide a good model for optical communication channels. For these
infinite dimensional systems, like in classical case, an upper bound on the input
energy is necessary in order to properly define the classical capacity as a maximum
over all quantum input ensembles satisfying this constraint. A further restriction to
the input ensembles formed by Gaussian states results in Gaussian capacity. This
quantity appears to be very convenient because on the one hand, it is conjectured to
achieve the capacity and on the other hand, can be found by solving a Lagrangian
optimization problem. We have shown that for almost all Gaussian bosonic channels
it can be reduced the one defined for a particular channel whose parameters are
determined by the original channel. We propose a method for the evaluation of the
Gaussian capacity discuss its additivity and elucidate its interesting properties as
functions of the channel parameters.
15
Statistics of reflection eigenvalues in chaotic cavities
with non-ideal leads
Eugene Kanzieper
Holon Institute of Technology, Holon, Israel
E-mail: [email protected]
The scattering matrix approach is employed to determine a joint probability density
function of reflection eigenvalues for chaotic cavities coupled to the outside world
through both ballistic and tunnel point contacts. Derived under assumption of broken
time-reversal symmetry, this result is further utilised to (i) calculate the density and
correlation functions of reflection eigenvalues, and (ii) analyse fluctuations properties
of the Landauer conductance for the illustrative example of asymmetric chaotic
cavity. Further extensions of the theory are pinpointed.
Tunneling current in carbon nanotubes with deep impurities
Natalia Konobeeva
Volgograd State University, Volgograd, Russia
E-mail: [email protected]
We consider the Maxwell’s equations for electromagnetic field propagating in an
array of carbon nanotubes with a deep impurity. The tunneling current in contact
nanotube-metal was investigated. The dependence of current-voltage characteristic of
such contact on the band gap of the impurity was analysed.
Large quantum graph simulation for mesoscopic system modeling:
current–voltage characteristic of graphene nanoribbon
Igor Lobanov, Ekaterina Trifanova
Saint Petersburg National Research University of Informational Technologies,
Mechanics, and Optics, Saint Petersburg, Russia
E-mail: [email protected], [email protected]
We present a new solver capable of calculating transport and spectral properties of
quantum graphs consisting of hundred thousand bonds. The solver is based on divide
and conquer design paradigm applied to commonly used approach to quantum graph
analysis, which is to substitute bonds with energy dependent boundary conditions.
The underlying mathematics is adapted to better fit numerical analysis needs. We
give upper estimate of computational complexity of the method and show a number
of ways to lower the complexity using parallel computation. A specialized quantum
graph description language is introduced to make the solver accessible for nonprogrammers.
16
The solver has been used to model graphene nanoribbon of zig-zag type subjected to
electric field. We found that the solver is suitable for modeling of mesoscopic size
systems. As an example we display current-voltage characteristic of the graphene
ribbon and compare our result with first principles calculations.
Hartree-Fock approximation for interacting particles
in deformed nanolayer
Ivan Melikhov
Saint Petersburg National Research University of Informational Technologies,
Mechanics, and Optics, Saint Petersburg, Russia
E-mail: [email protected]
The paper deals with the problem of quantum particles storage in a nanolayered
structure. The system of a few electrons interacting by δ-potential is considered. The
particles are placed in two dimensional deformed waveguide. From the mathematical
point of view the bound state of the system means the existence of the eigenvalues of
the corresponding Hamiltonian. To treat a multi-particle problem the Hartree-Fock
approach and the finite element method are used. Three different types of the
perturbation are considered: deformation of the layer boundary, a small window in a
wall between two layers and a curved layer. The systems of 2-10 particles with
various total spin are studied. The dependence of the minimal deformation parameter
which keeps bound state on the number of particles is given. Comparison of the
storage efficiencies in those cases is made.
Inverse problems for the wave equation and two-velocity
system on a tree
Victor Mikhailov
Saint Petersburg Department of V.A. Steklov Institute of Mathematics
of the Russian Academy of Sciences, St. Petersburg, Russia
E-mail: [email protected]
We consider dynamical inverse problems for the systems governed by the wave and
two-velocity equations on the tree-like graph. The inverse data being the dynamical
Dirichlet-to-Neumann map. We make use of the Boundary Control method to recover
unknown parameters of the systems and the topology of the graph. The work is based
on joint work with Prof. S. A. Avdonin and Prof. G. Leugering
17
Boundary triplets and tunnel junction formula with applications
Hagen Neidhardt
WIAS Berlin, Berlin, Germany
E-mail: [email protected]
We invent a tunnel junction formula using the theory of boundary triplets and apply it
to the Schrödinger and Dirac operators acting on real axis with a tunnel junction.
Joint work with Masao Hirokawa (Okayama, Japan).
Electrophysical properties of carbon nanotubes.
The method of linear augmented cylindrical wave
Anastasia Pak
Volgograd State University, Volgograd, Russia
E-mail: [email protected]
In this paper, density of states for conducting and semiconducting carbon nanotubes
(CNTs) of different diameters were obtained by using the method of linear
augmented cylindrical waves. Based on these data, we calculated the tunnel
characteristics between CNTs and quantum dot system with periodic structure.
Gaps opening and split band edges for quantum waveguides
with periodic perturbations
Konstantin Pankrashkin
University Paris-Sud, Orsay, France
E-mail: [email protected]
We consider small perturbations of the Laplace operator in a multi-dimensional
cylindrical domain by second order differential operators with periodic coefficients.
We show that under certain non-degeneracy conditions such perturbations can open a
gap in the continuous spectrum and give the leading asymptotic terms for the gap
edges. We also estimate the values of quasi-momentum at which the spectrum edges
are attained. The general machinery is illustrated by several new examples in twoand three-dimensional structures.
18
Benchmark solutions for Stokes flow algorithm testing
Anton Popov
Saint Petersburg National Research University of Informational Technologies,
Mechanics, and Optics, Saint Petersburg, Russia
E-mail: [email protected]
There are numerical algorithms for Earth mantle flow calculations in geophysics. In
case of strongly non-homogeneous mantle (there are high viscosity inclusions) some
numerical algorithms become working in wrong way. According to the mathematical
point of view, the problem reduces to the Stokes equation with high viscosity
contrast. Thus, we solve more general problem which describes not only geophysical
flow but also many other physical systems, particularly, nanotube flow. In present
work we suggest methods for algorithm checking. We solve the problem by two
ways. On the one hand, we find exact analytical solution. On the other hand, we solve
this problem by numerical algorithms. Then we compare results and estimate the
quality of the numerical algorithm.
Model of the point sources electromagnetic field interaction
with metamaterials
Konstantin Pravdin
Saint Petersburg National Research University of Informational Technologies,
Mechanics, and Optics, Saint Petersburg, Russia
E-mail: [email protected]
We study the linear phenomenological Maxwell’s equations in the presence of a
polarizable and magnetizable medium. For a dispersive, nonabsorptive, medium with
equal permittivity ε(ω) and permeability μ(ω), the latter can assume the value of –1
(+1 is their vacuum value) for a discrete set of frequencies, i.e., for these frequencies
the medium behaves as a negative index material (NIM). We consider a set of parallel
layers filled by the material. We study its interaction with the electromagnetic field
created by a point source.
The Peierls - Onsager substitution in the gauge covariant magnetic
pseudodifferential calculus
Radu Purice
Simion Stoilow Institute of Mathematics of the Romanian Academy,
Bucharest, Romania
E-mail: [email protected]
We oresent some recent results concerning the Peierls - Onsager substitution in a
non-adiabatic approach using the magnetic pseudodifferential calculus.
19
Soliton induced flow in nanotube
Olga Rodygina
Saint Petersburg National Research University of Informational Technologies,
Mechanics, and Optics, Saint Petersburg, Russia
E-mail: [email protected]
Flows in nanostructures (nanotubes, nanolayers) have many specific features that
can’t be observed for classical flows. At present there is no theory of nanoflows. In
the present paper a variant of nano-hydrodynamical model is suggested. Flow in
nanotube depends on many characteristics: chemical composition, structure, wall
morphology, etc. Moreover, it is known that there appear some structures in liquid in
nanotube due to confinement. In some cases there are dynamical structures induced
by flow eddies. It should be stressed that boundary condition, wall structure and
profile play crucial role. Namely, it is necessary to take into account vibration and
waves in molecular chains forming the nanotube wall. There are experimental
evidences of such wall vibration. The most interesting is solitary wave. The flow
induced by this wave is considered. The picture of the flow is obtained.
Modeling of characteristics of the photo-electric converter of solar
energy with an anti-reflecting coating on the basis of porous silicon
Galina Skoptsova
Ryazan State University named by S. Esenin, Ryazan, Russia
E-mail: [email protected]
The photo-electric converter of solar energy on the basis of silicon diffusive p-n-of
transition with an anti-reflecting coating is investigated. The coating represents
multilayered structure of porous silicon. The model of the reflection surface spectra is
presented. Influence of parameters of an anti-reflecting coating profile on
characteristics of the photo-electric converter is analyzed. Results of the
mathematical modeling are confirmed by an experiment.
The spectrum of the chain quantum graph with Y-branching
Pavel Smirnov
Saint Petersburg National Research University of Informational Technologies,
Mechanics, and Optics, Saint Petersburg, Russia
E-mail: [email protected]
Model of branching chain of rings with delta-coupling is constructed. The spectral
equation is obtained by using of the transfer matrix method. The existence of bound
states of the Schrödinger operator for the graph is proved. The spectrum of the
Schrödinger operator for the system is described.
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Electronic structure of disordered grapheme
and Green’s function approach
Jan Smotlacha
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,
Dubna, Russia (Czech Republic)
E-mail: [email protected]
The Green functions play a big role in the calculation of the local density of states of
the carbon nanostructures. We investigate their nature for the variously oriented and
disclinated graphene-like surface. Next, we investigate the case of a small
perturbation generated by two heptagonal defects and from the character of the local
density of states in the border sites of these defects we derive their minimal and
maximal distance on the perturbed cylindrical surface. For this purpose, we transform
the given surface into a chain using the Haydock recursion method. We will suppose
only the nearest-neighbor interactions between the atom orbitals, in other words, the
calculations suppose the short-range potential.
Hamiltonian with delta-potentials and with infinite
number of eigenvalues
Oleg Sokolov
Saint Petersburg National Research University of Informational Technologies,
Mechanics, and Optics, Saint Petersburg, Russia
E-mail: [email protected]
The infinite chain of delta-potentials in three-dimensional space is built, that is
described by a Hamiltonian with infinite number of eigenvalues below the border of
continuous spectrum. The model is based on the theory of self-adjoint extensions of
symmetric operators.
Kraus representation of conditional evolution operators
Alexander Trifanov
Saint Petersburg National Research University of Informational Technologies,
Mechanics, and Optics, Saint Petersburg, Russia
E-mail: [email protected]
The conditional evolution of the quantum system interacting with another one is
under investigation. The common state is entangled and the Kraus operators
formalism is used to study the evolution of the partially reduced system state. As an
example of the interacting subsystems we choose the quantum intracavity mode of
electromagnetic field and two level atom. Two cases are under investigation. In the
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first one we deal with the system is governed by Schrödinger equation and unitary
evolution. In the second one we study the situation subject to atom interacting with
the environment. For both cases we find the Kraus representations of the
corresponding evolution superoperators and estimate the information characteristics
of the indirect measurement process of quantum cavity mode.
Multiple-band superconductors: extended Ginzburg-Landau
formalism by a systematic expansion in small deviation
from the critical temperature
Alexei Vagov
Bayreuth University, Bayreuth, Germany
E-mail: [email protected]
We derive the extended Ginzburg-Landau (GL) theory for a multiple-band
superconductor employing a systematic expansion of the free-energy functional in
powers of the small deviation from the critical temperature. The two lowest orders of
this expansion yield the equation for the critical temperature and the reconstructed
GL theory. It is shown that in agreement with previous studies, the reconstructed
many-band GL theory effectively maps onto the single-band GL one and thus fails to
describe the difference in the spatial profiles of the condensates. We prove that
except for some very special cases, this difference appears in the leading correction to
the GL theory, which constitutes the extended GL formalism. We derive linear
differential equations that determine the leading corrections to the band order
parameters and magnetic field, discuss the validity of these equations, and consider
examples of an important interplay between the band condensates. Finally, we
present numerical results for the thermodynamic critical magnetic field and
temperature-dependent band gaps (at zero field), which are in a very good agreement
with the full BCS solution in a wide temperature range. The extended GL formalism
is applied for the studies of the interaction between vortices in the system. Analysis
of the possibility of the type 1.5 superconductors is presented.
Charge transport through the planar array of metal nanoparticles
Tigran Vartanyan
Saint Petersburg National Research University of Informational Technologies,
Mechanics, and Optics, Saint Petersburg, Russia
E-mail: [email protected]
An array of metal nanoparticles spread over the plane surface of a dielectric support
was obtained via vacuum thermal deposition followed by surface diffusion,
nucleation and growth of granular metal film. Two parallel electrodes made of
continuous metal films were painted over the particle array. The width of the gap
between the electrodes was in the range of several millimeters. The samples were
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annealed at 60 degrees centigrade up to a moment when their resistance jumps from
several kOhm to the values above TOhm. The large value of the resistance testifies
for the absence of the direct metallic paths connecting the electrodes. In this state the
samples demonstrate nonohmicity and hysteresis of current-voltage characteristics.
An interpretation of the observed phenomena requires the development of a theory
that accounts for the complex distribution of the currents and potential gradients
among the metal nanoparticles.
Phase diagram of the electron transition in the spin-polarized state
in one- dimensional quantum dot
Alexander Vasilchenko
Kuban State Technological University, Krasnodar, Russia
E-mail: [email protected]
Density functional theory is used to study the transition to the spin-polarized state of
electrons in quasi-one-dimensional quantum dot. We developed and implemented on
a computer efficient algorithm for the numerical solution of the Kohn-Sham
equations. Calculation results have shown that the transition to the spin-polarized
state of the electrons in the quantum dot happens at higher densities compared to the
same transition in the quantum wire. Phase diagram of the electron transition into the
spin-polarized state is constructed in the coordinates of length l and width d of the
quantum dot. In particular, the triplet state is energetically favorable for l > 30 nm for
GaAs quantum dot with two electrons and d = 10 nm.
Multiplexing of classical and quantum channels in optical
communication systems
Dmitrii Vavulin
Saint Petersburg National Research University of Informational Technologies,
Mechanics, and Optics, Saint Petersburg, Russia
E-mail: [email protected]
This paper investigates the transmission of quantum cryptographic key in optical
fiber communications. We consider the possibility of multiplexing the classical and
quantum signals in an optical fiber, taking into account emerging with the nonlinear
effects.
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A non-equilibrium system with repeated interactions
Valentin Zagrebnov
Aix-Marseille University, Marseille, France
E-mail: [email protected]
We consider a beam of randomly excited atoms that pass one-by-one through a onemode cavity. In the case of the ideal cavity (no leaking of photons) the pumping by
the beam leads to an unlimited increase in the photon number. For a leaky cavity we
prove that the mean photon number in it stabilizes in time. The limiting state of the
cavity exists and it is independent of the initial state. We calculate the characteristic
functional of this non-quasi-free non-equilibrium state. We also calculate the total
energy variation in both the ideal and the open cavities as well as the entropy
production in the ideal cavity.
Tunneling through a smooth potential barrier and localized states
in graphene monolayer with mass gap
Victor Zalipaev
Krylov Scientific Research Centre, St.Petersburg, Russia
E-mail: [email protected]
We present a semiclassical analysis of Dirac electron-hole tunneling in graphene
monolayer with mass gap through a smooth potential barrier in ballistic regime. This
1D scattering problem is formulated in terms of transfer matrix and treated in WKB
approximation. For a skew electron incidence this WKB approximation deals with
four turning points. Between the first and the second, the third and the fourth turning
points two tunnelling domains are observed. Scattering through a smooth barrier in
graphene resembles scattering through a double barrier for 1D Schrödinger operator
that is 1D Fabry-Perot resonator. The main results of the paper are very simple WKB
formulas for the entries of the barrier transfer matrix which explain the mechanism of
total transmission through the barrier for some resonance values of energy of a skew
incident electron. Moreover, we show an existence of modes localized within the
barrier and exponentially decaying away from it and its behaviour depending on mass
gap. These are two sets of discrete complex (with small imaginary part) and real
energy eigenlevels determined by Bohr-Sommerfeld quantization condition, above
and below the cut-off energy, respectively.
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Mathematical Challenge of Quantum Transport in Nanosystems
International Conference
Book of Abstracts
В авторской редакции
Компьютерный набор и верстка Блинова И.В., Попов И.Ю.
Дизайн обложки Попов И.Ю.
Тираж 60 экз.
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