1 Dynamics of Cold Atoms in Moving Optical Lattices by Nadal Sarkytbayev, BSc Physics Supervisor: Dr Alexander Balanov Senior Lecturer in Physics
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1
Dynamics of Cold Atoms in Moving
Optical Lattices
by
Nadal Sarkytbayev, BSc Physics
Supervisor: Dr Alexander Balanov
Senior Lecturer in Physics
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Content
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Theory 2.1 de Broglie Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2 Band Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Bloch Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Kronig and Penny Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Methods of Cooling 3.1 Compton Scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12 3.2 Laser Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.14 3.3 MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 15
4 Result and Discussion
4.1 Cold Atom Transport through miniband . . . . . . . . . . . . . . . . 16
4.1.1 Free Space model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16 4.1.2 Tilted Mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
4.2 Optical Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21 4.2.1 Atom Trajectory: Linear dispersion. . . . . . . . . . . . . . .
21 4.2.2 High Field regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3
4.2.3 Bloch Oscillation regime . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 Velocity of Optical Lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Abstract
This thesis studies the dynamics of cold atom and its transport in optical lattices. Two related but distinct systems are considered: periodic potential driven by an external force
(i.e. electric force and gravitational force); and optical lattice with additional moving potential caused by detuning of two counter propagating laser beams. For the second problem we investigated dynamics in terms of: displacement, average velocity and
momentum by varying the strength of the field and velocity of the optical lattice. We show fundamental cold atom dynamics, in particular, the transition between wave
dragging regime and Bloch oscillation regime which is extremely sensitive to field variations as it alters cold atomβs orientation.
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1 Introduction
This thesis is devoted to theoretical study of the dynamics of non-interacting cold atoms and their transport in periodic potential and optical lattice. Trapping and controlling cold
atoms in optical lattice with use of modern day lasers provides flexible quantum environment which makes it interesting subject for large number of theoretical and experimental purposes such as atom interferometry [1], quantum information processing
[2], Bose-Einstein condensation [3] and simulations of condensed matter systems [1,4]. The ability to control and manipulate the trapped cold atoms opens a venue in making
high precision measurements of length of displacement in terms of wavelength and as considered in Ref. [4] by using Bloch oscillations, which emerge from periodic potential, can be used as an instrument to make accurate measurement of gravity that depends on
number of Bloch oscillations detected. In this paper we study semi-classical particle dynamics inside the lattice with the
potential generated by several methods. Our problem in this thesis consists of investigation of dynamics of cold atoms and explore methods of their transport in optically driven lattice Dynamical behaviour of cold atoms under an influence of constant
potential were generated by applying uniform electric field and by tilting the system in which case gravity acts as constant force. On the other hand lasers were used as a tool for
both cooling method (see Section 3.1) and creating the counter propagating beams for periodic optical potential. The semi-classical approach has been taken solve the quantum behaviour of such atoms as the particles essentially demonstrate quantum tunnelling in
between the minibands (see Section 2.1) through the propagating potential. Also the Bloch oscillations, that corresponds to periodic motion thorough space for specific values
of the field applied, have been observed and investigated in details. We show that in spite the fact that atomic motion even in one-dimensional model can have complex behaviour we are able to predict its path and transport accordingly. Thus if we are able to control the
transport of cold atoms according to our requirements this contributes to multiple applications mentioned above [1,2,4].
In recent years there has been greater theoretical discussion and experimental interest into transport and dynamical examination of cold atoms in moving optical lattices [1-6], which makes the problem that we dealing with increasingly relevant to current
research areas of cold atoms and Bloch oscillations. Furthermore, in the case of the periodic optical potential mode its been discovered that the dynamics is identical to the
acoustically driven superlattice (see Ref. [9,25,26]) which suggest that the study of such problem provides a particle transport model applicable to multiple cases.
The contents of this thesis are briefly as follows. In the Section 2 (Theory) we
give an explanation to some fundamental theories and ideas that are used in this work when reflecting up on the effect of periodic potential on dynamics of the quantum
particles. Therefore explanation of Band theory will be helpful tool in understanding some basic features related to propagation of the wave and origins of Bloch oscillations in the system. Section 3 reveals the essential characteristics and methods used to create a
cold atom by optical molasses from basic theory of Compton scattering. Furthermore
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Section 4 is where the results are studied in details for those two different problems. Finally, Section 5 summarizes our conclusions with regards of our research.
2 Theory
Dynamical behaviour of ultra cold atoms under two different conditions where quantum
particle such as electron were under influence of constant potential generated by external force (such as electric force and gravity) and moving optical lattices in one-dimension by
two counter-propagating laser beams. The reason we consider an electron for our model of cold atom is because they both demonstrates many of the identical phenomena related to period potential [4].
2.1 de Broglie Hypothesis
We will start with some fundamental theory proposed by de Broglie, relating the energy
and momentum of a particle in discrete form, in order to move on to the Band theory. In the first quarter of the 20th century, French physics graduate student Louis de
Broglie, proposed that wave properties of a particle were characteristic of matter (i.e. electron) as well as the radiation [7]. De Broglie expressed mathematical equation that described frequency and wavelength relationship of an electron, know as de Broglie
relations:
π = πΈ
β (2.1-1)
π = β
π (2.1-2)
where E is the total energy, p is the momentum, π de Broglie wavelength and h is known as Plankβs constant. This is useful relation as we mostly dealing with electrons and
photons, which are massless bosons of integer spin, propagating in free space with speed of light. Photons have a momentum component that is used to relate to energy. From Einsteinβs quantization of radiation E=hf and for particle of zero rest energy E=pc
follows [7]:
πΈ = ππ = βπ = βπ
π (2.1-3)
where c is the speed of light in free space. As we are dealing with dual behaviour of matter and wave, classically it must have solutions of the wave equation. It is known that harmonic oscillatorβs properties:
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amplitude, frequency f and period T. Frequency and period also have useful relation with angular frequency and wave number respectively:
π = 2π
π (2.1-4)
and π = 2π
π (2.1-5)
A system in which phase velocity π£π is correlated with frequency (π/π not constant) is
known as dispersion relation that describe the change of angular frequency π on wave
number π. The velocity of maximum amplitude is known as group velocity: [8]
π£π = ΞΟ
π₯π =
dΟ
ππ (2.1-7)
since Ο=ππ£π
π£π = dΟ
ππ =
d(ππ£π)
ππ = π£π + π
d(π£π)
ππ
= π£π β πdπ£π
ππ (2.1-8)
where we have used de Broglie relationship for wavelength (1.1-5).
There are three cases for which group velocity π£π defines the mode of dispersion:
a) System where phase velocity π£π is not related to frequency (π/π is constant)
therefore π£π = π£π, for example propagation in free space (no dispersion).
b) Normal dispersion - where dπ£π
ππ is positive, hence π£π>π£π.
c) Abnormal dispersion β where dπ£π
ππ is negative, hence π£π<π£π.
At certain conditions of the applied field we were able to obtain dispersion relations for
atoms in optical lattices described in Section 4.
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2.2 Band Theory
From quantum theory, we know that electrons occupy quantized orbits and therefore form discrete energy levels. When the large numbers of atoms are brought close together to form a solid or crystal lattice, number of orbitals becomes inversely proportional to the
distance of near by energy levels making it very small. Therefore, energy levels are thought to form of continuous bands. [9]
The focal point of modern understanding of electronic and optical properties of solid materials is based on the Band theory, which provides the description of key features of electrons in periodic lattice [8]. Moreover Band theory provides understanding
of material properties as to why some materials are able to conduct electricity and others do not by considering the effect of the lattice on the electron energy levels. The band gap
or energy gap πΈπ is the difference in energy between lowest point of the conduction band
and the highest point of the valance band [10]. As the temperature increased, free
electrons are thermally excited from valance band to the conduction band (Figure 1).
Figure 1: Band scheme of for intrinsic conductivity in semiconductors. (Diagram taken from [11])
Both the electrons in the conduction and holes left behind in the valance band contribute
to the electrical conductivity, which determines the properties of a specific material. At temperature around 0 K the conductivity is zero because all the all the states in the valance band are full and all the states in the conduction band are empty.
Propagation of the wave in crystals is characterised by Bragg reflection of electron wave, which is caused by energy gap Eg. This is also know to be a forbidden zone where
electron can not stay due to quantum theory of discrete energy spectrum. These energy gaps are significant in determining whether a solid is an insulator or a conductor. For an
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insulator the valance band is fully filled so when electric field is applied it wonβt accelerate electrons, as there are no vacant valance bands available.[10]
On the other hand, in a conductor valance and conduction bands are only partly filled
electrons could be transmitted on passing through the energy gap.
Figure 2: Schematic electron occupancy of allowed energy gap for a metal, semiconductor and insulator. Blue areas indicate allowed energy regions; Red areas
indicate the regions with filled electrons; Fermi energy (πΈπΉ) is minimum amount of
energy required. (Diagram taken from [12]) When the electron channel through to the next available band it leaves a βholeβ behind
which allow the following electron to fill it up. Same process take place in the metal, as it is know that these bands overlap with each other constructing one single energy band
(Figure 2), which is only partially filled. Therefore, when electric field is applied there are sufficient amount of states available for electrons to accelerate to, meaning it is highly conductive.
Fermi energy is much greater than the average energy of the lattice ions, so electron
energy distribution in fact will not differ from that of temperature at absolute zero (T= 0 K), which is governed by Pauli exclusion principle that states only a single fermion can occupy a single state.
The following relation gives Fermi energy:
πΈπΉ = ππ΅ππΉ (2.2-1)
where ππΉ is Fermi temperature and ππ΅ is Boltzmann constant.
All the low-energy states are filled to certain Fermi energy and depend on density of that particular gas. Fermi energy can be equally thought as the increase of the ground-state
energy when a single fermion (e.g. electron) is added to the system. So by increasing the temperature, some of the electrons excited. But taking into account the fact that empty
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states are only available above the Fermi energy means that only s with energies close to πΈπΉ can be excited. Therefore some states above πΈπΉ, in a narrow energy range β ππ΅ππΉ, will
be occupied and others below πΈπΉ depleted.
2.3 Bloch Theorem
One other favourable result of quantum theory apart from Band theory described in
Section 2.2 is the solution to the problem of electrical transport in metals, semiconductors and insulators.[8]
The distribution of lattices across the space that are generated by potential V(r) is created by atoms in solid or by propagating laser beams in opposite direction in an optical lattice.
We use three translational vectors ππ, ππ and ππ named primitive lattice vectors to
describe the location of each lattice point relative to any other point in spatial lattice. [9] Combined vectors are given by:
πΉ = π’ππ + π£ππ + π ππ (2.3-1)
where u, v and s are integers. Bloch theory states that all the measurable properties of its wavefunction must show symmetrical behaviour for translational lattice because quantum
particles experiences periodic potential[10]. Therefore probability of locating a particle at particular point r must have the following equality:
|π(π + πΉ)|2 = |π(π)|2 (2.3-2)
from this it follows that:
π(π + πΉ) = πππ·(πΉ)π(r) (2.3-3)
where Ξ² is the linear function of R and can also be formulated as:
Ξ²(R) = ππ₯ π π₯+ ππ¦π π¦ +ππ§π π§ = k.R (2.3-4)
which is dimensionless, real and arbitrary function of R. Combing equations (2.2-3) and (2.2-4) gives us Blochβs theorem with wavefunction property:
ππ(π + πΉ) = πππ.(π+πΉ)π’π(π + πΉ) = πππ.(π)π’π(π)πππ.(πΉ)
= ππ (π)πππ.πΉ (2.3-5)
Bloch has proved the important theorem for a periodic potential by solving the SchrΓΆdingerβs equation and discovered it must be of particular form. Therefore product of
the plane wave πππ.π and function π’π(r), which has the identical periodicity as the applied
periodic potential gives us eigenfunction of the SchrΓΆdinger equation [8]:
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ππ(π) = π’π(π)πππ.π (2.3-6)
where π’π(r) is the function the periodicity that also has the symmetry in translation of the
lattice.
2.4 Kronig-Penny Model
One of the practical models for understanding how electrons conduct inside the materials
is the Kronig-Penny model. Using this model and Blochβs theorem we can compute the energy of the electrons as the wave number parameter k.
The Kronig-Penny model appears as the background periodic potential for the electrons in the crystals. When electrons travel through solid (e.g. a semimetal or metal), they run
across sequence of potential barriers produced by the atoms or ions (π +). These ions are
located in the centre of each successive barrier with a constant distance separating them. Assuming that distance between two ions is a, one-dimensional lattice array of ions looks like Figure 3 [8.2].
Figure 3: A one-dimensional periodic array of potential barriers formed by atoms or ions located along a crystal lattice. π+ indicate ions and a is the distance between ions
(Diagram taken from [13]).
The wave function for electron is derived using Bloch functions and the behaviour of individual electrons is determined using the Kronig-Penny Model, which adjusts curved
one-dimensional periodic array shown on Figure 3 to rectangular periodic series of potential well of finite depth represented on Figure 4. Kronig and Penney were able to
show how behaviour of electrons motion by making the wells deeper and reducing separation between them. [14]
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Once again, cold atoms in optical lattice duplicate many of the properties related with
electrons in periodic potential [D]. Kronig-Penny model for electrons with the constant periodic potential have simple approximate form represented below.
Figure 4 A series of finite potential wells used by Kronig and Penny as first approximation. The distance between the potential wells is a, thickness of the well is b
and its height is π0 (potential). (Diagram taken from [15]).
In Figure 4 the distance between the potential wells is a, thickness of the well is b and its height is π0 (potential) with multiple miniband. The problem can be considered as the
particle moving in in a square potential well of width a with its energy being less than
potential E<V. Inside the well the potential is zero and fixed height corresponds to the value of potential V.
Wave function is expected to have the form of an integral number of de Broglie
half wavelengths inside the well with addition of exponentially decaying penetration into the wall on both sides. [9]
Using the SchrΓΆdingerβs equations it is possible to solve the problem and find the results for all three regions of the well: 1) 0<xβ€a; 2) x>a and 3) x<0. *
ββ 2
2π
β2 π(π₯)
βπ₯2 +π(π₯)π = ππ (2.4-1)
where π(π₯) is the wave function of the quantum system, V(x) is the potential, m is the
mass of the particle, Ξ΅ is the energy eigenvalue and β is the reduced Plankβs constant
(β = β
2π = 1.054Γ10β34 J/s).
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This method of approximation is fairly accurate and simple to calculate with. Itβs been used in many cases in order to analyse: the barrier problem [16], 1D monatomic gas [17]
and other forms of the periodic potential.
*Solutions to the following cases can be found in the Appendix A
3 Methods of Cooling
In order to explore the dynamics of cold atom in periodic potential and optical lattice we have to lower the temperature of the atom down drastically, to about 1 nK (10β9K). How
is it done? This is possible to achieve by using two main technics described below, laser cooling and magneto-optical trap (MOT), which also something that is used to achieve
Bose-Einstein condensation [1] but we are not considering it here. The following two methods assumed to have been applied for all the cold atom results that we obtained and studied in Section 4.
However first we would like to explain how the process of photon (laser beam) and electron interaction allows us decrease the temperature of electron essentially by
slowing it down.
3.1 Compton Scattering
In order to explain the following Section 3.2 we would like to introduce the Compton Scattering effect to explain the nature of momentum transfer of particles and waves as well as the conservation of momentum.
In 1923 Arthur H. Compton have worked with the concept of photon concept to explain the results of his experimental data on scattering X-ray by electrons. Compton
came to a realization that during scattering only a fraction of the total energy of the photon is transferred to an electron during the collision. In contrast results shown in photoelectric-effect experiment where all the energy is transferred to the electron. As
stated by classical theory, if electromagnetic wave of frequency ππ is incident on material
containing free charges (e.g. metal) these electrons will oscillate with that same frequency and emit electromagnetic wave of the identical frequency. [9]
The assumption to these emitted waves as scattered photons and conservation of
energy and moment from classical mechanics, underline the behaviour of scattering process during photon-electron collision. During the collision the electron would absorb energy from incoming photon and recoil at a certain angle (see Figure 5). The scattered
photon ππ would have less energy and according to Planckβs relationship, lower
frequency and higher wavelength than incident photon ππ. Compton scattering along with Doppler cooling (Section 3.2) is the very useful technic for cooling the atom to ΞΌK.
From equation (2.1-3, Section 2.1) we know that momentum and wavelength is related by:
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π = β
π ; ππ =
β
ππ. (2.1-3)
Applying conservation of momentum to the collision gives:
ππ = ππ + ππ (3.1-1)
where ππ is momentum of incident photon, ππ momentum of electron after collision and ππ is momentum of recoil photon.
Figure 5: The scattering of light by an electron. The scattered photon has less energy
therefore a longer wavelength than incident photon (ππ > ππ). (Diagram taken from [18])
At the initial condition electron is at rest (Figure 5) therefore zero momentum. Rearranging equation (2.4-1) for electron momentum and taking the dot product each
side with itself gives:
ππ2 = ππ
2 + ππ2 - 2ππππcos (π) (3.1-2)
where ΞΈ is the angle of the direction of scattered photon makes with direction of motion
of incident photon. Due to the fact that the kinetic energy of the electron after the collision is the
significant amount of it the rest energy, the relativistic relationship used to associate the total energy E and momentum ππ of an electron. [9] This gives us:
πΈ = βππ2π2 + (πππ)2 (3.1-3)
where ππ is the rest mass of the electron β 9.1Γ10β34 kg and c is the speed of light.
Applying conservation of energy law to the collision gives:
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πππ + πππ2 = πππ + βππ2π2 + (πππ)2 (3.1-4)
Using equation (2.4-2) in order to eliminate ππ2 produces:
1
ππβ
1
ππ=
1
πππ [1 β cos(π)] (3.1-5)
and finally by substituting ππ and ππ by using the equation (2.1-2) for momentum and
wavelength relation we obtain Compton equation:
π₯π = ππ β ππ=β
πππ [1 β cos(π)] (3.1-6)
The main characteristic of the Compton effect is that change in wavelength π₯π does not
depend on initial wavelength of the photon ππ but rather on the angle of the scattering
[19].
3.2 Laser Cooling
First a sample of rubidium (π π87 ) [2] vapour is illuminated by circular polarized laser
beams from six different directions (optical molasses). Lasers, which essentially are
stream of massless photons each carrying a momentum, collide with rubidium atoms and slow them down. Polarized light pushes the atoms in opposite direction that provides
restoring force that allows us to trap the atoms (Figure 6). With in a few seconds millions of atoms are confined in small volume defined by the intersecting laser beams, of about 0.5-1cm in radius. The temperature of the laser-cooled sample is near 1mK (10β3 K).
[20]
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Figure 6: Experimental setup of laser cooling and MOT. Red arrows represent the laser
beams and blue rings represent magnetic coils that produce uniform magnetic field. Red cylinder in the middle is trapped atom (Diagram taken from [21])
Laser cooling can only be as low as Doppler limit temperature.
ππ· = βπΎ
2ππ΅ (3.2-1)
where h is the Planckβs constant, ππ΅ Boltzmann constant and Ξ³ is natural linewidth. Doppler cooling requires spontaneous emission of photons and therefore controlled by
natural life Ο of the excited state or equivalently is the inverse of natural line width (Ο =1/Ξ³). This is usually much higher than the recoil temperature ππ that relies on coherent
scattering of photons from first beam into the second with opposite direction and polarization. [22]
ππ = β 2
2πππ΅ π2 (3.2-2)
where ππ΅ Boltzmann constant, β is the reduced Plankβs constant, Ξ» is the wavelength of
light and m is the mass of an atom.
3.3 Magneto Optical Trap (MOT)
Doppler limit was overcome by applying non-uniform magnetic field. Magnetic trap is
used to control the enclosed area of trapped atoms. This technic cools the atoms down
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even lower than milli-Kelvin as it allows the atoms with higher kinetic energy (hotter) to escape leaving the ones with lower kinetic energy behind. Therefore evaporative cooling
takes place - cooling the remaining couple of thousands atoms down to 100 nK (10β9 K).
[20] Simpler example can also be though as everyday cup of hot tea in which water
molecules vaporising from top surface leaving the cold ones inside the cup. These remaining cold atoms move down into the lowest energy level (ground state) of finite potential and reach absolute zero. At that temperature new state called Bose-Einstein
Condensate, is achieved. The main reason of how MOT works lies in method of Doppler Cooling.
Polarized beams of light are tune to a certain frequency below atomic resonance frequency that allows photons from laser beams to be absorbed. The absorption and emission of photons transfer the momentum of the atom [1], which reduces the velocities
therefore lowering the temperature. These atoms have greater chance to absorb photons with energies that equals to the difference in their atomic energy levels [23]. For that
reason it is highly important to maintain the laser beams tuned at a certain frequency.
4 Results and Discussion
In this chapter we are using semi-classical approach to solve our problem and we obtain analytical expressions to the dynamics of cold atom. First we consider model of particle
transport in periodic potential driven by an external force such as electric force and examine its tunnelling through the potential. We then adjust the system by tilting it at an angle in which case gravitational force plays a crucial role as it generates new periodic
potential and we obtain a different periodic potential mode that generates oscillatory behaviour. We examined the dynamics by looking at momentum ππ₯ and displacement π₯
variation with time by using the Matlab simulations to verify the theoretical assumptions. Also we provide additional analysis of phase momentum variation with displacement.
After that we move on to the optical lattice, which is simply a set of standing waves counter-propagating laser beams that are detuned in such manner that it creates
driving motion. The same semiclassical approach to the Hamiltonian π»(π₯, ππ₯ ) as for the particle is applied for the cold atom problem due to their homogenous characteristics.
Kinetic energy πΈ(ππ₯ ) and potential energy π(π₯) correspondence change and we explore
dispersion relation and Bloch oscillations for different regimes by varying the applied
field ππ and velocity of optical lattice π£π parameters. In order to understand cold atomβs
transport we introduce average velocity < ππ > of the particle to quantitatively and
qualitatively relate it to the variation of the field ππ and optical lattice velocity π£π.
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4.1 Cold Atom Transport through miniband
By varying the speed or amplitude of the moving lattice produces Bloch oscillations along with high uniform electric field. From theoretical discussion two outcomes have verified Bloch oscillation and produced matching behaviour. In the results discussed
below one-dimensional propagation of the particle behaviour was considered.
4.1.1 Free Space model
Particle behaviour in free space corresponds to high velocity regime with kinetic energy given as:
πΈ(ππ₯) = π2
2π (4.1.1-1)
where πΈ(ππ₯ ) is the kinetic energy of the Hamiltonian π»(π₯, ππ₯ )= π(π₯) + πΈ(ππ₯ ) ππ₯ is the
momentum in the x-plane and m is the mass of the particle.
π(π₯) = βπΉππ₯ (4.1.1-2)
where π(π₯) is the potential energy of the Hamiltonian π»(π₯, ππ₯ )= π(π₯) + πΈ(ππ₯ ), πΉπ is the
electric force and π₯ is the displacement along x-axis.
Therefore Hamiltonian of the form:
π»(π₯, ππ₯) = ππ₯
2
2π β πΉππ₯ (4.1.1-3)
Free Space model for such Hamiltonian can visualized as Figure 7(a).
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Figure 7: (a) Transport of the particle through the miniband in 1D band structure. (b) Transport of the particle through the miniband in 1D band structure with a tilt due to
presence of gravitational force πΉπ. Where d is the lattice period, Ξ is the bandwidth and πΉπ
is the electric force. Figure 7(b) shows schematic of how the free particle model is now adjusted to have
gravity (πΉπ) acting when the system is being moved into sloping position. From Ref.[9] it
has been revealed that maximum output was achieved in the range of the tilt angle between 45 and 80 degrees. The potential energy is due to be invariable for this case π(π₯) = βπΉππ₯. On the other
hand kinetic energy πΈ(ππ₯ ) is proportional to sinusoidal configuration that is analysed in
Section 4.1.2. That change of kinetic energy is one of the aspects in determining precise measurement of gravitational acceleration π in Bloch oscillation [1].
Ξ
βπΉππ₯
Ξ
d (a)
(b)
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By adjusting the height of the energy minibands Ξ it is possible to control the rate at
which particle tunnels through each successive well [24]. But for the purpose of our investigation we left lattice period d and bandwidth Ξ values constant.
4.1.2 Tilted Mode
Conduction of the particle in the periodic potential has changed its behaviour and induced Bloch oscillations inside each miniband. Bloch oscillations arise from two nearby
potentials interactions with each other. From dispersion-relation for miniband, kinetic energy is of the form [25]:
πΈ(ππ₯) = π₯
2[1 β cos (
ππ₯π
β )] (4.1.2-1)
where E(ππ₯) is the kinetic energy of the Hamiltonian (H(x, ππ₯)= π(π₯) + πΈ(ππ₯ )), Ξ
bandwidth, β is reduced Plankβs constant (1.05Γ10β34 J s), ππ₯ is the momentum and d is
the distance between two walls of the band or just lattice period. Dispersion relation is of the form of Fourier series but only first term is significant enough for measurement so
equation (4.2.-1) is a rational approximation.
Potential energy was left unchanged and kept is it form still produced different
Hamiltonian to solve:
π»(π₯, ππ₯ ) = π₯
2[1 β cos (
ππ₯π
β )] β πΉππ₯ (4.1.2-2)
In order to analyse the theoretical prediction we initially solve the Hamiltonian as the
system of simultaneous equations for displacement and momentum such as:
ππ₯Μ = β βπ»
βπ₯ (4.1.2-3)
and
οΏ½ΜοΏ½ = β βπ»
βπ (4.1.2-4)
The momentum behaviour produces invariant results for free space and tilted modes:
π1 = π2 = π0 + πΉππ‘ (4.1.2-5)
where π0 is the initial momentum of an the particle.
But displacement of two regimes was certainly contrasting:
20
π₯1 = π₯0 + π1π‘
π (4.1.2-6)
where π₯1 corresponds displacement of the particle in free space model and π₯0 is the initial
position.
and
π₯2 = π₯0 + cos ( π0π
β ) β cos (
π2π
β ) (4.1.2-7)
where π₯2 corresponds displacement of the particle in tilted mode and π₯0 is the initial position.
Momentum was determined to have a linear relationship where as displacement appear to show sinusoidal shape. For simplicity displacement was reduced to dimensionless form
by interchanging variables (ππ΅ = ππΉπ
β , π =
π0π
β , A =
Ξ
2ππΉ and Ο = ππ΅ π‘) such as:
π₯2 = π₯0 β π΄cos (π + π) + π΄cos (π) (4.1.2-8)
This is of the form of simultaneous differential equations and has been worked out using
Matlab special solver ode45. This function evaluates the right hand side of the differential equation and solves the system of equation in the form yβ = f(t,y).
Figure 8: Numerical and Analytical graph for Momentum solved by ode45.
The increment of each step has been adjusted and produced two sets of data, numerical
and analytical predictions for both displacement π₯ and momentum ππ₯ variation with time
(figure 8 and figure 9). On the figure 8 we run the Matlab for two functions, one for numerical result π1 = π2 = π0 + ππΉπ‘ (equation 4.1.2-5) with atom being initially at rest
(π0 = 0) and equation (4.1.2-4) by using ode45 solver to obtain the analytical set of
results. Where as the figure 9 represents numerical and analytical results for displacement variation with time equations (4.1.2-7) and (4.1.2-3) respectively.
0 5 10 15 20 25 300
5
10
15
20
25
30
35
Time / [ms]
Mo
me
ntu
m P
x /
[k
g m
m m
s]
Numerical
Analytical
21
Figure 9: Numerical and Analytical graph for Momentum solved by ode45.
It is evident from two graphs that our analytical calculation is consistent with numerical curves that we predicted for displacement and momentum behaviour of and atom in periodic lattice. The displacement (figure 8) clearly perform Bloch oscillation with an
amplitude of its displacement A = 2 mm and period of oscillation t β 6 ms. Therefore increasing either the speed or amplitude of the moving lattice induces Bloch oscillation
with in the energy band πΈπ . By introducing an additional moving periodic potential the
dynamics of atoms through stationary optical lattice have been analysed in this paper,
which is discussed in the Section 4.2. [26]
0 5 10 15 20 25 300
0.5
1
1.5
2
Time / [ms]
Dis
pla
ce
men
t X
/ [
mm
]
Numerical
Analytical
22
4.2 Optical Lattice
In this section we are investigating the dynamics of cold atom in original optical lattice (free space model) but with an additional moving potential applied. This extra potential
has been a subject of calibrating two counter propagating laser beams.
4.2.1 Atom Trajectory: Linear dispersion
In the following Section 4.2.2, we look at the atomβs displacement, velocity and momentum behaviour in the low and high ππ regime.
Same argument as before are used to derive the following one-dimensional semiclassical
Hamiltonian [9]:
π»(π₯, ππ₯) = πΈ(ππ₯) + ππ(π₯, π‘) (4.2.1-1)
where πΈ(ππ₯ ) is the dispersion relation of the optical lattice and ππ (x,t) is the moving optical lattice potential. By solving the time independent Schrodinger equation we obtain
the form for kinetic energy πΈ(ππ₯ ) as equation(4.1.2-1):
πΈ(ππ₯) = π₯
2[1 β cos (
ππ₯π
β )] (4.2.1-2)
where π₯ is the bandwidth of the 1st energy band and d is the spatial period of the optical
lattice. The potential that atom will experience in the moving optical lattice ππ(π₯, π‘) is
time and coordinate dependent function and is:
ππ(π₯, π‘) = ππ
2 β
ππ
2sin (πππ₯ β πππ‘) (4.2.1-3)
where ππ = 2π
ππ is the wave number of the propagating lattice of wavelength ππ, ππ =
π£πππ is the frequency of the optical lattice (by which we determine the period of
oscillation) and ππ is the optical amplitude. Substituting both equations (4.2.1-2) and
(4.2.1-3) into the Hamiltonian gives:
π»(π₯, ππ₯) = π₯
2[1 β cos (
ππ₯π
β )] +
ππ
2(1 β sin(πππ₯ β πππ‘) (4.2.1-4)
which is of the same form as equation (4.1.2-2) but with extra optical lattice parameter.
23
Using equation (4.3-4) and applying the differential form of (4.2-3) and (4.2-4) we obtain equations of motion for the Hamiltonian as:
βππ₯
βπ‘=
ππππ
2cos(πππ₯ β πππ‘) (4.3-5)**
βπ₯
βπ‘=
π₯π
2β sin (
ππ₯π
β ) (4.3-6) **
Trajectories of cold atom were obtained by using numerical integration of Runge-Kutta routine (4th order) [27]. The numerical integrations of the simultaneous equations (4.3-5)
and (4.3-5) were solved by using Matlab ode45 solver (see Appendix C) that computed displacement and momentum trajectory. Figure 10 represents atomβs characteristics with
initial condition at time π‘ = 0 being stationary (π₯ = 0) and having no initial momentum π0 = 0 within low field regime ππ= 1 peV. We assume that there is no random jiggle of a
particle so overall movement is characterized by average velocity < π£π₯ > parameter
instead of drift velocity. For this particular system we have analysed momentum and displacement behaviour for the following parameters of: bandwidth π₯ = 24.35 peV
(β3.9Γ10β30 J) and lattice period d = 294.5 nm which were taken from experiment [28].
Figure 10.1: atom trajectory, average velocity and momentum respectively for ππ= 1
peV and π£π = 2.5 mm/s.
The figure 10.1 show how atom propagates through space is in dispersion mode. As a result atom is travelling in x-direction with positive gradient and particle starts to accelerate and increase its oscillatory average velocity and momentum with small
amplitude, < π£π₯ > β Β±0.2 mm/s and ππ₯=Β±0.1 kg m/s respectively. The particles dynamical components (displacement π₯, average velocity < π£π₯ > and momentum ππ₯)
highly dependent on the conditions we set for the field ππ and optical lattice velocity π£π .
Let us first see their relation with the field ππ and then we will study closer its
displacement correspondence to the optical lattice velocity π£π in Section 4.3.
**Solutions to the following equation can be found in the Appendix B
24
If we were to increase the field ππ by a magnitude of 40 (figure 10.2) particles
displacement, average velocity and momentum change substantially. In particular the displacement of the particle went from positive to negative slope meaning that it changed
its trajectory from one direction to another. Average velocity of the particle is periodically constant for brief amount of time and momentum starts to loose its smooth correspondence at such field (ππ=40 peV).
Figure 10.1: atom trajectory, average velocity and momentum respectively for ππ= 40
peV and π£π = 2.5 mm/s.
That being the case lets us look deeper to particles trajectory connection associated with
the magnitude of the field ππ . Increasing the field to the first critical point ππ1 β 2 peV
(figure 11) accelerates its average velocity and hence momentum to Β±0.5 mm/s and Β±0.2
kg m/s respectively. Also the amplitude of the displacement at that point rises reaching its maximum of 4 ΞΌm after 12 ms.
Figure 11: shows atom trajectory, average velocity and momentum respectively for ππ= 2 peV and π£π = 2.5 mm/s.
25
Increasing the field to the second critical point ππ2 β16 peV continues the trend of
increasing amplitude of displacement, velocity and momentum. But the oscillations of
average velocity < π£π₯ > start to take a new more complex shape (figure 12 (blue)).
Figure 12: atom trajectory, average velocity and momentum respectively for ππ= 16
peV and π£π = 2.5 mm/s.
Figure 13 show the distribution of average velocity < π£π₯ > depending on how ππ
changes for wavelength ππ= 20d and π£π = 2.5 mm/s.
Figure 13: the distribution of average velocity < π£π₯ > depending on how ππ changes for
wavelength ππ= 20d and velocity of optical lattice π£π = 2.5 mm/s.
The figure 13 shows that for very small values of ππ (from 0 to β2 peV) average < π£π₯ >
velocity behaves exponentially until it reaches its first critical value ππ1 β2 peV, at which
ππ1
ππ2
π1 π2
26
point < π£π₯ > = π£π = 2.5 mm/s which is consistent with what Greenaway has also
achieved [9]. As we increase the value of ππfurther the average velocity is being
unaffected and remains almost constant until ππ = ππ2 β 17 peV, second critical value.
After which average velocity decreases significantly. The reason for the particle having the same value of average velocity as the velocity of optical lattice (< π£π₯ > = π£π = 2.5
mm/s) is due to it being in wave dragging regime for ππ1 < ππ < ππ
2 . Potential generated by the moving optical lattice traps the atom (see Section 3.2) and hence propagates through stationary optical lattice at average velocity equal to velocity of the
potential (< π£π₯ > = π£π). Further increase of ππallows the particle to perform Bloch oscillations the high field regimes are indicated by peak values π1β 75 peV and π2β 150
peV, which were analysed and will be described in the Section 4.2.2.
4.2.2 High Field regime The Bloch oscillations only start show their shape after we raised the field beyond the
second crirtical point ππ > ππ2 . On top of that the behaviour of atom becomes more
sensitive to small variations of field ππ . We looked at the atomβs dynamics and
sensitivity near the first peak π1 > π > π1. From figure 13 it is clear that average velocity of the particle is decreasing its value until about ππβ 65 peV where curves start to pick
up a positive gradient again.
Figure 14: atom trajectory, average velocity and momentum respectively with the high field regime for ππ= 65 peV.
For that reason the gradient of the dispersion curve is now negative (Figure 14) and the
particle travels in the negative x-direction. This is due to Bragg reflection caused by energy gap Eg (see section 2.1). The particle continues to accelerate with negative
gradient achieving its maximum average velocity of < π£π₯ > β Β±2.1 mm/s. The increase
of velocity allows the atom perform more frequent oscillations but with smaller amplitude of displacement. Further increase of the field only decreases its amplitude and
period of oscillation of displacement until it generates high field regime and atom performs Bloch oscillations (see section 2.2).
27
Figure 15: atom trajectory, average velocity and momentum respectively with the high field regime for ππ= 75 peV.
It is evident from Figure 15 that characteristics of displacement π₯ and average velocity <π£π₯ > begin to take a compound form when the field is at the first critical point π1. The
gradient of dispersion curve for displacement becomes almost zero which suggest that atom travels in a straight line along the x-axis with amplitude of average velocity being <π£π₯ > = Β±2 mm/s with maximum momentum ππ₯= 10 kg mm/s.
Figure 16: distribution momentum ππ₯ versus displacement for optical velocity π£π=2.5
mm/s and field ππ=75peV (first peak value π1).
Another way of representing the correlation between the momentum and displacement is to plot phase trajectory graph (figure 16), which visualizes tangled oscillatory behaviour. The graph has quite a distinct shape, which is symmetrical about π₯β1.2 mm. It can also
28
be observed that along the displacement the trajectory is mostly in the positive side of the axis, which suggest that particle is travelling with positive gradient.
Figure 17 represents the first critical value of the field π1 that generates the Bloch oscillation that was deduced from Figure 13βs first peak. At this point the dynamics of
atom is easily affected to slight changes of the field. Increasing the field further by only 1 peV (ππ= 76 peV) changes the slope from zero to negative value hence particle travels
along the x-axis with the negative gradient.
Figure 17: atom trajectory, average velocity and momentum respectively with the high field regime for ππ= 76 peV.
Furthermore the slope of displacement curve constantly changes affecting the particleβs
dynamics for ππ β₯ 75 peV. The analysis for the Bloch oscillatory regime is discussed
and evaluated further in Section 4.2.3 for second critical value of the field π2 that has more rigorous fluctuations of displacement curve with time.
4.2.3 Bloch Oscillation regime
29
From figure 13 increasing the field ππ further towards the peak values π1 and π2 allows
the atom to perform Bloch oscillations in the high field regimes. Figure 18 shows even sharper alteration of oscillation for displacement of an atom over short period of time (12
ms). The fast oscillations break up by abrupt jumps. This pattern is precisely the one found for the particle Bloch oscillations driven by an acoustic wave case in Greenawayβs thesis [9] and consistent with one described by Stockhofe and Schmelcher [29].
Figure 18: atom trajectory with the Bloch oscillation regime with the velocity of optical lattice π£π = 2.5 mm/s for: [a] ππ= 145 peV and [b] ππ= 170 peV. Figure 18 correspond to Bloch regime around second peak value π2>P >π2. On Figure 20
[a] ππ= 145 peV atomβs trajectory generally have the same shape as [b] ππ= 170 peV.
Both perform Bloch oscillations with abrupt jumps. For ππ= 145 peV overall atom
spends more time with positive velocity than negative velocity which generates a forward jump with positive average velocity. On the other hand ππ= 170 peV we have average
velocity being negative, meaning that atom spends more time with negative velocity than positive velocity that generates a backward jump.
Once again another way of representing the correlation between the momentum and displacement is to plot phase trajectory graphs (figure 19 and figure 20). Using the
topology of that trajectory we can explain the forward and backward jumps that appeared
0 2 4 6 8 10 12
0
0.5
1
1.5
t / [ms]
X / [
mm
]
[a]
[b]
30
in figure 18. From figure 19 it can be seen that the particle shows identical behaviour to the distinct pattern that appeared in figure 16 but since the field ππ is almost doubled it
seems that the pattern repeated it self twice. It is symmetrical about π₯β0.1 mm and it is
evident that it spends slightly more time in the positive side of the displacement axis that suggest that the particle moves with the positive gradient, which is what is show on the
figure 18[a].
Figure 19: distribution momentum ππ₯ versus displacement for optical velocity π£π=2.5 mm/s and field ππ=145 peV (second peak value π2).
Figure 20: distribution momentum ππ₯ versus displacement for optical velocity π£π=2.5
mm/s and field ππ=170 peV (second peak value π2).
Similarly on the figure 20 the symmetry appears around π₯β-0.8 mm that means that
particleβs displacement marginally is in the negative side of the axis, which means that it moves in along the x-axis but with the negative slope. This was shown on figure 18[b].
4.3 Velocity of Optical Lattice ππ΄
31
We would now like to see the effect of velocity of optical lattice π£π on the system, which
can be adjusted by varying the frequencies of two counter propagating laser beams. We show that the interaction between the two lattices produce a large number of resonances
in transport properties of the cold atoms. By changing the parameter of optical lattice velocity π£π totally changes the characteristics of the field average velocity < π£π₯ >. The
figures 17-21 (next two pages) represent the behaviour for average velocity < π£π₯ > for: 2,3,4,6 and 8 mm/s with the exactly the same values of wavelength ππ= 20d, bandwidth
π₯ = 24.35 peV and lattice period d = 294.5 nm
Figure 17: distribution of average velocity < π£π₯ > depending on how ππ changes for
velocity of optical lattice π£π = 2 mm/s.
When initial value of optical lattice velocity π£π is being adjusted (figure 17 and figure 18)
the behaviour is not that dissimilar from when it was π£π = 2.5 mm/s other than having
more peaks for lower range (figure 17). In both cases atom accelerates exponentially reaching its maximum of average velocity < π£π₯ > = 2.5 mm/s which held constant for
some period of the field ππ for figure 17 its roughly Ξππβ20 peV and for figure 18 it is
Ξππβ5peV.
32
Figure 18: distribution of average velocity < π£π₯ > depending on how field ππ changes
for velocity of optical lattice π£π = 3 mm/s.
As the optical lattice velocity π£π increases the region of constant velocity becomes shorter (Ξππ β 0) and later as we keep on increasing velocity of the optical lattice π£π
further the average velocity< π£π₯ > starts to drop and it projects much smoother trajectory
(see figure 19-21).
Figure 19: distribution of average velocity < π£π₯ > depending on how field ππ changes
for velocity of optical lattice π£π = 4 mm/s.
33
On both figures 19-20 maximum average velocity < π£π₯ > decreases from initial value of
2.5 mm/s (when the optical velocity π£π = 2.5 mm/s) and as the field ππ increases from 0
to 200 peV the graphs show less peaks and hence transition from positive to negative gradient requires greater input of the field to achieve the next peak.
Figure 20: distribution of average velocity < π£π₯ > depending on how field ππ changes for velocity of optical lattice π£π = 6 mm/s.
When the optical lattice velocity raised to a value π£π = 8 mm/s maximum average
velocity is only about a third of its initial value < π£π₯ > β0.75 mm/s and clearly free from
any abrupt behaviour that it used to show before. Also on the previous three graphs (figure 19-21) there is no more wave dragging regime for which average velocity < π£π₯ >
has one to one correspondence with velocity at which optical wave is travelling (<π£π₯ >β π£π). This is due to the detuning of the counter propagating lasers that do not
Figure 21: distribution of average velocity < π£π₯ > depending on how field ππ changes
for velocity of optical lattice π£π = 8 mm/s.
produce previously shown constructive interference which lets us achieve maximum
possible average velocity of < π£π₯ > = 2.5 mm/s.
From above discussion we now have several ways of analysing the trajectory and treating the atom depending on how we would like it to behave. If we would like to transport the atom with a constant average velocity would be preferable to use low field
34
value 1peVβ€ππβ€10peV with velocity of optical lattice 2 mm/sβ€π£πβ€3 mm/s. On the other
hand if we want to suddenly accelerate or decelerate the atom we can use one of the peak value that will increase or decrease the resonant oscillations and speed up or down the
atom in a certain direction. Moreover when using the peak values to change the velocity profile of the atom we have to be careful with the precise value we are using because as show in Section 4.2 small variations of the field near the peak values could drastically
change the direction in which atom travels due to forward and backward jumps. For that reason, detuning counter propagating lasers beams in such manner that velocity of optical
lattice travelling at about 6 mm/sβ€π£πβ€8 mm/s will benefit the steady acceleration and deceleration profile of cold atom transport.
35
5 Conclusion
In this thesis, first system was studied as a simple model of particle in periodic potential driven by external force that helped us understand origins of the Bloch oscillations, which
then lead us to our main problem of transport and dynamics investigation of cold atoms under an influence of secondary periodic potentials. At first we have examined the effect
of periodic potential generated by the electric force and then tilted that system to induce the effect of gravitational force to the kinetic energy of the Hamiltonian. Analytical and numerical results showed the Bloch oscillations behaviour appearing in the conduction of
the particles through the miniband.
After that we investigated the dynamics of cold atom in original optical lattice but with
the additional moving potential. Secondary potential been attained by detuning two counter propagating laser beams. This time potential energy of the Hamiltonian has been change, which had to be, solved analytically using Matlab ode45 solver. The trajectory of
the cold atom has been analysed for large range of the field and also for some values of the optical lattice velocity. We have established certain conditions that allow us control
and manipulate cold atom trajectory for our purposes. We found that there are abrupt jumps in Bloch oscillation regimes where as wave dragging regime perform steady evolution of displacement with time. The dynamics of the particle is very sensitive to
slight variations of the field that causes the atom to move in opposite directions. For instance concluded that we can transport the particle with constant velocity, which equals
to the velocity of the propagating optical lattice that we applied and then we can also accelerate or decelerate it further all by changing the field to some critical values.
Therefore this supplies us with a technique for transporting cold atoms that allows us to
move the particle with the precise velocities and directs it to particular locations. This is in fact what is necessary in quantum information processing [2], condensed matter
simulations [1,4] and Bose-Einstein condensation [3].
Finally the dynamics analysed in this paper is analogues to those calculated for electrons in acoustically driven superlattice [2,9,14,15]. Therefore the cold-atom system that we
have described also supplies a quantum simulator for acoustic modes of the wave.
Acknowledgments
I would like to gratefully appreciate A. Balanov for his guidance and intellectual stimulation through my final project, with whom I had a pleasure to collaborate throughout the past few years at Loughborough.
36
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semiconductors/britney.gif [12] Web Image: http://upload.wikimedia.org/wikipedia/commons/thumb/0/0b/ Band_gap_comparison.svg/2000pxBand_gap_comparison.svg.png
[13] Web Image: http://en.wikipedia.org/wiki/Particle_in_a_one-dimensional_lattice [14] H. J. Pain. βThe Physics of Vibrations and Wavesβ. 6th Edition.
[15] Web Image: http://en.wikipedia.org/wiki/Particle_in_a_one-dimensional_lattice [16]Bruce H. J. McKellar and G. J. Stephenson, Jr. βKlein paradox and the Dirac-Kronig-Penney model. Physical Review A (1987)
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http://massey.dur.ac.uk/resources/grad_skills/LaserCooling.pdf
37
[23] Eric S. Muckley βConstructing A Magneto-Optical Trap For Cold Atom Trappingβ. California Polytechnic State University 2009.
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[25] M. T. Greenway, A. G. Balanov, D. Flower, A. J. Kent and T. M. Fromhold. Physical Review B β Using acoustic wave to induce high-frequency current oscillations in superlatticesβ 2010.
[26] M. T. Greenway, A. G. Balanov and T. M. Fromhold. Physical Review A βResonant of cold-atom transport through two optical lattices with a constant relative
speedβ 2013. [27] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical recipies in C. Cambridge University Press, 2nd edition, 1996.
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38
Appendix A
ββ 2
2π
β2π(π₯)
βπ₯π2 +π(π₯)π = ππ (2.3-1)
1) In the region 0<x<a where π = 0 the eigenfunction is a linear combination,
π = π΄ππππ₯ + π΅πβπππ₯ (A-1)
with energy Ξ΅ = β 2π2
2π (A-2)
2) In the region x>a where π = π0
π = πΆπβππ₯ (A-3)
with energy π0 β Ξ΅ = β 2π2
2π (A-4)
3) In the region x<a where π = π0
π = π·πππ₯ (A-5)
with energy π0 β Ξ΅ = β 2π2
2π (A-6)
39
Appendix B
βππ₯
βπ‘=
ππππ
2cos(πππ₯ β πππ‘) (4.3-5)
βπ₯
βπ‘=
π₯π
2β sin (
ππ₯π
β )
(4.3-6)
Using substitution:
π =ππ₯π
β (B-1)
π = πππ₯ β πππ‘ (B-2)
π = πππ‘ (B-3)
All three substitutions are in dimensionless units.
Therefore using (B-1) and (B-3) equation (4.3-5):
dππ₯
dπ‘=
dπ
dπ‘
β
π=
dπ
dπ
β ππ
π=
ππππ
2cos(πππ₯ β πππ‘)
hence dπ
dπ=
πππππ
2β ππcos(πππ₯ β ππ π‘)
and since π£π = ππ
ππ and using (B-2) we get
dπ
dπ=
πππ
2β π£πcos(π)
(B-4)
So consequently using (B-2):
dπ₯
dt=
d(π+πππ‘)
dt ππ=
d(π)
dt
1
ππ +
ππ
ππ =
π₯π
2β sin (
ππ₯π
β )
using (B-1) :
dπ
dπ ππ
ππ=
π₯π
2β sin(π) β
ππ
ππ
40
dπ
dπ =
π₯πππ
2β ππsin(π) β 1
since π£π = ππ
ππ
dπ
dπ =
π₯ π
2β π£πsin(π) β 1 (B-5)
Two simultaneous equations used to calculate the displacement and momentum dynamics using Matlab function ode45:
dπ
dπ=
πππ
2β π£πcos(π)
(B-4)
dπ
dπ =
π₯ π
2β π£πsin(π) β 1 (B-5)
41
Appendix C
Function βSopticβ is of the for of simultaneous equations (4.3-5) and (4.3-6).
function dS = Soptic(t,S) dS = zeros(2,1);
h = 6.62606957e-34; % J
hbar = h/(2*pi); % J eVJ = 1.602176565e-19; % J delta = 24.35e-12*eVJ ; % J
d = 294.5e-9 ; % m Vm = 25e-4; % 2.5 mm/s
lambdaM = 20*d; % m Km = 2*pi/lambdaM; % /m Wm = Vm*Km; % /s
Um = 76e-12*eVJ; % J % tau = Wm*t;
Ix = delta*d/(2*hbar*Vm); Yp = d*Um/(2*hbar*Vm);
dS(1)= Ix*sin(S(2))-1; % Displacement dS(2)= Yp*cos(S(1)); % Momentum
end The second script is the one which uses ode45 solve to run the function βSopticβ.
tspan = 0:1e-4:32; %Wm - few Wm's...
options = odeset('RelTol',1e-10,'AbsTol',1e-16); [tau,S] = ode45(@Soptic, tspan, [0 , 0], options);
% ode45 - better
%ode23s, ode23t, ode23tb - takes a very long time %ode15s - okay
figure; subplot(3,1,1);
plot(tau*1000/Wm,((S(:,1)+tau)/Km)*10^6,'-g'); ylabel ('X / [\mum]'); subplot(3,1,2);
plot(tau*100/Wm, (delta*d/(2*hbar*Vm)*sin(S(:,2))), '-b'); ylabel('<V> / [mm/s]')
subplot(3,1,3); plot(tau*100/Wm, S(:,2),'-r');
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ylabel('Px / [Ns]') xlabel (' t / [ms]'); % 10-3
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