Theses supervised by Doç. Dr. Mehmet Özgür Oktel
12 theses · İhsan Doğramacı Bilkent University
Optik örgülerde parçacığın adiyabatik ve adiyabatik olmayan davranışı
The cold atom experiments provide a clean and controlled environment for realizing many body systems. Recent realizations of artificial gauge fields and adjustable optical lattices paved the way for the study of effectively charged particles with neutral atoms in various lattice and continuum systems. Moreover, it is possible to precisely control the external system parameters, i.e. the artificial gauge fields much faster or slower than the time scales associated with atomic motion in the lattice. It still needs further analysis to fully understand how the adiabatic and non-adiabatic changes affect the stationary and dynamical behavior of the system. We first investigate the effect of the adiabatic changes in the artificial gauge fields, and focus on the famous problem: A charged particle in a periodic potential under magnetic field. This simple system leads a complicated and involved self-similar energy spectrum, the Hofstadter butterfly. The whole structure of this energy spectrum is determined by the lattice geometry as well as the external field. In this regard, we consider all possible Bravais lattices in two dimensions and investigate the structure of the Hofstadter butterfly as the different point symmetry groups of the lattices are adiabatically deformed from one into another. We find that each 2D Bravais lattice is uniquely mapped to a fractal energy spectrum and it is possible to understand the interplay between the point symmetry groups and the energy spectrum. This beautiful spectrum, in addition, consists of infinitely many topologically distinct regions as a function of magnetic flux and gap number. The topological character of energy bands are determined through their Chern numbers. We calculate the Chern numbers of the major gaps and Chern number transfer between bands during the topological transitions. In the second part, we investigate the dramatic effect of the non-adiabatic changes in the artificial gauge fields. In a synthetic lattice, the precise control over the hopping matrix elements makes it possible to change this artificial magnetic field non-adiabatically even in the quench limit. We consider such a magnetic-flux quench scenario in synthetic dimensions. Sudden changes have not been considered for real magnetic fields as such changes in a conducting system would result in large induced currents. Hence we first study the difference between a time varying real magnetic field and an artificial magnetic field using a minimal six-site model which leads to gauge dependent results. This model proves the relation between the gauge dependant dynamics and the absence of scalar potential terms connecting different gauge potentials. In this context, we secondly search for clear indication of the gauge dependent dynamics through magnetic flux quenches of wave packets in two- and three-leg synthetic ladders. We show that the choice of gauge potentials have tremendous effect on the post-quench dynamics of wave packets. Even trivially distinct two vector potentials by an additive constant can produce observable effects, we investigate the effects on the Landau levels and the Laughlin wave function for a filling factor $\nu = 1/q$. We also show that edge solutions in a wide synthetic ladder are protected under a flux quench only if there is another edge state solution in the quenched Hamiltonian.
Karadelik lazerinden sonic Hawking ışınımı
The quantum thermal radiation from a black hole (BH) known as "Hawking Radiation" or "Black Hole Evaporation" results from studying quantum fields in the curved space-time of the horizon of a BH. Experimentally, the radiation is difficult if not impossible to be detected from a real black hole with a mass much higher than that of our sun, since the Radiation temperature is substan- tially below that of microwave background radiation. However, in 1981 Unruh showed an analogy between the propagation of sound waves in any convergent fluid flow and that of the quantum field in a gravitational field. He showed that if the background fluid is accelerated to higher than the speed of sound then it can develop a horizon (point of no re- turn) for the sound waves. This is the so-called the sonic BH. This horizon will emit thermal radiation in terms of sound wave quanta (phonons) in an analogy to the thermal radiation of black holes (Analogue Hawking Radiation) (AHR). Bose-Einstein Condensates (BECs) can be used as a background fluid develop- ing a sonic horizon for the phonon modes propagating through its background due to the very low temperature of the BEC. Recently in 2014, Steinhauer has reported the observation of self-Amplifying Hawking radiation from the realization of an accelerated BEC. The experiment reported an exponentially growing signal of modes trapped between a BH and white hole (WH) horizon, where the white hole is the point where sound cannot enter. Experimental signatures of AHR are a growing oscillating perturbation of the condensate mean density and a characteristic pattern in density-density correlation functions. However, the former mentioned oscillations may result from the dynamical instabilities of the classical mean field density. iii iv In this work, we were able to reproduce the experimental results of density modulations in the mean field, and thus without AHR, using only the mean field Gross-Pitaevskii equation (GPE) for the BEC. Furthermore, we include the quantum fluctuation to study the density-density correlation function that is in qualitative agreement with the experiment using the truncated Wigner ap- proximation (TWA). Finally, we then calculate the One Body Density Matrix (OBDM) to distinguish condensed from non-condensed atoms using the Penrose Onsager criterion. We are able to contribute to a discussion in the literature re- garding the quantum field or mean field origin of the mean density oscillations in the experiment.
Hofstadter kelebeğinin ayarlanabilir optik örgülerdeki değişimi
There are a limited number of exact solutions for quantum mechanical systems. It is critical to obtain solutions for complex systems. One of these unsolved equations was the famous Harper's equation, which was proposed in 1955. It investigates the behavior of a particle in a periodic potential under a uniform magnetic field in two dimensions. Douglas Hofstadter, in 1976, obtained a numerical solution for the first time, discovering a non-trivial energy spectrum as a function of magnetic flux. The spectrum is a fractal structure, the Hofstadter butterfly, and depends purely on the lattice geometry. In other words, primitive lattice vectors and basis vectors determine the fractal energy spectrum under a uniform magnetic field. The experimental demonstration of such an energy spectrum requires a magnitude of thousands of Teslas magnetic field in the solid state systems since the area of a unit cell is on the order of a few square nanometers. Recently, two main developments in cold atom physics led the way to the realization of the Hofstadter butterfly energy spectrum. The first one as the creation and manipulation of optical lattices. It provides a controllable environment with lattice constants up to a few hundred nanometers, which means the required magnetic field is now within experimental capabilities. The second development is the realization of synthetic gauge fields on optical lattices. One recent development we focus in this thesis is the creation of an adjustable lattice geometry. The self-similar energy spectra for a uniform magnetic field depends purely on the lattice geometry. Recently, the Zurich group presented a unique chance to examine the connection between them. Particularly, we calculate the Hofstadter butterfly for all lattice parameters which can be obtained by the Zurich group. We then investigate the transition of the Hofstadter butterfly from a checkerboard lattice to a honeycomb lattice, which includes the observation of the change in topological invariants, the Chern numbers of the self-similar energy spectra. For this purpose, we first present the theoretical building blocks utilized throughout the research. We show the step-by-step procedure to obtain the Hofstadter butterfly, starting from the continuous Hamiltonian and projection onto a tight-binding Hamiltonian. We explicitly demonstrate the butterflies for the square lattice and the honeycomb lattice. Next, we concentrate on the experiment carried out by the Zurich group, and obtain the Hofstadter butterflies for all lattice geometries. The Hofstadter butterflies are analysed in detail. There are three different regimes. In the first regime the spectrum is formed by two stacked square lattice Hofstadter butterflies separated by a large energy gap. As the optical lattice evolves from the checkerboard to the honeycomb geometry, the second regime begins with the emergence of Dirac points for particular rational magnetic flux values Phi = p/q, where p,q are mutually prime integers. In the third regime infinitely many sequential closings of adjacent bands around zero energy give the honeycomb lattice Hofstadter butterfly as a limit. This closing process can be probed with current setups. We show that the existence of Dirac points at zero magnetic field does not imply its existence at a finite field. The topological properties of the energy spectrum can change with the applied magnetic field. We calculate the Chern numbers of the major gaps in the spectra and examine the exchange and the transfer of these topological invariants during the evolution of the lattice geometry. An analytic formula to determine the critical value for the emergence of Dirac points around zero energy is obtained in Eq. 5.2.
Silindirik simetrik tuzakta çift kutuplu Bose-Einstein yoğuşması
Bose-Einstein Condensate (BEC) and particularly its stability dynamics has been a subject to many investigations since the first realization of this new condensed state in alkali atoms interacting via short range potential. Short range or contact interactions account for a great number of physical properties ranging from formation of quantum vortices to the superfluid character of cold gases. In this thesis, dipolar Bose-Einstein condensate, which inherently possess long-range and anisotropic potential for the interaction of the constituent particles, is studied and its stability depending on the geometry of the system is investigated. The dipolar Bose gas is confined to a cylindrically symmetric harmonic trap and the dipoles within the gas is initially oriented along the symmetry axis of the confining prolate trap. In the condensed state, the condensate is observed to be elongated along harmonic trap symmetry axis as long as the axis corresponds to weak confinement direction. This elongation is understood to be resulting from the energy minimization of the system by adding the dipoles head to tail along the center of the trap, thereby determining the nature of the long-range interaction to be attractive and the condensate is liable to collapse. Below a certain value for the ratio of the dipolar and contact interactions, the condensate is stable, while above this value it undergoes collapse. In the opposite case where the trap axis is the strong confinement direction (oblate trap), the elongation occurs perpendicularly to the symmetry axis of the confining trap (with highly oblate geometry) with the energetically most favorable configuration being the alignment of the dipoles side by side implying mostly repulsive interactions in which case the condensate is always stable. To further understand the effect of the geometry on the stability, the dipoles are finally oriented at an angle from the trap axis by tuning the external field and elongation direction of the condensate is calculated; stable, metastable and unstable states of the condensate are observed in this new geometry.
Bose-Einstein yoğuşmalarındaki karanlık solitonların kütlelerinin Gelfand-Yaglom metodu ile hesaplanması
Nonlinear excitations of Bose-Einstein condensates (BEC) play important role in understanding the dynamics of BECs. Solitons, shape preserving wave packets, are the most fundamental nonlinear excitations of BECs. They exhibit particle-like behaviors since their characteristic features do not change during their oscillations and collisons. Moreover, their effective masses are calculated. We are interested in dark solitons which have their density minima at the center. In literature, the mass of dark soliton is obtained with Gross-Pitaevskii approximation. As a result of the contributions of quantum fluctuations to the ground state energy, a correction term is added to the effective mass. The dispersion relation of these fluctuations are derived from Bogoliubov de Gennes equations. However, with familiar analytical approaches, only a few modes can be taken into account. In order to include all the modes and find an exact expression for ground state energy, we obtain free energy from partition function. The partition function is equivalent to an imaginary-time coherent state Feynman path integral on which periodic boundary conditions are applied. The partition function is in the form of infinite dimensional Gaussian integral, therefore, it is proportional to the determinant of the functional in the integrand. We use Gelfand Yaglom method to calculate the corresponding determinant. Gelfand Yaglom method is a specialized formulation of using zeta functions and contour integrals in calculation of the functional determinant for one-dimensional Schrödinger operators. In this study, we formulate a new technique through this method to calculate ground state energy of stationary dark solitons up to the Bogoliubov order exactly.
Yüklü-yüksüz süperakışkan karışımlar
Motivated by the developments of artificial magnetic fields (AMFs) enabling coupling to the neutral particles of ultracold quantum gases, we have theoretically studied charged-neutral mixtures in various settings. The techniques that have been used to manufacture these AMFs are highly sensitive to the internal degrees of freedom of the atoms, resulting in unequal coupling to the components of a mixture. We demonstrate the possible consequences of this unequal coupling by considering two different systems. First, we examine an impurity problem in a fermion background under an AMF coupling selectively to the impurity in a ring trap. We calculate the response of the system exactly by using Bethe Ansatz and argue that the AMF can be employed as a probe to analyze polaron formation. Secondly, we explore Bardeen-Cooper-Schrieffer theory of superconductivity in the presence of a charge imbalance under an AMF. We analytically calculate the gap equation for any degree of asymmetry between the Landau level spectra of up and down spin particles, and show that the system displays reentrant superconductivity both in magnetic field and temperature. Apart from mixtures, we also investigate the non-equilibrium Hall response of a topological system. The strength of an AMF applied on a optical lattice can be suddenly changed without creating Eddy currents, allowing us to quench the system across a topological phase boundary. We report a fractional Hall response for the resulting non-equilibrium system and discuss possible implementations for cold atom experiments.
Dönen ikili Bose Hubbard merdiveni
We analyze two leg Bose Hubbard model under uniform magnetic field within various methods. Before studying the model, we discuss the background on rotating Bose Einstein condensates, Bose Hubbard model and superfluid Mott insulator transition. We give a general overview of Density Matrix Renormalization Group (DMRG) theory and show some of the applications. Introducing two leg system Hamiltonian, we solve the single particle problem and find distinct structures above and belove a critical magnetic field $\alpha_c=0.21\pi$. Above this value of the field, it is found that system has travelling wave solutions. To see the effects of interactions, we use Gross Pitaevskii approximation. Spectrum of the system below the critical field and the change of $\alpha_c$ with the interaction strength are obtained for small interactions, i.e $Un/t<1$. To specify Mott insulator boundary, variational mean field theory and strong coupling perturbation (SCP) theories are used. The travelling wave solutions found in single particle spectrum above $\alpha_c$ is found to be persistent in mean field description. On the other hand, comparing with the strong coupling expansion results, it has been found that the mean field theory gives poor results, because of the one dimensional structure of the system. The change of the tip of the lobe where BKT transition takes place is found as a function of magnetic field by SCP. Finally we use DMRG to obtain the exact shape of the phase diagram. It is found that second order strong coupling perturbation theory gives very good results. System is found to display reenterant phase to Mott insulator. Looking at the infinite onsite interaction limit via DMRG, the critical value of the magnetic field is found to be exactly equal to the single particle solution. We have calculated the particle-hole gap for various fillings and different magnetic fields and found Fractional Quantum Hall like behaviors
Dolanıklık: Belirsizlikler aracılığıyla nicelenmesi ve optik örgülerdeki ultrasoğuk bozonlar için araştırılması
In the first part of the Thesis, the known measures of entanglement for finite dimensional systems are reviewed. Both the simplest case of pure states that belong to bipartite systems and more general case of mixed states are discussed. The multipartite extensions are also mentioned. In addition to the already existing ones, we propose a new measure of entanglement for pure states of bipartite systems. It is based on the dynamical symmetry group approach to quantum systems. The new measure is given in terms of the total uncertainty of basic observables for the corresponding state. Unlike conventional measures concurrence and 3-tangle, which measure the amount of entanglement of different groups of correlated parties, our measure gives the total amount of multipartite entanglement in a specific state.In the second part of the Thesis, the trapping of bosonic atoms in optical lattices is reviewed. The band structure together with Bloch functions and Wannier basis are discussed for this system. In relation with that, the corresponding Bose-Hubbard model and by the use of this model, the resulting superfluid to Mott-insulator quantum phase transition is summarized. In this regard, the Bose-Hubbard Hamiltonian of a specific system, namely ultracold spin-1 atoms with coupled ground states in an optical lattice is considered. For this system we examine particle entanglement, that is characterized by pseudo-spin squeezing both for the superfluid and Mott-insulator phases in the case of ferromagnetic and antiferromagnetic interactions. The role of a small but nonzero angle between the polarization vectors of counterpropagating lasers forming the optical lattice on quantum correlations is investigated as well.
Bose-Einstein yoğuşması üzerinde ışığın yayılımı ve kuantum dolaşıklığı
We investigate the optical response of coherent media, aBose-Einstein condensate (BEC), to intense laser pump stimulationsand weak probe pulse propagation.First, we adopt the coherence in sequential superradiance (SR) asa tool for continuous-variable (CV) quantum entanglement of twocounter-propagating pulses from the two end-fire modes. In thefirst-sequence the end-fire and side mode are CV entangled. In thesecond sequence of SR, this entanglement is swapped in between thetwo opposite end-fire modes.Second, we investigate the photonic bands of an atomic BEC with atriangular vortex lattice. Index contrast between the vortex coresand the bulk of the condensate is achieved through the enhancementof the index via atomic coherence. Frequency dependent dielectricfunction is used in the calculations of the bands. We adopt aPoynting vector method to distinguish the photonic band gaps fromabsorption/gain regimes.
Dönen optik örgülerde kuvantum gazları
The thesis is structured into two main parts so as to cover bosons and fermions in rotating optical lattices separately. In the first part, after a brief introduction to ultracold atoms in optical lattices, we review the single-particle physics for the lowest (s) band of a periodic potential under an artificial magnetic field created by rotation. Next, we discuss rotational effects on the first excited (p) band of the lattice, extending the methods available for the lowest band. We conclude the first part with a discussion of many-body physics in rotating lattice systems using a mean-field approach and investigate how thetransition boundary between superfluid and Mott insulator phases is affected by the single-particle spectrum. In this context, we also examine a possible coexistent phase of Mott insulator and bosonic fractional quantum Hall states, appearing for certain system parameters near the Mott insulator lobes in the phase diagram.The second part starts with the proposal of a realization and detection scheme for the so-called topological Hofstadter insulator, which basically reveals the single-particle spectrum discussed before. The scheme depends on a measurement of the density profile for noninteracting fermions in a rotatingoptical lattice with a superimposed harmonic trapping potential. This method also allows one to measure the quantized Hall conductance, a feature which appears when the Fermi energy lies in an energy gap of the lattice potential. Finally, we explore the Bardeen-Cooper-Schrieffer type of pairing of fermionicatoms in optical lattices under an artificial magnetic field by paying special attention to single-particle degeneracies and present our results for the vortex lattice structure of the paired fermionic superfluid phase.
Yapay manyetik alan altında yüklü-nötr karışımlarda eşlenme
Bose-Einstein condensations (BEC), pairing behaviour, vortex formations in superconductivity and superfluidity are just a few examples of fascinating features of ultracold gases. In this thesis, we study charge-neutral cold atom mixtures which are obtained by placing a neutral mixture under an artificial magnetic field coupling only one of the components. We begin with two distinguishable (charged-neutral) particles on a ring trap. Charge particle gains angular momentum due to a magnetic field along the axis of the ring and we see that there is a big angular momentum transfer to neutral particle in orders of h. This work is set forth to guide us in the many body problem of vortex transformation in charged-neutral superfluid mixtures. In the main part of the thesis, we examine charge-neutral fermion mixtures. Thanks to artificial magnetic fields, Cooper pairs whose only one component coupling to magnetic field can be created now. We calculate the gap equation for this system and solve for the critical temperature. We show that critical temperature decreases for the increasing magnetic field.
Yüklü-yüksüz üstünakışkan karışımlarında girdap aktarımı
Bose-Einstein Condensation (BEC) was introduced by Einstein 1925. It took 70 years to con rm BEC by experiments. BEC creates a suitable environment to observe macroscopic-quantum behavior. Condensates consist of ultracold atoms allow physicists to create super uids and also they allow to manipulate these quantum structures easily. One of the main tool needed to manipulate these structures is synthetic magnetic eld. Under the light of these experimental achievements we studied the angular momentum transfer in the N-body systems. First of all, to develop physical intuition, we solved 2-body problem. This problem can be de ned as: The system consist of two particles and con ned in a ring. Particles interact with each other and charged one coupled to the magnetic eld. We used two approaches to solve the system and compared these approaches in the small limit of inter-particle interaction. Finally, we studied N-body systems and vortex transfer in the two-component super uid mixtures via Gross-Pitaevski equation and Bogoulibov equations. We observed that for various parameters neutral-neutral mixtures do not possess vortex transfer, yet charged-neutral mixtures coupled to the magnetic eld experience vortex transfer.