Browsing by Subject "Phase space methods"
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Item Open Access Canonical-covariant Wigner function in polar form(OSA - The Optical Society, 2000) Hakioǧlu, T.The two-dimensional Wigner function was investigated in polar canonical coordinates. The covariance properties under the action of affine canonical transformations were derived. The polar canonical phase-space representations were considered important for paraxial optical systems as well as other systems in which a rotational symmetry around a particular axis was present.Item Open Access Degrees of freedom of optical systems and signals with applications to sampling and system simulation(Optical Society of America, 2013) Oktem F.S.; Özaktaş, Haldun M.We study the degrees of freedom of optical systems and signals based on space-frequency (phase space) analysis. At the heart of this study is the relationship of the linear canonical transform domains to the space-frequency plane. Based on this relationship, we discuss how to explicitly quantify the degrees of freedom of first-order optical systems with multiple apertures, and give conditions for lossless transfer. Moreover, we focus on the degrees of freedom of signals in relation to the space-frequency support and provide a sub-Nyquist sampling approach to represent signals with arbitrary space-frequency support. Implications for simulating optical systems are also discussed. © 2013 Optical Society of America.Item Open Access Equivalence of linear canonical transform domains to fractional Fourier domains and the bicanonical width product: a generalization of the space-bandwidth product(Optical Society of America, 2010-07-30) Oktem, F. S.; Özaktaş, Haldun M.Linear canonical transforms (LCTs) form a three-parameter family of integral transforms with wide application in optics. We show that LCT domains correspond to scaled fractional Fourier domains and thus to scaled oblique axes in the space-frequency plane. This allows LCT domains to be labeled and ordered by the corresponding fractional order parameter and provides insight into the evolution of light through an optical system modeled by LCTs. If a set of signals is highly confined to finite intervals in two arbitrary LCT domains, the space-frequency (phase space) support is a parallelogram. The number of degrees of freedom of this set of signals is given by the area of this parallelogram, which is equal to the bicanonical width product but usually smaller than the conventional space-bandwidth product. The bicanonical width product, which is a generalization of the space-bandwidth product, can provide a tighter measure of the actual number of degrees of freedom, and allows us to represent and process signals with fewer samples.Item Open Access The fractional fourier transform(IEEE, 2001) Özaktas, Haldun M.; Kutay, M. A.A brief introduction to the fractional Fourier transform and its properties is given. Its relation to phase-space representations (time- or space-frequency representations) and the concept of fractional Fourier domains are discussed. An overview of applications which have so far received interest are given and some potential application areas remaining to be explored are noted.Item Open Access The Fractional Fourier transform and harmonic oscillation(Springer, 2002) Kutay, M. A.; Özaktaş, Haldun M.The ath-order fractional Fourier transform is a generalization of the ordinary Fourier transform such that the zeroth-order fractional Fourier transform operation is equal to the identity operation and the first-order fractional Fourier transform is equal to the ordinary Fourier transform. This paper discusses the relationship of the fractional Fourier transform to harmonic oscillation; both correspond to rotation in phase space. Various important properties of the transform are discussed along with examples of common transforms. Some of the applications of the transform are briefly reviewed.Item Open Access Linear canonical transforms, degrees of freedom, and sampling in optical signals and systems(IEEE, 2014) Özaktaş, Haldun M.; Öktem, F. S.We study the degrees of freedom of optical systems and signals based on space-frequency (phase-space) analysis. At the heart of this study is the relationship of the linear canonical transform domains to the space-frequency plane. Based on this relationship, we discuss how to explicitly quantify the degrees of freedom of first-order optical systems with multiple apertures, and give conditions for lossless transfer. Moreover, we focus on the degrees of freedom of signals in relation to the space-frequency support and provide a sub-Nyquist sampling approach to represent signals with arbitrary space-frequency support. Implications for simulating optical systems are also discussed.Item Open Access Non-local, non-commutative picture in quantum mechanics and distinguished continuous canonical maps(IOP Science, 2002) Hakioglu, T.It is shown that continuous classical nonlinear canonical (Poisson) maps have a distinguished role in quantum mechanics. They act unitarily on the quantum phase space and generate h-independent quantum nonlinear canonical maps. It is also shown that such maps act in the non-commutative phase space under the classical covariance. A crucial result of the work is that under the action of Poisson maps a local quantum mechanical picture is converted onto a non-local picture which is then represented in a non-local Hilbert space. On the other hand, it is known that a non-local picture is equivalent by the Weyl map to a non-commutative picture which, in the context of this work, corresponds to a phase space formulation of the theory. As a result of this equivalence, a phase space Schrödinger picture can be formulated. In particular, we obtain the *-genvalue equation of Fairlie [Proc. Camb. Phil. Soc., 60, 581 (1964)] and Curtright, Fairlie and Zachos [Phys. Rev., D 58, 025002 (1998)]. In a non-local picture entanglement becomes a crucial concept. The connection between the entanglement and non-locality is explored in the context of Poisson maps and specific examples of the generation of entanglement from a local wavefunction are provided by using the concept of generalized Bell states. The results obtained are also relevant for the non-commutative soliton picture in the non-commutative field theories. We elaborate on this in the context of the scalar non-commutative field theory.Item Open Access Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transforms(Elsevier BV * North-Holland, 1995-10-15) Aytür, O.; Özaktaş, Haldun M.It is customary to define a phase space such that position and momentum are mutually orthogonal coordinates. Associated with these coordinates, or domains, are the position and momentum operators. Representations of the state vector in these coordinates are related by the Fourier transformation. We consider a continuum of "fractional" domains making arbitrary angles with the position and momentum domains. Representations in these domains are related by the fractional Fourier transformation. We derive transformation, commutation, and uncertainty relations between coordinate multiplication, differentiation, translation, and phase shift operators making arbitrary angles with each other. These results have a simple geometric interpretation in phase space and applications in quantum optics.Item Open Access Phase-space window and degrees of freedom of optical systems with multiple apertures(Optical Society of America., 2013) Özaktaş, Haldun M.; Oktem, F. S.We show how to explicitly determine the space-frequency window (phase-space window) for optical systems consisting of an arbitrary sequence of lenses and apertures separated by arbitrary lengths of free space. If the space-frequency support of a signal lies completely within this window, the signal passes without information loss. When it does not, the parts that lie within the window pass and the parts that lie outside of the window are blocked, a result that is valid to a good degree of approximation for many systems of practical interest. Also, the maximum number of degrees of freedom that can pass through the system is given by the area of its space-frequency window. These intuitive results provide insight and guidance into the behavior and design of systems involving multiple apertures and can help minimize information loss.Item Open Access Quantum canonical transformations in star-product formalism(IOP, 2013) Dereli, T.; Hakioğlu, Tuğrul; Temen, A.We study construction of the star-product version of three basic quantum canonical transformations which are known as the generators of the full canonical algebra. By considering the fact that star-product of c-number phase-space functions is in complete isomorphism to Hilbert-space operator algebra, it is shown that while the constructions of gauge and point transformations are immediate, generator of the interchanging transformation deforms this isomorphism. As an alternative approach, we study all of them within the deformed form. How to transform any c-number function under linear-nonlinear transformations and the intertwining method are shown within this argument as the complementary subjects of the text.Item Open Access Quantum Dynamics of Long-Range Interacting Systems Using the Positive-P and Gauge-P Representations(American Physical Society, 2017) Wüster, S.; Corney, J. F.; Rost, J. M.; Deuar, P.We provide the necessary framework for carrying out stochastic positive-P and gauge-P simulations of bosonic systems with long-range interactions. In these approaches, the quantum evolution is sampled by trajectories in phase space, allowing calculation of correlations without truncation of the Hilbert space or other approximations to the quantum state. The main drawback is that the simulation time is limited by noise arising from interactions. We show that the long-range character of these interactions does not further increase the limitations of these methods, in contrast to the situation for alternatives such as the density matrix renormalization group. Furthermore, stochastic gauge techniques can also successfully extend simulation times in the long-range-interaction case, by making using of parameters that affect the noise properties of trajectories, without affecting physical observables. We derive essential results that significantly aid the use of these methods: estimates of the available simulation time, optimized stochastic gauges, a general form of the characteristic stochastic variance, and adaptations for very large systems. Testing the performance of particular drift and diffusion gauges for nonlocal interactions, we find that, for small to medium systems, drift gauges are beneficial, whereas for sufficiently large systems, it is optimal to use only a diffusion gauge. The methods are illustrated with direct numerical simulations of interaction quenches in extended Bose-Hubbard lattice systems and the excitation of Rydberg states in a Bose-Einstein condensate, also without the need for the typical frozen gas approximation. We demonstrate that gauges can indeed lengthen the useful simulation time.Item Open Access Stride-to-stride energy regulation for robust self-stability of a torque-actuated dissipative spring-mass hopper(A I P Publishing LLC, 2010) Ankarali, M. M.; Saranli, U.In this paper, we analyze the self-stability properties of planar running with a dissipative spring-mass model driven by torque actuation at the hip. We first show that a two-dimensional, approximate analytic return map for uncontrolled locomotion with this system under a fixed touchdown leg angle policy and an open-loop ramp torque profile exhibits only marginal self-stability that does not always persist for the exact system. We then propose a per-stride feedback strategy for the hip torque that explicitly compensates for damping losses, reducing the return map to a single dimension and substantially improving the robust stability of fixed points. Subsequent presentation of simulation evidence establishes that the predictions of this approximate model are consistent with the behavior of the exact plant model. We illustrate the relevance and utility of our model both through the qualitative correspondence of its predictions to biological data as well as its use in the design of a task-level running controller. © 2010 American Institute of Physics.Item Open Access Wigner-related phase spaces for signal processing and their optical implementation(Optical Society of America, 2000) Mendlovic, D.; Zalevsky, Z.; Özaktaş, Haldun M.Phase spaces are different ways to represent signals. Owing to their properties, they are often used for signal compression and recognition with high discrimination abilities. We present several recently introduced Wigner-related sets of representations that have improved signal processing performance, and we introduce an optical implementation. This study deals with the generalized Wigner spaces, the fractional Fourier transform, and the x-p and the r-p representations. The optical implementations are demonstrated and discussed. © 2000 Optical Society of America.