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Journal of the Indian Institute of Science

Symplectic techniques in mechanics and optics

Narasimhaiengar Mukunda

Published Aug 1, 2013• 0 citations
OpenAlex

Abstract

We give a brief introduction to the real symplectic groups and the geometry of canonical variables that they describe. The role of the metaplectic groups in the context of quantum mechanics is highlighted. Several applications of symplectic techniques to classical ray and wave optics, noise matrices and the uncertainty relations, squeezing, and Gaussian states are outlined.

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We give a brief introduction to the real symplectic groups and the geometry of canonical variables that they describe. The role of the metaplectic groups in the context of quantum mechanics is highlighted. Several applications of symplectic techniques to classical ray and wave optics, noise matrices and the uncertainty relations, squeezing, and Gaussian states are outlined.
In this thesis we study the roles played by symplectic geometry in quantum mechanics,in particular quantum dynamics and quantum information theory treated as two separate parts. The common ground for both parts is the geometrical formulation of quantum mechanics. In Chapter 2, we review the associated complex projective Hilbert space of quantum pure states, with symplectic and Riemannian structures and their roles in quantum dynamics and kinematics. In Chapter 3, we motivate the idea of information-theoretic constraint on the differentiable manifold of probability distributions through the maximum uncertainty principle and the linear Schrodinger equation by introduction of the wave function. In Chapter 4, we review both regular and singular symplectic reduction of a symplectic manifold, which is acted upon properly and symplectically by a compact Lie group. Chapter 5 contains the author's original contributions to the first part of the thesis. In this chapter, by using the same information-theoretic discussion of the Chapter 3 we propose a non-relativistic, spin-less, non-linear quantum dynamical equation,with the Fisher information metric replaced by the Jensen-Shannon distance information. Furthermore, we show that the non-linear Schrodinger equation is in fact a Hamiltonian dynamics, namely it preserves the symplectic structure of the complex Hilbert space. The projected dynamics on the corresponding projective Hilbert space is derived and its properties are highlighted in further details. Chapter 6 contains the author's primary contributions to the second part of the thesis. In particular, by using the singular symplectic reduction of the Chapter 4 we explicitly construct the space of entanglement types of three-qubit pure states with a specific (shifted) spectra of single-particle reduced density matrices. Moreover, we obtain the image of the symplectic quotient under the induced Hilbert map, by using local unitary invariant polynomials. Then the symplectic structure on the principal stratum of the symplectic quotient is derived. Finally, it is discussed that other lower dimensional strata are relative equilibria on the original manifold and their stability properties are investigated under compact subgroups of the local unitary transformations.
Optics has served as the expermental basis and model for much of science since Galielo. The minds at work in mechanics were so too in optics, in its early geometric aspects through Snell up to Newton, and its wave aspects since Huygens through Fourier and including Sommerfeld. Hamilton’s original application of generating functions for canonical transformations was in optics; only later were they recognized as a sharp tool for abstract mechanics through the introduction of the concept of a phase space with a symplectic metric [ 1 ]. This concept in embrionary form, served Heisenberg [ 2 ] and Weyl [ 3 ] for the introduction of quantum mechanics, where the Hamiltonian is quadratic in momentum. The history and scope of Sophus Lie’s work is well known to this audience, as is the fact that its manifold applications have been most successful when applied to systems with quadratic Hamiltonians [ 4 ]. Almost invariably, the kinetic energy term is p 2 /2m. This Hamiltonian is indeed also useful in optics, albeit in its Gaussian approximation, i.e., first-order optics which deals with linear transformations of phase space. The associated methods have been under research since the early seventies from the perspective of quantum mechanics [ 5 ] and groups of integral transforms [ 6 , 7 ].
Gaussian quantum states of bosonic systems are an important class of states. In particular, they play a key role in quantum optics as all processes generated by Hamiltonians up to second order in the field operators (i.e. linear optics and quadrature squeezing) preserve Gaussianity. A powerful approach to calculations and analysis of Gaussian states is using phase-space variables and symplectic transformations. The purpose of this note is to serve as a concise reference for performing phase-space calculations on Gaussian states. In particular, we list symplectic transformations for commonly used optical operations (displacements, beam splitters, squeezing), and formulae for tracing out modes, treating homodyne measurements, and computing fidelities.
Starting from the symplectic Lie groups Sp (2 n , ) of an n -mode system in classical and quantum mechanics we discuss, in particular, Sp (2, )~ SO (2,1) for a single mode and Sp (4, )~ SO (3,2) with some of its important subgroups such as SU (1,1)~ Sp (2, ) and SU (2). We then look at their applications in quantum optics (squeezing, phase states, beamsplitter and polarization). We explicitly construct the root scheme of the Lie algebra sp (4, ) in two different bases as it is an important means for the investigation of the group properties and the construction of the irreducible representations. Apart from the well known realizations of Sp (2, ) by single-mode and two-mode systems, we discuss some other realizations of this group by fractal combinations of boson operators.