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Optics and Spectroscopy

Symmetry approach to the analysis of coherent structures in quantum optics: Foundations and examples

V. P. Karassiov

Published Oct 1, 2001• 3 citations
Publisher

Abstract

A general symmetry method for the discovery and analysis of hidden coherent structures in quantum models with an invariant Hamiltonian is proposed. The method is based on the mathematical concept of dual algebraic pairs, which unifies the actions of invariance groups and algebras of dynamic symmetry. Its efficiency is illustrated by the examples of biphotons and the invariant treatment of unpolarized light quasi-spin clusters, and “optical atoms” in nonlinear quantum-optical models.

DOI
10.1134/1.1412668

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A general symmetry method for the discovery and analysis of hidden coherent structures in quantum models with an invariant Hamiltonian is proposed. The method is based on the mathematical concept of dual algebraic pairs, which unifies the actions of invariance groups and algebras of dynamic symmetry. Its efficiency is illustrated by the examples of biphotons and the invariant treatment of unpolarized light quasi-spin clusters, and “optical atoms” in nonlinear quantum-optical models.
A general algebraic approach incorporating both invariance groups and dynamical symmetry algebras is developed to reveal hidden coherent structures (closed complexes and configurations) in quantum manybody physics models due to symmetries of their Hamiltonians H . Its general ideas are manifested on some recent new examples: (1) G-invariant biphotons and a related SU (2)-invariant treatment of unpolarized light; (2) quasispin clusters in nonlinear models of quantum optics; and (3) construction of composite particles and (para)fields from G -invariant clusters due to internal symmetries.
In this tutorial I intend to present some of the results I obtained through my PhD work in the "quantum optics group of the University of Isfahan" under consideration Dr. R. Roknizadeh and Prof. S. Twareque Ali as my supervisor and advisor, respectively. I will revisit some of the pioneering proposals recently developed the concept of generalized CSs. As it can be observed the customary three generalization methods {\it (symmetry, algebraic and dynamical)} have never been considered in neither of them. Our intention in this work is at first to investigate the lost ring between the customary three methods and the recently developed ones, as possible. For this purpose it has been devised general analytic descriptions, which successfully demonstrate how different varieties of CSs (which are nonlinear in nature) can be obtained by two processes, first the {\it "nonlinear CSs"} method and second by {\it "basis transformations on an underlying Hilbert spaces"}. As a result, I will systematize the recently introduced generalized CSs in a clear and concise way. It will be clear also, that how our results can be considered as a first step in the generation process of the mathematical physics CSs in the context of quantum optics. Besides this, some new results emerge from our studies. I introduce a large classes of generalized CSs, namely the {\it "dual family"} associated with each set of early known CSs. But, in this relation, the previous processes for constructing the dual pair of Gazeau-Klauder CSs fail to work well, so I outlined a rather different method based on the {\it "temporal stability"} requirement of generalized nonlinear CSs.
1. Cooperative phenomena and processes in various many-body physical systems (atoms, molecules, nuclei etc.) are now the topic of great interest (see e.g. [ 1-6 ] and references therein). A natural and efficient tool for studying cooperative phenomena in quantum systems is the second quantization method [ 7 ], which is closely connected with the representation theory of Lie groups and supergroups [ 8 – 12 ]. Therefore, group-theoretical methods are very adequate to an analysis of different cooperative effects within the general approach of dynamic symmetries [ 13 ].
Simulation of quantum dynamics is a grand challenge of computational physics. In this work we investigate methods for reducing the demands of such simulation by identifying reduced-order models for dynamics generated by parameterized quantum Hamiltonians. In particular, we first formulate an algebraic condition that certifies the existence of invariant subspaces for a model defined by a parameterized Hamiltonian and an initial state. Following this we develop and analyze two methods to explicitly construct a reduced-order model, if one exists. In addition to general results characterizing invariant subspaces of arbitrary finite dimensional Hamiltonians, by exploiting properties of the generalized Pauli group we develop practical tools to speed up simulation of dynamics generated by certain spin Hamiltonians. To illustrate the methods developed we apply them to several paradigmatic spin models.
Coherent quantum physics is quantum physics in terms of a coherent space and an appropriate coherent product. The kinematical structure and the meaning of the quantities are given by the symmetries (invertible coherent maps) of the coherent space. If Z is a coherent manifold, a canonical symplectic form is often canonically determined by the coherent product through the coherent action principle discussed in Section 6.1. In a quantum mechanical context, Z is indeed a classical phase space or an extended phase space-typically a symplectic manifold, a Poisson manifold, or a circle or line bundle over such a manifold that incorporates the classical action variable (encoding the Berry phase under quantization). For example, the Aharonov-Bohm effect [3] needs the bundle formulation. A coherent space and its quantum space provide a classical and a quantum view of the same physical system, discussed in Section 6.2. Section 6.3 shows that symmetries of coherent spaces can be quantized, that is, promoted to linear operators on the quantum space. This leads to quantum dynamics, which in special (completely integrable) situations can be solved in closed form in terms of classical motions on the underlying coherent space, if the latter has a compatible manifold structure. Close relations to concepts from geometric quantization and Kähler manifolds are pointed out in Section 6.4. In order that the quantum space of a coherent space Z can describe physics, one needs not only a distinguished class of coherent states but a more general concept of state. In Sections 6.5-6.6, we give a general abstract setting for states in Lie ∗-algebras, emphasize the essential mathematical features and the close analogy between classical and quantum physics. We illustrate these generalities with a number of examples in Section 6.7. The coherent action principle is put to use in Section 6.8 for numerical quantum physics and an application to quantum chaos. Spectral issues can, in favorable cases, be handled in terms of dynamical Lie algebras, discussed in Section 6.9.
In this paper, we present various well-studied examples of quantum systems, and discuss how they might be analysed algebraically. We analyse the harmonic oscillator, hydrogenic atom, optical components, and particle spin using multiple algebraic methods, and highlight the central importance of algebraic relationships between operators in determining the physical character of the system. In particular, we show: the ubiquity of the Lie algebra so(2,1)≅su(1,1)\mathfrak{so}(2,1)\cong\mathfrak{su}(1,1)so(2,1)≅su(1,1) as the ``spectrum generating algebra''; the role of su(2)\mathfrak{su}(2)su(2) and su(1,1)\mathfrak{su}(1,1)su(1,1) in determining the action of optical components; that an elementary algebraic approach to particle spin offers novel insights into its nature versus the usual approaches; and how understanding the full (often hidden) symmetry of a problem enables it to be solved immediately without the need to solve differential equations. We hope this work will serve as a gentle introduction to algebraic methods in quantum mechanics for the interested reader --- we therefore signpost active areas of research within the community.