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arXiv:1904.12259 · Journal of Physics A Mathematical and Theoretical

S p ( 4 ; R ) squeezing for Bloch four-hyperboloid via the non-compact Hopf map

Kazuki Hasebe

Published Aug 20, 2019• 14 citations
arXivHTMLPDF

Abstract

Abstract We explore the hyperbolic geometry of squeezed states in the perspective of the non-compact Hopf map. Based on analogies between the squeeze operation and hyperbolic rotation, two types of the squeeze operators, the (usual) Dirac and the Schwinger types, are introduced. We clarify the underlying hyperbolic geometry and representations of the squeezed states along the line of the first non-compact Hopf map. Following the geometric hierarchy of the non-compact Hopf maps, we extend the analysis to —the isometry of a split-signature four-hyperboloid. We explicitly construct the squeeze operators in the Dirac and Schwinger types and investigate the physical meaning of the four-hyperboloid coordinates in the context of the Schwinger-type squeezed states. It is shown that the Schwinger-type squeezed one-photon state is equal to an entangled superposition state of two squeezed states and the corresponding concurrence has a clear geometric meaning. Taking advantage of the group theoretical formulation, basic properties of the squeezed coherent states are also investigated. In particular, we show that the squeezed vacuum naturally realizes a generalized squeezing in a 4D manner.

DOI
10.1088/1751-8121/ab3cda

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Abstract We explore the hyperbolic geometry of squeezed states in the perspective of the non-compact Hopf map. Based on analogies between the squeeze operation and hyperbolic rotation, two types of the squeeze operators, the (usual) Dirac and the Schwinger types, are introduced. We clarify the underlying hyperbolic geometry and representations of the squeezed states along the line of the first non-compact Hopf map. Following the geometric hierarchy of the non-compact Hopf maps, we extend the analysis to —the isometry of a split-signature four-hyperboloid. We explicitly construct the squeeze operators in the Dirac and Schwinger types and investigate the physical meaning of the four-hyperboloid coordinates in the context of the Schwinger-type squeezed states. It is shown that the Schwinger-type squeezed one-photon state is equal to an entangled superposition state of two squeezed states and the corresponding concurrence has a clear geometric meaning. Taking advantage of the group theoretical formulation, basic properties of the squeezed coherent states are also investigated. In particular, we show that the squeezed vacuum naturally realizes a generalized squeezing in a 4D manner.
Abstract The Bloch hyperboloid H 2 underlies the quantum geometry of the original SO (2, 1) squeezed states. In Hasebe (2020 J. Phys. A: Math. Theor. 53 055303), the author utilized a non-compact 2nd Hopf map and a Bloch four-hyperboloid H 2,2 to explore an SO (2, 3) extension of the squeezed states. In the present paper, we further pursue the idea to derive an SO (4, 1) version of squeezed vacuum based on the other Bloch four-hyperboloid H 4 . We show that the obtained SO (4, 1) squeezed vacuum is a particular four-mode squeezed state not quite similar to the previous SO (2, 3) squeezed vacuum. In view of the Schwinger’s formulation of angular momentum, the SO (4, 1) squeezed vacuum is interpreted as a superposition of an infinite number of maximally entangled spin-pairs of all integer spins. We clarify basic properties of the SO (4, 1) squeezed vacuum, such as von Neumann entropy of spin entanglement, spin correlations and uncertainty relations with emphasis on their distinctions to the original SO (2, 1) case.
Abstract Current definitions of both squeezing operator and squeezed vacuum state are critically examined on the grounds of consistency with the underlying su(1,1) algebraic structure. Accordingly, the generalized coherent states for su(1,1) in its Schwinger two-photon realization are proposed as squeezed states. The physical implication of this assumption is that two additional degrees of freedom become available for the control of quantum optical systems. The resulting physical predictions are evaluated in terms of quadrature squeezing and photon statistics, while the application to a Mach–Zehnder interferometer is discussed to show the emergence of nonclassical regions, characterized by negative values of Mandel’s parameter, which cannot be anticipated by the current formulation, and then outline future possible use in quantum technologies.
Four-mode squeezed states: two-field quantum systems and the symplectic group Sp(4,R)\mathrm {Sp}(4,{\mathbb {R}})Sp(4,R)
Abstract We construct the four-mode squeezed states and study their physical properties. These states describe two linearly-coupled quantum scalar fields, which makes them physically relevant in various contexts such as cosmology. They are shown to generalise the usual two-mode squeezed states of single-field systems, with additional transfers of quanta between the fields. To build them in the Fock space, we use the symplectic structure of the phase space. For this reason, we first present a pedagogical analysis of the symplectic group Sp(4,R)\mathrm {Sp}(4,{\mathbb {R}})Sp(4,R) Sp ( 4 , R ) and its Lie algebra, from which we construct the four-mode squeezed states and discuss their structure. We also study the reduced single-field system obtained by tracing out one of the two fields. This procedure being easier in the phase space, it motivates the use of the Wigner function which we introduce as an alternative description of the state. It allows us to discuss environmental effects in the case of linear interactions. In particular, we find that there is always a range of interaction coupling for which decoherence occurs without substantially affecting the power spectra (hence the observables) of the system.
We present a detailed discussion of a type of four-photon coherent state defined as right eigenstates of the operator â 4 where â is the usual annihilation operator. There are actually four sets of states that need to be considered, namely those containing as the lowest number states ¦0>, ¦1>, ¦2>, or ¦3>. These correspond to the possible unique superpositions of the ordinary coherent states ¦±α> and ¦±iα>. We discuss the nonclassical properties of these states such as photon antibunching and squeezing. The usual second order squeezing does not exist for these states but higher order squeezing and square field amplitude squeezing do exist. Also discussed are the quasiprobability distributions, namely the P-function, the Q-function and the Wigner function. Finally, a method of generating these states based on the competition between a four-photon parametric process and incoherent losses from four-photon absorption is presented.
A simple, and elegant geometrical representation is developed to describe the concept of coherence and squeezing for angular momentum operators. Angular momentum squeezed states were obtained by applying Bogoliubov transformation on the angular momentum coherent states in Schwinger representation [Phys. Rev. {\bf{A 51}} (1995)]. We present the geometrical phase space description of angular momentum coherent and squeezed states and relate with the harmonic oscillator. The unique feature of our geometric representation is the portraying of the expectation values of the angular momentum components accompanied by their uncertainties. The bosonic representation of the angular momentum coherent and squeezed states is compared with the conventional one mentioning the advantages of this representation of angular momentum in context of coherence and squeezing. Extension of our work on single mode squeezing to double mode squeezing is presented and compared with the single mode one. We also point out the possible applications of the geometrical representation in analyzing the accuracy of interferometers and in studying the behavior, dynamics of an ensemble of quantum-mechanical two-level systems and its interaction with radiation.