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Physical Review A

Uncertainties of Schwinger angular-momentum operators for squeezed radiation in interferometers

Abir Bandyopadhyay, Jagdish Rai

Published Feb 1, 1995• 7 citations
Publisher

Abstract

Atkins and Dobson [Proc. R. Soc. London Ser. A 321, 321 (1971)] have constructed coherent angular-momentum states using the Schwinger bosonic representation and operating the displacement (coherent) operator on both of the bosonic vacuum states. By performing a squeezing transformation on the bosonic coherent states we generalize the idea of Schwinger angular-momentum coherent (SAMC) states to Schwinger angular-momentum squeezed (SAMS) states. We show that these SAMS states, which are constructed from the bosonic squeezed coherent states, exhibit squeezing of Schwinger angular-momentum operators under a certain range of parameters. The properties of these states are studied and compared with their single-mode bosonic counterpart. Application of these states in increasing the accuracy of an interferometer is discussed.

DOI
10.1103/physreva.51.1597

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Atkins and Dobson [Proc. R. Soc. London Ser. A 321, 321 (1971)] have constructed coherent angular-momentum states using the Schwinger bosonic representation and operating the displacement (coherent) operator on both of the bosonic vacuum states. By performing a squeezing transformation on the bosonic coherent states we generalize the idea of Schwinger angular-momentum coherent (SAMC) states to Schwinger angular-momentum squeezed (SAMS) states. We show that these SAMS states, which are constructed from the bosonic squeezed coherent states, exhibit squeezing of Schwinger angular-momentum operators under a certain range of parameters. The properties of these states are studied and compared with their single-mode bosonic counterpart. Application of these states in increasing the accuracy of an interferometer is discussed.
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.
Abstract The work of Carruthers & Nieto on the harmonic oscillator coherent states is combined with Schwinger’s construction of angular momentum to produce the angular momentum coherent states. It is shown that these states become the vector representatives of angular momentum in the classical limit, and so are particularly useful for discussing the transition from quantum to classical angular momentum. The uncertainty relations for angle and angular momentum are described and are compatible with the classical limit. Under rotations the coherent states transform in a manner that in the classical limit is equivalent to the transformation of vectors, and in the same limit the root mean square variation of the expectation values of the components of angular momentum become negligible in comparison with the expectation values themselves. The coupling of two angular momenta in the classical limit is investigated: it is shown that although the product of two coherent states is not itself a coherent state, it does represent a packet similar to a true coherent state, and centred on the direction of the classical resultant of the two component vectors. The properties and implications of hyperbolic angular momentum space are discussed.
With the successes of the laser interferometer gravitational-wave observatory, we anticipate increased interest in working with squeezed states in the undergraduate and graduate quantum-mechanics classroom. Because squeezed-coherent states are minimum uncertainty states, their wavefunctions in position and momentum space must be Gaussians. But this result is rarely discussed in treatments of squeezed states in quantum textbooks or quantum optics textbooks. In this work, we show three different ways to construct the wavefunction for squeezed-coherent states: (i) a differential equation-based approach; (ii) an approach that uses an expansion in terms of the simple-harmonic oscillator wavefunctions; and (iii) a fully operator-based approach. We do this to illustrate that the concept of the wavefunction can be introduced no matter what methodology an instructor wishes to use. We hope that working with the wavefunction will help demystify the concept of a squeezed-coherent state.