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arXiv:quant-ph/9905066

su(1,1) coherent states and a normal extension of su(1,1) annihilation operator: squeezed states and the simultaneous measurement of Q^{-1}P+PQ^{-1} and Q^{-2}

Fuminori Sakaguchi, Masahito Hayashi

Published May 21, 1999• 0 citations
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Abstract

The the over-complete eigenvector system of the operator Q^{-1}P (Q:position, P:momentum) which consists of the squeezed states |0; s, t > with various s and t are investigated from the viewpoint of the annihilation and creation relations related to the algebra su(1,1). We derive a positive operator-valued measure(POVM) for the simultaneous measurement between the self-adjoint and anti-self-adjoint parts of PQ^{-1}.

DOI
10.48550/arxiv.quant-ph/9905066

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The the over-complete eigenvector system of the operator Q^{-1}P (Q:position, P:momentum) which consists of the squeezed states |0; s, t > with various s and t are investigated from the viewpoint of the annihilation and creation relations related to the algebra su(1,1). We derive a positive operator-valued measure(POVM) for the simultaneous measurement between the self-adjoint and anti-self-adjoint parts of PQ^{-1}.
Generalized q-boson operators associated with the simultaneous creation of several q-bosons are introduced. These operators give rise to a realization of the quantum Weyl-Heisenberg algebra and are applied to the construction of corresponding Holstein-Primakoff realizations of the quantum algebras SU q (2) and SU q (1,1). Coherent states of these algebras are defined in the various ways suggested by the equivalent definitions of the harmonic oscillator coherent states, and some of their properties are studied. Particular attention is devoted to the squeezing properties of the quadrature of the electromagnetic field in these states.
We present a correspondence between positive operator valued measures (POVMs) and sets of generalized coherent states. POVMs describe quantum observables and, similar to quantum states, quantum observables can also be mixed. We show how the formalism of generalized coherent states leads to a useful characterization of extremal POVMs. We prove that covariant phase-space observables related to squeezed states are extremal, while those related to number states are not extremal. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.
The generalized coherent states based on SU(1, 1) are applied to quantum-measuring processes, in which each detector consists of a single particle. Differing from the Glauber coherent states, possible interactions are quite restricted. New definitions of the classical limit and classical operators are presented, where any polynomial of the position and momentum operators becomes classical. Although quantum interference itself becomes very large as the time after the interaction is increased, the statistical operator approaches its mixed-state part in a relative sense.
The generalized coherent states based on SU(1, 1) are applied to quantum-measuring processes, in which each detector consists of a single particle. Differing from the Glauber coherent states, possible interactions are quite restricted. New definitions of the classical limit and classical operators are presented, where any polynomial of the position and momentum operators becomes classical. Although quantum interference itself becomes very large as the time after the interaction is increased, the statistical operator approaches its mixed-state part in a relative sense.
Lüders’ theorem states that two observables commute if measuring one of them does not disturb the measurement outcomes of the other. We study measurements which are described by continuous positive operator-valued measurements (or POVMs) associated with coherent states on Lie groups. In general, operators turn out to be invariant under the Lüders map if their P- and Q-symbols coincide. For a spin corresponding to SU(2), the identity is shown to be the only operator with this property. For a particle, a countable family of linearly independent operators is identified which are invariant under the Lüders map generated by the coherent states of the Heisenberg–Weyl group, H3. The Lüders map is also shown to implement the anti-normal ordering of creation and annihilation operators of a particle.
We derive a measurement operator corresponding to a quantum nondemolition (QND) measurement of an atomic ensemble. The quantum measurement operator takes the form of a positive operator valued measure (POVM) and is valid for arbitrary interaction times, initial coherent state amplitudes, and final photon measurement outcomes. We analyze the dependence on various parameters and show that the effect of the QND measurement for short interaction times is to apply a Gaussian modulation of the initial state wavefunction. We derive approximate expressions for the POVM in various limits, such as the short interaction time regime and projective measurement limit. Several examples are shown, which show how spin squeezing and Schrodinger cat states can be generated using the measurement.