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arXiv:quant-ph/9706028

Barut-Girardello coherent states for sp(N,C) and multimode Schrödinger cat states

D. A. Trifonov

Published Jun 13, 1997• 0 citations
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Abstract

Overcomplete families of states of the type of Barut-Girardello coherent states (BG CS) are constructed for noncompact algebras $u(p,q)$ and $sp(N,C)$ in quadratic bosonic representation. The $sp(N,C)$ BG CS are obtained in the form of multimode ordinary Schrödinger cat states. A set of such macroscopic superpositions is pointed out which is overcomplete in the whole $N$ mode Hilbert space (while the associated $sp(N,C)$ representation is reducible). The multimode squared amplitude Schrödinger cat states are introduced as macroscopic superpositions of the obtained $sp(N,C)$ BG CS.}

DOI
10.48550/arxiv.quant-ph/9706028

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Overcomplete families of states of the type of Barut-Girardello coherent states (BG CS) are constructed for noncompact algebras u(p,q)u(p,q)u(p,q) and sp(N,C)sp(N,C)sp(N,C) in quadratic bosonic representation. The sp(N,C)sp(N,C)sp(N,C) BG CS are obtained in the form of multimode ordinary Schrödinger cat states. A set of such macroscopic superpositions is pointed out which is overcomplete in the whole NNN mode Hilbert space (while the associated sp(N,C)sp(N,C)sp(N,C) representation is reducible). The multimode squared amplitude Schrödinger cat states are introduced as macroscopic superpositions of the obtained sp(N,C)sp(N,C)sp(N,C) BG CS.}
The Barut-Girardello (BG) coherent states (CS) representation is extended to the noncompact algebras and in (reducible) quadratic boson realizations. The BG CS take the form of multimode ordinary Schrödinger cat states. Macroscopic superpositions of CS ( canonical CS, n = 1, 2, ...) are pointed out which are overcomplete in the N -mode Hilbert space and the relation between the canonical CS and the BG-type CS representations is established. The sets of and BG CS and their discrete superpositions contain many states studied in quantum optics (even and odd N -mode CS, pair CS) and provide an approach to quadrature squeezing, alternative to that of intelligent states. New subsets of weakly and strongly nonclassical states are pointed out and their statistical properties (first- and second-order squeezing, photon number distributions) are discussed. For specific values of the angle parameters and small amplitude of the canonical CS components, these states approach multimode Fock states with one, two or three bosons/photons. It is shown that eigenstates of a squared non-Hermitian operator (generalized cat states) can exhibit squeezing of the quadratures of A .
The aim of this paper is to construct coherent states à la Perelomov and à la Barut–Girardello for a polynomial Weyl–Heisenberg algebra. This generalized Weyl–Heisenberg algebra, denoted by , depends on r real parameters and is an extension of the one-parameter algebra (Daoud and Kibler 2010 J. Phys. A: Math. Theor. 43 115303) which covers the cases of the su (1, 1) algebra (for κ > 0), the su (2) algebra (for κ < 0) and the h 4 ordinary Weyl–Heisenberg algebra (for κ = 0). For finite-dimensional representations of and , where is a truncation of order s of in the sense of Pegg–Barnett, a connection is established with k -fermionic algebras (or quon algebras). This connection makes it possible to use generalized Grassmann variables for constructing certain coherent states. Coherent states of the Perelomov type are derived for infinite-dimensional representations of and for finite-dimensional representations of and through a Fock–Bargmann analytical approach based on the use of complex (or bosonic) variables. The same approach is applied for deriving coherent states of the Barut–Girardello type in the case of infinite-dimensional representations of . In contrast, the construction of coherent states à la Barut–Girardello for finite-dimensional representations of and can be achieved solely at the price of replacing complex variables by generalized Grassmann (or k -fermionic) variables. Some of the results are applied to su (2), su (1, 1) and the harmonic oscillator (in a truncated or not truncated form). This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.
Quantum Schrodinger cat states are of great interest in quantum communications and quantum optics. These states are used in various scientific fields such as quantum computing, quantum error correction and high-precision measurements. The analysis of the Schrodinger cat states coherence is an important task for their complete practical application. Our developed approach makes it possible to estimate the coherence of the quantum Schrodinger cat state of arbitrary dimension, as well as to find the interference visibility of the state: an important optical characteristic. The obtained simple quantitative relationship between coherence and the Schmidt number, as well as the developed approach of reducing the multidimensional quantum cat state to a two-mode analog allow us to analyze macroscopic states formed by a large number of modes. Several explicit formulas for the reduced states that arise after measuring of some modes of the considered multimode system are obtained. The research results have significant application and can be used in the development of high-dimensional quantum information processing systems.
Schrödinger cat states, formed from superpositions of well-separated coherent states, are promising resources for bosonic quantum information. We construct parent Hamiltonians for entangled multimode cat states by separating the selection of local coherent-state branches from the correlations and symmetries that determine their global superposition. GHZ-, cluster-, and W-type cat states illustrate three distinct mechanisms for this construction. In the limit of well-separated branches, the resulting oscillator Hamiltonians reduce to familiar qubit parent models based on correlations, stabilizers, and exchange symmetry. The same constraint structure also provides a common basis for coherent preparation, dissipative stabilization, and state certification.
Quantum Schrodinger cat states are of great interest in quantum communications and quantum optics. These states are used in various scientific fields such as quantum computing, quantum error correction and high-precision measurements. The analysis of the Schrodinger cat states coherence is an important task for their complete practical application. Our developed approach makes it possible to estimate the coherence of the quantum Schrodinger cat state of arbitrary dimension, as well as to find the interference visibility of the state - an important optical characteristic. The obtained simple quantitative relationship between coherence and the Schmidt number, as well as the developed approach of reducing the multidimensional quantum cat state to a two-mode analog allow us to analyze macroscopic states formed by a large number of modes. Several explicit formulas for the reduced states that arise after measuring of some modes of the considered multimode system are obtained. The research results have significant application and can be used in the development of high-dimensional quantum information processing systems.
We define Schrödinger cat states as superpositions of q-deformed Barut–Girardello su (1, 1) coherent states with an adjustable angle φ in a q-deformed Fock space. We study the statistical properties of the q-deformed Barut–Girardello su (1, 1) coherent states and Schrödinger cat states. The statistical properties of photons are always sub-Poissonian for q-deformed Barut–Girardello su (1, 1) coherent states. For Schrödinger cat states in the cases φ = 0, π /2, π , the statistical properties of photons are always sub-Poissonian if φ = π /2, and the other cases are hard to determine because they depend on the parameters q and k. Moreover, we find some interesting properties of Schrödinger cat states in the limit | z | → 0, where z is the parameter of those states. We also derive that the statistical properties of photons are sub-Poissonian in the undeformed case where π /2 ≤ φ ≤ 3 π /2.