Two-Photon Algebra Eigenstates: A Unified Approach to Squeezing

Brif, Constantin

doi:10.1006/aphy.1996.0112

Cited by 43 publications

(63 citation statements)

References 61 publications

(165 reference statements)

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“…, j − 1, j. As we expect these su(2) RIS contain the set of standard SU(2) CS with maximal symmetry | ζ ′ ; j and this occurs when m = ±j with ζ ′ = −β − (ζ 3 ∓ b) −1 [28]. At β 3 = 0 the su(2) RIS coincide with the Schrödinger J 1 -J 2 IS considered in ref.…”

Section: Explicit Solutions For Su(1 1) and Su(2) Rismentioning

confidence: 64%

“…Unlike the even and odd CS | α ± the K jk -RIS (being eigenstates of combinations u jk a j a k + v jk a † j a † k ) can exhibit strong squeezing in quadratures of a j a k and therefor can be called multimode squared amplitude squeezed states in complete analogy to the well known case of multimode (amplitude) squeezed states. One mode squared amplitude squeezed states are constructed and discussed in [29,28,16].…”

Section: Ris Of Multimode Boson Systemmentioning

confidence: 99%

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Robertson intelligent states

Trifonov

1997

J. Phys. A: Math. Gen.

119

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Diagonalization of uncertainty matrix and minimization of Robertson inequality for n observables are considered. It is proved that for even n this relation is minimized in states which are eigenstates of n/2 independent complex linear combinations of the observables. In case of canonical observables this eigenvalue condition is also necessary. Such minimizing states are called Robertson intelligent states (RIS). It is shown that group related coherent states (CS) with maximal symmetry (for semisimple Lie groups) are particular case of RIS for the quadratures of Weyl generators. Explicit constructions of RIS are considered for operators of su(1, 1), su(2), h N and sp(N, R) algebras. Unlike the group related CS, RIS can exhibit strong squeezing of group generators. Multimode squared amplitude squeezed states are naturally introduced as sp(N, R) RIS. It is shown that the uncertainty matrices for quadratures of q-deformed boson operators a q,j (q > 0) and of any k power of a j = a 1,j are positive definite and can be diagonalized by symplectic linear transformations.

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Section: Explicit Solutions For Su(1 1) and Su(2) Rismentioning

confidence: 64%

Section: Ris Of Multimode Boson Systemmentioning

confidence: 99%

Robertson intelligent states

Trifonov

1997

J. Phys. A: Math. Gen.

119

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“…As discussed in [19], the variance of 1-local quadrature operators scales with the number of modes as O(N ) in each of the branches |α ⊗N and |−α ⊗N of |ECS N (α) , since these are product states. The lack of dependence on photon number of the variance of the 1-local quadrature operator in the branches may also be viewed as resulting from the fact that all quadratures have the same variance in a coherent state and that it is an intelligent state for the Heisenberg uncertainty relation [39], i.e., it saturates the lower bound in the Heisenberg uncertainty relation. The individual branches will then have values of N F which are "microscopic," i.e., in O(1).…”

Section: A Entangled Coherent Statesmentioning

confidence: 99%

Macroscopicity of quantum superpositions on a one-parameter unitary path in Hilbert space

Volkoff

Whaley

2014

Phys. Rev. A

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We analyze quantum states formed as superpositions of an initial pure product state and its image under local unitary evolution, using two measurement-based measures of superposition size: one based on the optimal quantum binary distinguishability of the branches of the superposition and another based on the ratio of the maximal quantum Fisher information of the superposition to that of its branches, i.e., the relative metrological usefulness of the superposition. A general formula for the effective sizes of these states according to the branch distinguishability measure is obtained and applied to superposition states of N quantum harmonic oscillators composed of Gaussian branches. Considering optimal distinguishability of pure states on a time-evolution path leads naturally to a notion of distinguishability time that generalizes the well known orthogonalization times of Mandelstam and Tamm and Margolus and Levitin. We further show that the distinguishability time provides a compact operational expression for the superposition size measure based on the relative quantum Fisher information. By restricting the maximization procedure in the definition of this measure to an appropriate algebra of observables, we show that the superposition size of, e.g., N00N states and hierarchical cat states, can scale linearly with the number of elementary particles comprising the superposition state, implying precision scaling inversely with the total number of photons when these states are employed as probes in quantum parameter estimation of a 1-local Hamiltonian in this algebra.

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“…The two-photon algebra can be used to generate a large zoo of squeezed and coherent states for (single mode) one and two-photon processes which have been recently analysed in [2] (in this respect, see also [1]). More explicitly, the following one-boson representation of shows that one-photon processes are algebraically encoded within the subalgebra h 4 and that sl(2, IR) contains the information concerning two-photon dynamics.…”

Section: Introductionmentioning

confidence: 99%

“…The two-photon algebra eigenstates [2] are given by the analytic eigenfunctions that fulfill (β 1 N + β 2 B − + β 3 B + + β 4 A − + β 5 A + )f (α) = λf (α).…”

Section: Introductionmentioning

confidence: 99%

QUANTUM TWO-PHOTON ALGEBRA FROM NON-STANDARD U_z(sl(2,ℝ)) AND A DISCRETE TIME SCHRÖDINGER EQUATION

1998

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The non-standard quantum deformation of the (trivially) extended sl(2, IR) algebra is used to construct a new quantum deformation of the two-photon algebra h 6 and its associated quantum universal R-matrix. A deformed oneboson representation for this algebra is deduced and applied to construct a first order deformation of the differential equation that generates the two-photon algebra eigenstates in Quantum Optics. On the other hand, the isomorphism between h 6 and the (1+1) Schrödinger algebra leads to a new quantum deformation for the latter for which a differential-difference realization is presented. From it, a time discretization of the heat-Schrödinger equation is obtained and the quantum Schrödinger generators are shown to be symmetry operators.

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Two-Photon Algebra Eigenstates: A Unified Approach to Squeezing

Cited by 43 publications

References 61 publications

Robertson intelligent states

Robertson intelligent states

Macroscopicity of quantum superpositions on a one-parameter unitary path in Hilbert space

QUANTUM TWO-PHOTON ALGEBRA FROM NON-STANDARD U_z(sl(2,ℝ)) AND A DISCRETE TIME SCHRÖDINGER EQUATION

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Two-Photon Algebra Eigenstates: A Unified Approach to Squeezing

Cited by 43 publications

References 61 publications

Robertson intelligent states

Robertson intelligent states

Macroscopicity of quantum superpositions on a one-parameter unitary path in Hilbert space

QUANTUM TWO-PHOTON ALGEBRA FROM NON-STANDARD Uz(sl(2,ℝ)) AND A DISCRETE TIME SCHRÖDINGER EQUATION

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QUANTUM TWO-PHOTON ALGEBRA FROM NON-STANDARD U_z(sl(2,ℝ)) AND A DISCRETE TIME SCHRÖDINGER EQUATION