Universal homodyne tomography with a single local oscillator

D’Ariano, Giacomo Mauro; Sacchi, Massimiliano F.; Kumar, Prem

doi:10.1103/physreva.61.013806

Cited by 35 publications

(42 citation statements)

References 19 publications

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“…Our results form a multimode generalization of the single-mode relations obtained by Vogel and Risken [9] and by Richter [23]. Notice also, that D'Ariano et al gave explicit formula for sampling function of twomode Husimi quasidistribution [32]. Several functions S N (ξ; −1) are plotted in Fig.…”

Section: A Sampling Of the Smoothed Wigner Functionsmentioning

confidence: 86%

“…It is convenient to work in the hyperspherical coordinates. The points {z j } lie on a surface of 2N -dimensional unit sphere and we parametrize them as [32] …”

Section: A Sampling Of the Smoothed Wigner Functionsmentioning

confidence: 99%

“…[32]. Well-known analytical formulas for single-mode sampling functions f mn involve products of regular and irregular eigenfunctions of the harmonic oscillator Hamiltonian [14,15].…”

Section: Density Matrix Elementsmentioning

confidence: 99%

“…The functions for sampling density matrix elements in Fock basis were found in [27,28,31], and those allowing direct reconstruction of two-mode correlation functions were obtained in [29][30][31] . Recently, a general multimode homodyne tomography with a single LO was considered and the sampling functions for density matrix elements were expressed in terms of integrals [32]. In this paper we shall derive various important sampling functions for multimode homodyne tomography with a single LO.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, increasing attention has been devoted to multimode homodyne tomography [26][27][28][29][30][31][32], because some of the most interesting quantum mechanical phenomena stem from correlations between several degrees of freedom. Let us mention just the EPR paradox and violation of Bell's inequalities [33].…”

Section: Introductionmentioning

confidence: 99%

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Sampling functions for multimode homodyne tomography with a single local oscillator

Fiurášek

2001

Phys. Rev. A

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We derive various sampling functions for multimode homodyne tomography with a single local oscillator. These functions allow us to sample multimode s-parametrized quasidistributions, density matrix elements in Fock basis, and s-ordered moments of arbitrary order directly from the measured quadrature statistics. The inevitable experimental losses can be compensated by proper modification of the sampling functions. Results of Monte Carlo simulations for squeezed three-mode state are reported and the feasibility of reconstruction of the three-mode Q-function and s-ordered moments from 10 7 sampled data is demonstrated. PACS number(s): 42.50. Dv,

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Section: A Sampling Of the Smoothed Wigner Functionsmentioning

confidence: 86%

“…It is convenient to work in the hyperspherical coordinates. The points {z j } lie on a surface of 2N -dimensional unit sphere and we parametrize them as [32] …”

Section: A Sampling Of the Smoothed Wigner Functionsmentioning

confidence: 99%

“…[32]. Well-known analytical formulas for single-mode sampling functions f mn involve products of regular and irregular eigenfunctions of the harmonic oscillator Hamiltonian [14,15].…”

Section: Density Matrix Elementsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Sampling functions for multimode homodyne tomography with a single local oscillator

Fiurášek

2001

Phys. Rev. A

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A new look at the quantum mechanics of the harmonic oscillator

Kastrup¹

2007

Ann. Phys.

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In classical mechanics the harmonic oscillator (HO) provides the generic example for the use of angle and action variables ϕ ∈ R mod 2π and I > 0 which played a prominent role in the "old" Bohr-Sommerfeld quantum theory. However, already classically there is a problem which has essential implications for the quantum mechanics of the (ϕ, I)-model for the HO: the transformation q = √ 2I cos ϕ, p = − √ 2I sin ϕ is only locally symplectic and singular for (q, p) = (0, 0). Globally the phase space {(q, p)} has the topological structure of the plane R 2 , whereas the phase space {(ϕ, I)} corresponds globally to the punctured plane R 2 − (0, 0) or to a simple cone with the tip deleted. From the properties of the symplectic transformations on that phase space one can derive the functions h0 = I, h1 = I cos ϕ and h2 = −I sin ϕ as the basic coordinates on {(ϕ, I)} , where their Poisson brackets obey the Lie algebra of the symplectic group of the plane. This implies a qualitative difference as to the quantum theory of the phase space {(ϕ, I)} compared to the usual one for {(q, p)} : In the quantum mechanics for the (ϕ, I)-model of the HO the three hj correspond to the self-adjoint generators Kj, j = 0, 1, 2, of certain irreducible unitary representations of the symplectic group or one of its infinitely many covering groups, the representations being parametrized by a (Bargmann) index k > 0. This index k determines the ground state energy E k,n=0 = ω k of the (ϕ, I)-Hamiltonian H( K) = ω K0. For an m-fold covering the lowest possible value for k is k = 1/m , which can be made arbitrarily small by choosing m accordingly! This is not in contradiction to the usual approach in terms of the operators Q and P which are now expressed as functions of the Kj, but keep their usual properties. The richer structure of the Kj quantum model of the HO is "erased" when passing to the simpler (Q, P )-model! This more refined approach to the quantum theory of the HO implies many experimental tests: Mullikentype experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE-condensates, with charged HOs in external electric fields and the (Landau) levels of charged particles in external magnetic fields, with the propagation of light in vacuum, passing through strong external electric or magnetic fields. Finally it may lead to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant.

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A new look at the quantum mechanics of the harmonic oscillator

Kastrup¹

2007

Annalen der Physik

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In classical mechanics the harmonic oscillator (HO) provides the generic example for the use of angle and action variables and I > 0 which played a prominent role in the “old” Bohr‐Sommerfeld quantum theory. However, already classically there is a problem which has essential implications for the quantum mechanics of the (φ,I)‐model for the HO: the transformation is only locally symplectic and singular for (q,p) = (0,0). Globally the phase space {(q,p)} has the topological structure of the plane ℝ2, whereas the phase space {(φ,I)} corresponds globally to the punctured plane ℝ2 ‐(0,0) or to a simple cone with the tip deleted. From the properties of the symplectic transformations on that phase space one can derive the functions h0 = I, h1 = Icos φ and h2 = ‐Isin φ as the basic coordinates on {(φ,I)}, where their Poisson brackets obey the Lie algebra of the symplectic group of the plane. This implies a qualitative difference as to the quantum theory of the phase space {(φ,I)} compared to the usual one for {(q,p)}: In the quantum mechanics for the (φ,I)‐model of the HO the three hj correspond to the self‐adjoint generators Kj, j = 0,1,2, of certain irreducible unitary representations of the symplectic group or one of its infinitely many covering groups, the representations being parametrized by a (Bargmann) index k > 0. This index k determines the ground state energy of the (φ,I)‐Hamiltonian . For an m‐fold covering the lowest possible value for k is k = 1/m, which can be made arbitrarily small by choosing m accordingly! This is not in contradiction to the usual approach in terms of the operators Q and P which are now expressed as functions of the Kj, but keep their usual properties. The richer structure of the Kj quantum model of the HO is “erased” when passing to the simpler (Q,P)‐model! This more refined approach to the quantum theory of the HO implies many experimental tests: Mulliken‐type experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE‐condensates, with charged HOs in external electric fields and the (Landau) levels of charged particles in external magnetic fields, with the propagation of light in vacuum, passing through strong external electric or magnetic fields. Finally it may lead to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant.

show abstract

Universal homodyne tomography with a single local oscillator

Cited by 35 publications

References 19 publications

Sampling functions for multimode homodyne tomography with a single local oscillator

Sampling functions for multimode homodyne tomography with a single local oscillator

A new look at the quantum mechanics of the harmonic oscillator

A new look at the quantum mechanics of the harmonic oscillator

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