Quantum States and Number-Phase Uncertainty Relations Measured by Optical Homodyne Tomography

Raymer, Michael G.; Smithey, D. T.; Beck, M.; Cooper, J.

doi:10.12693/aphyspola.86.71

Cited by 14 publications

(4 citation statements)

References 12 publications

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“…All other proofs of canonical regularizations can be made in analogy to this proof which is here given as an example for such kind of proofs. The meaning of (P(1/x)) (1) in the inversion formulas for the Radon transform is therefore that we have to substitute it by −R(1/x 2 ) considered in the sense of Eq.(2.15). One may think that this does not give the right sign but it is not so.…”

Section: Two-dimensional Radon and Fourier Transforms And Their Invermentioning

confidence: 99%

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Radon transform and pattern functions in quantum tomography

Wünsche¹

1997

Journal of Modern Optics

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The two-dimensional Radon transform of the Wigner quasiprobability is introduced in canonical form and the functions playing a role in its inversion are discussed. The transformation properties of this Radon transform with respect to displacement and squeezing of states are studied and it is shown that the last is equivalent to a symplectic transformation of the variables of the Radon transform with the contragredient matrix to the transformation of the variables in the Wigner quasiprobability. The reconstruction of the density operator from the Radon transform and the direct reconstruction of its Fock-state matrix elements and of its normally ordered moments are discussed. It is found that for finite-order moments the integration over the angle can be reduced to a finite sum over a discrete set of angles. The reconstruction of the Fock-state matrix elements from the normally ordered moments leads to a new representation of the pattern functions by convergent series over even or odd Hermite polynomials which is appropriate for practical calculations. The structure of the pattern functions as first derivatives of the products of normalizable and nonnormalizable eigenfunctions to the number operator is considered from the point of view of this new representation.

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Section: Two-dimensional Radon and Fourier Transforms And Their Invermentioning

confidence: 99%

“…They possess all properties of genuine probability densities and can be measured since recent time in quantum optics of the radiation field by homodyne detection [1][2][3][4]. The field of problems of the reconstruction of the density operator from such or similar data is called quantum tomography.…”

Section: Introductionmentioning

confidence: 99%

Radon transform and pattern functions in quantum tomography

Wünsche¹

1997

Journal of Modern Optics

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“…In analogy to equation (2), such a Wigner function gives rise to a spatial density pattern PO(x) = J + dpW0(x, p) . (4) Phase-space tomography relies on the fact that equation (4) can be inverted, i .e . that the Wigner function can be reconstructed from the continuous set of density patterns Pe(x) [8],…”

Section: Introductionmentioning

confidence: 99%

“…This method was applied in a theoretical work to measure amplitude and phase structures of optical pulses [2] . It was also experimentally applied to reconstruct the Wigner function of light modes [3][4][5] and vibrational states of molecules [6] . In atom optics, a tomographic procedure was proposed to reconstruct the Wigner function of the transverse motion of an atom beam [7] .…”

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confidence: 99%

Tomography of Atom Beams

Janicke

Wilkens

1995

Journal of Modern Optics

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We investigate the accuracy of an experimental reconstruction of the Wigner function of the transverse motion in an atom beam using numerical wave packet simulations . For the example of a superposition of two Gaussians (the outcome of a double slit, for example) we study in detail the influence of experimental restraints on the quality of the reconstruction . For the potential candidate, metastable helium, we demonstrate that an accurate reconstruction of the Wigner function, especially of its negative parts, is well within experimental reach . . IntroductionThe density matrix encodes all information which can be obtained about a quantum mechanical system . A particular representation of the density matrix is the Wigner function which is defined in phase-space in close analogy to a classical phase-space distribution . In contrast to the latter, the Wigner function is not confined to positive values and therefore it cannot be interpreted as a probability distribution . Likewise, in contrast to classical phase-space distributions, the Wigner function cannot be measured directly because its arguments refer to the eigenvalues of non-commuting observables .The marginal distributions of the Wigner function, which we will refer to as , quantum-shadows' t, are, however, experimentally accessible . In most experiments with atom beams, quantum-shadows are formed by the spatial density distributions of the atoms which include information about the state of the atomic centre-of-mass motion . For example, the position distribution of the atoms in the far field yields information about the momentum distribution . In atom interferometry, spatial interference patterns encode phase information about the atomic de Broglie waves .Recently, a method was presented for reconstructing the full Wigner function from its quantum-shadows only utilizing a tomographic procedure [1] . The method is based on a rotation of the Wigner function in phase-space and the observation of the marginal distributions of one phase-space variable . This method was applied in a theoretical work to measure amplitude and phase structures of optical pulses [2] . It was also experimentally applied to reconstruct the Wigner function of light modes [3][4][5] and vibrational states of molecules [6] . In atom optics, a tomographic procedure was proposed to reconstruct the Wigner function of the transverse motion of an atom beam [7] . In this case, the rotation is performed by the interaction of the beam with a thin lens and the free evolution following the interaction .In this paper we provide a detailed analysis of the proposal in [7] for the fi This name was suggested by U . Leonhardt in analogy to Plato's famous parable .

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