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arXiv:0806.3661 · Physical Review A

Full quantum reconstruction of vortex states

Ioannes Rigas, Luis Lorenzo Sánchez-Soto, A. B. Klimov, Jaroslav Řeháček, Z. Hradil

Published Dec 16, 2008• 22 citations
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Abstract

We propose a complete tomographic reconstruction of vortex states carrying orbital angular momentum. The scheme determines the angular probability distribution of the state at different times under free evolution. To represent the quantum state, we introduce a bona fide Wigner function defined on the discrete cylinder, which is the natural phase space for the pair angle-angular momentum. The feasibility of the proposal is addressed.

DOI
10.1103/physreva.78.060101

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It is pointed out that the bona fide Wigner distribution recently introduced [Phys. Rev. A 78, 060101(R) (2008)] is in fact the rotational Wigner function earlier derived and thoroughly studied [Phys. Rev. A 49, 3255 (1994); 71, 069901(E) (2005)], so the latter remains indeed the natural phase-space representation for the rotation-angle and angular-momentum pair.

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We propose a complete tomographic reconstruction of vortex states carrying orbital angular momentum. The scheme determines the angular probability distribution of the state at different times under free evolution. To represent the quantum state, we introduce a bona fide Wigner function defined on the discrete cylinder, which is the natural phase space for the pair angle-angular momentum. The feasibility of the proposal is addressed.
The problem of constructing physically and mathematically well-defined Wigner functions for the canonical pair angle $\ensuremath{\theta}$ and angular momentum ppp is solved. While a key element for the construction of Wigner functions for the planar phase space ${(q,p)\ensuremath{\in}{\mathbb{R}}^{2}}$ is the Heisenberg-Weyl group, the corresponding group for the cylindrical phase space ${(\ensuremath{\theta},p)\ensuremath{\in}{S}^{1}\ifmmode\times\else\texttimes\fi{}\mathbb{R}}$ is the Euclidean group E(2)E(2)E(2) of the plane and its unitary representations. Here the angle $\ensuremath{\theta}$ is replaced by the pair $(cos\ensuremath{\theta},sin\ensuremath{\theta})$, which corresponds uniquely to the points on the unit circle. The main structural properties of the Wigner functions for the planar and the cylindrical phase spaces are strikingly similar. A crucial role is played by the sincsincsinc function, which provides the interpolation for the discontinuous quantized angular momenta in terms of the continuous classical ones, in accordance with the famous Whittaker cardinal function well known from interpolation and sampling theories. The quantum mechanical marginal distributions for the angle (continuous) and angular momentum (discontinuous) are, as usual, uniquely obtained by appropriate integrations of the $(\ensuremath{\theta},p)$ Wigner function. Among the examples discussed is an elementary system of simple cat states.
We establish a reconstruction approach for the rotational quantum state of linear molecules possessing a magnetic manifold. Our approximate method contains an iteration with generalized matrix inverses, processing the tomographic integral of the time-dependent molecular-axis distribution in a polar angle. As shown in a simulated example for an alignment state, the density matrix is determined in a high fidelity. An analytic tomographic formula is also derived for the symmetric top rotation. The state coherent in the quantum space of both the angular momentum and its magnetic projection can be approximately retrieved from the observable time-resolved solid-angle distribution of the molecular axis.
We establish a reconstruction approach for the rotational quantum state of linear molecules possessing a magnetic manifold.Our approximate method contains an iteration with generalized matrix inverses,processing the tomographic integral of the time-dependent molecular-axis distribution in a polar angle.As shown in a simulated example for an alignment state,the density matrix is determined in a high fidelity.An analytic tomographic formula is also derived for the symmetric top rotation.The state coherent in the quantum space of both the angular momentum and its magnetic projection can be approximately retrieved from the observable time-resolved solid-angle distribution of the molecular axis.
We present a filtered backprojection algorithm for reconstructing the Wigner function of a system of large angular momentum j from Stern–Gerlach-type measurements. Our method is advantageous over the full determination of the density matrix in that it is insensitive to experimental fluctuations in j , and allows for a natural elimination of high-frequency noise in the Wigner function by taking into account the experimental uncertainties in the determination of j , its projection m and the quantization axis orientation. No data binning and no arbitrary smoothing parameters are necessary in this reconstruction. Using recently published data (Riedel et al 2010 Nature 464 1170), we reconstruct the Wigner function of a spin-squeezed state of a Bose–Einstein condensate of about 1250 atoms, demonstrating that measurements along quantization axes lying in a single plane are sufficient for performing this tomographic reconstruction. Our method does not guarantee positivity of the reconstructed density matrix in the presence of experimental noise, which is a general limitation of backprojection algorithms.