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arXiv:quant-ph/0007049 · Physical Review A

Reconstruction of SU(1,1) states

G. S. Agarwal, J. Banerji

Published Jul 17, 2001• 21 citations
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Abstract

We show how group symmetries can be used to reconstruct quantum states. The method we propose is presented in the context of the two-mode SU(1,1) states of the radiation field. In our scheme for SU(1,1) states, the input field passes through a nondegenerate parametric amplifier and one measures the probability of finding the output state with a certain number (usually zero) of photons in each mode. The density matrix in the Fock basis is retrieved from the measured data by the least-squares method after singular value decomposition of the design matrix followed by Tikhonov regularization. Several illustrative examples involving the reconstruction of a pair coherent state, a Perelomov coherent state, and a coherent superposition of pair coherent states are considered.

DOI
10.1103/physreva.64.023815

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We show how group symmetries can be used to reconstruct quantum states. The method we propose is presented in the context of the two-mode SU(1,1) states of the radiation field. In our scheme for SU(1,1) states, the input field passes through a nondegenerate parametric amplifier and one measures the probability of finding the output state with a certain number (usually zero) of photons in each mode. The density matrix in the Fock basis is retrieved from the measured data by the least-squares method after singular value decomposition of the design matrix followed by Tikhonov regularization. Several illustrative examples involving the reconstruction of a pair coherent state, a Perelomov coherent state, and a coherent superposition of pair coherent states are considered.
We consider how the conventional spectroscopic and interferometric schemes can be rearranged to serve for reconstructing quantum states of physical systems possessing SU (2) symmetry. The discussed systems include a collection of two-level atoms, a two-mode quantized radiation field with a fixed total number of photons, and a single laser-cooled ion in a two-dimensional harmonic trap with a fixed total number of vibrational quanta. In the proposed rearrangement, the standard spectroscopic and interferometric experiments are inverted. Usually one measures an unknown frequency or phase shift using a system prepared in a known quantum state. Our aim is just the inverse, i.e., to use a well-calibrated apparatus with known transformation parameters to measure unknown quantum states.
We propose a scheme to reconstruct the state of a two-mode Bose-Einstein condensate, with a given total number of atoms, using an atom interferometer that requires beam splitter, phase shift, and nonideal atom counting operations. The density matrix in the number-state basis can be computed directly from the probabilities of different counts for various phase shifts between the original modes, unless the beam splitter is exactly balanced. Simulated noisy data from a two-mode coherent state are produced and the state is reconstructed, for 49 atoms. The error can be estimated from the singular values of the transformation matrix between state and probability data.
We construct the Perelomov number coherent states for an arbitrary su(1, 1) group operation and study some of their properties. We introduce three operators which act on Perelomov number coherent states and close the su(1, 1) Lie algebra. By using the tilting transformation we apply our results to obtain the energy spectrum and eigenfunctions of the non-degenerate parametric amplifier. We show that these eigenfunctions are the Perelomov number coherent states of the two-dimensional harmonic oscillator.
We study two types of coherent states for two-mode realizations of the direct product group $\mathrm{SU}(1,1)\ensuremath{\bigotimes}\mathrm{SU}(1,1)$ [which is locally isomorphic to SO(2,2)] constructed from the coupling of two single-mode realizations of SU(1,1). The basis states for the relevant representations of $\mathrm{SU}(1,1)\ensuremath{\bigotimes}\mathrm{SU}(1,1)$ are constructed from the SU(1,1) Clebsch-Gordon coefficients. From these, Perelomov and Barut-Giradello coherent states are constructed. Various properties of the states are discussed, and methods for generating them are proposed. Some of the states can be generated by the operation of beam splitters with two-mode squeezed vacuum states or pair coherent states as inputs. We show that a competitive two-channel two-photon process gives rise to states of the Barut-Girardello type as the steady-state solutions of the associated master equation for appropriate initial conditions.