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Quantum polarization properties of two-mode number states

Anita Sehat, Jonas Söderholm, Gunnar G. Bjork, P.B. Espinoza, A. B. Klimov, Luis Lorenzo Sánchez-Soto

Published Jul 23, 2004• 0 citations
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Abstract

Quantum polarization properties of two-mode states have been discussed in the context of quantum states with ``hidden polarization''. In this paper we demonstrate that all pure two-photon and three-photon states have unit degree of quantum polarization. We also show that pure states belonging to groups of discrete symmetry have unit degree of quantum polarization for arbitrary number of photons. For a certain class of two-photon states, we substantiate the possibility of generating the full basis set (three orthogonal states) by using only combinations of the SU(2) generators, i.e., by using geometrical rotations and differential phase shifts. Finally, we generate the whole basis set for pure reflection-symmetrical three-photon states by means of SU(2) generators.

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Quantum polarization properties of two-mode states have been discussed in the context of quantum states with ``hidden polarization''. In this paper we demonstrate that all pure two-photon and three-photon states have unit degree of quantum polarization. We also show that pure states belonging to groups of discrete symmetry have unit degree of quantum polarization for arbitrary number of photons. For a certain class of two-photon states, we substantiate the possibility of generating the full basis set (three orthogonal states) by using only combinations of the SU(2) generators, i.e., by using geometrical rotations and differential phase shifts. Finally, we generate the whole basis set for pure reflection-symmetrical three-photon states by means of SU(2) generators.
We show that any pure, two-mode, NNN-photon state with NNN odd or equal to two can be transformed into an orthogonal state using only linear optics. According to a recently suggested definition of polarization degree, this implies that all such states are fully polarized. This is also found to be true for any pure, two-mode, energy eigenstate belonging to a two-dimensional SU(2) orbit. Complete two- and three-photon bases whose basis states are related by only phase shifts or geometrical rotations are also derived.
We examine whether the Stokes parameters of a two-mode electromagnetic field results from the superposition of the spins of the photons it contains. To this end we express any n-photon state as the result of the action on the vacuum of n creation operators generating photons which can have may different polarization states in general. We find that the macroscopic polarization holds as sum of the single-photon Stokes parameters only for the SU(2) orbits of photon-number states. The states that lack this property are entangled in every basis of independent field modes, so this is a class of entanglement beyond the reach of SU(2) transformations.
We experimentally investigate various quantum polarization features of three-photon quantum states, including product and entangled states with varying purity. The three-photon quantum states are categorized into six classes based on the rotation symmetry of mean, variance, and skewness of the polarization distribution. The representative three-photon quantum states in each category are prepared from double-pair emission from pulsed spontaneous parametric down-conversion and quantum interferometry. We demonstrate that the three-photon quantum states show interesting quantum polarization properties, such as maximum sum-uncertainty and hidden polarizations.
Linearly polarized classical light can be expressed in a vertical and a horizontal component. Geometrically rotating vertically polarized light by 90 degrees will convert it to the orthogonal horizontal polarization. We have experimentally generated a two-photon state of light which evolves into an orthogonal state upon geometrical rotation by 60 degrees. Rotating this state by an additional 60 degrees will yield a state which is mutually orthogonal to the first two states. Generalizing this procedure, one can generate N+1 mutually orthogonal N-photon states that cyclicly evolve from one to another upon a geometric rotation by 180/(N+1) degrees.
The SU(2) group is used in two different fields of quantum optics, the quantum polarization and quantum interferometry.Quantum degrees of polarization may be based on distances of a polarization state from the set of unpolarized states.The maximum polarization is achieved in the case where the state is pure and then the distribution of the photon-number sums is optimized.In quantum interferometry, the SU(2) intelligent states have also the property that the Fisher measure of information is equal to the inverse minimum detectable phase shift on the usual simplifying condition.Previously, the optimization of the Fisher information under a constraint was studied.Now, in the framework of constraint optimization, states similar to the SU(2) intelligent states are treated.
The degree of polarization of a quantum field can be defined as its distance to an appropriate set of states. When we take unpolarized states as this reference set, the states optimizing this degree for a fixed average number of photons N‾\overline{N}N present a fairly symmetric, parabolic photon statistic, with a variance scaling as N‾2{\overline{N}}^{2}N2. Although no standard optical process yields such a statistic, we show that, to an excellent approximation, a highly squeezed vacuum can be taken as maximally polarized. We also consider the distance of a field to the set of its SU(2) transformed, finding that certain linear superpositions of SU(2) coherent states make this degree to be unity.