Coherent States for the Rotation Group of Three-Dimensional Space

Sergey A. Podoshvedov

Weizhao Lu, Jianfeng Qiu

In this paper, a multiplexing technique of coherent polarization states based on three-dimensional rotation group is proposed. The technique applies two Stokes parameters in one polarized light as signal and local oscillator (LO), and combines them in the single spatial mode. Three-dimensional rotation group SO(3) is used to describe the transformation of Stokes parameters. SO(3) is decomposed into three subgroups, with each subgroup represents a rotation along different axes. A multiplexer is designed with three electro-optical crystals to realize the function of three-dimensional rotation group. The multiplexer achieves an ergodicity of polarization states. Signal and LO propagates in the same spatial mode, multiplexing of polarization states is realized. Finally, the multiplexer is tested in a free space quantum communication system and its feasibility in quantum communication is confirmed.

K. Goyal Sandeep

Understanding the geometry of state space is of fundamental interest in the field of quantum information, in particular to study entanglement. The states of a d dimensional system form a convex set in the d2−1 dimensional Euclidean space Rd2−1. For a two dimensional system this convex set is nothing but a solid unit sphere (Bloch sphere) with centre at the origin. The surface S of this unit sphere consists of all the pure states of the system. Every unitary transformation U ∈ SU(2) of the Hilbert space corresponds to an associated SO(3) rotation of this space. This kind of correspondence becomes much richer when we go to higher dimension. For example, in the d = 3 case the corresponding unitary group is the eight parameter group SU(3) whereas the rotation group in R is the 28 parameter group SO(8). Therefore, not every point on or inside the sphere S in R will correspond to a state of the system. So, not every rotation in R will correspond to a valid unitary transformation of the 3 dimensional Hilbert space. This gives us a fairly complex convex object as the set of all states. In order to gain insight into the structure of this state space, we study its two dimensional and three dimensional cross-sections. In particular, we describe in detail the discrete symmetries of a non-trivial 3-section of the convex set of states (density matrices) for three dimensional systems. This group turns out to be the same as that of a tetrahedron.

Lei Ma, Zhong Feng Tang, Yong-de Zhang

Hongqi Li

Based on the rotation transformation in the phase space and the technique of integration within an ordered product of operators,the coherent state representations of the rotation operator in quantum phase space,the Fourier transform operator and the parity operator are derived. Then the three-mode rotation operator is introduced.By the three-mode rotation operator,the Hamilton of the three-dimensional coordinate-momentum coupled harmonic oscillators is simply decoupled. Then its exact energy spectrums and eigenstates are obtained.

M. I. Monastyrskiǐ, A. M. Perelomov