Group-related Coherent States Research Articles

Quantizations on the circle and coherent states

We present a possible construction of coherent states on the unit circle as configuration space. Our approach is based on Borel quantizations on S1 including the Aharonov–Bohm-type quantum description. Coherent states are constructed by Perelomov’s method as group-related coherent states generated by Weyl operators on the quantum phase space . Because of the duality of canonical coordinates and momenta, i.e. the angular variable and the integers, this formulation can also be interpreted as coherent states over an infinite periodic chain. For the construction, we use the analogy with our quantization and coherent states over a finite periodic chain where the quantum phase space was . The coherent states constructed in this work are shown to satisfy the resolution of unity. To compare them with canonical coherent states, some of their further properties are also studied demonstrating similarities as well as substantial differences.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

Dihedral symmetry of periodic chain: quantization and coherent states

Our previous work on quantum kinematics and coherent states over finite configuration spaces is extended: the configuration space is, as before, the cyclic group Zn of arbitrary orders n = 2, 3, …, but a larger group—the non-Abelian dihedral group Dn—is taken as its symmetry group. The corresponding group-related coherent states are constructed and their overcompleteness is proved. Our approach based on geometric symmetry can be used as a kinematic framework for matrix methods in quantum chemistry of ring molecules.

Multicomponent Group-Related Coherent States for Lie Group Chain G⊃K**The project supported by National Natural Science Foundation of China

The multicomponent group-related coherent states for the Lie group chain G⊃K are introduced. Their coupling coefficients are presented. The relations between these coupling coefficients and those (in the usual sense) of the irreducible representation bases labelled by G⊃K are obtained. The generalized Racah's factorization lemma about the coupling coefficients of such coherent states is given. As an example, the multicomponent group-related coherent states for the Lie group chain Sp(4)⊃SO(3)1⊗SO(3)2 are found. The uncertainty relation and the squeezing property are discussed.

Coupling Between the Group-Related Coherent States

When two representations of the Lie algebra are coupled, the coupling integral kernels are presented to relate the coupled to uncoupled group-related coherent states. These kernels have a connection with usual coupling coefficients. The explicit expressions of these kernels for and are given. When the direct product of three representations is formed in two ways, the recoupling integral kernels relating to the coupled group-related coherent states corresponding to two different schemes are introduced, and the relations between these kernels and the general recoupling coefficients are obtained. The properties of these kernels are discussed.

Exact solutions for the general nonstationary oscillator with a singular perturbation

Three linearly independent Hermitian invariants for the nonstationary generalized singular oscillator (SO) are constructed and their complex linear combination is diagonalized. The constructed family of eigenstates contains as subsets all previously obtained solutions for the SO and includes all Robertson and Schrodinger intelligent states for the three invariants. It is shown that the constructed analogues of the SU(1,1) group-related coherent states for the SO minimize the Robertson and Schrodinger uncertainty relations for the three invariants and for every pair of them simultaneously. The squeezing properties of the new states are briefly discussed.

Characteristic uncertainty relations

New uncertainty relations for n observables are established. The relations take the invariant form of inequalities between the characteristic coefficients of order r, rD 1; 2;:::;n , of the uncertainty matrix and the matrix of mean commutators of the observables. It is shown that the second- and third-order characteristic inequalities for the three generators of SU.1;1/ and SU.2/ are minimized in the corresponding group-related coherent states with maximal symmetry.

On the Squeezed States for nObservables

Three basic properties (eigenstate, orbit and intelligence) of thecanonical squeezed states (SS) are extended to the case of n arbitrary observables. The SS for n observablesXi can be constructed as eigenstates of theirlinear complex combinations or as states which minimize the Robertsonuncertainty relation. When Xi close a Lie algebra L the generalized SS could also be introduced as orbit of Aut (LC). It is shown that for the nilpotent algebrahN the three generalizations are equivalent. Forthe simple su(1, 1) the family of eigenstates ofuK- + vK+(K± being lowering and raising operators)is a family of idealK1 – K2 SS, but it cannotbe represented as an Aut (suC(1, 1) orbit althoughthe SU(1, 1) group related coherent states (CS) with symmetryare contained in it.Eigenstates | z, u, v, w; k ⟩ of general combination of uK- + vK+ + wK3 the three generators Kj of SU(1, 1) in the representations with Bargman index k = 1/2, 1, ... , and k = 1/4, 3/4 are constructed and discussed ingreater detail. These are ideal SS for K1,2,3. In the case of the one mode realization of su(1, 1) the nonclassical properties (sub-Poissonian statistics,quadrature squeezing) of the generalized even CS | z, u, v; + ⟩ are demonstrated.The states | z, u, v, w; k = 1/4, 3/4 ⟩ can exhibit strong both linear and quadratic squeezing.

Robertson intelligent states

Diagonalization of uncertainty matrix and minimization of Robertson inequality for n observables are considered. It is proved that for even n this relation is minimized in states which are eigenstates of n/2 independent complex linear combinations of the observables. In case of canonical observables this eigenvalue condition is also necessary. Such minimizing states are called Robertson intelligent states (RIS). The group related coherent states (CS) with maximal symmetry (for semisimple Lie groups) are particular case of RIS for the quadratures of Weyl generators. Explicit constructions of RIS are considered for operators of su(1,1), su(2), h_N and sp(N,R) algebras. Unlike the group related CS, RIS can exhibit strong squeezing of group generators. Multimode squared amplitude squeezed states are naturally introduced as sp(N,R) RIS. It is shown that the uncertainty matrices for quadratures of q-deformed boson operators a_{q,j} (q > 0) and of any k power of a_j = a_{1,j} are positive definite and can be diagonalized by symplectic linear transformations. PACS numbers: 03.65.Fd, 42.50.Dv

Coherent state approach to the N-representability problem

The N-representability problem is reviewed and formulated for the first time in the context of group-related coherent states. The particular family of states chosen is based on the group of special unitary transformations of one-particle space. Density operators are analyzed in terms of their P and Q symbols associated with these states, and the holomorphic representations of quantum mechanics that they induce. The time-independent Schrödinger equation for the ground state of an electronic system is expressed as a variational problem over a set of positive distributions. This set of distributions solves in an abstract sense the N-representability problem. The computational feasibility of the solution is discussed.