On the representation theory of quantum Heisenberg group and algebra

We show that the quantum Heisenberg groupH _q (1) and its *-Hopf algebra structure can be obtained by means of contraction from quantumSU _q (2) group. Its dual Hopf algebra is the quantum Heisenberg algebraU _q (h(1)). We derive left and right regular representations forU _q (h(1)) as acting on its dualH _q (1). Imposing conditions on the right representation, the left representation is reduced to an irreducible holomorphic representation with an associated quantum coherent state. Realized in the Bargmann-Hilbert space of analytic functions the unitarity of regular representation is also shown. By duality, left and right regular representations for quantum Heisenberg group with the quantum Heisenberg algebra as representation module are also constructed. As before reduction of group left representations leads to finite dimensional irreducible ones for which the intertwinning operator is also investigated.     

Method of orbits for SUq(N)

Quantum dressing orbits are described forSU _q (N) acting on its solvable dual. Particularly, the case is considered where the corresponding classical orbit coincides with a Grassmann manifold. Quantum local coordinates are introduced. Finite-dimensional irreducible representations of are derived from the *-algebra structure of quantum functions living on the orbit.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June, 1992.     

基于p1-p2与a1+a2共同本征态的相干纠缠态表象的构建及性质

利用有序算符乘积内的积分技术（IWOP）,建立了一种称之为相干纠缠态的两粒子体系的新表象,研究了这种新表象的性质,从理论上探讨了这种相干纠缠态的产生方法.结果表明：本文建立的这种p₁-p₂与a₁+a₂的共同本征态|p,β〉,既具有相干态的特性,又体现了纠缠态的特征,具有超完备性,完全可以作为一个表象使用. 物理上可以用光分束器来实现|< 关键词： IWOP技术 相干纠缠态表象 分束器     

A unification of boson expansion theories: (I). Functional representations of fermion states

The method for obtaining boson expansion by representing the fermion states as holomorphic functions of many complex variables is presented. Such functional representation is explicitly constructed for each space which is the carrier space of an irreducible representation of a semisimple compact Lie group. This is achieved by proving the unity resolution in terms of holomorphically parametrized Perelomov's generalized coherent states. The functional images of fermion states are polynomials of complex variables, while those of fermion operators are differential operators of finite order with polynomial coefficients.     

Quantum dressing orbits on compact groups

The quantum double is shown to imply the dressing transformation on quantum compact groups and the quantum Iwasawa decompositon in the general case. Quantum dressing orbits are described explicitly as *-algebras. The dual coalgebras consisting of differential operators are related to the quantum Weyl elements. Besides, the differential geometry on a quantum leaf allows a remarkably simple construction of irreducible *-representations of the algebras of quantum functions. Representation spaces then consist of analytic functions on classical phase spaces. These representations are also interpreted in the framework of quantization in the spirit of Berezin applied to symplectic leaves on classical compact groups. Convenient coherent states are introduced and a correspondence between classical and quantum observables is given.     

Quantum Grassmann manifolds

Orbits of the quantum dressing transformation forSU _q (N) acting on its solvable dual are introduced. The case is considered when the corresponding classical orbits coincide with Grassmann manifolds. Quantization of the Poisson bracket on a Zariski open subset of the Grassmann manifold yields a *-algebra generated by the quantum coordinate functions. The commutation relations are written in a compact form with the help of theR-matrix. Finite-dimensional irreducible representations ofU _h are derived from the *-algebra structure.     

Infinitive Operator-Sum Representation for Damping in a Squeezed Heat Reservoir via the Thermo Entangled State Approach

Using the thermo entangled state approach, we successfully solve the master equation of a damped harmonic oscillator affected by a linear resonance force in a squeezed heat reservoir, and obtain the analytical evolution formula for the density operator in the infinitive Kraus operator-sum representation. Interestingly, the Kraus operators M_l,m,n,r and \(\mathfrak {M}_{l,m,n,r}^{\dag }\) are not Hermite conjugate, but they are still trace-preserving quantum operations because of the normalization condition. We also investigate the evolution for an initial coherent state for damping in a squeezed heat reservoir, which shows that the initial coherent state decays to a complex mixed state as a result of damping and thermal noise.     

Unitary representations of semi-direct product groups with infinite dimensional Abelian normal subgroup

We generalize Mackey's theory of induced representations to groups which are semidirect products of a locally compact group H by an infinite dimensional Abelian group A. A theorem on inducing in stages is proved and conditions ensuring conservation of the irreducibility of the representation through the inducing process are given. When A is a nuclear space or the Hilbert space A_O of a Gelfand triplet A_O ? A ? A′_O, a cylindrical ergodic measure on the dual space A′ of A (respectively A′_O) is associated to each irreducible representation of G, and it is proved that the representation is induced when the measure is concentrated on an orbit. In the last part, A is supposed to be a Hilbert space and sufficient conditions are given for the preceding ergodic measures to be concentrated on an orbit.     

量子相空间分布函数与压缩相干态表示间的变换关系

基于Husimi算符具有压缩相干态投影子形式, 首先介绍了一个新的量子算符表示, 即压缩相干态表示.当高斯展宽参数κ = 1时, 该函数约化为通常的P函数. 作为例子, 研究了热态的压缩相干态表示, 通过图示说明了压缩相干态表示与P函数的区别. 为更好地在量子光学问题中使用该表示, 我们揭示了压缩相干态表示与Wigner函数、Q函数以及Husimi函数间的积分变换关系.     

Fundamental representations of quantum groups at roots of 1

To every finite-dimensional irreducible representation V of the quantum group U(g) where is a primitive lth root of unity (l odd) and g is a finite-dimensional complex simple Lie algebra, de Concini, Kac and Procesi have associated a conjugacy class C _V in the adjoint group G of g. We describe explicitly, when g is of type A _n , B _n , C _n , or D _n , the representations associated to the conjugacy classes of minimal positive dimension. We call such representations fundamental and prove that, for any conjugacy class, there is an associated representation which is contained in a tensor product of fundamental representations.     

Kraus Operator-Sum Solution to the Master Equation Describing the Single-Mode Cavity Driven by an Oscillating External Field in the Heat Reservoir

Exploiting the thermo entangled state approach, we successfully solve the master equation for describing the single-mode cavity driven by an oscillating external field in the heat reservoir and then get the analytical time-evolution rule for the density operator in the infinitive Kraus operator-sum representation. It is worth noting that the Kraus operator M _{l, m} is proved to be a trace-preserving quantum operation. As an application, the time-evolution for an initial coherent state ρ _|β〉 = |β〉〈β| in such an environment is investigated, which shows that the initial coherent state decays to a new mixed state as a result of thermal noise, however the coherence can still be reserved for amplitude damping.     

On representations of the elliptic quantum groupE τ,η(sl 2)

We describe representation theory of the elliptic quantum groupE _,(sl ₂). It turns out that the representation theory is parallel to the representation theory of the YangianY(sl ₂) and the quantum loop group .We introduce basic notions of representation theory of the elliptic quantum groupE _,(sl ₂) and construct three families of modules: evaluation modules, cyclic modules, one-dimensional modules. We show that under certain conditions any irreducible highest weight module of finite type is isomorphic to a tensor product of evaluation modules and a one-dimensional module. We describe fusion of finite dimensional evaluation modules. In particular, we show that under certain conditions the tensor product of two evaluation modules becomes reducible and contains an evaluation module, in this case the imbedding of the evaluation module into the tensor product is given in terms of elliptic binomial coefficients. We describe the determinant element of the elliptic quantum group. Representation theory becomes special ifN=m+l, whereN,m,l are integers. We indicate some new features in this case.The authors were supported in part by NSF grants DMS-9400841 and DMS-9501290.     

利用双参变形自旋相干态计算量子代数SU(2)q,s的Clebsch-Gordan系数

利用量子代数SU(2)_q,s的双参数变形自旋相干态,通过引入量子代数SU(2)_q,s的不可约表示张量积空间的Bargmann表示,导出了这一表示中不可约表示基底、双参数变形自旋相干态以及算符的表达式.最后导出了量子代数SU(2)_q,s在双参数变形自旋相干态下的Clebsch－Gordan系数.     

Inhomogeneous inverse differential realization of multimodeSU(2) group

The generators and irreducible representation coherent state of the multimodeSU(2) group are constructed by using the inverse operators of the multimode bosonic harmonic oscillator, and the inhomogeneous inverse differential realization of the multimodeSU(2) group are derived.     

Quantum Invariant Measures

We derive an explicit expression for the Haar integral on the quantized algebra of regular functions ℂ _q [K] on the compact real form K of an arbitrary simply connected complex simple algebraic group G. This is done in terms of the irreducible ✶-representations of the Hopf ✶-algebra ℂ _q [K]. Quantum analogs of the measures on the symplectic leaves of the standard Poisson structure on K which are (almost) invariant under the dressing action of the dual Poisson algebraic group K ^✶ are also obtained. They are related to the notion of quantum traces for representations of Hopf algebras. As an application we define and compute explicitly quantum analogs of Harish-Chandra c-functions associated to the elements of the Weyl group of G. Received: 26 January 2001 / Accepted: 31 May 2001     

The cyclic representations of the quantum superalgebra U q osp(2, 1) withq a root of unity

By generalizing De Concini and Kac's cyclic representation theory of quantum groups at roots of unity, the cyclic representations of the quantum superalgebra U _q osp(2, 1) are constructed in three classes: irreducible representations with single multiplicities, irreducible representations with the multiplicities larger than one, and indecomposable representations.This work is supported in part by the National Sciene Foundation in China.     

Quantum Tunneling of Spinor Bose–Einstein Condensates in an Optical Lattice

We have studied tunneling of spinor Bose–Einstein condensate in an optical lattice. It is found that, when the system being prepared in a squeezed coherent state, there exist the quantum tunneling between lattices l and l+1, l and l−1, respectively. In particular, when the optical lattice is infinitely long and the spin excitations are in the long-wavelength limit, quantum tunneling disappear between lattices l and l+1, and that l and l−1, in this case the magnetic soliton appears.     

Effect of Phase-Sensitive Reservoir on the Decoherence of Pair-Cat Coherent States

We study the decoherence of a superposition of four coherent states under the action of a phase sensitive reservoir. We verify that the decoherence times _k,l, k,l=1,2,3,4, between any two coherent states of the superposition can be controlled through the reservoir parameters. The decoherence time between two components of any pair, for instance _1,2 or _3,4, can be significantly increased, compared with the decoherence time when the state is acted by a thermal reservoir. However, this occurs at the expense of decreasing the decoherence time between the ``cat states" (1,2) and (3,4). This can be useful in quantum computation.     

Invariant star products and representations of compact semisimple Lie groups

Starting from the work by F. A. Berezin, and earlier paper by the author defined an invariant star product on every nonexceptional Kähler symmetric space. In this Letter a recursion formula is obtained to calculate the corresponding invariant Hochschild 2-cochains for spaces of types II and III. An invariant star product is defined on every integral symplectic (Kähler) homogeneous space of simply-connected compact Lie groups (on every integral orbit of the coadjoint representation). The invariant 2-cochains are obtained from the Bochner-Calabi function of the space. The leading term of the lth-2-cochain is determined by the l-power of the Laplace operator.