Angular Momentum Coherent States Research Articles

New angular momentum coherent states via the nonlinear Bose SU(2) generators and their nonclassical behaviors

I theoretically construct a kind of new angular momentum coherent states by virtue of the nonlinear Bose SU(2) generators, and investigate their nonclassicality by analyzing the sub-Poissonian distributions, photon number distributions, entanglement entropy and Wigner phase-space distributions. The results show that the new states always have the sub-Poissonian distributions for all of the photon number sum [Formula: see text] of two modes, the photon number distributions are subjected to the parameter q, and the entanglement always increases quickly and then decreases slowly with increasing [Formula: see text], and the new states for odd q possess more stronger nonclassicality than for even q.

Tilted-axis wobbling in odd-mass nuclei

A triaxial rotor Hamiltonian with a rigidly aligned high-$j$ quasiparticle is treated by a time-dependent variational principle, using angular momentum coherent states. The resulting classical energy function has three unique critical points in a space of generalized conjugate coordinates, which can minimize the energy for specific ordering of the inertial parameters and a fixed angular momentum state. Because of the symmetry of the problem, there are only two unique solutions, corresponding to wobbling motion around a principal axis and, respectively, a tilted axis. The wobbling frequencies are obtained after a quantization procedure and then used to calculate $E2$ and $M1$ transition probabilities. The analytical results are employed in the study of the wobbling excitations of $^{135}\mathrm{Pr}$ nucleus, which is found to undergo a transition from low angular momentum transverse wobbling around a principal axis toward a tilted-axis wobbling at higher angular momentum.

Coherent3j-symbol representation for the loop quantum gravity intertwiner space

We introduce a new technique for dealing with the matrix elements of the Hamiltonian operator in loop quantum gravity, based on the use of intertwiners projected on coherent states of angular momentum. We give explicit expressions for the projections of intertwiners on the spin coherent states in terms of complex numbers describing the unit vectors which label the coherent states. Operators such as the Hamiltonian can then be reformulated as differential operators acting on polynomials of these complex numbers. This makes it possible to describe the action of the Hamiltonian geometrically, in terms of the unit vectors originating from the angular momentum coherent states, and opens up a way towards investigating the semiclassical limit of the dynamics via asymptotic approximation methods.

Squeezing in su(2) intelligent states

Angular momentum intelligent states are defined to satisfy . They also share with angular momentum coherent states a number of features. In this paper we describe and illustrate the squeezing properties of angular momentum intelligent states. We analyze three classes of state: never squeezed, always squeezed and sometimes squeezed. Our conclusions are applicable for a broad range of definitions of the standard quantum limit for angular momentum systems.

Husimi distribution for the linear rigid rotator

We find the Husimi distribution for the linear rigid rotator. The Husimi function can be viewed in a variety of ways. In particular, we obtain it as the expectation value of the density operator in a suitable basis of coherent states. We employ the Schwinger's oscillator model for the construction of the angular momentum coherent states.

Classicality of spin states

We extend the concept of classicality in quantum optics to spin states. We call a state ``classical'' if its density matrix can be decomposed as a weighted sum of angular momentum coherent states with positive weights. Classical spin states form a convex set C, which we fully characterize for a spin-1/2 and a spin-1. For arbitrary spin, we provide ``non-classicality witnesses''. For bipartite systems, C forms a subset of all separable states. A state of two spins-1/2 belongs to C if and only if it is separable, whereas for a spin-1/2 coupled to a spin-1, there are separable states which do not belong to C. We show that in general the question whether a state is in C can be answered by a linear programming algorithm.

Quantum-limited metrology with product states

We study the performance of initial product states of $n$-body systems in generalized quantum metrology protocols that involve estimating an unknown coupling constant in a nonlinear $k$-body $(k\ensuremath{\ll}n)$ Hamiltonian. We obtain the theoretical lower bound on the uncertainty in the estimate of the parameter. For arbitrary initial states, the lower bound scales as $1/{n}^{k}$, and for initial product states, it scales as $1/{n}^{k\ensuremath{-}1/2}$. We show that the latter scaling can be achieved using simple, separable measurements. We analyze in detail the case of a quadratic Hamiltonian $(k=2)$, implementable with Bose-Einstein condensates. We formulate a simple model, based on the evolution of angular-momentum coherent states, which explains the $O({n}^{\ensuremath{-}3/2})$ scaling for $k=2$; the model shows that the entanglement generated by the quadratic Hamiltonian does not play a role in the enhanced sensitivity scaling. We show that phase decoherence does not affect the $O({n}^{\ensuremath{-}3/2})$ sensitivity scaling for initial product states.

Manifestations of quantum holonomy in interferometry

Abelian and non-Abelian geometric phases, known as quantum holonomies, have attracted considerable attention in the past. Here, we show that it is possible to associate nonequivalent holonomies to discrete sequences of subspaces in a Hilbert space. We consider two such holonomies that arise naturally in interferometer settings. For sequences approximating smooth paths in the base (Grassmann) manifold, these holonomies both approach the standard holonomy. In the one-dimensional case the two types of holonomies are Abelian and coincide with Pancharatnam's geometric phase factor. The theory is illustrated with a model example of projective measurements involving angular momentum coherent states.

Quasiprobability distribution functions for periodic phase spaces: I. Theoretical aspects

An approach featuring s-parametrized quasiprobability distribution functions is developed for situations where a circular topology is observed. For such an approach, a suitable set of angle–angular momentum coherent states must be constructed in an appropriate fashion.

Indecomposable Representations of the Square-Root Lie Algebras of Vector Type and Their Boson Realizations**The project supported by National Natural Science Foundation of China (19905005), Major State Basic Research Development Program (G2000077400 and G2000077604) and Tsinghua Natural Science Foundation (985 Program, 98JC079)

The explicit expressions for indecomposable representations of nine square-root Lie algebras of vector type, , are obtained on the space of universal enveloping algebra of two-state Heisenberg–Weyl algebra, the invariant subspaces and the quotient spaces. From Fock representations corresponding to these indecomposable representations, the inhomogeneous boson realizations of are given. The expectation values of in the angular momentum coherent states are calculated as well as the corresponding classical limits.

The K-rotational labeling problem for eigenvectors from internal rotor calculations: Application to energy levels of acetaldehyde below the barrier

The problem of attaching K rotational quantum number labels to computer-generated numerical eigenvectors with extensive basis set mixing is considered for the internal-rotationoverall-rotation problem in molecules with one methyl top. Quantum number labeling problems arise physically because the torsional and the rotational degrees of freedom both pass from one limiting case to another as the torsional energy moves from below the top of the internal rotation barrier to above it, i.e., the torsional degree of freedom changes from a vibration to an internal rotation, while the rotational degree of freedom moves its direction of quantization from a principal axis to an axis depending also on angular momentum generated by the methyl top rotation. Since the choice of axis system, basis set, and computational scheme all influence the eigenfunction labeling procedure, consideration is limited to a commonly used two-step matrix-diagonalization scheme and to acetaldehyde as a numerical example. Torsional labels vt=0, 1, and 2 for eigenfunctions generated in the first diagonalization step are assigned in order of increasing eigenvalues, and rotational energy surfaces are constructed for the A and E symmetry components of these three torsional states. Projections of rotational eigenvectors over angular momentum coherent states are used to show that classical trajectories on the appropriate rotational energy surface give faithful representations of the quantum wave functions. Irregularities in the rotational energy level structure within a given torsional state can then be understood in terms of extra separatrices in the classical phase space and unusual wave function densities in the coherent state projections. Results are used to devise an automatic K labeling scheme for numerically obtained torsion-rotation levels belonging to torsional states below the top of the internal rotation barrier.

Clones and other interference effects in the evolution of angular-momentum coherent states

The aim of this paper is to present the interference effects that occur during the time evolution of simple angular wave packets (WP's) which can be associated with a diatomic rigid molecule (heteronuclear) or with a quantum rigid body with axial symmetry like a molecule or a nucleus. The time evolution is understood entirely within the framework of fractional revivals discovered by Averbukh and Perelman (Phys. Lett. A 39, 449 (1989); Usp. Fiz. Nauk 161, 41 (1991) [Sov. Phys. Usp. 37, 572 (1991)]), since the energy spectrum is exactly quadratic. Our objectives are to study how these interference effects differ when there is a change of the initial WP. For this purpose we introduce a two-parameter set of angular-momentum coherent states. On the one hand, this set emerges quite naturally from the three-dimensional coherent states of the harmonic oscillator; on the other hand, this set is shown to be built from intelligent spin states. By varying one parameter $(\ensuremath{\eta}),$ a scenario of interferences occurs on the sphere at fractional parts of the revival time that strongly depend on $\ensuremath{\eta}.$ For $\ensuremath{\eta}=\ifmmode\pm\else\textpm\fi{}1$ the WP, which coincides with a WP found by Mostowski [Phys. Lett. A 56, 369 (1976)], is a superposition of Bloch [Phys. Rev. 70, 460 (1946)] or Radcliffe [J. Phys. A 4, 313 (1971)] states, and clone exactly in time according to a scenario found for the infinite square well in one dimension, and also for a two dimensional rotor. In the context of intelligent spin states it is also natural to study the evolution by changing $\ensuremath{\eta}.$ For $\ensuremath{\eta}=0$ the WP is called linear, and in time produces a set of rings with axial symmetry over the sphere. The WP's for other values of $\ensuremath{\eta}$ are called elliptic, and sets of fractional waves are generated which make a transition between two symmetries. We call these fractional waves ``mutants.'' For specific times a clone is produced that stands among the mutants. Therefore the change in $\ensuremath{\eta}$ produces a change in the quantum spread on the sphere. We have also constructed simple coherent states for a symmetric rotor which are applicable to molecules and nuclei. Their time evolution also shows a cloning mechanism for the rational ratio of moments of inertia. For irrational values of this ratio, the scenario of partial revivals completed by Bluhm, Kostelecky, and Tudose [Phys. Lett. A 222, 220 (1996)] is valid.

Design of tailored microwave pulses to create rotational coherent states for an asymmetric-top molecule

Tailored microwave pulses, to guide asymmetric-top molecules from selected rotational states belonging to the vibronic ground state to generalized angular-momentum coherent states, are designed by using optimal control theory. Characteristics that the molecules have to fulfill in order to achieve the goal with feasible pulses are discussed. Properties of the pulses are discussed as well. The further dephasing among the components of the wave packet which, for the simplest coherent state, is a form of dynamical tunneling, can be locked by exploiting the changes that energy levels and eigenfunctions undergo in the presence of an external static electric field with appropriate intensity. For the special case with $M=0$, periodic fields are more flexible in avoiding dephasing. This is shown by examining properties of quasienergies and dressed states resulting from the diagonalization of a truncated Floquet matrix.

The Clebsch-Gordan problem of the angular momentum coherent states

The angular momentum coherent states are used to construct a Hilbert space of analytic functions, which is basically a subspace of Bargmann's well-known Hilbert space of analytic functions. The scalar product, the principal vector, and a complete orthonormal set in this Hilbert space are constructed and used for a complete solution of the Clebsch-Gardan problem in the coherent state basis. In this realization the product state is essentially the principal vector and therefore the coupled state itself is the Clebsch-Gordan coefficient.

The Clebsch-Gordan problem of the angular momentum coherent states: Debabrata Basu and T. K. Kar. Department of Physics, Indian Institute of Technology, Kharagpur 721302, West Bengal, India

The Clebsch-Gordan problem of the angular momentum coherent states: Debabrata Basu and T. K. Kar. Department of Physics, Indian Institute of Technology, Kharagpur 721302, West Bengal, India

The Lipkin model and coherent states

It is shown that the Bloch or angular momentum coherent states furnish a particularly efficacious basis for a discussion of various aspects of the Lipkin model of the ’’nucleus.’’ The Hartree–Fock description (as well as its projected version) is elegantly obtained in this framework. It is demonstrated that the ’’transition probability’’ between the first excited and ground states is proportional to the square of the number of ’’nucleons,’’ representing (in contrast to what obtains in the random phase approximation) a cooperativity of the ’’super-radiant’’ type. The extension of the model through the introduction of bosons permits, with the use of Bloch and Glauber coherent states, a succinct description of the phenomenon of boson condensation.

Transformations of angular momentum coherent states under coordinate rotations

Using the invariance of coherent states of the two harmonic oscillators under the spinor transformations of their boson operators a, a +, b, b + the transformations of angular momentum coherent states under coordinate system rotations are introduced.

Angular momentum coherent states

The work of Carruthers & Nieto on the harmonic oscillator coherent states is combined with Schwinger’s construction of angular momentum to produce the angular momentum coherent states. It is shown that these states become the vector representatives of angular momentum in the classical limit, and so are particularly useful for discussing the transition from quantum to classical angular momentum. The uncertainty relations for angle and angular momentum are described and are compatible with the classical limit. Under rotations the coherent states transform in a manner that in the classical limit is equivalent to the transformation of vectors, and in the same limit the root mean square variation of the expectation values of the components of angular momentum become negligible in comparison with the expectation values themselves. The coupling of two angular momenta in the classical limit is investigated: it is shown that although the product of two coherent states is not itself a coherent state, it does represent a packet similar to a true coherent state, and centred on the direction of the classical resultant of the two component vectors. The properties and implications of hyperbolic angular momentum space are discussed.