Spin-j System Research Articles

Quantized-Energy Equation for N-Level Atom in the Probability Representation of Quantum Mechanics

For Hermitian and non-Hermitian Hamiltonian matrices H, we present the Schr¨odinger equation for qudit (spin-j system, N-level atom) with the state vector |ψ〉 in a new form of the linear eigenvalue equation for the matrix ℋ = (H ⊗ 1N) and the probability eigenvector |p〉 identified with quantum states in the probability representation of quantum mechanics. We discuss the possibility to experimentally detect the difference between the system states described by the solutions, corresponding to the Schrodinger equation with Hermitian and non-Hermitian Hamiltonians, by measuring the probabilities of artificial spin-1/2 projections m = ± 1/2, sets of which are identified with qudit states. We show that different symmetries of systems, including 𝒫𝒯 -symmetry and broken 𝒫𝒯 -symmetry, are determined by a set of N complex eigenvalues of the Hamiltonian matrix H.

Hidden Correlations and Information-Entropic Inequalities in Systems of Qudits†

We present the results of our study of correlations in noncomposite systems like qudits (N-level atom or spin-j system). We show that they correspond to correlations in composite systems like bipartite or multipartite systems. We establish the correspondence between correlations in noncomposite systems and composite ones using the map of indices labeling the random variables in classical probability theory or matrix elements of density matrices in quantum mechanics. We obtain new information-entropic inequalities for density matrices of noncomposite systems. Also we present Bayes’ formula for noncomposite classical system with one random variable. Finally we discuss the possibility to use hidden correlations and entanglement of a single-qudit state.

Geometric multiaxial representation of N -qubit mixed symmetric separable states

Study of an N qubit mixed symmetric separable states is a long standing challenging problem as there exist no unique separability criterion. In this regard, we take up the N-qubit mixed symmetric separable states for a detailed study as these states are of experimental importance and offer elegant mathematical analysis since the dimension of the Hilbert space reduces from 2^N to N + 1. Since there exists a one to one correspondence between spin-j system and an N-qubit symmetric state, we employ Fano statistical tensor parameters for the parametrization of spin density matrix. Further, we use geometric multiaxial representation(MAR) of density matrix to characterize the mixed symmetric separable states. Since separability problem is NP hard, we choose to study it in the continuum limit where mixed symmetric separable states are characterized by the P-distribution function \lambda(\theta; \phi). We show that the N-qubit mixed symmetric separable state can be visualized as a uniaxial system if the distribution function is independent of \theta and \phi. We further choose distribution function to be the most general positive function on a sphere and observe that the statistical tensor parameters characterizing the N-qubit symmetric system are the expansion coefficients of the distribution function. As an example for the discrete case, we investigate the MAR of a uniformly weighted two qubit mixed symmetric separable state. We also observe that there exists a correspondence between separability and classicality of states.

Quantum Geometric Phase in Majorana’s Stellar Representation: Mapping onto a Many-Body Aharonov-Bohm Phase

The (Berry-Aharonov-Anandan) geometric phase acquired during a cyclic quantum evolution of finite-dimensional quantum systems is studied. It is shown that a pure quantum state in a (2J+1)-dimensional Hilbert space (or, equivalently, of a spin-J system) can be mapped onto the partition function of a gas of independent Dirac strings moving on a sphere and subject to the Coulomb repulsion of 2J fixed test charges (the Majorana stars) characterizing the quantum state. The geometric phase may be viewed as the Aharonov-Bohm phase acquired by the Majorana stars as they move through the gas of Dirac strings. Expressions for the geometric connection and curvature, for the metric tensor, as well as for the multipole moments (dipole, quadrupole, etc.), are given in terms of the Majorana stars. Finally, the geometric formulation of the quantum dynamics is presented and its application to systems with exotic ordering such as spin nematics is outlined.

Fisher information in a quantum-critical environment

We consider a process of parameter estimation in a spin-j system surrounded by a quantum-critical spin chain. Quantum Fisher information lies at the heart of the estimation task. We employ Ising spin chain in a transverse field as the environment which exhibits a quantum phase transition. Fisher information decays with time almost monotonously when the environment reaches the critical point. By choosing a fixed time or taking the time average, one can see the quantum Fisher information presents a sudden drop at the critical point. Different initial states of the environment are considered. The phenomenon that the quantum Fisher information, namely, the precision of estimation, changes dramatically can be used to detect the quantum criticality of the environment. We also introduce a general method to obtain the maximal Fisher information for a given state.

Geometry of Pure States of N Spin-J System

We present the geometry of pure states of an ensemble of N spin-J systems using a generalisation of the Majorana representation. The approach is based on Schur-Weyl duality that allows for simple interpretation of the state transformation under the action of general linear and permutation groups. We show an exemplary application in the theory of decoherence-free subspaces and noiseless subsystems.

SPIN-J GEOMETRIC PHASE DRIVEN BY A CLASSICAL FLUCTUATING FIELD

The authors investigate the geometric phase of a spin-J system interacting with a classical magnetic field which contains a fluctuating component. Assuming an Ornstein–Uhlenbeck process for the fluctuations, we find that the distribution due to the fluctuating component for the geometric phase is a Gaussian. We calculate in detail the variance of geometric and dynamic phases. The results clearly show that the geometric phase can reduce the fluctuations, but the dynamic phase can not.

Quantum reference frames and the classification of rotationally invariant maps

We give a convenient representation for any map that is covariant with respect to an irreducible representation of SU(2), and use this representation to analyze the evolution of a quantum directional reference frame when it is exploited as a resource for performing quantum operations. We introduce the moments of a quantum reference frame, which serve as a complete description of its properties as a frame, and investigate how many times a quantum directional reference frame represented by a spin-j system can be used to perform a certain quantum operation with a given probability of success. We provide a considerable generalization of previous results on the degradation of a reference frame, from which follows a classification of the dynamics of spin-j system under the repeated action of any covariant map with respect to SU(2).

SQUEEZING IN MULTIVARIATE SPIN SYSTEMS

In contrast to the canonically conjugate variates q, p representing the position and momentum of a particle in the phase space distributions, the three Cartesian components, Jx, Jy, Jz of a spin-j system constitute the mutually non-commuting variates in the quasi-probabilistic spin distributions. It can be shown that a univariate spin distribution is never squeezed and one needs to look into either bivariate or trivariate distributions for signatures of squeezing. Several such distributions result if one considers different characteristic functions or moments based on various correspondence rules. As an example, discrete probability distribution for an arbitrary spin-1 assembly is constructed using Wigner-Weyl and Margenau-Hill correspondence rules. It is also shown that a trivariate spin-1 assembly resulting from the exposure of nucleus with non-zero quadrupole moment to combined electric quadrupole field and dipole magnetic field exhibits squeezing in certain cases.

Degradation of a quantum reference frame

We investigate the degradation of reference frames (RFs), treated as dynamical quantum systems, and quantify their longevity as a resource for performing tasks in quantum information processing. We adopt an operational measure of an RF's longevity, namely, the number of measurements that can be made against it with a certain error tolerance. We investigate two distinct types of RF: a reference direction, realized by a spin-j system, and a phase reference, realized by an oscillator mode with bounded energy. For both cases, we show that our measure of longevity increases quadratically with the size of the reference system and is therefore non-additive. For instance, the number of measurements for which a directional RF consisting of N parallel spins can be put to use scales as N2. Our results quantify the extent to which microscopic or mesoscopic RFs may be used for repeated, high-precision measurements, without needing to be reset—a question that is important for some implementations of quantum computing. We illustrate our results using the proposed single-spin measurement scheme of magnetic resonance force microscopy.

Classification of cyclic initial states and geometric phase for the spin-j system

Quantum states which evolve cyclically in their projective Hilbert space give rise to a geometric (or Aharonov-Anandan) phase. An aspect of primary interest is stable cyclic behaviour as realized under a periodic Hamiltonian. The problem has been handled by use of time-dependent transformations treated along the lines of Floquet's theory as well as in terms of exponential operators with a goal to examine the variety of initial states exhibiting cyclic behaviour. A particular case of special cyclic initial states is described which is shown to be important for nuclear magnetic resonance experiments aimed at the study of the effects of the geometric phase. An example of arbitrary spin j in a precessing magnetic field and spin j=1 subject to both axially symmetric quadrupolar interaction and a precessing magnetic field are presented. The invariant (Kobe's) geometric phase is calculated for special cyclic states.

Formally exact solution and geometric phase for the spin-j system

The formally exact solution for the spin- j system is obtained by means of the Lewis-Riesenfeld theory. It is then pointed out that for the spin- j system the Berry phase, the Aharonov-Anandan phase and the phase difference associated with non-cyclic evolution can be calculated by making use of the formally exact solution.

Cyclic quantum evolution and Aharonov-Anandan geometric phases in SU(2) spin-coherent states.

We show that cyclic quantum evolution can be realized and the Aharonov-Anandan (AA) geometric phase can be determined for any spin-j system driven by periodic fields. Two methods are extended for the study of this problem: the generalized spin-coherent-state technique and the Floquet quasienergy approach. Using the former approach, we have developed a generalized Bloch-sphere model and presented a SU(2) Lie-group formulation of the AA geometric phase in the spin-coherent state. We show that the AA phase is equal to j times the solid angle enclosed by the trajectory traced out by the tip of a generalized Bloch vector. General analytic formulas are obtained for the Bloch vector trajectory and the AA geometric phase in terms of external physical parameters. In addition to these findings, we have also approached the same problem from an alternative but complementary point of view without recourse to the concept of coherent-state terminology. Here we first determine the Floquet quasienergy eigenvalues and eigenvectors for the spin-j system driven by periodic fields. This in turn allows the construction of the time-evolution propagator, the total wave function, and the AA geometric phase in a more general fashion.

Expansion technique for the dipole interaction of a spin-J system with a monochromatic electromagnetic field

An expansion technique for the dipole interaction of a spin-J system with a quantized monochromatic electromagnetic field is developed. The energy levels are equally spaced and the rotating-wave approximation is used. Particular results of the application of the method are presented.