Perelomov Coherent States Research Articles

Photon-Added Deformed Peremolov Coherent States and Quantum Entanglement

In the present article, we build the excitedcoherent states associated with deformed su(1,1) algebra (DSUA), called photon-added deformed Perelomov coherent states (PA-DPCSs). The constructed coherent states are obtained by using an alterationof the Holstein–Primakoff realization (HPR) for DSUA. A general method to resolve of the problem of the unitary operator was developed for these kinds of quantum states. The Mandel parameter is considered to examine the statistical properties of PA-DPCSs. Furthermore, we offer a physical method to generate the PA-DPCSs in the framework of interaction among fields and atoms. Finally, we introduce the concept of entangled states for PA-DPCSs and examine the entanglement properties for entangled PA-DPCSs.

Characterizing SU(1,1) nonclassicality via variance

We quantify the nonclassicality of quantum states associated with the Lie group SU(1,1) by regarding states as observables and considering their variances in the SU(1,1) Perelomov coherent states. Combining the resolution of identity induced by the SU(1,1) Perelomov coherent states, we propose a quantifier for nonclassicality of a state based on the average uncertainty (variance) of the state (regarded as an observable) in the SU(1,1) Perelomov coherent states. This quantifier is easy to calculate and possesses several operational interpretations. We reveal its basic properties and illustrate it by several prototypical examples.

Pullback Coherent States, Squeezed States and Quantization

In this semi-expository paper, we define certain Rawnsley-type coherent and squeezed states on an integral K\"ahler manifold (after possibly removing a set of measure zero) and show that they satisfy some properties which are akin to maximal likelihood property, reproducing kernel property, generalised resolution of identity property and overcompleteness. This is a generalization of a result by Spera. Next we define the Rawnsley-type pullback coherent and squeezed states on a smooth compact manifold (after possibly removing a set of measure zero) and show that they satisfy similar properties. Finally we show a Berezin-type quantization involving certain operators acting on a Hilbert space on a compact smooth totally real embedded submanifold of $U$ of real dimension $n$, where $U$ is an open set in ${\mathbb C}{\rm P}^n$. Any other submanifold for which the criterion of the identity theorem holds exhibit this type of Berezin quantization. Also this type of quantization holds for totally real submanifolds of real dimension $n$ of a general homogeneous K\"ahler manifold of real dimension $2n$ for which Berezin quantization exists. In the appendix we review the Rawnsley and generalized Perelomov coherent states on ${\mathbb C}{\rm P}^n$ (which is a coadjoint orbit) and the fact that these two types of coherent states coincide.

Coherent states for dispersive pseudo-Landau-levels in strained honeycomb lattices

Dirac fermions in graphene may experiment dispersive pseudo-Landau levels due to a homogeneous pseudomagnetic field and a position-dependent Fermi velocity induced by strain. In this paper, we study the (semi-classical) dynamics of these particles under such a physical context from an approach of coherent states. For this purpose we use a Landau-like gauge to built Perelomov coherent states by the action of a non-unitary displacement operator $D(\alpha)$ on the fundamental state of the system. We analyze the time evolution of the probability density and the generalized uncertainty principle as well as the Wigner function for the coherent states. Our results show how $x$-momentum dependency affects the motion periodicity and the Wigner function shape in phase space.

Path integral renormalization in loop quantum cosmology

A coarse graining technique akin to block spin transformations that groups together fiducial cells in a homogeneous and isotropic universe has been recently developed in the context of loop quantum cosmology. The key technical ingredient was an SU(1, 1) group and Lie algebra structure of the physical observables as well as the use of Perelomov coherent states for SU(1, 1). It was shown that the coarse graining operation is completely captured by changing group representations. Based on this result, it was subsequently shown that one can extract an explicit renormalisation group flow of the loop quantum cosmology Hamiltonian operator in a simple model with dust-clock. In this paper, we continue this line of investigation and derive a coherent state path integral formulation of this quantum theory and extract an explicit expression for the renormalisation-scale dependent classical Hamiltonian entering the path integral for a coarse grained description at that scale. We find corrections to the non-renormalised Hamiltonian that are qualitatively similar to those previously investigated via canonical quantisation. In particular, they are again most sensitive to small quantum numbers, showing that the large quantum number (spin) description captured by so called "effective equations" in loop quantum cosmology does not reproduce the physics of many small quantum numbers (spins). Our results have direct impact on path integral quantisation in loop quantum gravity, showing that the usually taken large spin limit should be expected not to capture (without renormalisation, as mostly done) the physics of many small spins that is usually assumed to be present in physically reasonable quantum states.

A nonlinear interaction between SU(1,1) quantum system and a three-level atom in different configurations with damping term

The effect of nonlinear medium and phase damping is investigated on an atom of three levels in different forms interacting with the SU(1,1) Lie algebra. The SU(1,1) system starts the interaction from the Perelomov coherent state, while the atom starts the interaction from excited-most state. The time dependent wave function is calculated by using the Schrödinger equation. The density matrix is obtained. The influences of damping, nonlinearity (Kerr-like medium), Perelomov coherent parameter and the Bargmann index on some non-classical properties, such as: revivals-collapses phenomenon, entanglement and atomic variables squeezing, are investigated.

Renormalisation with SU(1, 1) coherent states on the LQC Hilbert space

We present an analytic computation of an explicit renormalisation group flow for cosmological states in loop quantum gravity. A key ingredient in our analysis are Perelomov coherent states for the Lie group SU(1, 1) whose representation spaces are embedded into the standard loop quantum cosmology (LQC) Hilbert space. The SU(1, 1) group structure enters our analysis by considering a classical set of phase space functions that generates the Lie algebra . We implement this Poisson algebra as operators on the LQC Hilbert space in a non-anomalous way. This task requires a rather involved ordering choice, whose existence is one of the main results of the paper. As a consequence, we can transfer recently established results on coarse graining cosmological states from direct quantisations of the above Poisson algebra to the standard LQC Hilbert space and full theory embeddings thereof. We explicitly discuss how the representation spaces used in this latter approach are embedded into the LQC Hilbert space and how the representation label sets a lower cut-off for the loop quantum gravity spins (=U(1) representation labels in LQC). Our results provide an explicit example of a non-trivial renormalisation group flow with a scale set by the representation label and interpreted as the minimally resolved geometric scale.

Binary Communication with Gazeau-Klauder Coherent States.

We investigate advantages and disadvantages of using Gazeau–Klauder coherent states for optical communication. In this short paper we show that using an alphabet consisting of coherent Gazeau–Klauder states related to a Kerr-type nonlinear oscillator instead of standard Perelomov coherent states results in lowering of the Helstrom bound for error probability in binary communication. We also discuss trace distance between Gazeau–Klauder coherent states and a standard coherent state as a quantifier of distinguishability of alphabets.

Photon-added entangled Barut–Girardello coherent states: non-classicality and generation

In continuation of previous papers where the addition of photons into the two-mode entangled quantum states was studied, we proposed a scheme to establish and generate photon-added ‘entangled Barut–Girardello coherent states’ (PAEBGCSs), by applying photon creation operators on the two-mode EBGCSs, where the latter is introduced by Hach et al. (JOSA B 35:2433, 2018), as a new class of quantum states connected to the simple harmonic oscillator and generated through a Mach–Zehnder interferometer, and the dynamics of their entanglement is analyzed in details. In addition, the EBGCSs include strong non-classical properties; then, to gain insight into the effectiveness of photon addition to them and comparing with the case already discussed as the photon-added entangled Klauder–Perelomov coherent states, we present a general analysis of non-classical properties such as photon statistics and squeezing and photon number distribution. We also derive the concurrence measure to quantify the entanglement of these states and look for conditions that provide information on whether these become maximally entangled. We show that the photon addition plays an important role in non-classical effects, and this operation can be applied to enhance the entanglement of the PAEBGCSs. For instance, entanglement preserving against quantum polarization is occurred by adding photons to the EBGCSs. Finally, an experimental procedure for generating the PAEBGCSs in the framework of cavity quantum electrodynamics is established.

Covariant integral quantization of the unit disk

We implement a SU(1, 1) covariant integral quantization of functions on the unit disk. The latter can be viewed as the phase space for the motion of a “massive” test particle on (1+1)-anti-de Sitter space-time, and the relevant unitary irreducible representations of SU(1, 1) corresponding to the quantum version of such motions are found in the discrete series and its lower limit. Our quantization method depends on the choice of a weight function on the phase space in such a way that different weight functions yield different quantizations. For instance, the Perelomov coherent states quantization is derived from a particular choice. Semi-classical portraits or lower symbols of main physically relevant operators are determined, and the statistical meaning of the weight function is discussed.

Coarse graining as a representation change

We discuss how SU(1,1) coherent states from the discrete series allow for a natural coarse graining operation. The physical application is quantum theories based on a set of three extensive observables whose Poisson algebra is isomorphic to su(1,1). In particular, we show that a Perelomov coherent state with representation label Nj0 and spinor label z encodes the physics of N independent subsystems with labels j0,z. This property is suggested by existing results for the expectation values and variances of the observables. We prove that it holds for all higher moments. Our results in particular apply to a recent quantum cosmology model that has been derived using SU(1,1) group theoretic quantisation techniques. For it, it follows that a certain notion of fiducial cell independence holds exactly at the quantum level when using the coherent states.

Engineering SU(1, 1)\u2009\u2297\u2009SU(1, 1) vibrational states

We propose an ideal scheme for preparing vibrational SU(1, 1) ⊗ SU(1, 1) states in a two-dimensional ion trap using red and blue second sideband resolved driving of two orthogonal vibrational modes. Symmetric and asymmetric driving provide two regimes to realize quantum state engineering of the vibrational modes. In one regime, we show that time evolution synthesizes so-called SU(1, 1) Perelomov coherent states, that is separable squeezed states and their superposition too. The other regime allows engineering of lossless 50/50 SU(2) beam splitter states that are entangled states. These ideal dynamics are reversible, thus, the non-classical and entangled states produced by our schemes might be used as resources for interferometry.

Sensitivity measurement for asymmetric two two-level atoms interacting with field obeys SU(1,1) Lie group via atomic inversion

ABSTRACTA quantum optics model of the asymmetric case of the interaction between two two-level atoms and quantum field, which obeys SU(1,1) Lie group, is proposed. The atom-atom interaction and the rotating wave approximation are suggested in the Hamiltonian operator. Our aim is to obtain the time-dependent wave function of asymmetric case analytically. The analytical method is based on the eigenvalues and corresponding eigenvectors of the coefficient matrix of the interaction Hamiltonian operator. The SU(1,1) quantum system is initially in the Perelomov coherent state. Therefore, the atomic inversion is obtained and discussed for different values of model parameters such as initial atomic angles, Perelomov coherent, the Bargmann index and the detuning parameters. Observe that the quantum optics model is sensitive to the variation in both the Perelomov coherent parameter and the Bargmann index. In addition, there are nonclassical properties of the proposed quantum model in the presence the detuning parameter changes.

Airy wavepackets are Perelomov coherent states

Accelerating non-spreading wavepackets in a nonrelativistic free-particle system, with probability distribution having an Airy function profile, were discovered by Berry and Balazs [Am. J. Phys. 47(3), 264–267 (1979)], and have been subsequently realised in several optics experiments. It is shown that these wavepackets are actually Perelomov coherent states. It is found that the Galilean invariance of the Schrödinger equation plays a key role in making these states unique and gives rise to their unusual propagation properties.

Generalized Weyl-Heisenberg Algebra, Qudit Systems and Entanglement Measure of Symmetric States via Spin Coherent States.

A relation is established in the present paper between Dicke states in a d-dimensional space and vectors in the representation space of a generalized Weyl–Heisenberg algebra of finite dimension d. This provides a natural way to deal with the separable and entangled states of a system of symmetric qubit states. Using the decomposition property of Dicke states, it is shown that the separable states coincide with the Perelomov coherent states associated with the generalized Weyl–Heisenberg algebra considered in this paper. In the so-called Majorana scheme, the qudit (d-level) states are represented by N points on the Bloch sphere; roughly speaking, it can be said that a qudit (in a d-dimensional space) is describable by a N-qubit vector (in a N-dimensional space). In such a scheme, the permanent of the matrix describing the overlap between the N qubits makes it possible to measure the entanglement between the N qubits forming the qudit. This is confirmed by a Fubini–Study metric analysis. A new parameter, proportional to the permanent and called perma-concurrence, is introduced for characterizing the entanglement of a symmetric qudit arising from N qubits. For (), this parameter constitutes an alternative to the concurrence for two qubits. Other examples are given for and 5. A connection between Majorana stars and zeros of a Bargmmann function for qudits closes this article.

SO*(2N) coherent states for loop quantum gravity

A SU(2) intertwiner with N legs can be interpreted as the quantum state of a convex polyhedron with N faces (when working in 3D). We show that the intertwiner Hilbert space carries a representation of the non-compact group SO*(2N). This group can be viewed as the subgroup of the symplectic group Sp(4N,R) which preserves the SU(2) invariance. We construct the associated Perelomov coherent states and discuss the notion of semi-classical limit, which is more subtle than we could expect. Our work completes the work by Freidel and Livine [J. Math. Phys. 51, 082502 (2010) and J. Math. Phys. 52, 052502 (2011)], which focused on the U(N) subgroup of SO*(2N).

Linear entropy and squeezing of the interaction between two quantum system described by su (1, 1) and su(2) Lie group in presence of two external terms

A Hamiltonian, that describes the interaction between a two-level atom (su(2) algebra) and a system governed by su(1,1) Lie algebra besides two external interaction, is considered. Two canonical transformations are used, which results into removing the external terms and changing the frequencies of the interacting systems. The solution of the equations of motion of the operators is obtained and used to discuss the atomic inversion, entanglement, squeezing and correlation functions of the present system. Initially the atom is considered to be in the excited state while the other systems is in the Perelomov coherent state. Effects of the variations in the coupling parameters to the external systems are considered. They are found to be sensitive to changing entanglement, variance and entropy squeezing.

Entanglement of Moving and Non-Moving Two-Level Atoms

In this paper we study the dynamics of the atomic inversion, von Neumann entropy and entropy squeezing for moving and non-moving two-level atoms interacting with a Perelomov coherent state. The final state of the system using specific initial conditions is obtained. The effects of Perelomov and detuning parameters are examined in the absence and presence of the atomic motion. Important phenomena such as the collapse and revival are shown to be very sensitive to the variation of the Perelomov parameter in the presence of detuning parameter. The results show that the Perelomov parameter is very useful in generating a high amount of entanglement due to variation of the detuning parameter.

SU(1, 1) and SU(2) Perelomov number coherent states: algebraic approach for general systems

We study some properties of the SU(1, 1) Perelomov number coherent states. The Schrödinger's uncertainty relationship is evaluated for a position and momentum-like operators (constructed from the Lie algebra generators) in these number coherent states. It is shown that this relationship is minimized for the standard coherent states. We obtain the time evolution of the number coherent states by supposing that the Hamiltonian is proportional to the third generator K0 of the su(1, 1) Lie algebra. Analogous results for the SU(2) Perelomov number coherent states are found. As examples, we compute the Perelomov coherent states for the pseudoharmonic oscillator and the two-dimensional isotropic harmonic oscillator.

The construction of the Gilmore–Perelomov coherent states for the Kratzer–Fues anharmonic oscillator with the use of the algebraic approach

By applying the algebraic approach and the displacement operator to the ground state, the unknown Gilmore–Perelomov coherent states for the rotating anharmonic Kratzer–Fues oscillator are constructed. In order to obtain the displacement operator the ladder operators have been applied. The deduced SU(1, 1) dynamical symmetry group associated with these operators enables us to construct this important class of the coherent states. Several important properties of these states are discussed. It is shown that the coherent states introduced are not orthogonal and form complete basis set in the Hilbert space. We have found that any vector of Hilbert space of the oscillator studied can be expressed in the coherent states basis set. It has been established that the coherent states satisfy the completeness relation. Also, we have proved that these coherent states do not possess temporal stability. The approach presented can be used to construct the coherent states for other anharmonic oscillators. The coherent states proposed can find applications in laser-matter interactions, in particular with regards to laser chemical processing, laser techniques, in micro-machinning and the patterning, coating and modification of chemical material surfaces.