Boson Operators Research Articles

A new class of f-deformed charge coherent states and their nonclassical properties

Two-mode charge (pair) coherent states have been introduced previously by using the 〈η| representation. In this paper, we reobtain these states by a rather different method. Then, using the nonlinear coherent state approach and based on a simple manner by which the representation of two-mode charge coherent states is introduced, we generalize the bosonic creation and annihilation operators to the f-deformed ladder operators and construct a new class of f-deformed charge coherent states. Unlike the (linear) pair coherent states, our presented structure has the potential to generate a large class of pair coherent states with various nonclassicality signs and physical properties which are of interest. For this purpose, we use a few well-known nonlinearity functions associated with particular quantum systems as some physical appearances of our presented formalism. After introducing the explicit form of the above correlated states in the two-mode Fock space, several nonclassicality features of the corresponding states (as well as the two-mode linear charge coherent states) are numerically investigated by calculating quadrature squeezing, the Mandel parameter, the second-order correlation function, the second-order correlation function between the two modes and the Cauchy–Schwartz inequality. Also, the oscillatory behaviour of the photon count and the quasi-probability (Husimi) function of the associated states will be discussed.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

A symmetry adapted approach to the dynamic Jahn‐Teller problem: Application to mixed‐valence polyoxometalate clusters with keggin structure

AbstractIn this article, we present a symmetry‐adapted approach aimed to the accurate solution of the dynamic vibronic problem in large scale Jahn‐Teller (JT) systems. The algorithm for the solution of the eigen‐problem takes full advantage of the point symmetry arguments. The system under consideration is supposed to consist of a set of electronic levels mixed by the active JT and pseudo JT vibrational modes. Applying the successive coupling of the bosonic creation operators, we introduce the irreducible tensors that are called multivibronic operators. Action of the irreducible multivibronic operators on the vacuum state creates the vibrational symmetry adapted basis that is subjected to the Gram‐Schmidt orthogonalization at each step of evaluation. Finally, the generated vibrational basis is coupled to the electronic one to get the symmetry adapted electron‐vibrational (vibronic) basis within which the full matrix of the JT Hamiltonian is blocked according to the irreducible representations (irreps) of the point group. The proposed approach is a part of our study of the nanosized mixed valence (MV) clusters with large number of delocalized electrons that are at the border line between quantum and classical objects. Here, we illustrate in detail the developed technique by the application to the 2e‐reduced MV dodecanuclear Keggin anion in which the electronic pair is delocalized over 12 sites (overall symmetry Td) giving rise to the (1T2+1E+1A1)⊗(e+t2) (3T1+3T2)⊗(e+t2) combined JT/pseudo JT problems for the spin‐singlet and spin‐triplet states. © 2012 Wiley Periodicals, Inc.

A NEW DESCRIPTION OF MOTION OF THE FERMIONIC SO(2N+2) TOP IN THE CLASSICAL LIMIT UNDER THE QUASI-ANTICOMMUTATION RELATION APPROXIMATION

-1 The boson images of fermion SO(2N+1) Lie operators have been given together with those of SO(2N+2) ones. The SO(2N+1) Lie operators are generators of rotation in the (2N+1)-dimensional Euclidean space (N: number of single-particle states of the fermions). The images of fermion annihilation–creation operators must satisfy the canonical anticommutation relations, when they operate on a spinor subspace. In the regular representation space we use a boson Hamiltonian with Lagrange multipliers to select out the spinor subspace. Based on these facts, a new description of a fermionic SO(2N+2) top is proposed. From the Heisenberg equations of motions for the boson operators, we get the SO(2N+1) self-consistent field (SCF) Hartree–Bogoliubov (HB) equation for the classical stationary motion of the fermion top. Decomposing an SO(2N+1) matrix into matrices describing paired and unpaired modes of fermions, we obtain a new form of the SO(2N+1) SCF equation with respect to the paired-mode amplitudes. To demonstrate the effectiveness of the new description based on the bosonization theory, the extended HB eigenvalue equation is applied to a superconducting toy-model which consists of a particle–hole plus BCS-type interaction. It is solved to reach an interesting and exciting solution which is not found in the traditional HB eigenvalue equation due to the unpaired-made effects. To complete the new description, the Lagrange multipliers must be determined in the classical limit. For this aim a quasi-anticommutation relation approximation is proposed. Only if a certain relation between an SO(2N+1) parameter z and the N is satisfied, unknown parameters K and l in the Lagrange multipliers can be determined without any inconsistency.

Dual squeezed states in an atom-photon cluster and their manifestations

The general kinetic equation for an isolated two-level atom and a high-Q cavity mode in a heat bath exhibiting quantum correlations (entangled bath) is applied to the analysis of the squeezed states of the collective system. Two types of collective operators are introduced for the analysis: one is based on bosonic commutation relations, and the other, on the commutation relations of the algebra obtained by a polynomial deformation of the angular momentum algebra. On the basis of these relations, formulas for observables are constructed that identify squeezed states in the system. It is shown that, under certain conditions, the collective system exhibits dual squeezing within the relations for boson operators, as well as for the operators constructed from the angular momentum algebra. Such squeezing is demonstrated under a projective measurement of an atom and for an entanglement swapping protocol. In the latter case, when measuring two initially independent atomic systems, depending on the type of measurement, two cavity modes collapse into a nonseparable state, which is described either by a nonseparability relation based on boson operators or by a relation based on the operators of the algebra of the quasimomentum of the collective system consisting of these two modes.

Finite-temperature phase diagram of two-component bosons in a cubic optical lattice: Three-dimensionalt-Jmodel of hard-core bosons

We study the three-dimensional bosonic t-J model, i.e., the t-J model of "bosonic electrons", at finite temperatures. This model describes the $s={1 \over 2}$ Heisenberg spin model with the anisotropic exchange coupling $J_{\bot}=-\alpha J_z$ and doped {\it bosonic} holes, which is an effective system of the Bose-Hubbard model with strong repulsions. The bosonic "electron" operator $B_{r\sigma}$ at the site $r$ with a two-component (pseudo-)spin $\sigma (=1,2)$ is treated as a hard-core boson operator, and represented by a composite of two slave particles; a "spinon" described by a Schwinger boson (CP$^1$ boson) $z_{r\sigma}$ and a "holon" described by a hard-core-boson field $\phi_r$ as $B_{r\sigma}=\phi^\dag_r z_{r\sigma}$. By means of Monte Carlo simulations, we study its finite-temperature phase structure including the $\alpha$ dependence, the possible phenomena like appearance of checkerboard long-range order, super-counterflow, superfluid, and phase separation, etc. The obtained results may be taken as predictions about experiments of two-component cold bosonic atoms in the cubic optical lattice.

Quantum quenches and driven dynamics in a single-molecule device

The nonequilibrium dynamics of molecular devices is studied in the framework of a generic model for single-molecule transistors: a resonant level coupled by displacement to a single vibrational mode. In the limit of a broad level and in the vicinity of the resonance, the model can be controllably reduced to a form quadratic in bosonic operators, which in turn is exactly solvable. The response of the system to a broad class of sudden quenches and ac drives is thus computed in a nonperturbative manner, providing an asymptotically exact solution in the limit of weak electron-phonon coupling. From the analytic solution, we are able to (1) explicitly show that the system thermalizes following a local quantum quench, (2) analyze in detail the time scales involved, (3) show that the relaxation time in response to a quantum quench depends on the observable in question, and (4) reveal how the amplitude of long-time oscillations evolves as the frequency of an ac drive is tuned across the resonance frequency. Explicit analytical expressions are given for all physical quantities and all nonequilibrium scenarios under study.

Simulation of two-dimensional many-particle hardcore bosons by using the quantum Monte Carlo method

In this paper, the stochastic series expansion quantum Monte Carlo method is employed to investigate the thermodynamic properties of hardcore Bose-Hubbard model in two-dimensional space. The two-dimensional hardcore Bose-Hubbard model can be mapped into the two-dimensional antiferromagnetic quasi-Heisenberg model under transform of bosonic operators. There is an additional term which is proportional to the total number of sites compared with real Heisenberg model and it is difficult for simulation. Using a nonlocal operator-loop update, it allows one to simulate thousands of sites. Our simulation results show that, first, energy decreases with the increase of density of particles in a range from 0 to 0.5, and finally approaches to a fixed value. Moreover, with the size of square lattice increasing, energy also increases. Second, when we fix the system size, energy and magnetization increase with temperature, but not with of chemical potential. When we increase the system size, energy increases, while, the magnetization decreases. Third, specific heat is independent of chemical potential, but it dramatically increases with temperature and approaches to a peak, then decreases slowly. According to Landau theory of superfluidity, the tends of curve for energy and specific heat fit the research of He II in the Landau two-fluid model. Fourth, different square lattice linear system sizes have a little influence on tiny differences to the reciprocal of uniform susceptibility. There are small fluctuations in a range from 0 to 0.5(J/kB), where J is the coupling energy, kB is the Boltzmann constant, but the reciprocal of uniform susceptibility increases with temperature increasing in a range from 0.5 to 2(J/kB). The tends of curve are similar to those of Kondo effect.

Critique of generalised purity as an entanglement measure: Explicit failure in simple separable Bose–Hubbard models

The SU(2) and SU(3) Lie algebras lend themselves naturally to studies of two- and three-well Bose–Einstein condensates, with the group operators being expressed in terms of bosonic annihilation and creation operators at each site. The success of these representations has led to the purities associated with these algebras to be promoted as a measure of entanglement in these systems. In this work, we show that these purities do not provide an unambiguous measure of entanglement between wells, but instead give results which depend on the quantum statistical states of the atomic ensembles in each well. Using the example of totally uncoupled wells where the atoms in one have never interacted with the atoms in the other, we quantify these purities for different states and show that completely separable states can give values which have been claimed to indicate the presence of entanglement. We also consider claims that the generalised purities measure particle rather than mode entanglement, with emphasis on the case of indistinguishable bosons, as found in these systems.

Boundary CFT from holography

We explore some aspects of holographic dual of Boundary Conformal Field Theory (BCFT). In particular we study asymptotic symmetry of geometries which provide holographic dual of BCFTs. We also compute two-point functions of certain bosonic and fermionic operators in the dual BCFT by making use of AdS/BCFT correspondence.

How to Decompose Arbitrary Continuous-Variable Quantum Operations

We present a general, systematic, and efficient method for decomposing any given exponential operator of bosonic mode operators, describing an arbitrary multimode Hamiltonian evolution, into a set of universal unitary gates. Although our approach is mainly oriented towards continuous-variable quantum computation, it may be used more generally whenever quantum states are to be transformed deterministically, e.g., in quantum control, discrete-variable quantum computation, or Hamiltonian simulation. We illustrate our scheme by presenting decompositions for various nonlinear Hamiltonians including quartic Kerr interactions. Finally, we conclude with two potential experiments utilizing offline-prepared optical cubic states and homodyne detections, in which quantum information is processed optically or in an atomic memory using quadratic light-atom interactions.

Recursive formulation of the multiconfigurational time-dependent Hartree method for fermions, bosons and mixtures thereof in terms of one-body density operators

The multiconfigurational time-dependent Hartree method (MCTDH) [H.-D. Meyer, U. Manthe, L.S. Cederbaum, Chem. Phys. Lett. 165, 73 (1990); U. Manthe, H.-D. Meyer, L.S. Cederbaum, J. Chem. Phys. 97, 3199 (1992)] is celebrating nowadays entering its third decade of tackling numerically-exactly a broad range of correlated multi-dimensional non-equilibrium quantum dynamical systems. Taking in recent years particles’ statistics explicitly into account, within the MCTDH for fermions (MCTDHF) and for bosons (MCTDHB), has opened up further opportunities to treat larger systems of interacting identical particles, primarily in laser-atom and cold-atom physics. With the increase of experimental capabilities to simultaneously trap mixtures of two, three, and possibly even multiple kinds of interacting composite identical particles together, we set up the stage in the present work and specify the MCTDH method for such cases. Explicitly, the MCTDH method for systems with three kinds of identical particles interacting via all combinations of two- and three-body forces is presented, and the resulting equations-of-motion are briefly discussed. All four possible mixtures (Fermi–Fermi–Fermi, Bose–Fermi–Fermi, Bose–Bose–Fermi and Bose–Bose–Bose) are presented in a unified manner. Particular attention is paid to represent the coefficients’ part of the equations-of-motion in a compact recursive form in terms of one-body density operators only. The recursion utilizes the recently proposed Combinadic-based mapping for fermionic and bosonic operators in Fock space [A.I. Streltsov, O.E. Alon, L.S. Cederbaum, Phys. Rev. A 81, 022124 (2010)], successfully applied and implemented within MCTDHB. Our work sheds new light on the representation of the coefficients’ part in MCTDHF and MCTDHB without resorting to the matrix elements of the many-body Hamiltonian with respect to the time-dependent configurations. It suggests a recipe for efficient implementation of the schemes derived here for mixtures which is suitable for parallelization.

On the deformed squeezed states on noncommutative plane

Abstract A new kind of deformed boson operators is proposed to be consistent with the large noncommutativity parameters on noncommutative plane when noncommutativity of momentum spaces is considered. Using this kind of deformed boson operators, the coherent states and squeezed states are constructed, and their properties are discussed in detail.

A study of vibrational excitations of HCN in the framework of an algebraic model

The vibrational description of the hydrogen cyanide molecule is presented in the framework of the U(2) × U(3) × U(2) algebraic model. In our approach the Hamiltonian is first written in configuration space, and thereafter expressed in an algebraic form by introducing a realization of the coordinates and momenta in terms of the familiar bosonic operators for the harmonic oscillator. An anharmonization procedure is then applied to the algebraic Hamiltonian, which consists in replacing the bosonic operators by a new set of operators belonging to the su(2) algebra for the stretches and the su(3) algebra for the bends. This procedure is equivalent to starting the description in terms of interacting Morse oscillators for the stretching degrees of freedom, without any analog with the bends. This method of obtaining the algebraic representation of the Hamiltonian provides in natural form the connection between the spectroscopic parameters and the structure and force constants. The analysis of the vibrational excitations of HCN is considered, where a description of 73 vibrational levels is carried with a standard deviation of rms = 0.91 cm−1, up to now the best description using an algebraic approach. This fit is used to estimate the potential energy surface.

Bosonic representation of quantum magnets with large single-ion easy-plane anisotropy

We introduce a new representation of an integer spin $S$ via bosonic operators which is useful in describing the paramagnetic phase and transitions to magnetically ordered phases in magnetic systems with large single-ion easy-plane anisotropy $D$. Considering the exchange interaction between spins as a perturbation and using the diagram technique we derive the elementary excitation spectrum and the ground state energy in the third order of the perturbation theory. In the special case of S=1 we obtain these expressions also using simpler spin representations some of which were introduced before. Comparison with results of previous numerical studies of 2D systems with S=1 demonstrates that our approach works better than other analytical methods applied before for such systems. We apply our results for the elementary excitation spectrum analysis obtained experimentally in $\rm NiCl_2$-$\rm 4SC(NH_2)_2$ (DTN). It is demonstrated that a set of model parameters (exchange constants and $D$) which has been used for DTN so far describes badly the experimentally obtained spectrum. A new set of parameters is proposed using which we fit the spectrum and values of two critical fields of DTN.

Virial Theorem for Angular Displacement and Torque

The usual Virial theorem is expressed through the coordinate and the force, \(2\langle T\rangle =\langle X\frac{dV}{dX}\rangle =-\langle XF\rangle \), \(F=-\frac{dV}{dX}\), XF is the work done by the force F, T is the kinetic energy. In this paper we extend the usual discussion on the Virial theorem about coordinate-force variables to the case of angular displacement-torque variables. By virtue of introducing the entangled state representation and the bosonic operator realization of the Hamiltonian of quantum pendulum system we derive the Virial theorem for angular variable and torque.

Cavity quantum electrodynamics in the ultrastrong coupling regime

We revisit the mathematical formulation of the famous Jaynes–Cummings–Paul Hamiltonian, which describes the interaction of a two-level atom with a single mode of an electromagnetic cavity reservoir. We rigorously show that under the condition of ultrastrong coupling between the atom and cavity in which the transition frequency is comparable to the coupling frequency, the bosonic field operators undergo non-sinusoidal time variations. As a result, the well-known solutions to the Jaynes–Cummings–Paul model are no longer valid, even when the rotating wave approximation is not used. We show how a correct mathematical solution could be found instead.

Capacities of Grassmann channels

A new class of quantum channels called Grassmann channels is introduced and their classical and quantum capacity is calculated. The channel class appears in a study of the two-mode squeezing operator constructed from operators satisfying the fermionic algebra. We compare Grassmann channels with the channels induced by the bosonic two-mode squeezing operator. Among other results, we challenge the relevance of calculating entanglement measures to assess or compare the ability of bosonic and fermionic states to send quantum information to uniformly accelerated frames.

Violation of the area law and long-range correlations in infinite-dimensional-matrix product states

We propose to construct a family of states of one-dimensional spin chains by replacing the finite dimensional matrices in matrix product states by infinite dimensional operators constructed from bosonic annihilation and creation operators. The resulting states are demonstrated to violate the area law and to exhibit long range correlations. In addition, we propose an efficient way to prepare the states experimentally, in which the spins interact sequentially with an ancilla system.

Twist deformations of the supersymmetric quantum mechanics

Abstract The $$ \mathcal{N} $$-extended Supersymmetric Quantum Mechanics is deformed via an abelian twist which preserves the super-Hopf algebra structure of its Universal Enveloping Superalgebra. Two constructions are possible. For even $$ \mathcal{N} $$ one can identify the 1D $$ \mathcal{N} $$-extended superalgebra with the fermionic Heisenberg algebra. Alternatively, supersymmetry generators can be realized as operators belonging to the Universal Enveloping Superalgebra of one bosonic and several fermionic oscillators. The deformed system is described in terms of twisted operators satisfying twist-deformed (anti)commutators. The main differences between an abelian twist defined in terms of fermionic operators and an abelian twist defined in terms of bosonic operators are discussed.

On the Lupaş q-transform

The Lupaş q - transform emerges in the study of the limit q-Lupaş operator . The latter comes out naturally as a limit for a sequence of the Lupaş q -analogues of the Bernstein operator. Lately, it has been studied by several authors from different perspectives in mathematical analysis and approximation theory. This operator is closely related to the q -deformed Poisson probability distribution, which is used widely in the q -boson operator calculus. Given q ∈ ( 0 , 1 ) , f ∈ C [ 0 , 1 ] , the q-Lupaş transform of f is defined by: ( Λ q f ) ( z ) ≔ 1 ( − z ; q ) ∞ ⋅ ∑ k = 0 ∞ f ( 1 − q k ) q k ( k − 1 ) / 2 ( q ; q ) k z k . In this paper, we study some analytic properties of ( Λ q f ) ( z ) . In particular, we examine the conditions under which Λ q f can either be an entire function, or a rational one.