Boson Creation Operators Research Articles

Mathematical Aspects of <i>SU</i> (2) and <i>SO</i>(3,R) Derived from Two-Mode Realization in Coordinate-Invariant Form

Some mathematical aspects of the Lie groups SU (2) and in realization by two pairs of boson annihilation and creation operators and in the parametrization by the vector parameter instead of the Euler angles and the vector parameter c of Fyodorov are developed. The one-dimensional root scheme of SU (2) is embedded in two-dimensional root schemes of some higher Lie groups, in particular, in inhomogeneous Lie groups and is represented in text and figures. The two-dimensional fundamental representation of SU (2) is calculated and from it the composition law for the product of two transformations and the most important decompositions of general transformations in special ones are derived. Then the transition from representation of SU (2) to of is made where in addition to the parametrization by vector the convenient parametrization by vector c is considered and the connections are established. The measures for invariant integration are derived for and for SU (2) . The relations between 3D-rotations of a unit sphere to fractional linear transformations of a plane by stereographic projection are discussed. All derivations and representations are tried to make in coordinate-invariant way.

Analytical approach to the Bose polaron via a q -deformed Lie algebra

We present a novel approach to the Bose polaron based on the notion of quantum groups, also known as $q$-deformed Lie algebras. In this approach, a mobile impurity can be depicted as a deformation of the Lie algebra of the bosonic creation and annihilation operators of the bath, in which the impurity is immersed. Accordingly, the Bose polaron can be described as a bath of noninteracting $q$-deformed bosons, which allows us to provide an analytical formulation of the Bose polaron at arbitrary couplings. Particularly, we derive its ground state energy in the phonon branch of the Bogoliubov dispersion and demonstrate that the previously observed transition from a repulsive to an attractive polaron occurs at the vicinity where the quantum group symmetry is broken. Furthermore, our approach has the potential to open up new avenues in polaron physics by connecting it with seemingly unrelated research topics where quantum groups play an essential role, such as anyons.

Nonclassicality of Photon Added Bipartite Glauber Lachs State

Mixture of coherent state with thermal noise is known as the Glauber Lachs state, which is taken into consideration. Nonlocal photon addition is imposed on the two mode Glauber Lachs state by the application of bosonic creation operators. This nonlocal photon addition induces nonclassicality, revealed in the single mode and two mode squeezing properties. The ratio of the coherent noise to thermal noise is the key factor in observing the nonclassical features. The tolerance limit of the noise for the nonclassicality is discussed in single mode squeezing and two mode squeezing quadrature operators. The results show that the squeezing properties are less pronounced even for the low level of thermal noise.

Dynamical Symmetries in Shell and Collective Models of Nuclear Structure

The dynamical symmetries in the algebraic shell model and the collective Interacting Vector Boson Model (IVBM) realized in terms of fermion and boson creation and annihilation operators are investigated. The obtained analytic eigen-energies in both models are compared with the experimental ones and the strengths of the corresponding terms of the model Hamiltonians are evaluated in both cases. In the algebraic realization of the Pairing plus Quadrupole Shell Model the correlations and transition between the quadrupole and pairing phases—dynamical symmetries are investigated in application to nuclear systems in the first few light shells. In the symplectic extension of the IVBM the spectra of heavy even-even nuclei with transitional between rotational and vibrational character is well reproduced. The algebraic connections between dynamical symmetries of the nuclear collective spectra and the ordering of the low-lying states with fixed angular momentum permits a reasonable and experimentally proved prediction of the position of the 0+ band heads of the collective bands. The models based on dynamical symmetries give an elegant and simple way to describe the complex spectra of nuclei with different shapes.

Irreversible quantum evolution with quadratic generator: Review

We review results on GKSL-type equations with multi-modal generators which are quadratic in bosonic or fermionic creation and annihilation operators. General forms of such equations are presented. The Gaussian solutions are obtained in terms of equations for the first and the second moments. Different approaches for their solutions are discussed.

Nonlinear coherent states of the para-Bose oscillator and their non-classical features

The construction of nonlinear coherent states via a unitary displacement operator is possible only for a few quantum-mechanical systems. In this paper, we define two non-unitary and a unitary displacement operators with the help of corresponding f -deformed bosonic annihilation and creation operators. While the action of the non-unitary displacement type operators on the vacuum state of field results in two new families of nonlinear coherent states (NLCSs), the unitary displacement operator reproduces Wigner-Heisenberg coherent states of the Gilmore-Perelomov type. We prove that the introduced NLCSs satisfy the resolution of the identities through positive definite measures. We also examine the non-classical properties of the obtained NLCSs by evaluating Klyshko’s criterion, Mandel’s parameter, quadrature squeezing and Wigner quasi-probability distribution function, in detail. Finally, we propose a simple scheme for the physical generation of the introduced NLCSs of the Gilmore-Perelomov type.

Out-of-time-ordered correlators in short-range and long-range hard-core boson models and in the Luttinger-liquid model

We study out-of-time-ordered correlators (OTOC) in hard-core boson models with short-range and long-range hopping and compare the results to the OTOC in the Luttinger liquid model. For the density operator, a related `commutator function' starts at zero and decays back to zero after the passage of the wavefront in all three models, while the wavefront broadens as $t^{1/3}$ in the short-range model and shows no broadening in the long-range model and the Luttinger liquid model. For the boson creation operator, the corresponding commutator function shows saturation inside the light cone in all three models, with similar wavefront behavior as in the density-density commutator function, despite the presence of a nonlocal string in terms of Jordan-Wigner fermions. For the long-range model and the Luttinger liquid model, the commutator function decays as power law outside the light cone in the long time regime when following different fixed-velocity rays. In all cases, the OTOCs approach their long-time values in a power-law fashion, with different exponents for different observables and short-range vs long-range cases. Our long-range model appears to capture exponents in the Luttinger liquid model (which are found to be independent of the Luttinger parameter in the model). This conclusion also bears on the OTOC calculations in conformal field theories, which we propose correspond to long-ranged models.

Squeezing and time evolution of magnon states under perpendicular pumping

The coherent magnon state representation is employed to investigate the quantum-statistical behavior of the nonlinear excitation of magnons in ferromagnets. Both the long-range magnetic dipole–dipole and short-range exchange interactions are included, along with a static longitudinal applied field and a microwave pumping field in the perpendicular orientation. Within a microscopic (or Hamiltonian-based) approach the total Hamiltonian is transformed from spin operators to a normal-mode set of boson creation and annihilation operators. When the three-magnon interactions are included, it is found that the microwave pumping field may be used to control the nonlinear statistical properties of the system. From a study of the time evolution of the system we deduce the average number of magnons, the super-Poissonian statistical behavior, and the occurrence of magnon squeezing. We also compare the results with the case where the microwave pumping field is in the parallel orientation, and it is found that there are important differences in the time dependence.

Squeezed States of Damon–Eshbach-Like Modes in Ferromagnetic Films Under Microwave Pumping

A theoretical account is given for the non-classical properties of magnons in ultra-thin ferromagnetic films with in-plane magnetization under the conditions of parallel pumping. A microscopic or Hamiltonian-based approach with a lattice of effective spins interacting through the magnetic dipole–dipole and exchange interactions is used. With the Zeeman energies of the static applied field and the microwave pumping field included, the total Hamiltonian is transformed from spin operators to boson creation and annihilation operators. Using a representation of coherent magnon states, we obtain the explicit results for the temporal evolution of the magnon occupation number as well as the magnon squeezing. In particular, it is shown that the manipulation of the collapse-and-revival phenomena for the average number of magnons and the properties of the magnon squeezing is made possible by controlling the nonlinearity in the system through the microwave pumping field.

A functional directional derivative in infinite dimensional spaces and its application to [formula omitted]-equations

A functional directional derivative composed of the Boson creation operators and the Boson annihilation operators is considered. And we apply some results on it to solving ∂¯-equations.

Bosonic quadratic Hamiltonians

We discuss self-adjoint operators given formally by expressions quadratic in bosonic creation and annihilation operators. We give conditions when they can be defined as self-adjoint operators, possibly after an infinite renormalization. We also discuss explicit formulas for their infimum. Our main motivation comes from local quantum field theory, which furnishes interesting examples of bosonic quadratic Hamiltonians that require an infinite renormalization.

Why you should not use the electric field to quantize in nonlinear optics.

We show that using the electric field as a quantization variable in nonlinear optics leads to incorrect expressions for the squeezing parameters in spontaneous parametric down-conversion (SPDC) and conversion rates in frequency conversion. This observation is related to the fact that if the electric field is written as a linear combination of bosonic creation and annihilation operators one cannot satisfy Maxwell's equations in a nonlinear dielectric.

Strategies for extending geminal-based wavefunctions: Open shells and beyond

We discuss some strategies for extending recent geminal-based methods to open-shells by replacing the geminal-creation operators with more general composite boson creation operators, and even creation operators that mix fermionic and bosonic components. We also discuss the utility of symmetry-breaking and restoration, but using a projective (not a variational) approach. Both strategies—either together or separately—give a pathway for extending geminals-based methods to open shells, while retaining the computational efficiency and conceptual simplicity of existing geminal product wavefunctions.

Quantum statistics and squeezing for a microwave-driven interacting magnon system

Theoretical studies are reported for the statistical properties of a microwave-driven interacting magnon system. Both the magnetic dipole–dipole and the exchange interactions are included and the theory is developed for the case of parallel pumping allowing for the inclusion of the nonlinear processes due to the four-magnon interactions. The method of second quantization is used to transform the total Hamiltonian from spin operators to boson creation and annihilation operators. By using the coherent magnon state representation we have studied the magnon occupation number and the statistical behavior of the system. In particular, it is shown that the nonlinearities introduced by the parallel pumping field and the four-magnon interactions lead to non-classical quantum statistical properties of the system, such as magnon squeezing. Also control of the collapse-and-revival phenomena for the time evolution of the average magnon number is demonstrated by varying the parallel pumping amplitude and the four-magnon coupling.

New Class of Quantum Error-Correcting Codes for a Bosonic Mode

We construct a new class of quantum error-correcting codes for a bosonic mode which are advantageous for applications in quantum memories, communication, and scalable computation. These 'binomial quantum codes' are formed from a finite superposition of Fock states weighted with binomial coefficients. The binomial codes can exactly correct errors that are polynomial up to a specific degree in bosonic creation and annihilation operators, including amplitude damping and displacement noise as well as boson addition and dephasing errors. For realistic continuous-time dissipative evolution, the codes can perform approximate quantum error correction to any given order in the timestep between error detection measurements. We present an explicit approximate quantum error recovery operation based on projective measurements and unitary operations. The binomial codes are tailored for detecting boson loss and gain errors by means of measurements of the generalized number parity. We discuss optimization of the binomial codes and demonstrate that by relaxing the parity structure, codes with even lower unrecoverable error rates can be achieved. The binomial codes are related to existing two-mode bosonic codes but offer the advantage of requiring only a single bosonic mode to correct amplitude damping as well as the ability to correct other errors. Our codes are similar in spirit to 'cat codes' based on superpositions of the coherent states, but offer several advantages such as smaller mean number, exact rather than approximate orthonormality of the code words, and an explicit unitary operation for repumping energy into the bosonic mode. The binomial quantum codes are realizable with current superconducting circuit technology and they should prove useful in other quantum technologies, including bosonic quantum memories, photonic quantum communication, and optical-to-microwave up- and down-conversion.

A new kind of nonlinear coherent states and their properties

A new kind of nonlinear coherent states, i.e. the k eigenstates of the powers (k ≥ 3) of the generalized annihilation operator of f-oscillators, are constructed using the inverse operators of the boson annihilation and creation operators. The set of these states may form a complete quantum mechanics representation. Using newly defined higher order squeezing and an antibunching, the properties of the Mth-order squeezing and the antibunching effect of the k states are investigated. The results show that the Mth-order [M = (n + 1/2)k; n = 0, 1, 2, …] squeezing effects exist in all of the k states when k is even and the antibunching effect exists in all of the k states.

Time-Dependent Correlation Functions for the Bimodal Bose–Hubbard Model

The bimodal Bose–Hubbard model is studied. The application of the Quantum Inverse Method allows us to calculate the time-dependent correlation functions of the model. Form-factors of the bosonic creation and annihilation operators in the wells are expressed in the determinantal form.

DiracQ: A Package for Algebraic Manipulation of Non-Commuting Quantum Variables

In several problems of quantum many body physics, one is required to handle complex expressions originating in the non-commutative nature of quantum operators. Their manipulation requires precise ordering and application of simplification rules. This can be cumbersome, tedious and error prone, and often a challenge to the most expert researcher. In this paper we present a software package DiracQ to facilitate such manipulations. The package DiracQ consists of functions based upon and extending considerably the symbolic capabilities of Mathematica . With DiracQ, one can proceed with large scale algebraic manipulations of expressions containing combinations of ordinary numbers or symbols (the c-numbers) and arbitrary sets of non-commuting variables (the q-numbers) with user defined properties. The DiracQ package is user extendable and comes encoded with the algebraic properties of several standard operators in popular usage. These include Fermionic and Bosonic creation and annihilation operators, spin operators, and canonical position and momentum operators. An example book is provided with some suggestive calculations of large-scale algebraic manipulations.

Construction of basis vectors for symmetric irreducible representations of O(5) $ \supset$ O(3)

A recursive method for the construction of symmetric irreducible representations of $ O(2l+1)$ in the $ O(2l+1)\supset O(3)$ basis for identical boson systems is proposed. The formalism is realized based on the group chain $ U(2l+1)\supset U(2l-1)\otimes U(2)$ , of which the symmetric irreducible representations are simply reducible. The basis vectors for symmetric irreducible representations of the $ O(2l+1)\supset O(2l-1)\otimes U(1)$ can easily be constructed from those of U(2l + 1) $ \supset$ U(2l - 1) ⊗ U(2) $ \supset$ O(2l - 1) ⊗ U(1) with no l -boson pairs, namely with the total boson number exactly equal to the seniority number in the system, from which one can construct symmetric irreducible representations of $ O(2l+1)$ in the $ O(2l-1)\otimes U(1)$ basis when all symmetric irreducible representations of O(2l - 1) are known. As a starting point, basis vectors of symmetric irreducible representations of O(5) are constructed in the $ O_{1}(3)\otimes U(1)$ basis, where $ O_{1}(3)\equiv O(2l-1)$ , when l = 2 , which is generated not by the angular momentum operators of the d -boson system, but by the operators constructed from d -boson creation (annihilation) operators $ d^{\dagger}_{\mu}$ ( $ d_{\mu}$ with $ \mu=1$ , 0, -1 . Matrix representations of $ O(5)\supset O_{1}(3)\otimes U(1)$ , together with the elementary Wigner coefficients, are presented. After the angular momentum projection, a three-term relation in determining the expansion coefficients of the $ O(5)\supset O(3)$ basis vectors, where the O(3) group is generated by the angular momentum operators of the d -boson system, in terms of those of $ O_{1}(3)\otimes U(1)$ is derived. The eigenvectors of the projection matrix with zero eigenvalues constructed according to the three-term relation completely determine the basis vectors of $ O(5)\supset O(3)$ . Formulae for evaluating the elementary Wigner coefficients of $ O(5)\supset O(3)$ are derived explicitly. Analytical expressions of some elementary Wigner coefficients of $ O(5)\supset O(3)$ for the coupling $ (\tau 0)\otimes(1 0)$ with resultant angular momentum quantum number L = 2 $ \tau$ + 2 - k for $ k=0,2,3,\ldots,6$ with a multiplicity 2 case for k = 6 are presented.

Classical mapping for Hubbard operators: application to the double-Anderson model.

A classical Cartesian mapping for Hubbard operators is developed to describe the nonequilibrium transport of an open quantum system with many electrons. The mapping of the Hubbard operators representing the many-body Hamiltonian is derived by using analogies from classical mappings of boson creation and annihilation operators vis-à-vis a coherent state representation. The approach provides qualitative results for a double quantum dot array (double Anderson impurity model) coupled to fermionic leads for a range of bias voltages, Coulomb couplings, and hopping terms. While the width and height of the conduction peaks show deviations from the master equation approach considered to be accurate in the limit of weak system-leads couplings and high temperatures, the Hubbard mapping captures all transport channels involving transition between many electron states, some of which are not captured by approximate nonequilibrium Green function closures.