P. Shanta, S. Chaturvedi, V. Srinivasan
Eigenstates of the linear combinations $a^2+βa^{\dagger2}$ and $ab+βa^\dagger b^\dagger$ of two boson creation and annihilation operators are presented. The algebraic procedure given here is based on the work of Shanta et al. [Phys. Rev. Lett. {\bf 72}, 1447, 1994] for constructing eigenstates of generalized annihilation operators.
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We examine nonclassical properties of the field states generated by applying the photon annihilation-then-creation operation (AC) and creation-then-annihilation operation (CA) to the thermal and coherent states. Effects of repeated applications of AC and of CA are also studied. We also discuss experimental schemes to realize AC and CA with a cavity system using atom-field interactions.
We give the normal and anti-normal order expressions of the number operator to the power $k$ by using the commutation relation between the annihilation and creation operators. We use those expressions to give general formulae for functions of the number operator in normal and anti-normal order.
A quantum mechanical description of vortices in two dimensional superfluid $^4$He flims is presented. Single vortex creation and annihilation operators are defined and wavefunctions for these states are explicitly constructed. A hamitonian for these states is then proposed.
We develop a theory based on Bohmian mechanics in which particle world lines can begin and end. Such a theory provides a realist description of creation and annihilation events and thus a further step towards a "beable-based" formulation of quantum field theory, as opposed to the usual "observable-based" formulation which is plagued by the conceptual difficulties--like the measurement problem--of quantum mechanics.
The ability of implementing quantum operations plays fundamental role in manipulating quantum systems. Creation and annihilation operators which transform a quantum state to another by adding or subtracting a particle are crucial of constructing quantum description of many body quantum theory and quantum field theory. Here we present a quantum algorithm to perform them by the linear combination of unitary operations associated with a two-qubit ancillary system. Our method can realize creation and annihilation operators simultaneously in the subspace of the whole system. A prototypical experime…
Defining coherent states: why must they be eigenstates of the annihilation operator?
本文从物理动机出发，通过两条等价路径证明谐振子相干态当且仅当为湮灭算符本征态时具有最小不确定性和最小能量差，并解释了退相干过程选择相干态为最经典态的原因。
Creation and annihilation operators for 2D non-abelian anyons
该研究通过分析任意子图解形式的代数结构，定义二维非阿贝尔任意子的产生湮灭算符，显式构造斐波那契任意子的对应算符并明确多融合通道需额外算符，还表示了相关可观测量与哈密顿量，为模拟方法开发提供基础。
We propose a scheme for measurements of anomalous moments of creation and annihilation operators requiring only one beam splitter, only one photodetector and not many measurements with phase-shift $φ$ in homodyning. This has advantageous over scheme of Shchukin and Vogel [Phys. Rev. A 72, 043808 (2005)] which requires many photodetectors, many beam splitters and a large number of measurements with phase-shift $φ$ as a Fourier transform is calculated integrating over $φ$.
We present a systematic analysis on coherent states of composite bosons consisting of two distinguishable particles. By defining an effective composite boson (coboson) annihilation operator, we derive its eigenstate and commutator. Depending on the elementary particles comprising the composite particles, we gauge the resemblance between this eigenstate and traditional coherent states through typical measures of nonclassicality, such as quadrature variances and Mandel's Q parameter. Furthermore, we show that the eigenstate of the coboson annihilation operator is useful in estimating the maximum…
The Klein-Gordon equation with scalar potential is considered. In the Feshbach-Villars representation the annihilation operator for a linear potential is defined and its eigenstates are obtained. Although the energy levels in this case are not equally-spaced, depending on the eigenvalues of the annihilation operator, the states are nearly coherent and squeezed. The relativistic Poschl-Teller potential is introduced. It is shown that its energy levels are equally-spaced. The coherence of time evolution of the eigenstates of the annihilation operator for this potential is evaluated.
Quantum model of hydrogen-like atoms in hilbert space by introducing the creation and annihilation operators
本研究提出利用产生与湮灭算符的纯代数技术在希尔伯特空间求解类氢原子径向方程，该方法比传统级数解析法更完备，可作为第三个量子基本模型，无需额外假设就能得到能量量子化结果与归一化径向波函数。
The photon creation and annihilation operators are cornerstones of the quantum description of the electromagnetic field. They signify the isomorphism of the optical Hilbert space to that of the harmonic oscillator and the bosonic nature of photons. We perform complete experimental characterization (quantum process tomography) of these operators. By measuring their effect on coherent states, we obtain their process tensor in the Fock basis, which explicitly shows the "raising" and "lowering" properties of these operators with respect to photon number states. This is the first experimental demon…
Interplay of simultaneous creation, annihilation, propagation, and relaxation of an excitation in molecular condensates interacting with an ultrashort quantum optical pulse is studied in general and specialized to a dimer. A microscopic model appropriate for such systems (with strong exciton-phonon coupling) is presented. It also incorporates effects of (quantum) noise in the optical field. A variety of new features in the initial stage of excitation dynamics (when it is being created) is revealed; a strong influence of the coherent excitation propagation on the processes of excitation creatio…
We introduced fermionic variables in complex modules over real Clifford algebras of even dimension which are analog of the Witt basis. We built primitive idempotents which are a set of equivalent Clifford vacuums. It is shown that the modules are decomposed into direct sum of minimal left ideals generated by these idempotents and that the fermionic variables can be considered as more fundamental mathematical objects than spinors.
We construct two commuting sets of creation and annihilation operators for the PT-symmetric oscillator. We then build coherent states of the latter as eigenstates of such annihilation operators by employing a modified version of the normalization integral that is relevant to PT-symmetric systems. We show that the coherent states are normalizable only in the range (0, 1) of the underlying coupling parameter $α$.
Motivated by the readout scheme in interferometric gravitational-wave detectors, we consider the device which measures the expectation value of the photon annihilation and creation operators for output optical field from the main interferometer. As the result, the eight-port homodyne detection is rediscovered as such a device. We evaluate the noise spectral density in this measurement. We also briefly discuss on the application of our results to the readout scheme of gravitational-wave detectors. We call this measurement scheme to measure these expectation values as "double balanced homodyne d…
We derive Heisenberg equations for arbitrary high order moments of creation and annihilation operators in the case of the quantum master equation with a multimode generator which is quadratic in creation and annihilation operators and obtain their solutions. Based on them we also derive similar equations for the case of the quantum master equation, which occur after averaging the dynamics with a quadratic generator with respect to the classical Poisson process. This allows us to show that dynamics of arbitrary finite-order moments of creation and annihilation operators is fully defined by fini…
The annihilation-creation operators of the harmonic oscillator, the basic and most important tools in quantum physics, are generalised to most solvable quantum mechanical systems of single degree of freedom including the so-called `discrete' quantum mechanics. They admit exact Heisenberg operator solution. We present unified definition of the annihilation-creation operators (a^{(\pm)}) as the positive/negative frequency parts of the exact Heisenberg operator solution.
The exact solvability and impressive pedagogical implementation of the harmonic oscillator creation and annihilation operators make it a problem of great physical relevance and the most fundamental one in quantum mechanics. So would be the position dependent mass oscillator for the position dependent mass quantum mechanics. We, hereby, construct the position dependent mass creation and annihilation operators for the position dependent mass oscillator via two different approaches. First, via von Roos position dependent mass Hamiltonian and show that the commutation relation between the position…
We consider a Lindblad equation that for particular initial conditions reduces to an asymmetric simple exclusion process with additional loss and gain terms. The resulting Lindbladian exhibits operator-space fragmentation and each block is Yang-Baxter integrable. For particular loss/gain rates the model can be mapped to free fermions. We determine the full quantum dynamics for an initial product state in this case.
Various examples of exactly solvable `discrete' quantum mechanics are explored explicitly with emphasis on shape invariance, Heisenberg operator solutions, annihilation-creation operators, the dynamical symmetry algebras and coherent states. The eigenfunctions are the (q-)Askey-scheme of hypergeometric orthogonal polynomials satisfying difference equation versions of the Schrödinger equation. Various reductions (restrictions) of the symmetry algebra of the Askey-Wilson system are explored in detail.
The annihilation-creation operators $a^{(\pm)}$ are defined as the positive/negative frequency parts of the exact Heisenberg operator solution for the `sinusoidal coordinate'. Thus $a^{(\pm)}$ are hermitian conjugate to each other and the relative weights of various terms in them are solely determined by the energy spectrum. This unified method applies to most of the solvable quantum mechanics of single degree of freedom including those belonging to the `discrete' quantum mechanics.
In this article we introduce a simple physical model which realizes the algebra of orthofermions. The model is constructed from a cylinder which can be filled with some balls. The creation and annihilation operators of orthofermions are related to the creation and annihilation operators of balls in certain positions in the cylinder. Relationship between this model and topological symmetries in quantum mechanics is investigated.
A proof is given that an invertible and a unitary operator can be used to reproduce the effect of a q-deformed commutator of annihilation and creation operators. In other words, the original annihilation and creation operators are mapped into new operators, not conjugate to each other, whose standard commutator equals the identity plus a correction proportional to the original number operator. The consistency condition for the existence of this new set of operators is derived, by exploiting the Stone theorem on 1-parameter unitary groups. The above scheme leads to modified equations of motion …
We develop a star-product scheme of symbols defined by the normally ordered powers of the creation and annihilation photon operators, (a†)^m a^n. The corresponding phase space is a two-dimensional lattice with nodes (m,n) given by pairs of nonnegative integers. The star-product kernel of symbols on the lattice and intertwining kernels to other schemes are found in explicit form. Analysis of peculiar properties of the star-product kernel results in new sum relations for factorials. Advantages of the developed star-product scheme for describing dynamics of quantum systems are discussed and time …
Supersymmetry in Quantum Mechanics by Generalized Uncertainty Principle
本文基于广义不确定性原理构造量子力学中的广义超对称性，利用微扰理论建立产生湮灭算符关系，并通过解析求解薛定谔方程生成新势。
We investigate how particle pair creation and annihilation, within the quantum transverse XY model, affects the non-equilibrium steady state (NESS) and Liouvillian gap of the stochastic Totally Asymmetric Exclusion Process (TASEP). By utilising operator quantization we formulate a perturbative description of the NESS. Furthermore, we estimate the Liouvillian gap by exploiting a Majorana canonical basis as the basis of super-operators. In this manner we show that the Liouvillian gap can remain finite in the thermodynamic limit provided the XY model anisotropy parameter remains non-zero. Additio…
A new integral representation is obtained for the star product corresponding to the $s$-ordering of the creation and annihilation operators. This parametric ordering convention introduced by Cahill and Glauber enables one to vary the type of ordering in a continuous way from normal order to antinormal order. Our derivation of the corresponding integral representation is based on using reproducing formulas for analytic and antianalytic functions. We also discuss a different representation whose kernel is a generalized function and compare the properties of this kernel with those of the kernels …
In this paper combinatorial aspects of normal ordering arbitrary words in the creation and annihilation operators of the q-deformed boson are discussed. In particular, it is shown how by introducing appropriate q-weights for the associated ``Feynman diagrams'' the normally ordered form of a general expression in the creation and annihilation operators can be written as a sum over all q-weighted Feynman diagrams, representing Wick's theorem in the present context.
A fully developed Bose-Einstein condensate, containing macroscopically large number of bosons can under certain conditions be considered as a generalized vacuum state. Applying the annihilation operator to the condensate, hole states can be defined. Infinite ladders of such hole states can be considered as generalized coherent states of the creation operator. Dedicated to the memory of V. N. Gribov.
With a recently introduced geometric momentum that depends on the extrinsic curvature and offers a proper description of momentum on two-dimensional sphere, we show that the annihilation operators whose eigenstates are coherent states on the sphere take the expected form αx+iβp, where α and β are two operators that depend on the angular momentum and x and p are the position and the geometric momentum, respectively. Since the geometric momentum is manifestly a consequence of embedding the two-dimensional sphere in the three-dimensional flat space, the coherent states reflects some aspects beyon…
The the over-complete eigenvector system of the operator Q^{-1}P (Q:position, P:momentum) which consists of the squeezed states |0; s, t > with various s and t are investigated from the viewpoint of the annihilation and creation relations related to the algebra su(1,1). We derive a positive operator-valued measure(POVM) for the simultaneous measurement between the self-adjoint and anti-self-adjoint parts of PQ^{-1}.
Quantum physics has revealed many interesting formal properties associated with the algebra of two operators, A and B, satisfying the partial commutation relation AB-BA=1. This study surveys the relationships between classical combinatorial structures and the reduction to normal form of operator polynomials in such an algebra. The connection is achieved through suitable labelled graphs, or "diagrams", that are composed of elementary "gates". In this way, many normal form evaluations can be systematically obtained, thanks to models that involve set partitions, permutations, increasing trees, as…
The problem of measurement in quantum mechanics is that the quantum particle in the course of evolution, as described by the linear Schrodinger equation, exists in all of its possible states, but in measuring, the particle is always detected in only one of its states. This property is called the "collapse of the wave function" and was formulated by Von Neumann as one of the postulates of quantum mechanics. However, it remains unclear at what point in time and under what laws this transition occurs. This article demonstrates that the collapse of the wave function may be due to the creation or a…
Interior-boundary conditions (IBCs) have been suggested as a possibility to circumvent the problem of ultraviolet divergences in quantum field theories. In the IBC approach, particle creation and annihilation is described with the help of linear conditions that relate the wave functions of two sectors of Fock space: $ψ^{(n)}(p)$ at an interior point $p$ and $ψ^{(n+m)}(q)$ at a boundary point $q$, typically a collision configuration. Here, we extend IBCs to the relativistic case. To do this, we make use of Dirac's concept of multi-time wave functions, i.e., wave functions $ψ(x_1,...,x_N)$ depen…
Singularity theory of Weyl-point creation and annihilation
本文揭示了Weyl点合并与奇点理论的基本联系，证明Weyl相图的相界点可按流形间映射的奇点类分类，并预测了相图的普适几何特征。
A displacement operator d_ζis introduced, verifying commutation relations [d_ζ, a_f^\dagger]=[d_ζ, a_f]=ζ(f)d_ζwith field creation and annihilation operators that verify [a_f,a_g]=0, [a_f,a_g^\dagger]=(g,f), as usual. f and g are test functions, ζis a Poincare invariant real-valued function on the test function space, and (g,f) is a Poincare invariant Hermitian inner product. The *-algebra generated by all these operators, and a state defined on it, nontrivially extends the *-algebra of creation and annihilation operators and its Fock space representation. If the usual requirement for linearit…
We consider an atomic beam reservoir as a source of quantum noise. The atoms are modelled as two-state systems and interact one-at-a-time with the system. The Floquet operators are described in terms of the Fermionic creation, annihilation and number operators associated with the two-state atom. In the limit where the time between interactions goes to zero and the interaction is suitably scaled, we show that we may obtain a causal (that is, adapted) quantum stochastic differential equation of Hudson-Parthasarathy type, driven by creation, annihilation and conservation processes. The effect of …
We use the approach of "transitionless quantum driving" proposed by Berry to construct shortcuts to the population transfer and the creation of maximal entanglement between two $Λ$-type atoms based on the cavity quantum electronic dynamics (CQED) system. An effective Hamiltonian is designed by resorting to an auxiliary excited level, a classical driving field and an extra cavity field mode to supplement or substitute the original reference Hamiltonian, and steer the system evolution along its instantaneous eigenstates in an arbitrarily short time, speeding up the rate of population transfer an…
An exactly solvable Quantum Field Theory (QFT) model of Lee-type is constructed to study how neutrino flavor eigenstates are created through interactions and how the localization properties of neutrinos follows from the parent particle that decays. The two-particle states formed by the neutrino and the accompanying charged lepton can be calculated exactly as well as their creation probabilities. We can show that the coherent creation of neutrino flavor eigenstates follows from the common negligible contribution of neutrino masses to their creation probabilities. On the other hand, it is shown …
In this paper we consider the interaction of electrons in bilayer graphene with a constant homogeneous magnetic field which is orthogonal to the bilayer surface. Departing from the energy eigenstates of the effective Hamiltonian, the corresponding coherent states will be constructed. For doing this, first we will determine appropriate creation and annihilation operators in order to subsequently derive the coherent states as eigenstates of the annihilation operator with complex eigenvalue. Then, we will calculate some physical quantities, as the Heisenberg uncertainty relation, the probabilitie…
Generalized boson and fermion operators with a maximal total occupation property
本文提出一种包含正整数参数 $p$ 的广义（反）对易关系，在保留玻色子与费米子对称性的同时使系统总占据数最大值为 $p$，并基于 Fock 空间算符作用建立群论框架以对应已知粒子统计。
The difference between a Klein-Gordon random field and the complex Klein-Gordon quantum field is characterized, explicitly comparing the roles played by negative frequency modes of test functions in creation and annihilation operator presentations of the two theories. The random field and the complex quantum field can both be constructed from the same creation and annihilation operator algebra, making them equivalent in that sense.
Based on the Gaussian wave packet solution for the harmonic oscillator and the corresponding creation and annihilation operators, a generalization is presented that also applies for wave packets with time-dependent width as they occur for systems with different initial conditions, time-dependent frequency or in contact with a dissipative environment. In all these cases the corresponding coherent states, position and momentum uncertainties and quantum mechanical energy contributions can be obtained in the same form if the creation and annihilation operators are expressed in terms of a complex v…
We show that the Bohmian approach in terms of persisting particles that move on continuous trajectories following a deterministic law can be literally applied to QFT. By means of the Dirac sea model -- exemplified in the electron sector of the standard model neglecting radiation -- we explain how starting from persisting particles, one is led to standard QFT employing creation and annihilation operators when tracking the dynamics with respect to a reference state, the so-called vacuum. Since on the level of wave functions, both formalisms are mathematically equivalent, this proposal provides f…
Hybrid Dirac fields are fields that are general superpositions of the annihilation and creation parts of four Dirac spin 1/2 fields, $ψ^{(\pm)}(x;\pm m)$, whose annihilation and creation parts obey the Dirac equation with mass $m$ and mass $-m$. We discuss a specific case of such fields, which has been called ``homeotic.'' We show for this case, as is true in general for hybrid Dirac fields (except the ordinary fields whose annihilation and creation parts both obey one or the other Dirac equation), that (1) any interacting theory violates both Lorentz covariance and causality, (2) the discrete…
In this article we are concerned with finite dimensional Fermions, by which we mean vectors in a finite dimensional complex space embedded in the exterior algebra over itself. These Fermions are spinless but possess the characterizing anticommutativity property. We associate invariant complex vector fields on the Lie group $\mathrm{Spin}(2n+1)$ to the Fermionic creation and annihilation operators. These vector fields are elements of the complexification of the regular representation of the Lie algebra $\mathfrak{so}(2n+1)$. As such, they do not satisfy the canonical anticommutation relations, …
In this paper, we will construct the coherent states for a Dirac electron in graphene placed in a constant homogeneous magnetic field which is orthogonal to the graphene surface. First of all, we will identify the appropriate annihilation and creation operators. Then, we will derive the coherent states as eigenstates of the annihilation operator, with complex eigenvalues. Several physical quantities, as the Heisenberg uncertainty product, probability density and mean energy value, will be as well explored.
We generalize Schwinger boson representation of SU(2) algebra to SU(N) and define coherent states of SU(N) using $2(2^{N-1}-1)$ bosonic harmonic oscillator creation and annihilation operators. We give an explicit construction of all (N-1) Casimirs of SU(N) in terms of these creation and annihilation operators. The SU(N) coherent states belonging to any irreducible representations of SU(N) are labelled by the eigenvalues of the Casimir operators and are characterized by (N-1) complex orthonormal vectors describing the SU(N) manifold. The coherent states provide a resolution of identity, satisfy…
A relativistic quantum mechanics is formulated in which all of the interactions are in the four-momentum operator and Lorentz transformations are kinematic. Interactions are introduced through vertices, which are bilinear in fermion and antifermion creation and annihilation operators, and linear in boson creation and annihilation operators. The fermion-antifermion operators generate a unitary Lie algebra, whose representations are fixed by a first order Casimir operator (corresponding to baryon number or charge). Eigenvectors and eigenvalues of the four-momentum operator are analyzed and exact…
On global dynamics for damped driven Jaynes-Cummings equations
本文研究了阻尼驱动Jaynes-Cummings方程的全局广义解构造，针对时间依赖泵浦情况，在Hilbert空间中非负Hermitian Hilbert-Schmidt算子上建立解，并利用有限维近似方法证明。
We describe an algebra G of diagrams which faithfully gives a diagrammatic representation of the structures of both the Heisenberg-Weyl algebra H - the associative algebra of the creation and annihilation operators of quantum mechanics - and U(L_H), the enveloping algebra of the Heisenberg Lie algebra L_H. We show explicitly how G may be endowed with the structure of a Hopf algebra, which is also mirrored in the structure of U(L_H). While both H and U(L_H) are images of G, the algebra G has a richer structure and therefore embodies a finer combinatorial realization of the creation-annihilation…
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