K. Kowalski, J. Rembielinski, L. C. Papaloucas
The coherent states for the quantum particle on the circle are introduced. The Bargmann representation within the actual treatment provides the representation of the algebra $[\hat J,U]=U$, where $U$ is unitary, which is a direct consequence of the Heisenberg algebra $[\hat ϕ, \hat J]=i$, but it is more adequate for the study of the circlular motion.
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In this paper, the Higgs-like approach is used to analyze the quantum dynamics of a harmonic oscillator constrained on a circle. We obtain the Hamiltonian of this system as a function of the Cartesian coordinate of the tangent line through the gnomonic projection and then quantize it in the standard way. We then recast the Hamiltonian in a shape-invariant form and derive the spectrum energy of the confined harmonic oscillator on the circle. With help of the f-deformed oscillator algebra, we construct the coherent states on the circle and investigate their quantum statistical properties. We fin…
We discuss some basic tools for an analysis of one-dimensionalquantum systems defined on a cyclic coordinate space. The basic features of the generalized coherent states, the complexifier coherent states are reviewed. These states are then used to define the corresponding (quasi)densities in phase space. The properties of these generalized Husimi distributions are discussed, in particular their zeros.Furthermore, the use of the complexifier coherent states for a semiclassical analysis is demonstrated by deriving a semiclassical coherent state propagator in phase space.
A careful study of the physical properties of a family of coherent states on the circle, introduced some years ago by de Bièvre and González in [DG 92], is carried out. They were obtained from the Weyl-Heisenberg coherent states in $L^2(\R)$ by means of the Weil-Brezin-Zak transformation, they are labeled by the points of the cylinder $S^1 \times \R$, and they provide a realization of $L^2(S^1)$ by entire functions (similar to the well-known Fock-Bargmann construction). In particular, we compute the expectation values of the position and momentum operators on the circle and we discuss the Heis…
We present a possible construction of coherent states on the unit circle as configuration space. In our approach the phase space is the product Z x S^1. Because of the duality of canonical coordinates and momenta, i.e. the angular variable and the integers, this formulation can also be interpreted as coherent states over an infinite periodic chain. For the construction we use the analogy with our quantization over a finite periodic chain where the phase space was Z_M x Z_M. Properties of the coherent states constructed in this way are studied and the coherent states are shown to satisfy the re…
We present a possible construction of coherent states on the unit circle as configuration space. Our approach is based on Borel quantizations on S^1 including the Aharonov-Bohm type quantum description. The coherent states are constructed by Perelomov's method as group related coherent states generated by Weyl operators on the quantum phase space Z x S^1. Because of the duality of canonical coordinates and momenta, i.e. the angular variable and the integers, this formulation can also be interpreted as coherent states over an infinite periodic chain. For the construction we use the analogy with…
The q-deformed coherent states for a quantum particle on a circle are introduced and their properties investigated.
We extend former results for coherent states on the circle in the literature in two ways. On the one hand, we show that expectation values of fractional powers of momentum operators can be computed exactly analytically by means of Kummer's confluent hypergeometric functions. Earlier, these expectation values have only been obtained by using suitable estimates. On the other hand, we consider the Zak transformation not only to map harmonic oscillator coherent states to coherent states on the circle as it has been discussed before, but we also use the properties of the Zak transformation to deriv…
A discrete completeness relation and a continuous one with a positive measure are found for the photon-added squeezed vacuum states. Extension to the photon-added squeezed one-photon states is considered. Photon-added coherent states on a circle are introduced. Their normalization and unity resolution relation are given.
The uncertainty relations for the position and momentum of a quantum particle on a circle are identified minimized by the corresponding coherent states. The sqeezed states in the case of the circular motion are introduced and discussed in the context of the uncertainty relations.
The phase space $S\times Z$ for a particle on a circle is considered. Displacement operators in this phase space are introduced and their properties are studied. Wigner and Weyl functions in this context are also considered and their physical interpretation and properties are discussed. All results are compared and contrasted with the corresponding ones for the harmonic oscillator in the $R \times R$ phase space.
The Dirac method is used to analyze the classical and quantum dynamics of a particle constrained on a circle. The method of Lagrange multipliers is scrutinized, in particular in relation to the quantization procedure. Ordering problems are tackled and solved by requiring the hermiticity of some operators. The presence of an additional term in the quantum Hamiltonian is discussed.
We review the definition of hypergeometric coherent states, discussing some representative examples. Then we study mathematical and statistical properties of hypergeometric Schrödinger cat states, defined as orthonormalized eigenstates of $k$-th powers of nonlinear $f$-oscillator annihilation operators, with $f$ of hypergeometric type. These "$k$-hypercats" can be written as an equally weighted superposition of hypergeometric coherent states $|z_l\rangle, l=0,1,\dots,k-1$, with $z_l=z e^{2πi l/k}$ a $k$-th root of $z^k$, and they interpolate between number and coherent states. This fact motiva…
It is demonstrated that in contrast to the well-known case with a quantum particle moving freely in a real line, the wave packets corresponding to the coherent states for a free quantum particle on a circle do not spread but develop periodically in time. The discontinuous changes during the course of time in the phase representing the position of a particle can be interpreted as the quantum jumps on a circle.
Coherent states for equally spaced, homogeneous waveguide arrays are defined, in the infinite, semiinfinite and finite cases, and resolutions of the identity are constructed, using different methods. In the infinite case, which corresponds to Euclidean coherent states, a resolution of the identity with coherent states on the circle and involving a nonlocal inner product is reviewed. In the semiinfinite case, which corresponds to London coherent states, various construction are given (restricting to the circle with a non-local scalar product, rescaling the coherent states, modifying them, or us…
Two-particle Hadamard walk on dynamically percolated line and circle
本文研究了双粒子在动态渗流线与圆上的Hadamard行走的渐近动力学，构建了随机酉演化吸引子空间的基并证明其完备性，发现共享渗流图能作为弱相互作用使初始不相关粒子产生关联。
We study a particle propagation on a circle in the presence of a point interaction. We show that the one-particle Feynman kernel can be written into the sum of reflected and transmitted trajectories which are weighted by the elements of the n-th power of the scattering matrix evaluated on a line with a point interaction. As a by-product we find three-parameter family of trace formulae as a generalization of the Poisson summation formula.
We apply the quantization scheme introduced in [arXiv:1204.2870] to a particle on a circle. We find that the quantum action functional restricted to appropriate coherent states can be expressed as the classical action plus $\hbar$-corrections. This result extends the examples presented in the cited paper.
A generalized Weyl quantization formalism for a particle on the circle investigated in \cite{1} is developed. A Wigner function for the state $\hat{\varrho}$ and the kernel $\mathcal{K}$ for a particle on the circle is defined and its properties are analyzed. Then it is shown how this Wigner function can be easily modified to give the number-phase Wigner function in quantum optics. Some examples of such number-phase Wigner function are considered.
Quantum superpositions of coherent states are produced both in microwave and optical domains, and are considered realizations of the famous "Schrödinger cat" state. The recent progress shows an increase in the number of components and the number of modes involved. Our work creates a theoretical framework for treatment of multicomponent two-mode Schrödinger cat states. We consider a class of single-mode states, which are superpositions of $N$ coherent states lying on a circle in the phase space. In this class we consider an orthonormal basis created by rotationally-invariant circular states (RI…
We find that existing multi-party quantum key agreement (MQKA) protocols designed for fairness of the key are, in fact, unfair. Our analysis shows that these protocols are sensitive to collusive attacks; that is, dishonest participants can collaborate to predetermine the key without being detected. In fact, the transmission structures of the quantum particles in those unfair MQKA protocols, three of which have already been analyzed, have much in common. We call these unfair MQKA protocols circle-type MQKA protocols. Likewise, the transmission structures of the quantum particles in MQKA protoco…
The quantum harmonic oscillator on a circle -- fragmentation of the algebraic method
研究圆圈上二次势量子粒子的能谱，发现其非谐振子特性但保留代数性质，探讨代数方法失效的原因及其物理意义。
Quantum mechanics on a circle with a finite number of α-uniformly distributed points
本文研究了具有有限个α-均匀分布点的圆上的量子力学，定义了角度和平移算符，利用离散角度表示构造了两类离散角动量算符和厄米哈密顿量，并计算了自由粒子的能级。
Operational criterion for Wigner function negativity
本文提出了一种操作性标准，用于在量子非破坏测量框架内识别任意量子态的Wigner函数负性，通过直接测量Wigner函数并确立相干态基底作为确定其负区域的优先基底。
In the preceding Comment (quant-ph/0209032) Trifonov disputes our recently proposed uncertainty relations for a quantum particle on a circle. He states that (i) the quantity $Δ^2(\hatϕ)$ introduced by us representing the uncertainty of the angle is not a proper measure of the position uncertainty and therefore the proposed inequalities can hardly be qualified as a relevant uncertainty relations on a circle; and that (ii) the most suitable uncertainty relations on a circle are those based on the Gram-Robertson matrix. It is shown by examples that both points are erroneous.
A review of discrete quantum walk with two particle is given. The use of different states encountered in identical particle, and the idea of entanglement and superposition is explored to explored the interesting dynamics of two particle quantum walk. Boundary conditions can specify certain dynamics and so a survey of periodic boundary condition (circle) is presented. A simulation for a Hadamard walk for different periods of a circle is considered and results are compared for various periods and for a case with absorbing boundaries.
The coherent states are reviewed with particular application to the free particle system. The didactic advantages of the formalism is emphasized. Several interesting features, like the relation of the coherent states with the Galilei group and with the Husimi distribution are presented. Keywords: quantum mechanics, coherent states, free particle.
Revivals of the coherent states of a deformed, adiabatically and cyclically varying oscillator Hamiltonian are examined. The revival time distribution is exactly that of Poincaré recurrences for a rotation map: only three distinct revival times can occur, with specified weights. A link is thus established between quantum revivals and recurrences in a coarse-grained discrete-time dynamical system.
We introduce a large class of holomorphic quantum states by choosing their normalization functions to be given by generalized hypergeometric functions. We call them generalized hypergeometric states in general, and generalized hypergeometric coherent states in particular, if they allow a resolution of unity. Depending on the domain of convergence of the generalized hypergeometric functions, we distinguish generalized hypergeometric states on the plane, the open unit disk, and the unit circle. All states are eigenstates of suitably defined lowering operators. We then study their photon number s…
The quantum dynamics of a free particle on a circle with point interaction is described by a U(2) family of self-adjoint Hamiltonians. We provide a classification of the family by introducing a number of subfamilies and thereby analyze the spectral structure in detail. We find that the spectrum depends on a subset of U(2) parameters rather than the entire U(2) needed for the Hamiltonians, and that in particular there exists a subfamily in U(2) where the spectrum becomes parameter-independent. We also show that, in some specific cases, the WKB semiclassical approximation becomes exact (modulo p…
A method is developed to construct the solutions of one and many variable, linear differential equations of arbitrary order. Using this, the $N$-particle Sutherland model, with pair-wise inverse sine-square interactions among the particles, is shown to be equivalent to free particles on a circle. Applicability of our method to many other few and many-body problems is also illustrated.
New position uncertainty (delocalization) measures for a particle on the circle are proposed and illustrated on several examples, where the previous measures (based on 2pi-periodic position operators) appear to be unsatisfactory. The new measures are suitably constructed using the standard multiplication angle operator variances. They are shown to depend solely on the state of the particle and to obey uncertainty relations of the Schroedinger-Robertson type.
The coherent states are constructed for a charged particle in a uniform magnetic field based on coherent states for the circular motion which have recently been introduced by the authors.
We discuss the possibility of observing quantum nonlocality using the so-called mode entanglement, analyzing the differences between different types of particles in this context. We first discuss the role of coherent states in such experiments, and we comment on the existence of coherent states in nature. The discussion of coherent states naturally raises questions about the role of particle statistics in this problem. Although the Pauli exclusion principle precludes coherent states with a large number of fermionic particles, we find that a large number of fermionic coherent states, each conta…
Quantum versions of cylindric phase space, like for the motion of a particle on the circle, are obtained through different families of coherent states. The latter are built from various probability distributions of the action variable. The method is illustrated with Gaussian distributions and uniform distributions on intervals, and resulting quantizations are explored.
Developing the analysis of the distribution of the particle's position-momentum dot product, the so-called \textit{posmom} $\mathbf{x}% \cdot \mathbf{p}$\textbf{,} to quantum states on a circular circle on two-dimensional Cartesian coordinates, we give its posmometry (introduced recently by Y. A. Bernard and P. M. W. Gill, Posmom: The Unobserved Observable, J. Phys. Chem. Lett. 1\textbf{(}2010\textbf{)}1254) for eigenstates of the free motion on the circle, i.e., $z$-axis component of the angular momentum. The posmom has two parity symmetries, specifically, invariant under two operations $m_{x…
By representing a quantum state and its evolution with the majorana stars on the Bloch sphere, the Majorana representation (MR) provide us an intuitive way to study a physical system with SU(2) symmetry. In this work, based on coherent states, we propose a method to establish generalization of MR for a general symmetry. By choosing a generalized coherent state as a reference state, we give a more general MR for both finite and infinite systems and the corresponding star equations are given. Using this method, we study the squeezed vacuum states for three different symmetries, Heisenberg-Weyl, …
Discrete-to-Continuum Approach for the Analytic Continuation of One-Particle Propagator on the Circle
针对圆上单粒子传播子显式数值表达式匮乏的问题，提出离散到连续的代数方法，推导格点正则化下自由离散传播子有限式，重构连续极限传播子并实现椭圆θ函数有限解析延拓，还验证了无限直线极限并应用于势场中波包动力学研究。
For a charged particle in a homogeneous magnetic field, we construct stationary squeezed states which are eigenfunctions of the Hamiltonian and the non-Hermitian operator $\hat{X}_Φ = \hat{X} \cos Φ+ \hat{Y} \sin Φ$, $\hat{X}$ and $\hat{Y}$ being the coordinates of the Larmor circle center and $Φ$ is a complex parameter. In the family of the squeezed states, the quantum uncertainty in the Larmor circle position is minimal. The wave functions of the squeezed states in the coordinate representation are found and their properties are discussed. Also, for arbitrary gauge of the vector potential we…
Covariant integral quantisation using coherent states for semidirect product groups is studied and applied to the motion of a particle on the circle. In the present case the group is the Euclidean group E$(2)$. We implement the quantisation of the basic classical observables, particularly the $2π$-periodic discontinuous angle function and the angular momentum, and compute their corresponding lower symbols. An important part of our study is devoted to the angle operator given by our procedure, its spectrum and lower symbol, its commutator with the quantum angular momentum, and the resulting Hei…
Though not so widely appreciated in the literature, supersymmetric quantum mechanics provides an ideal playground for studying non-Abelian geometric phase, because supersymmetry always guarantees degeneracies in energy levels. In this paper we first present a simple supersymmetric model for a free particle on a circle with point-like interactions that exhibits $\mathscr{N} = 2$ supersymmetry and doubly degenerate energy levels. We then show that Berry's connection in this model is given by the Wu-Yang-like magnetic monopole in SU(2) Yang-Mills gauge theory. This article is largely based on our…
We rederive uncertainty relations for the angular position and momentum of a particle on a circle by employing the exponential of the angle instead of the angle itself, which leads to circular variance as a natural measure of resolution. Intelligent states minimizing the uncertainty product under the constraint of a given uncertainty in angle or in angular momentum turn out to be given by Mathieu wave functions. We also discuss a number of physically feasible approximations to these optimal states. The theory is applied to the orbital angular momentum of a beam of photons and verified in an ex…
We show under which conditions a particle scatters coherently on a multi-particles system, working in the non relativistic framework. In a nutshell, in order to have coherent scattering, the incident particle has to not resolve the internal structure of the composite system. We show that the above condition is satisfied when the de Broglie length of the incident particle is much larger than the size of the system.
The method of integrals of motion is used to construct families of generalized coherent states of a nonrelativistic spinless charged particle in a constant electric field. Families of states, differing in the values of their standard deviations at the initial time, are obtained. Depending on the initial values of the standard deviations, and also on the electric field, it turns out to be possible to identify some families with semiclassical states.
We investigate how the coherent spreading of the center of mass wave function of a particle, such as an atom, molecule or ion, affects the particle's interaction with fields such as the electromagnetic field or a phonon field, in view also of possible applications to emerging quantum technologies. To this end, we develop a suitably generalized Unruh-deWitt model for the interaction between a delocalizing first quantized particle and a second quantized field. We study how the coherent spreading of the center of mass wave function of the particle affects emission and absorption rates and we find…
In the present paper we study the interaction of a photonic field with a particle in a magnetic field. We use formalism similar to the Born-Oppenheimer approximation of molecular physics and we eliminate the photonic field variables by using a method related to the Berry phase to obtain an effective Hamiltonian and then extract the propagator of the particle. As an initial photonic state we assume a coherent one.
Using the orthodox Weyl -- Wigner -- Stratonovich -- Cohen (WWSC) quantization rule we construct a time -- of -- arrival operator for a free particle on the circle. It is shown that this operator is self -- adjoint but the careful analysis of its properties suggests that it cannot represent a `physical' time -- of -- arrival observable. The problem of a time -- of -- arrival observable for the `waiting screen' is also considered. A method of avoiding the quantum Zeno effect is proposed and the positive operator valued measure (POV -- measure) or the generalized positive operator valued measure…
We study a relativistic quantum particle in cosmic string spacetime in the presence of a uniform magnetic field and a Coulomb-type scalar potential. It is shown that the radial part of this problem possesses the $su(1,1)$ symmetry. We obtain the energy spectrum and eigenfunctions of this problem by using two algebraic methods: the Schrödinger factorization and the tilting transformation. Finally, we give the explicit form of the relativistic coherent states for this problem.
In the present article we extend our study (BJP 45 (2015) 369) of generalized coherent states (GCS) of a one-dimensional particle considering such important physical system as a 3-dimensional charged particle in electric and magnetic fields. Constructing GCS in many-dimensional case, we meet nontrivial technical complications that make the consideration nontrivial and instructive. The GCS of a system under consideration are constructed. We study properties of these GCS such as completeness relations, minimization of uncertainty relations and so on. We point out which family of the obtained GCS…
We present a general approach to calculating the entanglement of formation for superpositions of two-mode coherent states, placed equidistantly on a circle in the phase space. We show that in the particular case of rotationally-invariant circular states the Schmidt decomposition of two modes, and therefore the value of their entanglement, are given by analytical expressions. We analyse the dependence of the entanglement on the radius of the circle and number of components in the superposition. We also show that the set of rotationally-invariant circular states creates an orthonormal basis in t…
A solution to a version of the Stieltjes moment problem is presented. Using this solution, we construct a family of coherent states of a charged particle in a uniform magnetic field. We prove that these states form an overcomplete set that is normalized and solves the identity. By the help of the coherent states we construct the Fock-Bergmann representation related to the particle quantization. This quantization procedure takes into account a circle topology of the classical motion.
We consider discrete-alphabet encoding schemes for coherent-state quantum key distribution. The sender encodes the letters of a finite-size alphabet into coherent states whose amplitudes are symmetrically distributed on a circle centered in the origin of the phase space. We study the asymptotic performance of this phase-encoded coherent-state protocol in direct and reverse reconciliation assuming both loss and thermal noise in the communication channel. In particular, we show that using just four phase-shifted coherent states is sufficient for generating secret key rates of the order of $4 \ti…
In this paper, by mapping datasets to a set of non-linear coherent states, the process of encoding inputs in quantum states as a non-linear feature map is re-interpreted. As a result of this fact that the Radial Basis Function is recovered when data is mapped to a complex Hilbert state represented by coherent states, non-linear coherent states can be considered as natural generalisation of associated kernels. By considering the non-linear coherent states of a quantum oscillator with variable mass, we propose a kernel function based on generalized hypergeometric functions, as orthogonal polynom…
The coherent states for a particle on a sphere are introduced. These states are labelled by points of the classical phase space, that is the position on the sphere and the angular momentum of a particle. As with the coherent states for a particle on a circle discussed in Kowalski K {\em et al} 1996 {\em J. Phys. A} {\bf 29} 4149, we deal with a deformation of the classical phase space related with quantum fluctuations. The expectation values of the position and the angular momentum in the coherent states are regarded as the best possible approximation of the classical phase space. The correctn…
The three approaches to relativistic generalization of coherent states are discussed in the simplest case of a spinless particle: the standard, canonical coherent states, the Lorentzian states and the coherent states introduced by Kaiser and independently by Twareque Ali, Antoine and Gazeau. All treatments utilize the Newton-Wigner localization and dynamics described by the Salpeter equation. The behavior of expectation values of relativistic observables in the coherent states is analyzed in detail and the Heisenberg uncertainty relations are investigated.
We construct coherent states of the massless and massive representations of the Poincaré group. They are parameterised by points on the classical state space of spinning particles. Their properties are explored, with special emphasis on the geometrical structures on the state space.
We investigate a quantum many-body system with particles moving on a circle and subject to two-body and three-body potentials. In this new class of models, that extrapolates from the celebrated Calogero-Sutherland model and a system with interactions among nearest and next-to-nearest neighbors, the interactions can be tuned as a function of range. We determine the exact ground state energy and wavefunction and obtain a part of the excitation spectrum.
We examine mathematical questions around angle (or phase) operator associated with a number operator through a short list of basic requirements. We implement three methods of construction of quantum angle. The first one is based on operator theory and parallels the definition of angle for the upper half-circle through its cosine and completed by a sign inversion. The two other methods are integral quantization generalizing in a certain sense the Berezin-Klauder approaches. One method pertains to Weyl-Heisenberg integral quantization of the plane viewed as the phase space of the motion on the l…
We study particle current and occupation distribution in a recently proposed model for coherent quantum transport. In this model a system connected to mesoscopic Fermi reservoirs (mesoreservoir) is driven out of equilibrium by the action of superreservoirs with prescribed temperatures and chemical potentials described by a simple dissipative mechanism with the Lindblad equation. We compare exact (numerical) results for the non-equilibrium steady state particle current with theoretical expectations based on the Landauer formula and show that the model reproduce the behavior of coherent quantum …
Squeezed coherent states for a free particle with time-varying mass
该研究针对质量随时间指数变化的自由粒子，推导了其压缩相干态的参数表达与半经典条件，证明质量随时间增长时存在最小不确定度相干态，并分析了Fock态到该压缩相干态的跃迁概率。
We describe coherent states and associated generalized Grassmann variables for a system of $m$ independent $q$-boson modes. A resolution of unity in terms of generalized Berezin integrals leads to generalized Grassmann symbolic calculus. Formulae for operator traces are given and the thermodynamic partition function for a system of $q$-boson oscillators is discussed.
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