In this work, we construct different classes of coherent states related to a quantum system, recently studied in [1], of an electron moving in a plane in uniform external magnetic and electric fields which possesses both discrete and continuous spectra. The eigenfunctions are realized as an orthonormal basis of a suitable Hilbert space appropriate for building the related coherent states. These latter are achieved in the context where we consider both spectra purely discrete obeying the criteria that a family of coherent states must satisfies.
In this paper, we construct the coherent states for a system of an electron moving on plane in uniform external magnetic and electric fields. These coherent states are built in the context of both discrete and continuous spectra and satisfy the Gazeau-Klauder coherent states properties [1].
Coherent states for general systems with discrete spectrum, such as the bound states of the hydrogen atom, are discussed. The states in question satisfy: (1) continuity of labeling, (2) resolution of unity, (3) temporal stability, and (4) an action identity. This set of reasonable physical requirements uniquely specify coherent states for the (bound state portion of the) hydrogen atom.
We predict and study the quantum-electrodynamical effect of parametric self-induced excitation of a molecule moving above the dielectric or conducting medium with periodic grating. In this case the radiation reaction force modulates the molecular transition frequency which results in a parametric instability of dipole oscillations even from the level of quantum or thermal fluctuations. The present mechanism of instability of electrically neutral molecules is different from that of the well-known Smith-Purcell and transition radiation in which a moving charge and its oscillating image create an…
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.
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…
Discrete time quantum walks are unitary maps defined on the Hilbert space of coupled two-level systems. We study the dynamics of excitations in a nonlinear discrete time quantum walk, whose fine-tuned linear counterpart has a flat band structure. The linear counterpart is, therefore, lacking transport, with exact solutions being compactly localized. A solitary entity of the nonlinear walk moving at velocity $v$ would therefore not suffer from resonances with small amplitude plane waves with identical phase velocity, due to the absence of the latter. That solitary excitation would also have to …
Discrete wave mechanics describes the evolution of classical or matter waves on a lattice, which is governed by a discretized version of the Schrödinger equation. While for a vanishing lattice spacing wave evolution of the continuous Schrödinger equation is retrieved, spatial discretization and lattice effects can deeply modify wave dynamics. Here we discuss implications of breakdown of exact Galilean invariance of the discrete Schrödinger equation on the bound states sustained by a smooth potential well which is uniformly moving on the lattice with a drift velocity $v$. While in the continuou…
The coherent and Fock states of a charge moving in varying homogeneous magnetic field are studied in the tomographic probability representation of quantum mechanics. The states are expressed in terms of quantum tomograms. The coherent states tomograms are shown to be described by normal distributions with varying dispersions and means. The Fock state tomograms are given in the form of probability distributions described by multivariable Hermite polynomials with time-dependent arguments.
The Casimir-Polder force is an attractive force between a polarizable atom and a conducting or dielectric boundary. Its original computation was in terms of the Lamb shift of the atomic ground state in an electromagnetic field (EMF) modified by boundary conditions along the wall and assuming a stationary atom. We calculate the corrections to this force due to a moving atom, demanding maximal preservation of entanglement generated by the moving atom-conducting wall system. We do this by using non-perturbative path integral techniques which allow for coherent back-action and thus can treat non-M…
We consider discrete spectra of bound states for non-relativistic motion in attractive potentials V_σ(x) = -|V_{0}| |x|^{-σ}, 0 < σ\leq 2. For these potentials the quasiclassical approximation for n -> \infty predicts quantized energy levels e_σ(n) of a bounded spectrum varying as e_σ(n) ~ -n^{-2σ/(2-σ)}. We construct collective quantum states using the set of wavefunctions of the discrete spectrum taking into account this asymptotic behaviour. We give examples of states that are normalizable and satisfy the resolution of unity, using explicit positive functions. These are coherent states in t…
Discrete coherent states for a system of $n$ qubits are introduced in terms of eigenstates of the finite Fourier transform. The properties of these states are pictured in phase space by resorting to the discrete Wigner function
We are continuing here the study of generalized coherent states of Barut-Girardello type for the oscillator-like systems connected with the given set of orthogonal polynomials. In this work we construct the family of coherent states associated with discrete $q$-Hermite polynomials of the II-type and prove the over-completeness of this family of states by constructing the measure for unity decomposition for this family of coherent states.
We put forward a method of constructing discrete coherent states for n qubits. After establishing appropriate displacement operators, the coherent states appear as displaced versions of a fiducial vector that is fixed by imposing a number of natural symmetry requirements on its Q-function. Using these coherent states we establish a partial order in the discrete phase space, which allows us to picture some n-qubit states as apparent distributions. We also analyze correlations in terms of sums of squared Q-functions.
We construct the systems of generalised coherent states for the discrete and continuous spectra of the hydrogen atom. These systems are expressed in elementary functions and are invariant under the $SO(3, 2)$ (discrete spectrum) and $SO(4, 1)$ (continuous spectrum) subgroups of the dynamical symmetry group $SO(4, 2)$ of the hydrogen atom. Both systems of coherent states are particular cases of the kernel of integral operator which interwines irreducible representations of the $SO(4, 2)$ group.
In a previous paper [{\it J. Phys. A: Math. Theor.} {\bf 40} (2007) 11105], we constructed a class of coherent states for a polynomially deformed $su(2)$ algebra. In this paper, we first prepare the discrete representations of the nonlinearly deformed $su(1,1)$ algebra. Then we extend the previous procedure to construct a discrete class of coherent states for a polynomial su(1,1) algebra which contains the Barut-Girardello set and the Perelomov set of the SU(1,1) coherent states as special cases. We also construct coherent states for the cubic algebra related to the conditionally solvable radi…
Using the example of the Jaynes-Cummings model, we present a comparison between time-delayed coherent feedback mediated by reservoirs with continuous and discrete mode structures and work out their qualitative differences. In contrast to the discrete-mode case, the continuous-mode case results in the well-known single-delay dynamics which can, e.g., stabilize Rabi oscillations. The discrete-mode case, however, shows population trapping, not present in the continuous-mode model. Given these differences, we discuss the cavity output spectra and show how these characteristic properties are spectr…
Recently, the coherent Ising machine (CIM) as a degenerate optical parametric oscillator (DOPO) network has been researched to solve Ising combinatorial optimization problems. We formulate a theoretical model for the CIM with discrete-time measurement feedback processes, and perform numerical simulations for the simplest network, composed of two degenerate optical parametric oscillator pulses with the anti-ferromagnetic mutual coupling. We evaluate the extent to which quantum coherence exists during the optimization process.
In this paper, the generalized coherent state for quantum systems with degenerate spectra is introduced. Then, the nonclassicality features and number-phase entropic uncertainty relation of two particular degenerate quantum systems are studied. Finally, using the Gazeau-Klauder coherent states approach, time evolution of some of the nonclassical properties of the coherent states corresponding to the considered physical systems are discussed.
Coherent state photon sources are widely used in quantum information processing. In many applications, such as quantum key distribution (QKD), a coherent state is functioned as a mixture of Fock states by assuming its phase is continuously randomized. In practice, such a crucial assumption is often not satisfied and, therefore, the security of existing QKD experiments is not guaranteed. To bridge this gap, we provide a rigorous security proof of QKD with discrete-phase-randomized coherent state sources. Our results show that the performance of the discrete-phase randomization case is close to …
The single-photon-added coherent state (SPACS), as an intermediate classical-to-purely-quantum state, was first realized recently by Zavatta \emph{$et al.$} (Science 306, 660 (2004)). We show here that the success probability of their SPACS generation can be enhanced by a simple method which leads to simultaneous creations of a discrete-variable entangled state and a SPACS or even a hybrid-variable entangled SPACS in two different channels. The impacts of the input thermal noise are also analyzed.
We analyze the behaviour of a coherent field driving a single mode optical cavity with one perfectly reflecting moving mirror and a partially reflecting fixed mirror, and show that this system's output exhibits optical bistability due to radiation pressure acting over the moving mirror.
We consider the phase space for a system of $n$ identical qudits (each one of dimension $d$, with $d$ a primer number) as a grid of $d^{n} \times d^{n}$ points and use the finite field $GF(d^{n})$ to label the corresponding axes. The associated displacement operators permit to define $s$-parametrized quasidistribution functions in this grid, with properties analogous to their continuous counterparts. These displacements allow also for the construction of finite coherent states, once a fiducial state is fixed. We take this reference as one eigenstate of the discrete Fourier transform and study …
The nonorthogonality of coherent states is a fundamental property which prevents them from being perfectly and deterministically discriminated. To circumvent this problem, we present an experimentally feasible protocol for the probabilistic orthogonalisation of a pair of coherent states, independent of their amplitude and phase. In contrast to unambiguous state discrimination, successful operation of our protocol is heralded without measuring the states, such that they remain suitable for further manipulation. As such, the resulting orthogonalised state may be used for further processing. Inde…
We show that in parametrically driven systems and, more generally, in systems in coherent states, off-resonant pumping can cause a transition from a continuum energy spectrum of the system to a discrete one, and result in quantum revivals of the initial state. The mechanism responsible for quantum revivals in the present case is different from that in the non-linear wavepacket dynamics of systems such as Rydberg atoms. We interpret the reported phenomena as an optical analog of Bloch oscillations realized in Fock space and propose a feasible scheme for inducing Bloch oscillations in trapped io…
Based on tensor network realizations of path integrals reducing exponential memory scaling to polynomial efficiency and a Liouville space implementation of a time-discrete quantum memory, we investigate a quantum system simultaneously exposed to two structured reservoirs. For this purpose, we employ a numerically exact quasi-2D tensor network combining both diagonal and off-diagonal system-reservoir interactions with a twofold memory for continuous and discrete retardation effects. As a possible example, we study the non-Markovian dynamical interplay between discrete photonic feedback and stru…
The problem of generating discrete superpositions of coherent states in the process of light propagation through a nonlinear Kerr medium, which is modelled by the anharmonic oscillator, is discussed. It is shown that under an appropriate choice of the length (time) of the medium the superpositions with both even and odd numbers of coherent states can appear. Analytical formulae for such superpositions with a few components are given explicitly. General rules governing the process of generating discrete superpositions of coherent states are also given. The maximum number of well distinguished s…
To reconstruct a mixed or pure quantum state of a spin s is possible through coherent states: its density matrix is fixed by the probabilities to measure the value s along 4s(s+1) appropriately chosen directions in space. Thus, after inverting the experimental data, the statistical operator is parametrized entirely by expectation values. On this basis, a symbolic calculus for quantum spins is developed, the `expectation-value representation.' It resembles the Moyal representation for SU(2) but two important differences exist. On the one hand, the symbols take values on a discrete set of points…
Adaptable Route to Fast Coherent State Transport via Bang-Bang-Bang Protocols
Institute of Atomic and Molecular Sciences, Academia Sinica（中央研究院原子与分子科学研究所）；Department of Physics and Center for Theoretical Physics, National Taiwan University（台湾大学物理系暨理论物理中心）；Center for Quantum Science and Engineering, National Taiwan University（台湾大学量子科学与工程中心）；Trapped-Ion Quantum Computing Laboratory, Hon Hai Research Institute（鸿海研究院 trapped-ion 量子计算实验室）；Physics Division, National Center for Theoretical Sciences（国家理论科学中心物理组）
Theoretical studies of coherent atom transport have as yet mainly been restricted to one-dimensional model systems with harmonic trapping potentials. Here we investigate this important phenomenon -- a prerequisite for a variety of quantum-technology applications based on cold neutral atoms -- under much more complex physical circumstances. More specificially yet, we study fast atomic transport in a moving {\em double-well optical lattice}, whose three-dimensional (anharmonic) potential is nonseparable in the $x-y$ plane. We first propose specific configurations of acousto-optic modulators that…
We study decoherence of the external degree of freedom of a tracer particle moving in a one dimensional dilute Boltzmann gas. We find that phase averaging is the dominant decoherence effect, rather than information exchange between tracer and gas particles. While a coherent superposition of two wave packets with different mean positions quickly turns into a mixed state, it is demonstrated that such superpositions of different momenta are robust to phase averaging, until the two wave packets acquire a different position due to the different velocity of each wave packet.
Role of quantum interference in the origin of higher-order nonclassical characteristics of radiation field has been probed vis-a-vis a discrete and a continuous superposition of coherent states. Specifically, the possibilities of observing higher-order nonclassical properties (e.g., higher-order antibunching (HOA), higher-order sub-Poissonian photon statistics (HOSPS), higher-order squeezing (HOS) of Hong-Mandel type and Hillery type) have been investigated using a shifted symmetric cat state that reduces to Yurke-Stoler, even and odd coherent states at various limits. This shifted symmetric c…
Discrete breathers (DBs) -- a spatial time-periodic localization of energy -- are predicted in a large variety of non-linear systems. Motivated by the conceptual bridging of the DBs phenomena in classical and quantum mechanical representation, we study their signatures in the dynamics of a quantum equivalent of a classical mechanical point in a phase space -- a coherent state. We show that in contrast to a point in a phase space that can exhibit either delocalized or localized motion, a coherent state can show signatures of a quantum equivalent of both localized and delocalized behavior. In cl…
We present a protocol for quantum key distribution using discrete modulation of coherent states of light. Information is encoded in the variable phase of coherent states which can be chosen from a regular discrete set ranging from binary to continuous modulation, similar to phase-shift-keying in classical communication. Information is decoded by simultaneous homodyne measurement of both quadratures and requires no active choice of basis. The protocol utilizes either direct or reverse reconciliation, both with and without postselection. We analyze the security of the protocol and show how to en…
Darboux transformation of both Barut-Girardello and Perelomov coherent states for the time-dependent singular oscillator is studied. In both cases the measure that realizes the resolution of the identity operator in terms of coherent states is found and corresponding holomorphic representation is constructed. For the particular case of a free particle moving with a fixed value of the angular momentum equal to two it is shown that Barut-Giriardello coherent states are more localized at the initial time moment while the Perelomov coherent states are more stable with respect to time evolution. It…
In this paper we study the quantum phase properties of {\it "nonlinear coherent states"} and {\it "solvable quantum systems with discrete spectra"} using the Pegg-Barnett formalism in a unified approach. The presented procedure will then be applied to few special solvable quantum systems with known discrete spectrum as well as to some new classes of nonlinear oscillators with particular nonlinearity functions. Finally the associated phase distributions and their nonclasscial properties such as the squeezing in number and phase operators have been investigated, numerically.
In this letter, the number-phase entropic uncertainty relation and the number-phase Wigner function of generalized coherent states associated to a few solvable quantum systems with nondegenerate spectra are studied. We also investigate time evolution of number-phase entropic uncertainty and Wigner function of the considered physical systems with the help of temporally stable Gazeau-Klauder coherent states.
We study the possibility of realizing perfect quantum state transfer in mesoscopic devices. We discuss the case of the Fano-Anderson model extended to two impurities. For a channel with an infinite number of degrees of freedom, we obtain coherent behavior in the case of strong coupling or in weak coupling off-resonance. For a finite number of degrees of freedom, coherent behavior is associated to weak coupling and resonance conditions.
We present an algorithmic approach to calculate the quantum-noise spectral density of photocurrents generated by optical fields with arbitrary discrete classical spectrum in coherent or squeezed states. The measurement scheme may include an arbitrary number of demodulations of the photocurrent. Thereby, our method is applicable to the general heterodyne detection scheme which is implemented in many experiments. For some of these experiments, e.g. in laser-interferometric gravitational-wave detectors, a reliable prediction of the quantum noise of fields in coherent and squeezed states plays a d…
