Quantum Dynamics of a Single Trapped Ion Interacting with Standing Laser Pulses

A classically chaotic system consisting of a Paul trapped ion and a sequences of standing laser pulses is treated quantum-mechanically. Under the circumstance of time-dependence, we derive the transition probability from the ion‘s motional state n to n‘, and find, in the first-order approximation, the classically chaotic character disappears.Theoretical analysis and numerical calculations show that by regulating the phase parameter φ we can control the transition probability. When φ reaches some specific values, the transition from the state n to n‘ is forbidden and, for some laser periods, resonance occurs, which leads to the corresponding transitions between different motional states.     

Quantum Dynamics of a Single Trapped Ion Interacting with Standing Laser Pulses

A classically chaotic system consisting of a Paul trapped ion and a sequences of standing laser pulses is treated quantum-mechanically. Under the circumstance of time-dependence, we derive the transition probability from the ion‘s motional state n to n‘, and find, in the first-order approximation, the classically chaotic character disappears.Theoretical analysis and numerical calculations show that by regulating the phase parameter Ф we can control thetransition probability. When Ф reaches some specific values, the transition from the state n to n‘ is forbidden and, for some laser periods, resonance occurs, which leads to the corresponding transitions between different motional states.The time-evolution of an initial motional state |ψz) just over one period is also studied in detail.     

Continued Fraction Approach to the Solution of a Trapped Ultracold Ion Interacting with a Standing Wave Laser without Rotating Wave Approximation

In coherent state representation and in the absence of rotating wave approximation, we present an approach to solve the interaction of a trapped ultracold ion with a standing wave laser by a continued fraction, within the Lamb-Dicke limit and the low excitation regime. It is shown that, the eigen-energies as well as the coefficients of the series of the eigenfunction can be obtained rapidly from the continued fraction.     

Analytical Solution for an Ultracold Trapped Ion at the Node of the Standing Wave Laser Without Rotating Wave Approximation

A method of solving an ultracold trapped ion at the node of the standing wave laser without rotating wave approximation is proposed and the analytical forms of the eigenfunctions and eigenenergies of the system are presented.     

Analytical Solution for an Ultracold Trapped Ion at the Node of the Standing Wave Laser Without Rotating Wave Approximation

A method of solving an ultracold trapped ion at the node of the standing wave laser without rotating wave approximation is proposed and the analytical forms of the eigenfunctions and eigenenergies of the system are presented.     

行波激光场中囚禁离子的布居反转

研究了以波激光场中单个囚禁离子布居反转的动力学特性，分别讨论了失谐参数、质心振动声子场的初始值以及Lamb-Dicke参数对离子布居反转的崩塌和回复现象的影响。     

Two-Level Atom in a Standing, Electromagnetic Wave

We study a two-level atom interacting with a standing electromagnetic wave, and work out the wave functions, the energy values and the momentum values of the atom.     

非对易相空间中角动量的分裂

非对易空间效应是一种在弦尺度下出现的物理效应. 本文首先介绍了在Schwinger表象中角动量的3个分量用产生--消灭算符的表示形式, 接着讨论了非对易相空间的量子力学代数; 然后用对易空间谐振子的产生-消灭算符表示出了在非对易情况下的角动量; 最后讨论了非对易相空间中角动量的分裂.     

Gauge Theory on a Discrete Noncommutative Space

In this paper, we apply Connes' noncommutative geometry and the Seiberg—Witten map to a discrete noncommutative space consisting of n copies of a given noncommutative space R ^m . The explicit action functional of gauge fields on this discrete noncommutative space is obtained.     

Relativistic Oscillators in a Noncommutative Space and in a Magnetic Field

In this work, we study the relativistic oscillators in a noncommutative space and in a magnetic field. It is shown that the effect of the magnetic field may compete withthat of the noncommutative space and that is able to vanish the effect of the noncommutative space.     

State-Vector Space and Canonical Coherent States in Noncommutative Plane

The structure of the state-vector space of identical bosons in noncommutative spaces is investigated. To maintain Bose-Einstein statistics the commutation relations of phase space variables should simultaneously include coordinate-coordinate non-commutativity and momentum-momentum non-commutativity, which leads to a kind of deformed Heisenberg-Weyl algebra. Although there is no ordinary number representation in this state-vector space, several set of orthogonal and complete state-vectors can be derived which are common eigenvectors of corresponding pairs of commuting Hermitian operators. As a simple application of this state-vector space, an explicit form of two-dimensional canonical coherent state is constructed and its properties are discussed.     

DKP Oscillator in a Noncommutative Space

We present the DKP oscillator model of spins 0 and 1, in a noncommutative space. In the case of spin 0, the equation is reduced to Klein Gordon oscillator type, the wave functions are then deduced and compared with the DKP spinless particle subjected to the interaction of a constant magnetic field. For the case of spin 1, the problem is equivalent with the behavior of the DKP equation of spin 1 in a commutative space describing the movement of a vectorial boson subjected to the action of a constant magnetic field with additional correction which depends on the noncommutativity parameter.     

Series Solution of a Trapped Ultracold Ion in a Raman—Type Configuration

The Raman interaction of a trapped ultracold ion with two traveling wave lasers is studied analytically by series expansion technique without the need of rotating wave approximation and the limitations of both the Lamb-Dicke limit and the weak excitation regime.As an example,a scheme for the preparation of Schrodinger-cat states in such a process is proposed beyond the weak excitation regime.     

DKP Oscillator in a Noncommutative Space

We present the DKP oscillator model of spins 0 and 1, in a noncommutative space. In the case of spin 0, the equation is reduced to Klein-Gordon oscillator type, the wave functions are then deduced and compared with the DKP spinless particle subjected to the interaction of a constant magnetic field. For the case of spin 1, the problem is equivalent with the behavior of the DKP equation of spin 1 in a commutative space describing the movement of a vectorial boson subjected to the action of a constant magnetic field with additional correction which depends on the noncommutativity parameter.     

Quantum Signatures of Chaos in Adiabatic Interaction Between a Trapped Ion and a Laser Standing Wave

We study quantum motion around a classical heteroclinic point of a single trapped ion interacting with a strong laser standing wave. We construct a set of exact coherent states of the quantum system and from the exact solutions reveal that quantum signatures of chaos can be induced by the adiabatic interaction between the trapped ion and the laser standing wave, where the quantum expectation values of position and momentum correspond to the classically chaotic orbit. The chaotic region on the phase space is illustrated. The energy crossing and quantum resonance in time evolution and the exponentially increased Heisenberg uncertainty are found. The results suggest a theoretical scheme for controlling the unstable regular and chaotic motions.     

Analytical Solutions of a Trapped Ultracold Ion Experiencing a Travelling Wave Laser without Rotating Wave Approximation

In coherent state representation and in the absence of rotating wave approximation,the analytical forms of the eigenenergies and eigenfunctions of a trapped ultracold ion interactingwith a travelling wave laser are presented for some particular detunings. The differencebetween the standing wave laser and the travelling one radiating a trapped ultracold ion isalso pointed out.     

Quantum Field Theory with a Minimal Length Induced from Noncommutative Space

From the inspection of noncommutative quantum mechanics, we obtain an approximate equivalent relation for the energy dependence of the Planck constant in the noncommutative space, which means a minimal length of the space. We find that this relation is reasonable and it can inherit the main properties of the noncommutative space. Based on this relation, we derive the modified Klein-Gordon equation and Dirac equation. We investigate the scalar field and φ4 model and then quantum electrodynamics in our theory, and derive the corresponding Feynman rules. These results may be considered as reasonable approximations to those of noncommutative quantum field theory. Our theory also shows a connection between the space with a minimal length and the noncommutative space.     

On Quantum Mechanics on Noncommutative Quantum Phase Space

In this work, we develop a general framework in which Noncommutative Quantum Mechanics (NCQM), characterized by a space noncommutativity matrix parameter θ=ε_ij^k θ_k and a momentum noncommutativity matrix parameter β=ε_ij^k β_k, is shown to be equivalent to Quantum Mechanics (QM) on a suitable transformed Quantum Phase Space (QPS). Imposing some constraints on this particular transformation, we firstly find that the product of the two parameters θ and β possesses a lower bound in direct relation with Heisenberg incertitude relations, and secondly that the two parameters are equivalent but with opposite sign, up to a dimension factor depending on the physical system under study. This means that noncommutativity is represented by a unique parameter which may play the role of a fundamental constant characterizing the whole NCQPS. Within our framework, we treat some physical systems on NCQPS : free particle, harmonic oscillator, system of two-charged particles, Hydrogen atom. Among the obtained results, we discover a new phenomenon which consists of a free particle on NCQPS viewed as equivalent to a harmonic oscillator with Larmor frequency depending on β, representing the same particle in presence of a magnetic field $\vec{B}=q^{-1}\vec{\beta}$. For the other examples, additional correction terms depending on β appear in the expression of the energy spectrum. Finally, in the two-particle system case, we emphasize the fact that for two opposite charges noncommutativity is effectively feeled with opposite sign.     

Scheme for Direct Measurement of Wigner Characteristic Function for Motion of a Single Trapped Ion

In this paper, a procedure is proposed for measuring Wigner characteristic function for motion of a single trapped ion. In the scheme, the trapped ion is driven by two laser beams with different frequencies tuned to the first lower and upper vibrational sidebands, respectively. Unlike other previous scheme, our scheme does not require the displacement of the external state before the ion-laser interaction and thus our scheme night be more simple and feasible. The scheme can also be used to prepare the motional Schroedinger cat states and entangled coherent states.