Effects of Spatial Noncommutativity on Energy Spectrum of a Trapped Bose-Einstein Condensate

In noncommutative space, we examine the problem of a noninteracting and harmonically trapped Bose- Einstein condensate, and derive a simple analytic expression for the effect of spatial noncommutatlvity on energy spectrum of the condensate, it indicates that the ground-state energy incorporating the spatial noncommutativity is reduced to a lower level, which depends upon the noncommutativity parameter 8. The gap between the noncommutative space and commutative one for the ground-state level of the condensate should be a signal of spatial noncommutativity.     

System of interacting harmonic oscillators in rotationally invariant noncommutative phase space

Rotationally invariant space with noncommutativity of coordinates and noncommutativity of momenta of canonical type is considered. A system of N interacting harmonic oscillators in uniform field and a system of N particles with harmonic oscillator interaction are studied. We analyze effect of noncommutativity on the energy levels of these systems. It is found that influence of coordinates noncommutativity on the energy levels of the systems increases with increasing of the number of particles. The spectrum of N free particles in uniform field in rotationally invariant noncommutative phase space is also analyzed. It is shown that the spectrum corresponds to the spectrum of a system of N harmonic oscillators with frequency determined by the parameter of momentum noncommutativity.     

Testing spatial noncommutativity via Rydberg atoms

The possibility of testing spatial noncommutativity via Rydberg atoms is explored. An atomic di-pole of a cold Rydberg atom is arranged in appropriate electric and magnetic fields, so that the motion of the dipole is constrained to be planar and rotationally symmetric. Spatial noncommutativity leads the canonical angular momentum to possess fractional values. In the limit of vanishing kinetic energy, the dominate value of the lowest canonical angular momentum takes variant Planck's over h/2. Furthermore, in the limit of eliminating the magnetic field, the dominate value of the lowest canonical angular momentum changes from variant Planck's over h/2 to variant Planck's over h/4. This result is a clear signal of spatial noncommutativity. An experimental verification of this prediction is suggested.     

Dynamics of Two-Level Trapped Ion in a Standing Wave Laser in Noncommutative Space

We study the dynamics of a two-level trapped ion in a standing wave electromagnetic field in two-dimensional （2D） noncommutative spaces in the Lamb-Dicke regime under the rotating wave approximation. We obtain the explicit analytical expressions for the energy spectra, energy eigenstates, unitary time evolution operator, atomic inversion, and phonon number operators. The Rabi oscillations, the collapse, and revivals in the average atomic inversion and the average phonon number are explicitly shown to contain the information of the parameter of the space noncommutativity, which sheds light on proposing new schemes based on the dynamics of trapped ion to test the noncommutativity.     

非对易相空间Dirac oscillator的研究

在极坐标系中研究了非对易相空间中的Dirac oscillator问题.研究显示:系统的波函数可以表示为合流超几何函数,而非对易相空间Dirac oscillator的量子行为类似于朗道问题.最后,对η=0和对易极限两种特殊情况进行了简单讨论.     

Landau problem in noncommutative quantum mechanics

The Landau problem in non-commutative quantum mechanics (NCQM) is studied. First by solving the Schrodinger equations on noncommutative (NC) space we obtain the Landau energy levels and the energy correction that is caused by space-space noncommutativity. Then we discuss the noncommutative phase space case, namely, space-space and momentum-momentum non-commutative case, and we get the explicit expression of the Hamiltonian as well as the corresponding eigenfunctions and eigenvalues.     

非对易时空下Gibbons-Maeda dilaton黑洞和Garfinkle-Horowitz-Strominger dilaton黑洞的热力学性质

以Gibbons-Maeda dilaton黑洞和Garfinkle-Horowitz-Strominger dilaton黑洞为例，研究空间的非对易性对黑洞热力学性质的影响.通过对比对易时空中Gibbons-Maeda dilaton黑洞和非对易时空中Garfinkle-Horowitz-Strominger dilaton黑洞的温度，得出如下结论：从对黑洞热力学性质产生影响这一角度来说，时空的非对易性和黑洞的荷(电荷或磁荷)有相似的作用.     

Influence of Noncommutativity on the Motion of Sun-Earth-Moon System and the Weak Equivalence Principle

Features of motion of macroscopic body in gravitational field in a space with noncommutativity of coordinates and noncommutativity of momenta are considered in general case when coordinates and momenta of different particles satisfy noncommutative algebra with different parameters of noncommutativity. Influence of noncommutativity on the motion of three-body Sun-Earth-Moon system is examined. We show that because of noncommutativity the free fall accelerations of the Moon and the Earth toward the Sun in the case when the Moon and the Earth are at the same distance to the source of gravity are not the same even if gravitational and inertial masses of the bodies are equal. Therefore, the Eotvos-parameter is not equal to zero and the weak equivalence principle is violated in noncommutative phase space. We estimate the corrections to the Eotvos-parameter caused by noncommutativity on the basis of Lunar laser ranging experiment results. We obtain that with high precision the ratio of parameter of momentum noncommutativity to mass is the same for different particles.     

Hydrogen atom spectrum and the lamb shift in noncommutative QED

We have calculated the energy levels of the hydrogen atom as well as the Lamb shift within the noncommutative quantum electrodynamics theory. The results show deviations from the usual QED both on the classical and the quantum levels. On both levels, the deviations depend on the parameter of space/space noncommutativity.     

Landau problem in noncommutative quantum mechanics

The Landau problem in non-commutative quantum mechanics (NCQM) is studied.First by solving the Schr(o)dinger equations on noncommutative (NC) space we obtain the Landau energy levels and the energy correction that is caused by space-space noncommutativity.Then we discuss the noncommutative phase space case,namely,space-space and momentum-momentum non-commutative case,and we get the explicit expression of the Hamfltonian as well as the corresponding eigenfunctions and eigenvalues.     

Hydrogen atom in rotationally invariant noncommutative space

We consider the noncommutative algebra which is rotationally invariant. The hydrogen atom is studied in a rotationally invariant noncommutative space. We find the corrections to the energy levels of the hydrogen atom up to the second order in the parameter of noncommutativity. The upper bound of the parameter of noncommutativity is estimated on the basis of the experimental results for 1s–2s$1s\text{–}2s$ transition frequency.     

非对易空间中耦合谐振子的能级分裂

非对易空间效应的出现引起了物理学界的广泛兴趣。 介绍了非对易空间中量子力学的代数关系,在所考虑的空间变量的对易关系中包含了坐标 坐标的非对易性, 并且把 Moyal-Weyl 乘法在非对易空间中通过一个Bopp变换转变成普通的乘法。 然后给出了非对易空间中耦合谐振子的能级分裂情况。 The effect of noncommutativity of space have caused the physical academic circles widespread interest. In this paper, the non commutative (NC) is introduced, which contain non commutative of coordinate coordinate, and find that the Moyal Weyl product in NC space can be replaced with a Bopp shift. Then, the energy splitting of the coupling harmonic oscillator in non commutative spaces are discussed.     

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.     

Nonperturbative deformation of D-brane states by the world sheet noncommutativity

By introducing the noncommutativity in the world sheet, we discuss a modification of the D-brane states in the closed string theory. In particular we show how the world sheet noncommutativity induces a nonperturbative effect to the D-brane states.     

On the deformed squeezed states on noncommutative plane

A new kind of deformed boson operators is proposed to be consistent with the large noncommutativity parameters on noncommutative plane when noncommutativity of momentum spaces is considered. Using this kind of deformed boson operators, the coherent states and squeezed states are constructed, and their properties are discussed in detail.     

Higher order theories and their relationship with noncommutativity

We present a relationship between noncommutativity and higher order time derivative theories using a perturbation method. We make a generalization of the Chern–Simons quantum mechanics for higher order time derivatives. This model presents noncommutativity in a natural way when we project to low-energy physical states without the necessity of taking the strong field limit. We quantize the theory using a Bopp's shift of the noncommutative variables and we obtain a spectrum without negative energies, under the perturbation limits. In addition, we extent the model to high order time derivatives and noncommutativity with variable dependent parameter.     

Parikh-Wilczek Tunneling as Massive Particles from Noncommutative Schwarzschild Black Hole

In this paper, we apply the tunneling of massive particle through the quantum horizon of a Schwarzschild black hole in noncommutative spacetime. The tunneling effects lead to modified Hawking radiation due to inclusion of back-reaction effects. Our calculations show also that noncommutativity effects cause the further modifications to the hermodynamical relations in black hole. We calculate the emission rate of the massive particles' tunneling from aSchwarzschild black hole which is modified on account of noncommutativity influences. The issues of information loss and possible correlations between emitted particles are discussed. Unfortunately even by considering noncommutativity view point, there is no correlation between different modes of evaporation at least at late-time. Nevertheless, as a result of spacetime noncommutativity, information may be conserved by a stable black hole remnant.     

Twist-Deformed Gravitational Quantum Well

In this article we define the gravitational quantum well model on twist-deformed space with two spatial directions commuting to time-dependent function f_κ(t). Further, we find the corresponding energy spectrum and by its comparision with the GRANIT experiment predictions, we obtain bounds on the noncommutativity function in the case of two first energy levels.

     

Accelerating universe in two-dimensional noncommutative dilaton cosmology

We show that the phase transition from the decelerating universe to the accelerating universe, which is of relevance to the cosmological coincidence problem, is possible in the semiclassically quantized two-dimensional dilaton gravity by taking into account the noncommutative field variables during the finite time. Initially, the quantum-mechanically induced energy from the noncommutativity among the fields makes the early universe decelerate and subsequently the universe is accelerating because the dilaton driven cosmology becomes dominant later.