Tunneling of an energy eigenstate through a parabolic barrier viewed from Wigner phase space

We analyze the tunneling of a particle through a repulsive potential resulting from an inverted harmonic oscillator in the quantum mechanical phase space described by the Wigner function. In particular, we solve the partial differential equations in phase space determining the Wigner function of an energy eigenstate of the inverted oscillator. The reflection or transmission coefficients R or T are then given by the total weight of all classical phase-space trajectories corresponding to energies below, or above the top of the barrier given by the Wigner function.     

On Virtual Phonons,Photons, and Electrons

A macroscopic realization of the peculiar virtual particles is presented. The classical Helmholtz and the Schrödinger equations are differential equations of the same mathematical structure. The solutions with an imaginary wave number are called evanescent modes in the case of elastic and electromagnetic fields. In the case of non-relativistic quantum mechanical fields they are called tunneling solutions. The imaginary wave numbers point to strange consequences: The waves are non-local, they are not observable, and they are described as virtual particles. During the last two decades QED calculations of the solutions with an imaginary wave number have been experimentally confirmed for phonons, photons, and electrons. The experimental proofs of the predictions of non-relativistic quantum mechanics and the Wigner phase time approach for the elastic, electromagnetic and Schrödinger fields will be presented in this article. The results are zero time in the barrier and an interaction time (i.e. a phase shift) at the barrier interfaces. The measured tunneling time scales approximately inversely with the particle energy. Actually, the tunneling time is given only by the barrier boundary interaction time, as zero time is spent inside a barrier.     

The quantum state of the universe from deformation quantization and classical-quantum correlation

In this paper we study the quantum cosmology of homogeneous and isotropic cosmology, via the Weyl–Wigner–Groenewold–Moyal formalism of phase space quantization, with perfect fluid as a matter source. The corresponding quantum cosmology is described by the Moyal–Wheeler-DeWitt equation which has exact solutions in Moyal phase space, resulting in Wigner quasiprobability distribution functions peaking around the classical paths for large values of scale factor. We show that the Wigner functions of these models are peaked around the non-singular universes with quantum modified density parameter of radiation.     

Quantum coherent versus classical coherent light

The paper shows that the Wigner distribution function of quantum optical coherent states, or of a superposition of such states, can be produced and measured with a classical optical set-up using classical coherent light fields. This measurement cannot be done directly in quantum optics since the quantum phase space variables correspond to non-commuting operators. As an example, the Wigner distribution function of Schrödinger cat states of light has been measured. It is also shown that the possibility of measuring the Wigner distribution function of quantum coherent states with classical coherent fields is unique in the sense that it cannot be extended to other quantum states, not even to the incoherent limit of the superposition of coherent states.     

Non-equilibrium approach to transport phenomena in quantum crystals

A pure dielectric quantum crystal subjected to an external mechanical force is described by non-equilibrium Green’s functions. In equilibrium the leading approximation leads to the definition of elementary excitations, the phonons in the renormalized harmonic approximation. Their temperature dependent energies are to be determined as solutions of an integral equation. For hydrodynamic disturbances a generalized transport equation for a phonon number density is derived. A similar approximation for the spectral function yields an integral equation for space and time dependent quasiparticle energies which are expressed as functionals of the displacement field and the phonon distribution. The Boltzmann equation for the latter includes the quasi-particle interaction.     

The classical limit for quantum mechanical correlation functions

For quantum systems of finitely many particles as well as for boson quantum field theories, the classical limit of the expectation values of products of Weyl operators, translated in time by the quantum mechanical Hamiltonian and taken in coherent states centered inx- andp-space around? ^?1/2 (coordinates of a point in classical phase space) are shown to become the exponentials of coordinate functions of the classical orbit in phase space. In the same sense,? ^?1/2 [(quantum operator) (t) — (classical function) (t)] converges to the solution of the linear quantum mechanical system, which is obtained by linearizing the non-linear Heisenberg equations of motion around the classical orbit.     

Weyl Ordering Symbol Method for Studying Wigner Function of the Damping Field

The symbolic method (including normal ordering. antinormal ordering and Weyl ordering symbol) is usually utilized to tackle miscellaneous operators which have different commutative relations. Considering the Weyl ordering symbol’s remarkable properties, we have efficiently and conveniently derived the Wigner distribution function for field damping in a squeezed bath and a vacuum bath respectively, and then examined the decoherence processes from the plots of Wigner function and its contour in quantum phase space. Alternatively, we can employ a general Wigner operator under phase space transform to calculate distribution function and discuss the damping process.     

声子对隧穿量子点分子辐射场系统量子相位的影响

用全量子理论导出隧穿量子点分子-辐射场相互作用系统状态满足的微分方程组, 在相干态辐射场和量子点分子处于隧穿激发态及基态的初始条件下, 应用Pegg-Barnett相位理论计算和分析了辐射场的相位概率分布及相位涨落, 研究了声子-量子点分子作用对辐射场相位的影响, 并与Husimi相位分布做了比较. 结果表明, 温度显著影响光场相位概率分布的时间演化规律, 声子既可以抑制也可以增强辐射场相位扩散和涨落, 取决于量子点分子的初态. Husimi相位分布和Pegg-Barnett相位分布符合度相当高. 关键词： 量子点分子 声子 量子相位 Q函数')" href="#">Q函数     

The Quantum-Classical Comparison of the Arrival-Time Distribution Through the Probability Current

We consider the arrival time distribution defined through the quantum probability current for a Gaussian wave packet representing free particles in quantum mechanics in order to explore the issue of the classical limit of arrival time. We formulate the classical analogue of the arrival time distribution for an ensemble of free particles represented by a phase space distribution function evolving under the classical Liouville's equation. The classical probability current so constructed matches with the quantum probability current in the limit of minimum uncertainty. Further, it is possible to show in general that smooth transitions from the quantum mechanical probability current and the mean arrival time to their respective classical values are obtained in the limit of large mass of the particles.     

A Wigner-function formulation of finite-state quantum mechanics

For a non-relativistic system with only continous degrees of freedom (no spin, for example), the original Wigner function can be used as an alternative to the density matrix to represent an arbitrary quantum state. Indeed, the quantum mechanics of such systems can be formulated entirely in terms of the Wigner function and other functions on phase space, with no mention of state vectors or operators. In the present paper this Wigner-function formulation is extended to systems having only a finite number of orthogonal states. The “phase space” for such a system is taken to be not continuous but discrete. In the simplest cases it can be pictured as an N×N array of points, where N is the number of orthogonal states. The Wigner function is a real function on this phase space, defined so that its properties are closely analogous to those of the original Wigner function. In this formulation, observables, like states, are represented by real functions on the discrete phase space. The complex numbers still play an important role: they appear in an essential way in the rule for forming composite systems.     

一维受迫谐振子量子运动的经典特性

对经典一维受迫谐振子量子化,求解量子化后体系的时间演化算符.应用相空间准概率分布函数,研究了体系的量子特性.研究结果表明,初始为真空态,经过时间演化,系统波函数是一个二维高斯波包;波包中心的振幅和相位受到作用力的调制,成为调幅、调相波,波包中心的运动与经典受迫谐振子的运动形式相同.     

三维谐振子Wigner函数的星乘解

Wigner函数作为相空间中的一个准概率分布函数,也是密度矩阵的特殊表示形式,具有十分重要的物理意义。首先介绍了Wigner函数的性质及其计算方法,然后利用星本征方程(Moyal方程)计算了三维谐振子的Wigner函数。最后讨论了在相空间中描述声子与电子(或光子)相互作用的方法,并得到了跃迁几率在相空间中所满足的方程。     

Quantum Dynamical Behavior of the Morse Oscillator: the Wigner Function Approach

We explore the quantum dynamical behavior of the Morse oscillator in the phase space using the Wigner function. For an initial wave packet excited with Gaussian probability distribution, we calculate the associated Wigner function and compute its time evolution. By calculating the marginal probabilities, we study the formation of quantum carpets both in the position space and in the momentum space. In addition, in view of these probabilities, we present the time evolution of the position and momentum expectation values. The structure of quantum carpets and the time-evolved expectation values mimic the emergence of quantum revivals and fractional revivals.     

量子扩散通道中Wigner算符的演化规律

众所周知,量子态的演化可用与其相应的Wigner函数演化来代替.因为量子态的Wigner函数和量子态的密度矩阵一样,都包含了概率分布和相位等信息,因此对量子态的Wigner函数进行研究,可以更加快速有效地获取量子态在演化过程的重要信息.本文从经典扩散方程出发,利用密度算符的P表示,导出了量子态密度算符的扩散方程.进一步通过引入量子算符的Weyl编序记号,给出了其对应的Weyl量子化方案.另外,借助于密度算符的另一相空间表示-Wigner函数,建立了Wigner算符在扩散通道中演化方程,并给出了其Wigner算符解的形式.本文推导出了Wigner算符在量子扩散通道中的演化规律,即演化过程中任意时刻Wigner算符的形式.在此结论的基础上,讨论了相干态经过量子扩散通道的演化情况.     

Quantum distributions for localized quantum states

The principle of ergodicity of the quantum theory has been used for elaboration of a new technique for numerical simulation of the Wigner function of open dissipative quantum systems. With this purpose the density matrix of a quantum system is represented via averaging over the ensemble of quantum states in time intervals instead of averaging over the ensemble of stochastic variables. It is shown that this approach leads to new approximate expressions for quantum distributions in the phase space, in particular, Wigner functions for systems localized in the region of classical phase trajectories. As an application, the Wigner functions are calculated for the process of intracavity second harmonic generation in the region of Hopf bifurcations.     

Decoherence effects in Bose–Einstein condensate interferometry I. General theory

The present paper outlines a basic theoretical treatment of decoherence and dephasing effects in interferometry based on single component Bose–Einstein condensates in double potential wells, where two condensate modes may be involved. Results for both two mode condensates and the simpler single mode condensate case are presented. The approach involves a hybrid phase space distribution functional method where the condensate modes are described via a truncated Wigner representation, whilst the basically unoccupied non-condensate modes are described via a positive P representation. The Hamiltonian for the system is described in terms of quantum field operators for the condensate and non-condensate modes. The functional Fokker–Planck equation for the double phase space distribution functional is derived. Equivalent Ito stochastic equations for the condensate and non-condensate fields that replace the field operators are obtained, and stochastic averages of products of these fields give the quantum correlation functions that can be used to interpret interferometry experiments. The stochastic field equations are the sum of a deterministic term obtained from the drift vector in the functional Fokker–Planck equation, and a noise field whose stochastic properties are determined from the diffusion matrix in the functional Fokker–Planck equation. The stochastic properties of the noise field terms are similar to those for Gaussian–Markov processes in that the stochastic averages of odd numbers of noise fields are zero and those for even numbers of noise field terms are the sums of products of stochastic averages associated with pairs of noise fields. However each pair is represented by an element of the diffusion matrix rather than products of the noise fields themselves, as in the case of Gaussian–Markov processes. The treatment starts from a generalised mean field theory for two condensate modes, where generalised coupled Gross–Pitaevskii equations are obtained for the modes and matrix mechanics equations are derived for the amplitudes describing possible fragmentations of the condensate between the two modes. These self-consistent sets of equations are derived via the Dirac–Frenkel variational principle. Numerical studies for interferometry experiments would involve using the solutions from the generalised mean field theory in calculations for the stochastic fields from the Ito stochastic field equations.     

EPR correlations and EPW distributions revisited

It is shown that Bell's proof of the violation of local realism in phase space is incorrect. Bell's experiment is based upon position measurements of free particles. A violation can be derived even for a nonnegative Wigner distribution, which in this case acts as a local classical model.     

Quantum damped harmonic oscillator on non-commuting plane

In the present paper we study the quantum damped harmonic oscillator on non-commuting two-dimensional space. We calculate the time evolution operator and we find the exact propagator of the system. We investigate as well the thermodynamic properties of the system using the standard canonical density matrix. We find the statistical distribution function and the partition function. We calculate the specific heat for the limiting case of critical damping, where the frequencies of the system vanish. Finally we study the state of the system when the phase space of the second dimension becomes classical. We find that these systems have some singularities and zeros for low temperatures.