Quantum Time Scales Research Articles

Certain Novel Dynamic Inequalities Applicable in the Theory of Retarded Dynamic Equations and Their Applications

In this article, we establish certain time-scale-retarded dynamic inequalities that contain nonlinear retarded integral equations on various time scales. These inequalities extend and generalize some significant inequalities existing in the literature to their more general forms. The qualitative and quantitative characteristics of solutions to various dynamic equations on time scales involving retarded integrals can be studied using these inequalities. The results presented in this manuscript furnish a powerful tool to analyze the boundedness of nonlinear integral equations with retarded integrals on several time scales. In the end, we also include numerical illustrations to signify the applicability of these results to power nonlinear retarded integral equations on real and quantum time scales.

Higher order differences on arbitrary discrete time scales and related generating functions

In this paper we uncover some fundamental relationships involving the weights one needs to calculate the n-th difference of a function on an arbitrary discrete time scale. One of several interesting results we obtain is a formula that allows one to calculate the value of any desired finite difference coefficient directly from the graininess function for the time scale under consideration. This foundational work beautifully combines analysis, algebra, and combinatorics to obtain this and other interesting results. Some of the other interesting results include (i) for a fixed n, the n-th finite difference coefficients sum to 0 and (ii) there are nice and useful generating functions that encode various sequences of finite difference coefficients. Throughout we show the results coincide with well-known results in the special cases where the time scales are either quantum time scales or time scales with constant graininess.

Quantum ammeter: Measuring full counting statistics of electron currents at quantum timescales

We present the theoretical model of the "quantum ammeter", a device that is able to measure the full counting statistics of an electron current at quantum time scales. It consists of an Ohmic contact, perfectly coupled to chiral quantum Hall channels, and of a quantum dot attached to one of the outgoing channels. At energies small compared to its charging energy, the Ohmic contact fractionalizes each incoming electron and redistributes it between outgoing channels. By monitoring the resonant tunneling current through the quantum dot, one gets an access to the moment generator of the current in one of the incoming channels at time scales comparable to its correlation time.

Almost Periodic Functions on the Quantum Time Scale and Applications

In this paper, we first propose two types of concepts of almost periodic functions on the quantum time scale. Secondly, we study some basic properties of almost periodic functions on the quantum time scale. Thirdly, based on these, we study the existence and uniqueness of almost periodic solutions of dynamic equations on the quantum time scale by Lyapunov method. Then, we give an equivalent definition of almost periodic functions on the quantum time scale. Finally, as an application, we propose a class of high-order Hopfield neural networks on the quantum time scale and establish the existence and global exponential stability of almost periodic solutions of this class of neural networks.

New Ostrowski and Ostrowski-Gruss type inequalities for double integrals on time scales involving a combination of $\Delta$-integral means

In 2014, some Ostrowski type inequalities for functions of a single variable were obtained in [Y. Jiang, H. R\"uzgar, W. J. Liu and A. Tuna: Some new generalizations of Ostrowski type inequalities on time scales involving combination of $\Delta$-integral means, J. Nonlinear Sci. Appl., {\bf{7}} (2014), 311--324]. In this paper, we extend some of the inequalities obtained in the above paper for double integrals. One of our results generalizes a result in the article [W. J. Liu, Q. A. Ng\^o and W. Chen: On new Ostrowski type inequalities for double integrals on time scales, Dyn. Syst. Appl., {\bf 19} (2010), 189--198]. We also apply our results to the continuous, discrete and quantum time scales to obtain some interesting inequalities.

Schwarzian derivative and Ermakov equation on a time scale

We introduce the Schwarzian, simplified Schwarzian, and Schwarzian Korteweg-de Vries equations on a time scale that are invariant under the fractional linear transformations. As an application, we derive their solutions and establish the invariant disconjugacy condition for second order dynamic equations on a time scale. Furthermore, we consider the Ermakov dynamic equation and the Ermakov-Lewis adiabatic invariant on a time scale. We discuss specific examples of discrete, quantum, and continuous time scales to compare our equations with the well-known ones. We also derive the linearization of the Ermakov equation and the corresponding Painleve equation on a time scale.

New time scale generalizations of the Ostrowski-Gr\xfcss type inequality for k points

Two Ostrowski-Grüss type inequalities for k points with a parameter lambdain[0, 1] are hereby presented. The first generalizes a recent result due to Nwaeze and Tameru, and the second extends the result of Liu et al. to k points. Many new interesting inequalities can be derived as special cases of our results by considering different values of λ and kinmathbb{N}. In addition, we apply our results to the continuous, discrete, and quantum time scales to obtain several novel inequalities in this direction.

Almost Automorphic Functions on the Quantum Time Scale and Applications

We first propose two types of concepts of almost automorphic functions on the quantum time scale. Secondly, we study some basic properties of almost automorphic functions on the quantum time scale. Then, we introduce a transformation between functions defined on the quantum time scale and functions defined on the set of generalized integer numbers; by using this transformation we give equivalent definitions of almost automorphic functions on the quantum time scale; following the idea of the transformation, we also give a concept of almost automorphic functions on more general time scales that can unify the concepts of almost automorphic functions on almost periodic time scales and on the quantum time scale. Finally, as an application of our results, we establish the existence of almost automorphic solutions of linear and semilinear dynamic equations on the quantum time scale.

Itô’s Formula, the Stochastic Exponential, and Change of Measure on General Time Scales

We provide an Itô formula for stochastic dynamical equation on general time scales. Based on this Itô’s formula we give a closed-form expression for stochastic exponential on general time scales. We then demonstrate Girsanov’s change of measure formula in the case of general time scales. Our result is being applied to a Brownian motion on the quantum time scale (q-time scale).

Photo-induced 1,3-cyclohexadiene ring opening reaction: Ab initio on-the-fly nonadiabatic molecular dynamics simulation

Abstract By a newly developed algorithm to compute global nonadiabatic switching probability only using electronic adiabatic potential energy surfaces and its gradients, we could able to perform on-the-fly trajectory surface hopping molecular dynamics at the SA-CASSCF(14, 8)/6-31G* quantum level to investigate the photo-induced 1,3-cyclohexadiene ring opening reaction from 1,3-cyclohexadiene (CHD) to 1,3,5-hexatriene (HT). We found four conical intersection zones within three singlet low-lying electronic states, in which 1B/2A and 1A/2A are well known to be the most important conical intersection zones to govern dynamics of ring opening reaction. The present simulation gained insight into nonadiabatic dynamics in which 516 sampling trajectories out of 600 can switch back and forth between two involved potential energy surfaces not only at 1B/2A, but also at 1A/2A. This insight is the key to influence quantum yields and time scales of the ring opening reaction. We estimated time constants as 46 fs, 82 fs, and 120 fs, respectively for crossing 1B/2A and 1A/2A, and excited-state lifetime in comparison with corresponding experimental values 56 fs, 80 fs and 130 fs. We estimated quantum yield 0.47/0.53 to HT/CHD in comparison to experiment values 0.30/0.70 in the gas-phase and 0.41/0.59 in the condense-phase. Moreover, the present simulation within 600 sampling trajectories indicates that there is no five membered ring forming in agreement with experiment results and with prediction from the previous wave-packet simulation as a rare case.

Investigation of onboard quantum time scale for orbital flight of a space radio telescope (the RadioAstron project)

Results of observing the operation of instrumentation of the ground-space complex of the Radio-Astron project during space flight conditions of the radio observatory are presented. The technology of quality evaluation of the data received from the space radio telescope (SRT) is considered. The dependence of readings of the onboard frame counter on SRT radial velocity and distance is determined. Technology of constructing a model of the ground-space atomic clocks and onboard quantum time scale based on the results of radio astronomic observations is tested. The method of measurement of the coherent cumulative navigation delay using the onboard quantum time scale is considered. The results of observation of the effect of relativistic and kinematic time dilation onboard the SRT are presented.

Landau damping and the onset of particle trapping in quantum plasmas

Using analytical theory and simulations, we assess the impact of quantum effects on non-linear wave-particle interactions in quantum plasmas. We more specifically focus on the resonant interaction between Langmuir waves and electrons, which, in classical plasmas, lead to particle trapping. Two regimes are identified depending on the difference between the time scale of oscillation tB(k)=m/eEk of a trapped electron and the quantum time scale tq(k)=2m/ℏk2 related to recoil effect, where E and k are the wave amplitude and wave vector. In the classical-like regime, tB(k) < tq(k), resonant electrons are trapped in the wave troughs and greatly affect the evolution of the system long before the wave has had time to Landau damp by a large amount according to linear theory. In the quantum regime, tB(k) > tq(k), particle trapping is hampered by the finite recoil imparted to resonant electrons in their interactions with plasmons.

On Dirac equation on a time scale

We consider the non-autonomous linear Dirac equation on a time scale containing important discrete, continuous, and quantum time scales. A representation of the solutions is established via an approximate solutions in terms of unknown phase functions with the error estimates. JWKB and other asymptotic representations are discussed. The adiabatic invariants of the Dirac equation are described by using a small parameter method. We also calculate the transition probabilities for the Dirac equation. Using the asymptotic solutions we show that the electron-positron transition probability during a long period of time is about 1/3. Since this probability is high, there is a simple explanation of the stability of the revolution of an electron about the proton only by the electromagnetic field. Indeed when the electron is far from the proton, it is attracted by the electromagnetic field of the proton. When the electron approaches closer to the proton, it turns to the positron which is repelling from the proton by the same electromagnetic field.

Positive solutions for second-order semipositone problems on time scales

We are concerned with the existence of positive solutions for second-order superlinear semipositone Sturm–Liouville boundary value problems on general time scales. In particular, we unify and generalize semipositone results of this type in the continuous and discrete cases to quantum and arbitrary time scales, and then introduce a related second-order multi-point semipositone problem that is new on all time scales.

Hamilton–Jacobi–Bellman equations on time scales

In this paper, we consider a class of optimal control problems on time scales without state constraints, target conditions or the fixed terminal time. We first present and show a time scale version of the Bellman optimality principle. On this basis, using a chain rule of multivariables on time scales, we will derive Hamilton–Jacobi–Bellman equations on a time scale for these kind of optimal control problems. Finally, the quantum time scale is considered as an example to illustrate our results.

Ostrowski Type Inequality for Double Integrals on Time Scales

We prove the Ostrowski type inequality for double integrals on time scales and thus unify corresponding continuous and discrete versions from the literature. We also apply the Ostrowski inequality for double integrals to the quantum time scales.

The different tongues of q-calculus

In this review paper we summarize the various dialects of q-calculus: quantum Calculus, time scales, and partitions. The close connection between Gamma q(x) functions on the one hand, and elliptic functions and theta functions on the other hand will be shown. The advantages of the Heine notation will be illustrated by the (q-)Euler reflection formula, q-Appell functions, Carlitz-AlSalam polynomials, and the so-called q-addition. We conclude with some short biographies about famous scientists in q-calculus.

Quantum time scales and the classical limit: analytic results for some simple systems.

We set up a semiclassical approximation which helps us clarify by means of several simple examples the rich variety of time scale in the quantum domain. The underlying structure of quantum and classical mechanics is so completly different that it is naive to expect to reach a classical regime by counting powers of the quantum scale variant Planck's over 2pi. We show although it is possible to define a time scale for nonclassical phenomena, but it is impossible to characterize quantum dynamics through a unique time scale, such as Ehrenfest's time. We use simple systems to critically discuss and illustrate these features of the quantum-classical limit.

Statistical thermodynamics of quantum Brownian motion: construction of perpetuum mobile of the second kind.

The Brownian motion of a quantum particle in a harmonic confining potential and coupled to harmonic quantum thermal bath is exactly solvable. Though this system presents at high temperatures a pedagogic example to explain the laws of thermodynamics, it is shown that at low enough temperatures the stationary state is non-Gibbsian due to an entanglement with the bath. In physical terms, this happens when the cloud of bath modes around the particle starts to play a nontrivial role, namely, when the bath temperature T is smaller than the coupling energy. Indeed, equilibrium thermodynamics of the total system, particle plus bath, does not imply standard equilibrium thermodynamics for the particle itself at low T. Various formulations of the second law are found to be invalid at low T. First, the Clausius inequality can be violated, because heat can be extracted from the zero point energy of the cloud of bath modes. Second, when the width of the confining potential is suddenly changed, there occurs a relaxation to equilibrium during which the entropy production is partly negative. In this process the energy put on the particle does not relax monotonically, but oscillates between particle and bath, even in the limit of strong damping. Third, for nonadiabatic changes of system parameters the rate of energy dissipation can be negative, and, out of equilibrium, cyclic processes are possible which extract work from the bath. Conditions are put forward under which perpetuum mobility of the second kind, having one or several work extraction cycles, enter the realm of condensed matter physics. Fourth, it follows that the equivalence between different formulations of the second law (e.g., those by Clausius and Thomson) can be violated at low temperatures. These effects are the consequence of quantum entanglement in the presence of the slightly off-equilibrium nature of the thermal bath, and become important when the characteristic quantum time scale variant Planck's over 2pi /k(B)T is larger than or comparable to other time scales of the system. They show that there is no general consensus between standard thermodynamics and quantum mechanics. The known agreements occur only due to the weak coupling limit, which does not pertain to low temperatures. Experimental setups for testing the effects are discussed.

Variational approach to the tunneling-time problem

Tunneling problems are characterized by different quantum time scales of motion. In this paper, we identify a tunneling time scale, which is based on a simple variational principle. The method utilizes the stationary eigenfunctions for a given one-dimensional potential structure, and it provides a truly local definition of the tunneling time, independent of the asymptotic shape of the potential. We express the minimum tunneling time in terms of the more common time scales obtained from the Larmor clock setup. Asymptotic formulas for both the extreme quantum and the semiclassical limit are presented. As an experimental verification of the variational approach we demonstrate that the minimum tunneling time governs the time a particle requires to traverse the barrier in a symmetric double-well structure.