Fresnel Operator for Deriving the Propagator of 2D Harmonic Oscillator with Cross Coupling

Using the identity of operator decomposition we obtain a normal ordered form of the time-evolution operator for cross coupling quantum harmonic oscillator Hamiltonian system in two dimensions, which is just a special two-mode Fresnel operator. The Feynman propagator for the Hamiltonian system is found by a direct calculation by means of the method deriving the matrix element of two-mode Fresnel operator in the entangled state representation. The technique of integration within an ordered product (IWOP) of operators is employed to derive the matrix elements of the operator in the coherent state and the entangled state representations.     

Quantum holonomy theory

We present quantum holonomy theory, which is a non‐perturbative theory of quantum gravity coupled to fermionic degrees of freedom. The theory is based on a ‐algebra that involves holonomy‐diffeo‐morphisms on a 3‐dimensional manifold and which encodes the canonical commutation relations of canonical quantum gravity formulated in terms of Ashtekar variables. Employing a Dirac type operator on the configuration space of Ashtekar connections we obtain a semi‐classical state and a kinematical Hilbert space via its GNS construction. We use the Dirac type operator, which provides a metric structure over the space of Ashtekar connections, to define a scalar curvature operator, from which we obtain a candidate for a Hamilton operator. We show that the classical Hamilton constraint of general relativity emerges from this in a semi‐classical limit and we then compute the operator constraint algebra. Also, we find states in the kinematical Hilbert space on which the expectation value of the Dirac type operator gives the Dirac Hamiltonian in a semi‐classical limit and thus provides a connection to fermionic quantum field theory. Finally, an almost‐commutative algebra emerges from the holonomy‐diffeomorphism algebra in the same limit.     

Influence of coloured noise on the diffusion of a quantum particle. Part II: Diffusion constant for dichotomic markovian fluctuations

In part I of this paper a new formalism for the calculation of stochastic moments in quantum mechanical particle motion has been developed. Now we use this formalism to obtain expressions for the mean square displacement within a model containing dichotomic Markovian fluctuations. A self-energy like quantity in the equation of motion for a contracted kernel or propagator determining the mean square displacement is replaced by its second order approximation in powers of the deterministic part of the Hamiltonian. This is the only approximation throughout the paper. In the one-dimensional case the contracted propagator itself is calculated. Instead, in the general case the mean square displacement is given in terms of a continued fraction. We compare our result to several previous ones and especially discuss the question of Anderson, localization in the static limit.     

Bose-Fermi systems and computer algebra

Coupled Bose-Fermi systems play a central role in quantum mechanics and solid state physics. We give an implementation of the mathematical properties of Bose-Fermi systems using computer algebra. As an application we consider among others a one-fermion one-boson system for magnetic elastic systems and a supersymmetric Hamilton operator.     

Ruelle resonances in kicked quantum spin chain

We study exponential decay of high temperature time correlation functions in a non-integrable quantum spin chain problem, namely Ising spin 1/2 chain kicked with tilted homogeneous magnetic field. For this purpose we define a master propagator over a suitable banach space of quantum observables (quantum many-body analogue of Perron–Frobenius operator) whose leading eigenvalue determines the asymptotic decay of correlations. This is demonstrated with explicit calculation for which a fast algorithm for the construction of the master propagator is developed.     

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.     

Un-particle effective action

We study un-particle dynamics in the framework of standard quantum field theory. We obtain the Feynman propagator by supplementing standard quantum field theory definitions with integration over the mass spectrum. Then we use this information to construct effective actions for scalar, gauge vector and gravitational un-particles.     

含时量子系统传播子的ABCD形式

将含时量子系统状态的演化看成物质波波束沿时间轴的传输,引入束宽、发散角、曲率半径、品质因子和束熵表征波束的传输.发现品质因子守恒的量子系统,其动量算子和位置算子呈线性演化,传播子可以表达成ABCD的形式,最小波包(品质因子为1)的形式不随时间改变.讨论了对常数势能系统和谐振子系统的应用 关键词： 物质波束 光束 谐振子     

Quantum Dynamics of Systems Connected by a Canonical Transformation

Quantum Hamiltonian systems corresponding to classical systems related by a general canonical transformation are considered. The differential equation to find the unitary operator, which corresponds to the canonical transformation and connects quantum states of the original and transformed systems, is obtained. The propagator associated with their wave functions is found by the unitary operator. Quantum systems related by a linear canonical point transformation are analyzed. The results are tested by finding the wave functions of the under-, critical-, and over-damped harmonic oscillator from the wave functions of the harmonic oscillator, free-particle system, and negative harmonic potential system, using the unitary operator to connect them, respectively.     

Connes distance of 2D harmonic oscillators in quantum phase space

We study the Connes distance of quantum states of two-dimensional (2D) harmonic oscillators in phase space. Using the Hilbert-Schmidt operatorial formulation, we construct a boson Fock space and a quantum Hilbert space, and obtain the Dirac operator and a spectral triple corresponding to a four-dimensional (4D) quantum phase space. Based on the ball condition, we obtain some constraint relations about the optimal elements. We construct the corresponding optimal elements and then derive the Connes distance between two arbitrary Fock states of 2D quantum harmonic oscillators. We prove that these two-dimensional distances satisfy the Pythagoras theorem. These results are significant for the study of geometric structures of noncommutative spaces, and it can also help us to study the physical properties of quantum systems in some kinds of noncommutative spaces.     

Operator Correlations and Quantum Regression Theorem in Non-Markovian Lindblad Rate Equations

Non-Markovian Lindblad rate equations arise from alternative microscopic interactions such as quantum systems coupled to composite reservoirs, where extra degrees of freedom mediate the interaction between the system and a Markovian reservoir, as well as from systems coupled to complex structured reservoirs whose action can be well approximated by a direct sum of Markovian sub-reservoirs (Budini in Phys. Rev. A 74:053815 [2006]). The purpose of this paper is two fold. First, for both kinds of interactions we find general expressions for the system operator correlations written in terms of the Lindblad rate propagator. Secondly, we find the conditions under which a quantum regression hypothesis is valid. We show that a non-Markovian quantum regression theorem can only be granted in a stationary regime, being a necessary condition the fulfillment of a detailed balance condition. This result is independent of the underlying microscopic interaction, providing a criterion for the validity of the regression hypothesis in non-Markovian Lindblad-like master equations. As an example, we study the correlations of a two-level system coupled to different kind of reservoirs.     

Lie algebraic approach to the time-dependent quantum general harmonic oscillator and the bi-dimensional charged particle in time-dependent electromagnetic fields

We discuss the one-dimensional, time-dependent general quadratic Hamiltonian and the bi-dimensional charged particle in time-dependent electromagnetic fields through the Lie algebraic approach. Such method consists in finding a set of generators that form a closed Lie algebra in terms of which it is possible to express a quantum Hamiltonian and therefore the evolution operator. The evolution operator is then the starting point to obtain the propagator as well as the explicit form of the Heisenberg picture position and momentum operators. First, the set of generators forming a closed Lie algebra is identified for the general quadratic Hamiltonian. This algebra is later extended to study the Hamiltonian of a charged particle in electromagnetic fields exploiting the similarities between the terms of these two Hamiltonians. These results are applied to the solution of five different examples: the linear potential which is used to introduce the Lie algebraic method, a radio frequency ion trap, a Kanai–Caldirola-like forced harmonic oscillator, a charged particle in a time dependent magnetic field, and a charged particle in constant magnetic field and oscillating electric field. In particular we present exact analytical expressions that are fitting for the study of a rotating quadrupole field ion trap and magneto-transport in two-dimensional semiconductor heterostructures illuminated by microwave radiation. In these examples we show that this powerful method is suitable to treat quadratic Hamiltonians with time dependent coefficients quite efficiently yielding closed analytical expressions for the propagator and the Heisenberg picture position and momentum operators.     

Exact Solvability and Quantum Integrability of a Derivative Nonlinear Schrödinger Model

Using a variant of quantum inverse scattering method (QISM) which is directly applicable to field theoretical systems, we derive all possible commutation relations among the operator valued elements of the monodromy matrix associated with an integrable derivative nonlinear Schrödinger (DNLS) model. From these commutation relations we obtain the exact Bethe eigenstates for the quantum conserved quantities of DNLS model. We also explicitly construct the first few quantum conserved quantities including the Hamiltonian in terms of the basic field operators of this model. It turns out that this quantum Hamiltonian has a new kind of coupling constant which is quite different from the classical one. This fact allows us to apply QISM to generate the spectrum of quantum DNLS Hamiltonian for the full range of its coupling constant.     

Nontrivial Q2-Dependence in the Gottfried Sum Induced by Nonperturbative Effect

In consideration of the lowest dimensional condensate contributions to the free quark propagator, we obtain the nonperturbative quark propagator with the nonvanishing vacuum average value for quark composite operator (|qq|), Using the -corrected quark propagator and basing on the simple quark-parton model, we discuss the nonperturbative effect in the nucleon structure function. It is shown that the nonperturbative effect modifies the conventional quark-parton model formula of the nucleon structure function at finite Q² and suggests a nontrivial Q²-dependence in the Gottfried sum.     

经典粒子和量子微观粒子的运动轨道

用量子力学传播子讨论了经典粒子和量子微观粒子沿着不同轨道对传播子的贡献．通过这样的计算可以清楚地看到为什么经典粒子只是沿着满足最小作用原理的轨道，即经典轨道运动，而对于量子微观粒子沿着许多条路径都有可能．这篇短文将一些教科书中的讨论用具体的数学表达出来，使得读者更容易理解．计算表明，对于质量较大的经典粒子，沿着非经典轨道对传播子的贡献和它周围的临近轨道有强烈的抵消．只有在经典轨道附近对传播子的贡献才不为零．对于量子微观粒子没有这种差别，它可以沿着多种可能的轨道运动．     

Fermionic quasiparticle representation of Tomonaga-Luttinger Hamiltonian

We find a unitary operator which asymptotically diagonalizes the Tomonaga-Luttinger Hamiltonian of one-dimensional spinless electrons. The operator performs a Bogoliubov rotation in the space of electron-hole pairs. If bare interaction of the physical electrons is sufficiently small this transformation maps the original Tomonaga-Luttinger system on a system of free fermionic quasiparticles. Our representation is useful when the electron dispersion deviates from linear form. For such situation we obtain non-perturbative results for the electron gas free energy and the density-density propagator.     

Density-Density Propagator of the Luttinger Model of Spin-Half Fermions after an Interaction Quench

Motivated by recent research achievement of quantum interacting systems in non-equilibrium, we consider a Luttinger model with a suddenly switched-on interaction proposed by Cazalilla [M.A. Cazalilla, Phys. Rev. Lett. 97 (2006) 156403]. In order to compare with real systems, we extend Cazalilla's scenario to the spinful system. To find the influence of initial states on the time evolution of some non-equilibrium systems, we mainly focus on the density-density propagator. By comparison and analysis, we discover the different behavior of this non-equilibrium system. Further, it is found that the propagator saves strong memory of initial state, and the effects of right-left interaction cancel out in total density-density propagator.     

Density-Density Propagator of the Luttinger Model of Spin-Half Fermions after an Interaction Quench

Motivated by recent research achievement of quantum interacting systems in non-equilibrium, we consider a Luttinger model with a suddenly switched-on interaction proposed by Cazalilla [M.A. Cazalilla, Phys. Rev. Lett. 97 (2006) 156403]. In order to compare with real systems, we extend Cazalilla's scenario to the spinful system. To find the influence of initial states on the time evolution of some non-equilibrium systems, we mainly focus on the density-density propagator. By comparison and analysis, we discover the different behavior of this non-equilibrium system. Further, it is found that the propagator saves strong memory of initial state, and the effects of right-left interaction cancel out in total density-density propagator.     

Pure absorption multiple quantum spectra of molecules dissolved in liquid crystalline solvents

The use of time reversal pulse sequences to obtain multiple quantum spectra of molecules in thermotropic liquid crystalline phases is described. Several studies have already demonstrated that in order to obtain multiple quantum coherence of high order in polycrystalline solids, it is necessary to utilize pulse trains that produce a preparation propagator that is the adjoint of the mixing propagator. Such pulse sequences produce multiple quantum powder spectra that are pure absortive, thus avoiding destructive phase interference that would occur if standard multiple quantum pulse sequences were used. However even in cases where all single quantum transitions are well-resolved, standard multiple quantum pulse sequences yield multiple quantum spectra of low signal-to-noise because single quantum coherent states are projected out of phase. Sensitivity may be improved by projecting the full two dimensional transform, but this may not be practical in cases involving moderately large numbers of strongly coupled spin one-half nuclei. If the time reversal sequences are used however single quantum coherent states are projected in phase and the full two dimensional transform need not be calculated. The pure absorption double quantum spectrum of oriented benzene has been obtained using time reversal pulse trains and demonstrates a considerable increase in sensitivity over standard methods. Practical aspects of applying multiple pulse sequences to thermotropic systems are considered.