Statistical mechanical transformation theory for quantum systems with discrete spectra

The superoperator transformation theory developed by the Brussels school is applied to quantum systems with discrete spectra. In the case of nondegeneracy of the spectra, there is no difficulty in obtaining explicit expressions for the most important superoperators in terms of the unitary operator which diagonalizes the Hamiltonian. The degenerate case presents special problems which are studied in detail.     

Distribution of degeneracies in simple quantum systems

In some simple quantum mechanical systems, the degeneracy of typical energy levels grows as a power of the energy or size. We ask whether, after dividing out this average growth, there is a well defined probability distribution of scaled degeneracies in the limit of large size or energy. The answer is yes, for a free particle in a sphere or cube. For the sphere, the distribution of scaled degeneracies is shown to follow a circular law. For the cube, a numerical study shows that the distribution rises linearly for low values of the scaled degeneracy and decays exponentially for large values.     

Rates of convergence to equilibrium in the Prigogine-Misra-Courbage theory of irreversibility

The rates of convergence to equilibrium in the Prigogine-Misra-Courbage theory of irreversibility, as developed by Goldstein, Misra, and Courbage, are examined. It is found that arbitrarily slow convergence to equilibrium should be present; in fact, in a certain precise sense, it should be the most abundant behavior. This is compared with the common beliefs in kinetic theory.Research supported in part by NSF Grant No. PHY 78-23952.     

大型脉冲功率系统真空功率传输问题

脉冲功率科学与技术广泛应用于X射线闪光照相、惯性约束聚变(ICF) 和高能量密度物理等研究领域,其基本特征是将能量在时间和空间上进行压缩来获得极高的功率密度,进而获得高温、高压、强辐射和高能量密度的极端条件。大型脉冲功率装置是获得高功率脉冲的载体,真空功率传输系统是此类装置的重要组成部分,是将储能系统的能量传递给负载的关键,也是决定大型脉冲功率系统技术可行性的重要方面。分析了大型脉冲功率系统真空功率传输的物理基础,综述了相关物理问题的国内外研究状况,结合Z箍缩聚变驱动器特定的需求,具体分析了技术现状与目标的差距,最后针对后续研究提出了建议和看法。     

The effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction-diffusion systems

Many oscillatory biological systems show periodic travelling waves. These are often modelled using coupled reaction-diffusion equations. However, the effects of different movement rates (diffusion coefficients) of the interacting components on the predictions of these equations are largely unknown. Here we investigate the ways in which varying the diffusion coefficients in such equations alters the wave speed, time period, wavelength, amplitude and stability of periodic wave solutions. We focus on two sets of kinetics that are commonly used in ecological applications: lambda-omega equations, which are the normal form of an oscillatory coupled reaction-diffusion system close to a supercritical Hopf bifurcation, and a standard predator-prey model. Our results show that changing the ratio of the diffusion coefficients can significantly alter the shape of the one-parameter family of periodic travelling wave solutions. The position of the boundary between stable and unstable waves also depends on the ratio of the diffusion coefficients: in all cases, stability changes through an Eckhaus (‘sideband’) instability. These effects are always symmetrical in the two diffusion coefficients for the lambda-omega equations, but are asymmetric in the predator-prey equations, especially when the limit cycle of the kinetics is of large amplitude. In particular, there are two separate regions of stable waves in the travelling wave family for some parameter values in the predator-prey scenario. Our results also show the existence of a one-parameter family of travelling waves, but not necessarily a Hopf bifurcation, for all values of the diffusion coefficients. Simulations of the full partial differential equations reveals that varying the ratio of the diffusion coefficients can significantly change the properties of periodic travelling waves that arise from particular wave generation mechanisms, and our analysis of the travelling wave families assists in the understanding of these effects.     

Padé approximant approach to singular properties of quantum gases: the ideal cases

In this paper, we show how to recover the low-temperature and high-density information of ideal quantum gases from the high-temperature and low-density approximation by the Padéapproximant. The virial expansion is a high-temperature and low-density expansion and in practice, often, only the first several virial coefficients can be obtained. For Bose gases, we determine the BEC phase transition from a truncated virial expansion. For Fermi gases, we recover the low-temperature and high-density result from the virial expansion.     

Analytic continuation in two-color QCD: new results on the critical line

We test the method of analytic continuation from imaginary to real chemical potential in two-color QCD, which is free from the sign problem. In particular, we consider the analytic continuation of the critical line to real values of the chemical potential.     

Schrieffer–Wolff transformation for quantum many-body systems

The Schrieffer–Wolff (SW) method is a version of degenerate perturbation theory in which the low-energy effective Hamiltonian is obtained from the exact Hamiltonian by a unitary transformation decoupling the low-energy and high-energy subspaces. We give a self-contained summary of the SW method with a focus on rigorous results. We begin with an exact definition of the SW transformation in terms of the so-called direct rotation between linear subspaces. From this we obtain elementary proofs of several important properties of such as the linked cluster theorem. We then study the perturbative version of the SW transformation obtained from a Taylor series representation of the direct rotation. Our perturbative approach provides a systematic diagram technique for computing high-order corrections to . We then specialize the SW method to quantum spin lattices with short-range interactions. We establish unitary equivalence between effective low-energy Hamiltonians obtained using two different versions of the SW method studied in the literature. Finally, we derive an upper bound on the precision up to which the ground state energy of the nth-order effective Hamiltonian approximates the exact ground state energy.     

Behavior of a quantum particle in contact with a classical heat bath

We investigate the behavior of a two-level quantum system in contact with a classical heat bath, e.g., a solute particle with internal degrees of freedom immersed in a solvent of massive particles. Using a combination of analytical and numerical methods, we obtain precise information about localization, time-displaced correlation functions, and the frequency-dependent susceptibility of such solute particles. We find that these quantities can have a strong dependence on the density of the solvent fluid, with the maximum changes from the behavior of the corresponding isolated quantum system occurring in many cases at very low densities. We compare the exact results with those obtained by path integral Monte Carlo. There is good agreement with the imaginary time correlations, but analytic continuation to real time proves elusive: even with the best numerical data on the former, we can only get very gross features of the latter.     

Time optimal control of two-level quantum systems

The objective of time-optimal control that helps to minimize relaxation losses, is the evolution of a quantum state from a given initial mixed state to a final target mixed state in minimum time. In this paper, we study a time-optimal control problem of the dynamic of a pure two-level system with unbounded control using Pontryagin's minimum principle and obtain the minimal time for some initial and final states. The results will apply to basically all qubit systems that one can consider such as NMR spectroscopy, trapped ions, superconducting qubits, etc. We also show that these results hold for pure states, and only the direction $\hat{n}$ is important in the evolution of a quantum state. In this work, the problem of computing minimum time to produce any unitary transformation $U_f\in SU(2)$ is reduced to finding the minimum time to steer the system from an initial to a final state.     

Noise and controllability: suppression of controllability in large quantum systems

A closed quantum system is defined as completely controllable if an arbitrary unitary transformation can be executed using the available controls. In practice, control fields are a source of unavoidable noise. Can one design control fields such that the effect of noise is negligible on the timescale of the transformation? Complete controllability in practice requires that the effect of noise can be suppressed for an arbitrary transformation. The present study considers a paradigm of control, where the Lie-algebraic structure of the control Hamiltonian is fixed, while the size of the system increases, determined by the dimension of the Hilbert space representation of the algebra. We show that for large quantum systems, generic noise in the controls dominates for a typical class of target transformation; i.e., complete controllability is destroyed by the noise.     

On quantum stability for systems under quasiperiodic perturbations

We prove that the spectrum defined in terms of the autocorrelation function of a harmonic subject to a quasiperiodic perturbation, is, at resonance, transient absolutely continuous, covering the whole line. In the nonresonant case, and under some supplementary Diophantine condition, it is pure point, coinciding with the spectrum of a special almost-periodic function.     

Isoperiodic classical systems and their quantum counterparts

One-dimensional isoperiodic classical systems have been first analyzed by Abel. Abel’s characterization can be extended for singular potentials and potentials which are not defined on the whole real line. The standard shear equivalence of isoperiodic potentials can also be extended by using reflection and inversion transformations. We provide a full characterization of isoperiodic rational potentials showing that they are connected by translations, reflections or Joukowski transformations. Upon quantization many of these isoperiodic systems fail to exhibit identical quantum energy spectra. This anomaly occurs at order O(?²) because semiclassical corrections of energy levels of order O(?) are identical for all isoperiodic systems. We analyze families of systems where this quantum anomaly occurs and some special systems where the spectral identity is preserved by quantization. Conversely, we point out the existence of isospectral quantum systems which do not correspond to isoperiodic classical systems.     

Kerr-Newman黑洞的谐振子模型及量子面积谱

利用Kerr-Newman黑洞的质量M，电量Q，角动量J和它们各自的对偶量，π_M,π_Q,π_J,构成的六维相空间，通过规范变换，首先建立黑洞的简谐振子模型；再利用该模型进一步研究Kerr-Newman黑洞的量子面积谱. 关键词： 黑洞 规范变换 简谐振子模型 量子面积谱     

Thermodynamic paths and stochastic order in open systems

The theorem of extremum entropy generation is related to the stochastic order of the paths inside the phase space; indeed, the system evolves, from an indistinguishable configuration to another one, on the most probable path in relation to the paths stochastic order. The result is that, at the stationary state, the entropy generation is maximal and, this maximum value is a consequence of the stochastic order of the paths in the phase space. Conversely, the stochastic order of the paths in the phase space is a consequence of the maximum of the entropy generation for the open systems at the stationary states.     

非线性耗散对亚稳态系统量子衰变速率的影响

发展了一种快速傅里叶变换路径积分方法，研究非线性耗散系统的量子衰变速率，得到了Bounce轨道的作用量S_B，即衰变速率的指数因子.在系统与环境存在非线性耦合f(x)=tanh［λ(x-x_b)］的情形下，发现其对衰变速率具有抑制作用.指数因子随温度T的关系不再满足S_B=a［1-b(T/T_c)²］法则；与通常的线性耗散情形相比，跨越温度T_c回升，即系统更早地进入穿透区域. 关键词： 量子衰变 非线性耦合 路径积分 快速傅里叶变换     

Proca Effect in Kerr–Newman Metric

The Proca effect of an electric field is studied in curved space. A Kerr–Newman metric with the photon rest mass can be presented by the analytic continuation (Xu, C. M. (....). General Relativity and Modern Cosmology, Nanking Normal University) in a short range. It yields the correction in the Kerr–Newman space.     

The geometric phase of the quantum systems with slow but finite rate of the external time-dependent field

With the help of the time-dependent gauge transformation technique, we have studied the geometric phase of a spin-half particle in a rotating magnetic field. We have found that the slow but finite frequency of the rotating magnetic field will make the difference between the adiabatic geometric phase and the exact geometric phase. When the frequency is much smaller than the energy space and the adiabatic condition is perfectly guaranteed, the adiabatic approximation geometric phase is exactly consistent with the adiabatic geometric phase. A simple relation for the accuracy of the adiabatic approximation is given in terms of the changing rate of the frequency of the rotating magnetic field and the energy level space.