无限深势阱中粒子的态密度

对于自由粒子在有限容器中的能态密度,热力学统计教材一般根据半经典量子图像,由驻波条件和德布罗意关系,以动量分立值为基础出发得到;然而根据量子理论,无限深势阱中的粒子存在能量本征态,而非动量本征态.本文以能量本征态为统计对象推导了有限体积中的自由粒子的能态密度,结果与教材一致.但是我们的处理方式显得更为自然.     

Fundamental concepts of quantum chaos

We review the fundamental concepts of quantum chaos in Hamiltonian systems. The quantum evolution of bound systems does not possess the sensitive dependence on initial conditions, and thus no chaotic behaviour occurs, whereas the study of the stationary solutions of the Schrödinger equation in the quantum phase space (Wigner functions) reveals precise analogy of the structure of the classical phase portrait. We analyze the regular eigenstates associated with invariant tori in the classical phase space, and the chaotic eigenstates associated with the classically chaotic regions, and the corresponding energy spectra. The effects of quantum localization of the chaotic eigenstates are treated phenomenologically, resulting in Brody-like level statistics, which can be found also at very high-lying levels, while the coupling between the regular and the irregular eigenstates due to tunneling, and of the corresponding levels, manifests itself only in low-lying levels.     

Quantum states far from the energy eigenstates of any local Hamiltonian

What quantum states are possible energy eigenstates of a many-body Hamiltonian? Suppose the Hamiltonian is nontrivial, i.e., not a multiple of the identity, and L local, in the sense of containing interaction terms involving at most L bodies, for some fixed L. We construct quantum states psi which are "far away" from all the eigenstates E of any nontrivial L-local Hamiltonian, in the sense that ||psi-E|| is greater than some constant lower bound, independent of the form of the Hamiltonian.     

Connections Between the Thermodynamics of Classical Electrodynamic Systems and Quantum Mechanical Systems for Quasielectrostatic Operations

The thermodynamic behavior is analyzed of a single classical charged particle in thermal equilibrium with classical electromagnetic thermal radiation, while electrostatically bound by a fixed charge distribution of opposite sign. A quasistatic displacement of this system in an applied electrostatic potential is investigated. Treating the system nonrelativistically, the change in internal energy, the work done, and the change in caloric entropy are all shown to be expressible in terms of averages involving the distribution of the position coordinates alone. A convenient representation for the probability distribution is shown to be the ensemble average of the absolute square value of an expansion over the eigenstates of a Schrödinger-like equation, since the heat flow is shown to vanish for each hypothetical state. Subject to key assumptions highlighted here, the demand that the entropy be a function of state results in statistical averages in agreement with the form in quantum statistical mechanics. Examining the very low and very high temperature situations yields Planck's and Boltzmann's constants. The blackbody radiation spectrum is then deduced. From the viewpoint of the theory explored here, the method in quantum statistical mechanics of statistically counting the states at thermal equilibrium by using the energy eigenvalue structure, is simply a convenient counting scheme, rather than actually representing averages involving physically discrete energy states.     

Evolution processes of coupled thermal qubits

We investigate the quantum speed limit time (QSLT) of quantum evolution before thermal equilibrium of two coupled qubits each of which is coupled to a separate thermal bath at the same temperature within the Born-Markov approximation. The evolution process in one particular initial state can change between speed-up and speed-down two times before reaching equilibrium. We call this double cusp behaviour. This behaviour is an anomalous phenomenon in evolution processes in the weak-coupling Markovian regime. We study QSLT corresponding to all pure initial energy eigenstates and categorise them. In addition, we also display the conditions for double cusp behaviour in terms of temperature, qubit interaction and frequency.     

Spectroscopy of a canonically quantized horizon

Deviations from Hawking's thermal black hole spectrum, observable for macroscopic black holes, are derived from a model of a quantum horizon in loop quantum gravity. These arise from additional area eigenstates present in quantum surfaces excluded by the classical isolated horizon boundary conditions. The complete spectrum of area unexpectedly exhibits evenly spaced symmetry. This leads to an enhancement of some spectral lines on top of the thermal spectrum. This can imprint characteristic features into the spectra of black hole systems. It most notably gives the signature of quantum gravity observability in radiation from primordial black holes, and makes it possible to test loop quantum gravity with black holes well above Planck scale.     

Quantum Monodromy in the Spectrum of Schrödinger Equation with a Decatic Potential

In this study the spectral problem of the two-dimensional Schrödinger equation with the cylindrically symmetrical decatic potential is carried out. The concept of quantum monodromy is introduced to give insight into the energy levels of system with this potential. It is shown that quantum monodromy occurs at = 0 in the distribution of eigenstates around a critical point on the spectrum at E = 0 with zero angular momentum, such that there can be no smoothly valid assignment of quantum number. Cases with the three-well and four-well potentials are presented to give rise to the double degeneracies with respect to energy except for the angular momentum m = 0.     

On the exact mean-field description of continuous quantum systems in equilibrium

The thermodynamic limit of free energy density is investigated for quantum systems of n particles obeying Boltzmann, Fermi and Bose statistics, interacting via spin-independent 2-body bounded separable potentials and confined to a bounded region Λ ? R^v. The technique used exploits the Feynman-Kac theorem in finite volume and the saddle-point method of Tindemans and Capel. It is shown that the limiting free energy density of such systems is equal to that of a system of noninteracting particles subject to a mean field which is equal to the averaged 2-body interaction. The equations for the mean field of n particles obeying Boltzmann, Fermi or Bose statistics represent self-consistent field problems and their forms comply with the well-known theorems on mean occupation numbers of single-particle eigenstates of ideal quantum gases at inverse temperature β.     

Helicity Basis and Parity

We study the theory of the (1/2,0) (0,1/2) representation in helicity basis. Helicity eigenstates are not the parity eigenstates. This is in accordance with the consideration of Berestetskii, Lifshitz, and Pitaevskii. Relations to the Gelfand-Tsetlin-Sokolik-type quantum field theory are discussed. Finally, a new form of the parity operator is proposed. It commutes with the Hamiltonian.     

Entanglement of a degenerate system in adiabatic process

Entanglement evolution of a degenerate system is studied. It is found that an entangled state becomes unentangled under an adiabatic variation of the system, and vice versa. The amount of entanglement varies with different adiabatic paths. It is concluded that the entanglement of complete set of eigenstates of a system at thermal equilibrium is independent on perturbations, but the entanglement of partial eigenstates may change with adiabatic perturbations. This can help us to control entanglement properties and is very useful in the quantum computations.     

Generalized Rotating-Wave Approximation for the Quantum Rabi Model with Optomechanical Interaction

The spectrum of energy and eigenstates of an hybrid cavity optomechanical system, where a cavity field mode interacts with a mechanical mode of a vibrating end mirror via radiation pressure and with a two level atom via electric dipole interaction are investigated. In the spirit of approximations developed for the quantum Rabi model beyond rotating-wave approximation (RWA), the so-called generalized RWA (GRWA) to diagonalize the tripartite Hamiltonian for arbitrary large couplings is implemented. Notably, the GRWA approach still allows to rewrite the hybrid Hamiltonian in a bipartite form, like a Rabi model with dressed atom-field states (polaritons) coupled to mechanical modes through reparametrized coupling strength and Rabi frequency. A more accurate energy spectrum for a wide range of values of the atom-photon and photon–phonon couplings, when compared to the RWA results is found. The fidelity between the numerical eigenstates and its approximated counterparts is also calculated. The degree of polariton-phonon entanglement of the eigenstates presents a non-monotonic behavior as the atom-photon coupling varies, in contrast to the characteristic monotonic increase in the RWA treatment.     

Measurement of energy eigenstates by a slow detector

We propose a method for a weak continuous measurement of the energy eigenstates of a fast quantum system by means of a slow detector. Such a detector is sensitive only to slowly changing variables, e.g., energy, while its backaction can be limited solely to decoherence of the eigenstate superpositions. We apply this scheme to the problem of detection of quantum jumps between energy eigenstates in a harmonic oscillator.     

Localization-delocalization transition in a system of quantum kicked rotors

The quantum dynamics of atoms subjected to pairs of closely spaced delta kicks from optical potentials are shown to be quite different from the well-known paradigm of quantum chaos, the single delta-kick system. We find the unitary matrix has a new oscillating band structure corresponding to a cellular structure of phase space and observe a spectral signature of a localization-delocalization transition from one cell to several. We find that the eigenstates have localization lengths which scale with a fractional power L approximately h(-0.75) and obtain a regime of near-linear spectral variances which approximate the "critical statistics" relation summation2(L) approximately or equal to chi(L) approximately 1/2 (1-nu)L, where nu approximately 0.75 is related to the fractal classical phase-space structure. The origin of the nu approximately 0.75 exponent is analyzed.     

Squeezed Number Eigenstate of XYZ Heisenberg Antiferromagnetics Under a Magnetic Field

By using an algebraic diagonalization method, the XYZ Heisenberg antiferromagnetics under an external magnetic field is studied in the framework of spin-wave theory. The energy eigenstates are shown to be squeezed number states and the energy eigenvalues are obtained in some cases. Some quantum properties of the energy eigenstates, and the connection of the model with the two-mode coupled harmonic oscillators are also discussed. PACS numbers: 42.50.Dv, 42.50.Ar, 03.65.Fd.     

On local thermal equilibrium given by an energy eigenstate in isolated quantum mechanical system

It is proved that almost all energy eigenstates of a certain isolated system have the following property: quantum expectation values of all local observables are equal to these statistical expectation values. Temperature is uniquely determined by energy density.     

由非绝热相关获得的形式量子数及其在高激发振动态的经典解释

我们利用非绝热相关方法 ,通过关闭所有的振动模式间的耦合项并追溯到零级本征态 ,以得到体系的形式量子数 ,将形式量子数对高激发振动态的能级谱图进行归属 ,并重构本征能级图谱 ,使本征能级以有序的方式排列。这有助于对高激发振动态的能级进行分类和归属。形式量子数是体现高激发振动态的重要特征 ,是高激发振动态的近似运动守恒量。我们将多维陪集相空间的经典方法应用于高激发态的研究 ,发现形式量子数对应的李雅普诺夫指数为零或最小 ,并且它对应于较大的相空间密度     

Protected quantum computation with multiple resonators in ultrastrong coupling circuit QED

We investigate theoretically the dynamical behavior of a qubit obtained with the two ground eigenstates of an ultrastrong coupling circuit-QED system consisting of a finite number of Josephson fluxonium atoms inductively coupled to a transmission line resonator. We show a universal set of quantum gates by using multiple transmission line resonators (each resonator represents a single qubit). We discuss the intrinsic "anisotropic" nature of noise sources for fluxonium artificial atoms. Through a master equation treatment with colored noise and many-level dynamics, we prove that, for a general class of anisotropic noise sources, the coherence time of the qubit and the fidelity of the quantum operations can be dramatically improved in an optimal regime of ultrastrong coupling, where the ground state is an entangled photonic "cat" state.     

Measuring scars of periodic orbits

The phenomenon of periodic orbit scarring of eigenstates of classically chaotic systems is attracting increasing attention. Scarring is one of the most important "corrections" to the ideal random eigenstates suggested by random matrix theory. This paper discusses measures of scars and in so doing also tries to clarify the concepts and effects of eigenfunction scarring. We propose a universal scar measure which takes into account an entire periodic orbit and the linearized dynamics in its vicinity. This measure is tuned to pick out those structures which are induced in quantum eigenstates by unstable periodic orbits and their manifolds. It gives enhanced scarring strength as measured by eigenstate overlaps and inverse participation ratios, especially for longer orbits. We also discuss off-resonance scars which appear naturally on either side of an unstable periodic orbit.