On exact master equation for an open system

An exact equation is obtained for the evolution of density matrix in the coherent state representation for the open system of a single harmonic oscillator coupled to a reservoir of N oscillators.     

The double Caldeira-Leggett model: Derivation and solutions of the master equations, reservoir-induced interactions and decoherence

In this paper we analyze the double Caldeira-Leggett model: the path integral approach to two interacting dissipative harmonic oscillators. Assuming a general form of the interaction between the oscillators, we consider two different situations: (i) when each oscillator is coupled to its own reservoir, and (ii) when both oscillators are coupled to a common reservoir. After deriving and solving the master equation for each case, we analyze the decoherence process of particular entanglements in the positional space of both oscillators. To analyze the decoherence mechanism we have derived a general decay function, for the off-diagonal peaks of the density matrix, which applies both to common and separate reservoirs. We have also identified the expected interaction between the two dissipative oscillators induced by their common reservoir. Such a reservoir-induced interaction, which gives rise to interesting collective damping effects, such as the emergence of relaxation- and decoherence-free subspaces, is shown to be blurred by the high-temperature regime considered in this study. However, we find that different interactions between the dissipative oscillators, described by rotating or counter-rotating terms, result in different decay rates for the interference terms of the density matrix.     

Reduced density matrices of a system of N coupled oscillators: 4. The eigenstructure of the p-particle matrices for the 2-temperature canonical ensemble

It is proved that the reduced p-particle matrix (p?N) corresponding to a 2-temperature equilibrium state of a system of N coupled oscillators (the two temperatures T_tr and T_re1 relating to the translational and relative motion of the latter) coincides with the density matrix of a two-temperature canonical ensemble of an effective system of p coupled oscillators, the effective temperature T?_re1 being equal to T_re1 while $\text{T}\text{?}{}_{\mathrm{tr}}\text{≠T}{}_{\mathrm{tr}}$, and in particular $\text{T}\text{?}{}_{\mathrm{tr}}\text{>0}$ for T_tr=0.     

Effects of pair resonant dipole-dipole interactions on the formation of band profiles in infrared absorption spectra

Resonant dipole-dipole interactions are shown to play a crucial role in forming the profiles of several bands in the fundamental and overtone regions of the vibrational spectrum for molecules with a relatively large first derivative of the dipole moment (P′>0.3 D). Analytic expressions are derived for the probability density of the matrix element for the interaction of one-and two-dimensional oscillators. We computed the band profiles in the infrared absorption spectrum for a pair of molecules at a fixed distance R ～ 5 Å, typical of condensed media. Comparison is made with experimental data for low-temperature condensed systems: matrices and solutions in liquefied gases.     

Phase transitions in a system of coupled Boson oscillators on a sphere

The solutions of mean-field equations for a system of coupled Boson oscillators on an infinite k-dimensional sphere are discussed in the low density - high temperature region and high density — low temperature region. It is shown that for k = 2 the system exhibits only spatial condensation, whereas for k ⩾ 3 both spatial condensation and Bose-Einstein condensation.     

Eigenstructure of the 1-particle density operator corresponding to the first excited state of two coupled oscillators

The eigenstructure of the 1-particle density operator ? corresponding to the first excited state of two coupled oscillators is investigated. It is shown that ? is strictly positive and thus may be considered as the canonical density matrix of some effective Hamiltonian system. The form of the eigenfunctions and their dependence on the eigenvalues is found. A number of properties of the spectrum are proved and a method allowing to find a finite number of exact terms of power expansions of the eigenvalues in a parameter λ∈(0, 1) is given.     

Density Matrix and Squeezed State for Two Harmonic Oscillators with a Kinetic Coupling

We study two harmonic oscillators with a kinetic coupling system. By taking a unitary transformation approach, we turn the system into the Fock space of two independent harmonic oscillators and derive the density matrix for it. The corresponding unitary operator U is characteristic of including frequency-jump squeezing transformation. By virtue of the technique of the integration within an ordered product of operators, we manifestly show that the ground state of the system is a squeezed state.     

A new formula to calculate Franck-Condon factors for displaced and distorted harmonic oscillators

A new formula has been derived to calculate Franck-Condon factors for two harmonic oscillators with different equilibrium positions and different vibrational frequencies. The derivation is simple and straightforward, and a new expression for calculating the matrix element 〈v|xⁿ|v′〉 has also been derived.     

Evaluation of the density matrix for an ensemble of anharmonic oscillators by a path integral approach

Considering the Feynman path integral representation for the configuration-space density matrix for an ensemble of anharmonic oscillators, we determine the stationary paths near which the integrand remains stationary. By taking the path integral to be saturated by contributions from the neighborhood of the path which maximizes the integrand we evaluate the density matrix explicitly in analytic form. This seems to be the first such evaluation of a path integral for a system not describable by a quadratic Hamiltonian. We also comment briefly on the question of analyticity with respect to the perturbation parameter.     

Spin and the Thermal Equilibrium Distribution of Wave Functions

Consider a quantum system S weakly interacting with a very large but finite system B called the heat bath, and suppose that the composite S∪B is in a pure state Ψ with participating energies between E and E+δ with small δ. Then, it is known that for most Ψ the reduced density matrix of S is (approximately) equal to the canonical density matrix. That is, the reduced density matrix is universal in the sense that it depends only on S’s Hamiltonian and the temperature but not on B’s Hamiltonian, on the interaction Hamiltonian, or on the details of Ψ. It has also been pointed out that S can also be attributed a random wave function ψ whose probability distribution is universal in the same sense. This distribution is known as the “Scrooge measure” or “Gaussian adjusted projected (GAP) measure”; we regard it as the thermal equilibrium distribution of wave functions. The relevant concept of the wave function of a subsystem is known as the “conditional wave function.” In this paper, we develop analogous considerations for particles with spin. One can either use some kind of conditional wave function or, more naturally, the “conditional density matrix,” which is in general different from the reduced density matrix. We ask what the thermal equilibrium distribution of the conditional density matrix is, and find the answer that for most Ψ the conditional density matrix is (approximately) deterministic, in fact (approximately) equal to the canonical density matrix.     

Mechanism for the metal-conducting behavior of a CaB4 single crystal

The temperature-dependent resistivity of a pure CaB₄ single crystal has been observed to demonstrate metal-conducting behavior. The normal Hall-coefficient measurement is indicative of the electrons as the conducting carriers with a room-temperature density of 4.7×10²¹ cm^?3. The observed metal-conducting behavior can be understood using a simple model, in which the Ca ions are treated as independent Einstein harmonic oscillators embedded in a Debye rigid boron network. The ab initio calculations of band structure and density of states corroborate the experimentally observed metal-conducting behavior of CaB₄, showing that it originates mainly from the p orbital of B atom and the hybridized d orbital of Ca atom.     

Density Matrix and Squeezed Vacuum State for General Coupling Harmonic Oscillator

By taking a unitary transformation approach, we study two harmonic oscillators with both kinetic coupling and coordluate coupling terms, and derive the density matrix of the system. The results show that the ground state of the system is a rotated two single-mode squeezed state.     

Density Matrix and Squeezed Vacuum State for General Coupling Harmonic Oscillator

By taking a unitary transformation approach, we study two harmonic oscillators with both kinetic coupling and coordinate coupling terms, and derive the density matrix of the system. The results show that the ground state of the system is a rotated two single-mode squeezed state.     

Unitary Operator and Density Matrix for a Ring of Coupled Oscillator Model

A unitary operator U which can directly diagonalize the Hamiltonian of a ring of N coupled oscillators is found. With use of the technique of integration within an ordered product of operators we see that U includes a squeezing transformation. For large N it turns out that, except a zero-mode, the ring Hamiltonian has the same spectrum as the one-dimensional monatomic lattice model plus the Born-Von-Karmen boundary condition. The density matrix of this model is calculated by the U transformation, which further leads us to derive the heat capacity of the coupled oscillators in the large N limit.     

On the theory of Boson peak in glasses

A universal mechanism of the Boson peak formation in glasses is proposed. The mechanism is based on the concept of interacting quasilocal oscillators. Even in the case of weak interaction, the low-frequency spectrum becomes unstable. Due to anharmonicity, the system undergoes a transition into a new stable configuration. As a result, below some characteristic frequency ω_c, proportional to the typical strength of interaction, the renormalized density of states becomes a universal function of ω with a Boson peak feature; i.e., the reduced density of states g(ω)/ω² has a maximum at a frequency ω_b?ω_c. We derive an analytic form of this function.     

基于路径积分的声传播微观特性的研究

运用量子力学的Feynman路径积分理论,尝试研究空气中声传播的计算方法及其微观特性。选取一小团空气(小体元)为研究对象,将声传播过程中一列振动的空气小体元近似为一组谐振子集合,用路径积分方法给出系统的能量方程及波函数,研究并分析声传播的某些机制和它的微观特性。在量子分子状态下,用密度矩阵将系统概率波幅(跃迁幅)与配分函数相关联,给出了谐振子处于能量E_n的概率p(E_n)和能量的平均值(?)。     

Liapunov spectra for infinite chains of nonlinear oscillators

We argue that the spectrum of Liapunov exponents for long chains of nonlinear oscillators, at large energy per mode, may be well approximated by the Liapunov exponents of products of independent random matrices. If, in addition, statistical mechanics applies to the system, the elements of these random matrices have a distribution which may be calculated from the potential and the energy alone. Under a certain isotropy hypothesis (which is not always satisfied), we argue that the Liapunov exponents of these random matrix products can be obtained from the density of states of a typical random matrix. This construction uses an integral equation first derived by Newman. We then derive and discuss a method to compute the spectrum of a typical random matrix. Putting the pieces together, we see that the Liapunov spectrum can be computed from the potential between the oscillators.     

Determination of the final nucleus density matrix by measuring the angular correlation of charged particles and γ-quanta in various planes

The measurements of the angular correlation function of charged particles and γ-quanta emitted by a final even-even nucleus in state J_f in different planes of γ-quantum ejection with respect to the reaction plane have been shown to offer the unique possibility of restoring all the even-rank components of the model-less density matrix in the transitions with J_f = L (L) is the multipolarity of the γ-transition). The minimum number of planes in which the angular correlation function should be measured to restore the density matrix has been found. The formulas relating the components of the model-less density matrix to the polarization characteristics of an even-even nucleus in a given state have been obtained.     

Flavouring the Gribov process

The exchange of flavour carrying trajectories is studied in the non-covariant parton interpretation of reggeon field theory. While pomeron exchange is described by wee partondensities, i.e. diagonal elements of the density matrix of a fast hadron, meson exchange is described by density matrix elements which are diagonal in parton number but off-diagonal in flavour. The reggeon field theory “hamiltonian” describes a markoffian evolution of this block-diagonal density matrix during a boost. This interpretation is possible both if there are two distinctf and ? trajectories and in case of ? identity. The meson trajectories are superpositions of odd and even signature trajectories.     

An approximate scheme to determine the vibratory density of states in a quasi-harmonic solid

In a new and realistic conception of solids, an analytical approach is presented to find the phonon density of states in a quasi-harmonic one-dimensional solid by starting from the fact that a solid with strong interatomic forces can be regarded as an atomic array of quantum anharmonic oscillators under a Morse potential. In fact, our quasi-harmonic approximation is derived by assuming a large enough solid with an involved anharmonic parameter as a sufficiently small quantity. In this context, a mathematical relationship between the above parameter and the matrix element relative to oscillator’s strength is obtained.