Mathematical Models for Unstable Quantum Systems and Gamow States

We review some results in the theory of non-relativistic quantum unstable systems. We account for the most important definitions of quantum resonances that we identify with unstable quantum systems. Then, we recall the properties and construction of Gamow states as vectors in some extensions of Hilbert spaces, called Rigged Hilbert Spaces. Gamow states account for the purely exponential decaying part of a resonance; the experimental exponential decay for long periods of time physically characterizes a resonance. We briefly discuss one of the most usual models for resonances: the Friedrichs model. Using an algebraic formalism for states and observables, we show that Gamow states cannot be pure states or mixtures from a standard view point. We discuss some additional properties of Gamow states, such as the possibility of obtaining mean values of certain observables on Gamow states. A modification of the time evolution law for the linear space spanned by Gamow shows that some non-commuting observables on this space become commuting for large values of time. We apply Gamow states for a possible explanation of the Loschmidt echo.     

多能级原子与多模光场的相互作用机理研究

研究了多能级原子与多模光场的相互作用.探讨了旋波近似下N能级原子与(N-1)模光场相互作用演化规律，给出了相互作用绘景中其相应的Schrdinger方程的解析通式，分析了三能级原子、四能级原子分别与双模、三模光场相互作用的演化规律.研究结果表明：当原子初始时刻处于基态时，三能级原子与四能级原子的基态概率幅具有相同的变化规律. 关键词： 多能级原子 多模光场 旋波近似     

Time reversible evolution via nonadiabatic coupling in adiabatic dark subspace

We propose a method for the creation of arbitrary superposition of N atomic states using generalized stimulated Raman adiabatic passage (STIRAP) techniques with laser fields coupling each one of N lower states to a single upper state in a (N+1)-level atomic system. (N-1) dark states that are composed of N lower states span a dark subspace. In the adiabatic limit, the dark and bright subspaces are decoupled, thus the nonadiabatic interaction within this dark subspace dominates the evolution of the system. Different from general methods to create our required coherent superposition state, in a reverse way, here we consider the required state as the starting point of evolution dynamics, and utilize laser fields to drive it into a single lower state step by step. Time reverse pulses of laser fields return the single lower state back to our required coherent superposition state based on time reversal symmetry. In principle, the computationally simple method allows the case with a large value of N. Based on the STIRAP techniques, it is robust against small variations of parameters of laser pulses and is immune to spontaneous radiation.     

Probability representation and quantumness tests for qudits and two-mode light states

Using the tomographic-probability representation of spin states, the quantum behavior of qudits is examined. For a general j-qudit state, we propose an explicit formula for quantumness witness whose negative average value is incompatible with classical statistical model. The probability representations of quantum and classical (2j + 1)-level systems are compared within the framework of quantumness tests. In view of the Jordan–Schwinger map, the method is extended for checking the quantumness of two-mode light states.     

Vector states for single and multiple-pole resonances

The formulation of quantum mechanics in rigged Hilbert spaces is used to study the vector states for resonance states or Gamow vectors. An important part of the work is devoted to the construction of Gamow vectors for resonances that appear as multiple poles on the analytic continuation of theS-matrix,S(E). The kinematical behavior of these vectors is also studied. This construction allow for generalized spectral decompositions of the Hamiltonian and the evolutionary semigroups, valid on certain locally convex spaces. Also a first attempt is made to define the resonance states as densities in an extension of the Liouville space, here called rigged Liouville space.     

Gamow Vectors in Exactly Solvable Models

Through two examples: the Friedrichs model and a particular case of central potential scattering, we illustrate the way of constructing Gamow vectors.     

2能级模型处理长脉冲圆偏振光与钠原子相互作用的合理性

 采用密度矩阵方法来处理长脉冲光与钠原子的相互作用,求解24能级钠精细结构的布洛赫方程,给出参与跃迁的每个态密度随时间的演化曲线。对于圆偏振情形,发现在长脉冲(大于100 ns)的持续时间内,最终的原子通过转移只在3S_1/2(2,2)到3P_/3/2(3,3)之间进行泵浦并且激发态达到稳态。进一步的2能级模型计算与24能级计算结果的对比表明：用2能级近似的速率方程方法取代24能级密度矩阵方法具有合理性,由此简化了模型,节约了后续研究的计算量。     

A remark on the optimal cloning of an N-level quantum system

We study quantum cloning machines (QCM) that act on an unknown N-level quantum state and make M copies. We give a formula for the maximum of the fidelity of cloning and exhibit the unitary transformations that realize this optimal fidelity. We also extend the results to treat the case of M copies from () identical N-level quantum systems. Received 21 September 1999     

Large-scale shell model calculations for exotic nuclei

Recent shell model calculations for the neutron-rich nuclei around the magic numbers N = 20, N = 28 and N = 40 are reviewed. We stress two points: i) The crucial role played by the monopole part of the effective interaction that determines the evolution of the spherical mean field. In particular, the reduction in the quasiparticle gaps at the magic numbers can erode or even erase the shell closures. ii) The rich variety of structures that can be found in these situations, with coexisting deformed and spherical states, rapid changes of behaviour with N or Z, and the massive occurrence of intruder states as ground states. Received: 21 March 2002 / Accepted: 16 May 2002 / Published online: 31 October 2002 RID="a" ID="a"e-mail: alfredo.poves@uam.es     

Discrete-time quantum walks on one-dimensional lattices

In this paper, we study discrete-time quantum walks on one-dimensional lattices. We find that the coherent dynamics depends on the initial states and coin parameters. For infinite size of lattices, we derive an explicit expression for the return probability, which shows scaling behavior P(0, t) ~ t ^-1 and does not depends on the initial states of the walk. In the long-time limit, the probability distribution shows various patterns, depending on the initial states, coin parameters and the lattice size. The time-averaged probability mixes to the limiting probability distribution in linear time, i.e., the mixing time M _ε is a linear function of N (size of the lattices) for large values of thresholds ϵ. Finally, we introduce another kind of quantum walk on infinite or even-numbered size of lattices, and show that by the method of mathematical induction, the walk is equivalent to the traditional quantum walk with symmetrical initial state and coin parameter.     

Systematic study of survival probability of excited superheavy nuclei

The stability of excited superheavy nuclei (SHN) with 100 ⩽ Z ⩽ 134 against neutron emission and fission is investigated by using a statistical model. In particular, a systematic study of the survival probability against fission in the 1n-channel of these SHN is made. The present calculations consistently take the neutron separation energies and shell correction energies from the calculated results of the finite range droplet model which predicts an island of stability of SHN around Z = 115 and N = 179. It turns out that this island of stability persists for excited SHN in the sense that the calculated survival probabilities in the 1n-channel of excited SHN at the optimal excitation energy are maximized around Z = 115 and N = 179. This indicates that the survival probability in the 1n-channel is mainly determined by the nuclear shell effects.

     

Truncated Gamow functions, α-decay and the exponential law

For a quantum mechanical two-bodys-wave resonance we prove that the evolution of square integrable approximations of the Gamow function is outgoing and exponentially damped. An error estimate is given in terms of resonance energy and explicity. We obtain the Breit-Wigner form. The results are used in an -decay model to prove general validity of the exponential decay law for periods of several lifetimes.     

Decay of a thermofield-double state in chaotic quantum systems

Scrambling in interacting quantum systems out of equilibrium is particularly effective in the chaotic regime. Under time evolution, initially localized information is said to be scrambled as it spreads throughout the entire system. This spreading can be analyzed with the spectral form factor, which is defined in terms of the analytic continuation of the partition function. The latter is equivalent to the survival probability of a thermofield double state under unitary dynamics. Using random matrices from the Gaussian unitary ensemble (GUE) as Hamiltonians for the time evolution, we obtain exact analytical expressions at finite N for the survival probability. Numerical simulations of the survival probability with matrices taken from the Gaussian orthogonal ensemble (GOE) are also provided. The GOE is more suitable for our comparison with numerical results obtained with a disordered spin chain with local interactions. Common features between the random matrix and the realistic disordered model in the chaotic regime are identified. The differences that emerge as the spin model approaches a many-body localized phase are also discussed.     

Bethe-ansatz solution of the groundstate of a model for mixed-valent Ce and Yb impurities in metals

We consider an impurity model consistent of the 4f ⁰-singlet and a multiplet of total angular momentumj of the 4f ¹ configuration hybridized with conduction states of the metal. The model is solved by Bethe-ansatz and related to theU limit of the Anderson model. If thef-level is far below the Fermi level the equations are reduced to the Bethe-ansatz solution of the Coqblin-Schrieffer model. Approximate analytical results valid in the mixed valence regime are given. A numerical solution for the impurity groundstate energy,f-level occupancy and charge susceptibility as a function of thef-level energy and in the absence of a magnetic field is presented.On a Heisenberg fellowship of the Deutsche Forschungsgemeinschaft     

Relativistic Gamow Vectors

Whereas in Dirac quantum mechanics and relativistic quantum field theory one uses Schwartz space distributions, the extensions of the Hilbert space that we propose uses Hardy spaces. The in- and out-Lippmann-Schwinger kets of scattering theory are functionals in two rigged Hilbert space extensions of the same Hilbert space. This hypothesis also allows to introduce generalized vectors corresponding to unstable states, the Gamow kets. Here the relativistic formulation of the theory of unstable states is presented. It is shown that the relativistic Gamow vectors of the unstable states, defined by a resonance pole of the S-matrix, are classified according to the irreducible representations of the semigroup of the Poincaré transformations (into the forward light cone). As an application the problem of the mass definition of the intermediate vector boson Z is discussed and it is argued that only one mass definition leads to the exponential decay law, and that is not the standard definition of the on-the-mass-shell renormalization scheme.     

A hierarchy of Hamilton operators and entanglement

We consider a hierarchy of Hamilton operators Ĥ _N in finite dimensional Hilbert spaces . We show that the eigenstates of Ĥ _N are fully entangled for N even. We also calculate the unitary operator U _N (t) = exp(—Ĥ _N t/ħ) for the time evolution and show that unentangled states can be transformed into entangled states using this operator. We also investigate energy level crossing for this hierarchy of Hamilton operators.