Ionization of Atoms in a Thermal Field

We study the stationary states of a quantum mechanical system describing an atom coupled to black-body radiation at positive temperature. The stationary states of the non-interacting system are given by product states, where the particle is in a bound state corresponding to an eigenvalue of the particle Hamiltonian, and the field is in its equilibrium state. We show that if Fermi's Golden Rule predicts that a stationary state disintegrates after coupling to the radiation field then it is unstable, provided the coupling constant is sufficiently small (depending on the temperature). The result is proven by analyzing the spectrum of the thermal Hamiltonian (Liouvillian) of the system within the framework of W ^*-dynamical systems. A key element of our spectral analysis is the positive commutator method.     

Exact calculation,using angular momentum,of combined Zeeman and quadrupolar interactions in NMR

The NMR of nuclei with spins greater than ½ is often strongly influenced by the quadrupole interaction. This combination of Zeeman and quadrupole terms can usually be treated using perturbation theory, but an exact calculation is also needed. We explain an exact approach that eliminates the evaluation of commutators of complicated operators. Instead, the calculation is based on matrix elements of the Liouvillian, the commutator with the Hamiltonian. The spectrum can then be calculated directly from the eigenvectors and eigenvalues of the Liouvillian. With the aid of angular momentum methods, it can be shown that the quadupole interaction for spin I is fully determined by only (2I ?1) reduced matrix elements—for spin 3/2, this means only two quantities. The exact nature of the various basis operators is not needed, since the calculation only needs the angular momentum quantum numbers. The full Liouvillian matrix can be calculated from selection rules and the Wigner-Eckart theorem. Furthermore, we present an expression for these reduced matrix elements which is valid for any spin. This theory covers the whole range from quadrupole-perturbed NMR spectra to Zeeman-perturbed nuclear quadrupole resonance.     

Variational quantum eigensolvers by variance minimization

The original variational quantum eigensolver (VQE) typically minimizes energy with hybrid quantum-classical optimization that aims to find the ground state. Here, we propose a VQE based on minimizing energy variance and call it the variance-VQE, which treats the ground state and excited states on the same footing, since an arbitrary eigenstate for a Hamiltonian should have zero energy variance. We demonstrate the properties of the variance-VQE for solving a set of excited states in quantum chemistry problems. Remarkably, we show that optimization of a combination of energy and variance may be more efficient to find low-energy excited states than those of minimizing energy or variance alone. We further reveal that the optimization can be boosted with stochastic gradient descent by Hamiltonian sampling, which uses only a few terms of the Hamiltonian and thus significantly reduces the quantum resource for evaluating variance and its gradients.     

Rapid state reduction of quantum systems using feedback control

We consider using Hamiltonian feedback control to increase the speed at which a continuous measurement purifies (reduces) the state of a quantum system, and thus to increase the speed of the preparation of pure states. For a measurement of an observable with equispaced eigenvalues, we show that there exists a feedback algorithm which will speed up the rate of state reduction by at least a factor of 2(N + 1)/3.     

A New Approach to the {\ast} -Genvalue Equation

We show that the eigenvalues and eigenfunctions of the star-genvalue equation can be completely expressed in terms of the corresponding eigenvalue problem for the quantum Hamiltonian. Our methods make use of a Weyl-type representation of the star-product and of the properties of the cross-Wigner transform, which appears as an intertwining operator.     

Quantum field theoretical description of unstable behavior of trapped Bose-Einstein condensates with complex eigenvalues of Bogoliubov-de Gennes equations

The Bogoliubov-de Gennes equations are used for a number of theoretical works on the trapped Bose-Einstein condensates. These equations are known to give the energies of the quasi-particles when all the eigenvalues are real. We consider the case in which these equations have complex eigenvalues. We give the complete set including those modes whose eigenvalues are complex. The quantum fields which represent neutral atoms are expanded in terms of the complete set. It is shown that the state space is an indefinite metric one and that the free Hamiltonian is not diagonalizable in the conventional bosonic representation. We introduce a criterion to select quantum states describing the metastablity of the condensate, called the physical state conditions. In order to study the instability, we formulate the linear response of the density against the time-dependent external perturbation within the regime of Kubo’s linear response theory. Some states, satisfying all the physical state conditions, give the blow-up and damping behavior of the density distributions corresponding to the complex eigenmodes. It is qualitatively consistent with the result of the recent analyses using the time-dependent Gross-Pitaevskii equation.     

Supersymmetric quantum hall liquid with a deformed supersymmetry

We construct a supersymmetric quantum Hall liquid with a deformed supersymmetry. One parameter is introduced in the supersymmetric Laughlin wavefunction to realize the original Laughlin wavefunction and the Moore—Read wavefunction in two extremal limits of the parameter. The introduced parameter corresponds to the coherence factor in the BCS theory. It is pointed out that the parameter-dependent supersymmetric Laughlin wavefunction enjoys a deformed supersymmetry. Based on the deformed supersymmetry, we construct a pseudopotential Hamiltonian whose groundstate is exactly the parameter-dependent supersymmetric Laughlin wavefunction. Though the SUSY pseudopotential Hamiltonian is parameter-dependent and non-Hermitian, its eigenvalues are parameter-independent and real.     

Quantum Gibbs Samplers: The Commuting Case

We analyze the problem of preparing quantum Gibbs states of lattice spin Hamiltonians with local and commuting terms on a quantum computer and in nature. Our central result is an equivalence between the behavior of correlations in the Gibbs state and the mixing time of the semigroup which drives the system to thermal equilibrium (the Gibbs sampler). We introduce a framework for analyzing the correlation and mixing properties of quantum Gibbs states and quantum Gibbs samplers, which is rooted in the theory of non-commutative \({\mathbb{L}_p}\) spaces. We consider two distinct classes of Gibbs samplers, one of them being the well-studied Davies generator modelling the dynamics of a system due to weak-coupling with a large Markovian environment. We show that their spectral gap is independent of system size if, and only if, a certain strong form of clustering of correlations holds in the Gibbs state. Therefore every Gibbs state of a commuting Hamiltonian that satisfies clustering of correlations in this strong sense can be prepared efficiently on a quantum computer. As concrete applications of our formalism, we show that for every one-dimensional lattice system, or for systems in lattices of any dimension at temperatures above a certain threshold, the Gibbs samplers of commuting Hamiltonians are always gapped, giving an efficient way of preparing the associated Gibbs states on a quantum computer.     

Variational method for lattice systems: General formalism and application to the two-dimensional Ising model in an external field

A simple variational approach to the eigenvalue problem of the transfer operator is proposed. After reducing the transfer operator according to the symmetries of the Hamiltonian, the leading eigenvalues of the irreducible blocks can be evaluated by elementary variational principles. Hence the thermodynamics and a large class of correlation functions of lattice systems can be calculated. Following a natural truncation scheme the results can be improved in a systematic way. The high accuracy and the convergence of the method is demonstrated by two-dimensional Ising model. As a first application, the thermodynamics of the two-dimensional Ising ferro-and antiferromagnet in an external field is studied. We show how the same method can be used to obtain zero-temperature properties of interacting quantum lattice systems.     

Time evolution of quasi-stationary states

In this paper we study the time evolution of prepared states in some quantum mechanical models, and discuss the probability of decay and the rate of energy dissipation and their dependence on the form of the interaction. First we consider solvable models with divergent matrix elements for the operatorH ², whereH is the Hamiltonian of the system. We study two specific examples, one with well-defined eigenvalues and the other with renormalizable interaction. The time development of the initial state in the latter case depends on the cut-off parameter. In the second part of the paper, we show the possibility of existence of decaying states with long lifetime, where the amplitude of the initial state decreases like a Bessel function. In the third part, we determine the time development of a prepared state in a simple many-boson problem. Finally we study the problem of penetration of a wave packet through two phase-equivalent potential barriers, and we conclude that from the scattering phase shifts alone, it is not possible to determine the lifetime or the mode of decay of an unstable particle uniquely.     

含时边界条件和Berry相位

对一具有含时边界条件的量子体系,有效哈密顿算符可用一种简单的方式去构造而不涉及任何几何处理。用这种方法构造了一个在半径随时间变化的球形盒子内的量子粒子的有效哈密顿算符,并用它来计算波函数的Berry相位。发现有效哈密顿算符与原哈密顿算符在形式上由一静态规范变换相关联。这两个哈密顿算符的量子态差Berry相位。 关键词：     

原子-异核-三聚物分子转化系统暗态的动力学不稳定性

研究了受激拉曼绝热过程中原子-异核-三聚物分子转化系统暗态的动力学稳定性.通过将量子哈密顿对应到经典哈密顿,并求解和分析线性化经典运动方程后得到的哈密顿-雅克比矩阵本征值,解析地得到了原子-三聚物暗态的动力学不稳定性发生的条件.并以异核原子⁸⁷Rb和⁴¹K混合凝聚体为例,数值地给出了系统发生动力学不稳定性的区域.研究发现,这种动力学不稳定性是由粒子之间的相互作用带来的.此外,还发现系统动力学不稳定性的发生不仅与哈密顿-雅克比矩阵是否出现实数或复数的本征值有关,还 关键词： 原子-异核-三聚物分子转化系统 暗态 受激拉曼绝热过程 动力学不稳定性     

Product-State Approximations to Quantum States

We show that for any many-body quantum state there exists an unentangled quantum state such that most of the two-body reduced density matrices are close to those of the original state. This is a statement about the monogamy of entanglement, which cannot be shared without limit in the same way as classical correlation. Our main application is to Hamiltonians that are sums of two-body terms. For such Hamiltonians we show that there exist product states with energy that is close to the ground-state energy whenever the interaction graph of the Hamiltonian has high degree. This proves the validity of mean-field theory and gives an explicitly bounded approximation error. If we allow states that are entangled within small clusters of systems but product across clusters then good approximations exist when the Hamiltonian satisfies one or more of the following properties: (1) high degree, (2) small expansion, or (3) a ground state where the blocks in the partition have sublinear entanglement. Previously this was known only in the case of small expansion or in the regime where the entanglement was close to zero. Our approximations allow an extensive error in energy, which is the scale considered by the quantum PCP (probabilistically checkable proof) and NLTS (no low-energy trivial-state) conjectures. Thus our results put restrictions on the possible Hamiltonians that could be used for a possible proof of the qPCP or NLTS conjectures. By contrast the classical PCP constructions are often based on constraint graphs with high degree. Likewise we show that the parallel repetition that is possible with classical constraint satisfaction problems cannot also be possible for quantum Hamiltonians, unless qPCP is false. The main technical tool behind our results is a collection of new classical and quantum de Finetti theorems which do not make any symmetry assumptions on the underlying states.     

Quantum deleting and cloning in a pseudo-unitary system

In conventional quantum mechanics, quantum no-deleting and no-cloning theorems indicate that two different and nonorthogonal states cannot be perfectly and deterministically deleted and cloned, respectively. Here, we investigate the quantum deleting and cloning in a pseudo-unitary system. We first present a pseudo-Hermitian Hamiltonian with real eigenvalues in a two-qubit system. By using the pseudo-unitary operators generated from this pseudo-Hermitian Hamiltonian, we show that it is possible to delete and clone a class of two different and nonorthogonal states, and it can be generalized to arbitrary two different and nonorthogonal pure qubit states. Furthermore, state discrimination, which is strongly related to quantum no-cloning theorem, is also discussed. Last but not least, we simulate the pseudo-unitary operators in conventional quantum mechanics with post-selection, and obtain the success probability of simulations. Pseudo-unitary operators are implemented with a limited efficiency due to the post-selections. Thus, the success probabilities of deleting and cloning in the simulation by conventional quantum mechanics are less than unity, which maintain the quantum no-deleting and no-cloning theorems.     

Dissipative quantum Church-Turing theorem

We show that the time evolution of an open quantum system, described by a possibly time dependent Liouvillian, can be simulated by a unitary quantum circuit of a size scaling polynomially in the simulation time and the size of the system. An immediate consequence is that dissipative quantum computing is no more powerful than the unitary circuit model. Our result can be seen as a dissipative Church-Turing theorem, since it implies that under natural assumptions, such as weak coupling to an environment, the dynamics of an open quantum system can be simulated efficiently on a quantum computer. Formally, we introduce a Trotter decomposition for Liouvillian dynamics and give explicit error bounds. This constitutes a practical tool for numerical simulations, e.g., using matrix-product operators. We also demonstrate that most quantum states cannot be prepared efficiently.     

Transitionless driving on local adiabatic quantum search algorithm

We apply the transitionless driving on the local adiabatic quantum search algorithm to speed up the adiabatic process.By studying quantum dynamics of the adiabatic search algorithm with the equivalent two-level system, we derive the transitionless driving Hamiltonian for the local adiabatic quantum search algorithm. We found that when adding a transitionless quantum driving term H_D(t) on the local adiabatic quantum search algorithm, the success rate is 1 exactly with arbitrary evolution time by solving the time-dependent Schr odinger equation in eigen-picture. Moreover, we show the reason for the drastic decrease of the evolution time is that the driving Hamiltonian increases the lowest eigenvalues to a maximum of ON~(1/2).     

Exact solutions of a new elliptic Calogero–Sutherland model

A quantum Hamiltonian describing N particles on a line interacting pairwise via an elliptic function potential in the presence of an external field is introduced. For a discrete set of values of the strength of the external potential, it is shown that a finite number of eigenfunctions and eigenvalues of the model can be exactly computed in an algebraic way.     

Quadratic and Coulomb Terms for the Spectrum of a Three-Electron Quantum Dot

We consider the Hamiltonian of a three-electron quantum dot composed of quadratic plus Coulomb terms and calculate the system’s spectra. We next apply the hyperradius to reduce the three-body Schrödinger equation into a one-variable differential equation that is solvable. To avoid the complexity, the Taylor expansion of the effective potential is enters the problem and thereby a solution is found for the eigenvalues of the corresponding three-body Schrödinger equation in terms of the Wigner parameter.     

Adiabatic switching approach to multidimensional supersymmetric quantum mechanics for several excited states

We present a time-dependent method for determining several approximate excited-state energies and wave functions using a vectorial approach to multidimensional supersymmetric quantum mechanics. First, a vectorial approach is used to generate the tensor sector two Hamiltonian, which is isospectral with the original scalar sector one Hamiltonian above the ground state of the sector one Hamiltonian. We construct a time-dependent Hamiltonian interpolating between the scalar sector one Hamiltonian and the tensor sector two Hamiltonian. Then, we can adiabatically switch from the ground state of the sector one Hamiltonian to the ground state of the sector two Hamiltonian by solving the time-dependent Schrödinger equation. In addition, by employing an initial wave packet orthogonal to that leading to the ground state of sector two, we also obtain the first-excited state of sector two. Construction of the orthogonal sector one states is trivial due to the tensor nature of sector two. The ground and first-excited states of the sector two Hamiltonian can be used with the charge operator to obtain the first two excited state wave functions of the sector one Hamiltonian. Excellent computational results are obtained for two-dimensional nonseparable degenerate and nondegenerate systems.     

Discrete representation of the spectral shift function and the multichannel <Emphasis Type="Italic">S</Emphasis>-matrix

A new method for solving the multichannel quantum scattering problem in a wide energy range based on the single diagonalization of the Hamiltonian matrix of the system in a finite-dimensional basis is briefly described. It has been shown that the interaction-matrix-induced shifts of the eigenvalues of the free Hamiltonian matrix in the continuous spectrum are directly related to the partial phase shifts. The two-channel scattering problem with shifted channel thresholds is considered for illustration.