The Quantum Geometric Tensor in a Parameter-Dependent Curved Space

We introduce a quantum geometric tensor in a curved space with a parameter-dependent metric, which contains the quantum metric tensor as the symmetric part and the Berry curvature corresponding to the antisymmetric part. This parameter-dependent metric modifies the usual inner product, which induces modifications in the quantum metric tensor and Berry curvature by adding terms proportional to the derivatives with respect to the parameters of the determinant of the metric. The quantum metric tensor is obtained in two ways: By using the definition of the infinitesimal distance between two states in the parameter-dependent curved space and via the fidelity susceptibility approach. The usual Berry connection acquires an additional term with which the curved inner product converts the Berry connection into an object that transforms as a connection and density of weight one. Finally, we provide three examples in one dimension with a nontrivial metric: an anharmonic oscillator, a Morse-like potential, and a generalized anharmonic oscillator; and one in two dimensions: the coupled anharmonic oscillator in a curved space.     

Born-Oppenheimer近似下谐振子场驱动电磁模系统的Berry相和Hannay角

研究了Born-Oppenheimer近似下谐振子场驱动电磁模系统的Berry相和Hannay角, 通过理论计算得到了其表达式, 并讨论了这二者之间的半经典关系.结果表明, 这一量子Born-Oppenheimer复合系统的Berry相包含两部分: 第一部分与通常几何相的定义相同, 另一项则是由耦合造成的有效规范式引入的.这一量子修正可以被看作一个等效的Aharonov-Bohm效应.不仅如此, 其对应经典系统的Hannay角的定义中也存在类似的现象. 由此可见, 这一复合系统的Berry相与Hannay角之间也存在半经典关系, 并与文献[16] 中通常情况下的半经典关系相同.此外, 上述理论也可以运用于解决产生中性原子的人造规范势等物理问题. 关键词： Berry相 Hannay角 量子经典对应 Born-Oppenheimer近似     

Stress Tensor Fluctuations and Passive Quantum Gravity

The quantum fluctuations of the stress tensor of a quantum field are discussed, as are the resulting space-time metric fluctuations. Passive quantum gravity is an approximation in which gravity is not directly quantized, but fluctuations of the space-time geometry are driven by stress tensor fluctuations. We discuss a decomposition of the stress tensor correlation function into three parts, and consider the physical implications of each part. The operational significance of metric fluctuations and the possible limits of validity of semiclassical gravity are discussed.     

Warm Inflation and Scalar Perturbations of the Metric

A second-order expansion for the quantum fluctuations of the matter field was considered in the framework of the warm inflation scenario. The friction and Hubble parameters were expanded by means of a semiclassical approach. The fluctuations of the Hubble parameter generates fluctuations of the metric. These metric fluctuations produce an effective term of curvature. The power spectrum for the metric fluctuations can be calculated on the infrared sector.     

谐振子系统的量子-经典轨道、Berry相及Hannay角

从量子-经典轨道和几何相两方面, 研究了二维旋转平移谐振子系统的量子-经典对应. 通过广义规范变换得到了Lissajous经典周期轨道和Hannay角. 另外, 使用含时规范变换解析推导了旋转平移谐振子系统Schrödinger方程的本征波函数和Berry相, 得出结论: 原规范中的非绝热Berry相是经典Hannay角的-n倍. 最后, 使用SU(2)自旋相干态叠加, 构造一稳态波函数, 其波函数的概率云很好地局域于经典轨道上, 满足几何相位和经典轨道同时对应.     

Riemannian structure on manifolds of quantum states

A metric tensor is defined from the underlying Hilbert space structure for any submanifold of quantum states. The case where the manifold is generated by the action of a Lie group on a fixed state vector (generalized coherent states manifold hereafter noted G.C.S.M.) is studied in details; the geometrical properties of some wellknown G.C.S.M. are reviewed and an explicit expression for the scalar Riemannian curvature is given in the general case. The physical meaning of such Riemannian structures (which have been recently introduced to describe collective manifolds in nuclear physics) is discussed. It is shown on examples that the distance between nearby states is related to quantum fluctuations; in the particular case of the harmonic oscillator group the condition of zero curvature appears to be identical to that of non dispersion of wave packets.Equipe de Recherche Associée au C.N.R.S.     

Dislocations and torsion in graphene and related systems

A continuum model to study the influence of dislocations on the electronic properties of condensed matter systems is described and analyzed. The model is based on a geometrical formalism that associates a density of dislocations with the torsion tensor and uses the technique of quantum field theory in curved space. When applied to two-dimensional systems with Dirac points like graphene we find that dislocations couple in the form of vector gauge fields similar to these arising from curvature or elastic strain. We also describe the ways to couple dislocations to normal metals with a Fermi surface.     

Bohmian trajectory perspective on strong field atomic processes

The interaction of an atom with an intense laser field provides an important approach to explore the ultrafast electron dynamics and extract the information of the atomic and molecular structures with unprecedented attosecond temporal and angstrom spatial resolution. To well understand the strong field atomic processes, numerous theoretical methods have been developed, including solving the time-dependent Schr ?dinger equation(TDSE), classical and semiclassical trajectory method, quantum S-matrix theory within the strong-field approximation, etc. Recently, an alternative and complementary quantum approach, called Bohmian trajectory theory, has been successfully used in the strong-field atomic physics and an exciting progress has been achieved in the study of strong-field phenomena. In this paper, we provide an overview of the Bohmian trajectory method and its perspective on two strong field atomic processes, i.e., atomic and molecular ionization and high-order harmonic generation, respectively.     

Finite Temperature Semiclassical Approach for Local Potentials

The nucleon distribution, kinetic energy density, free energy density and entropy density up to the second order in h are derived in a finite temperature semiclassical approach for two local potentials, harmonic oscillator and Woods-Saxon potentiale. The preaent results are compared with the exact quantum reaults for both potentials. The comparison justifies the extension of semiclassical approximation from zero temperature to finite temperature.     

Instability of flat spacetime in semiclassical gravity

The semiclassical theory of gravity is considered in which an asymptotically flat background metric is coupled to quantized matter. We show that, in general, there are modes with spacelike wave vectors for small metric fluctuations around flat spacetime. Besides the usual axioms of quantum field theory in flat spacetime, the proof rests on the existence of a hard trace anomaly in the energy-momentum tensor due to matter self-couplings. Two possible interpretations of the result are discussed.     

Harmonic oscillator in Snyder space: The classical case and the quantum case

The harmonic oscillator in Snyder space is investigated in its classical and quantum versions. The classical trajectory is obtained and the semiclassical quantization from the phase space trajectories is discussed. An effective cut-off to high frequencies is found. The quantum version is developed and an equivalent usual harmonic oscillator is obtained through an effective mass and an effective frequency introduced in the model. This modified parameters give us a modified energy spectrum also.     

Selected topics of quantum computing for nuclear physics

Nuclear physics,whose underling theory is described by quantum gauge field coupled with matter,is fundamentally important and yet is formidably challenge for simulation with classical computers.Quantum computing provides a perhaps transformative approach for studying and understanding nuclear physics.With rapid scaling-up of quantum processors as well as advances on quantum algorithms,the digital quantum simulation approach for simulating quantum gauge fields and nuclear physics has gained lots of attention.In this review,we aim to summarize recent efforts on solving nuclear physics with quantum computers.We first discuss a formulation of nuclear physics in the language of quantum computing.In particular,we review how quantum gauge fields(both Abelian and non-Abelian)and their coupling to matter field can be mapped and studied on a quantum computer.We then introduce related quantum algorithms for solving static properties and real-time evolution for quantum systems,and show their applications for a broad range of problems in nuclear physics,including simulation of lattice gauge field,solving nucleon and nuclear structures,quantum advantage for simulating scattering in quantum field theory,non-equilibrium dynamics,and so on.Finally,a short outlook on future work is given.     

Squeezed states of light as self-similar structures

A new approach to investigating a broad class of dynamic states for a quantum oscillator is suggested. It is based on an invariant transformation of the equation to a new time determined by the quantum dispersion of the corresponding state. The squeezed states of a quantum system generated by the ground-state wave function are constructed. In coordinate representation, these states are described by a self-similar wave function localized near a classical trajectory. The statistics of the squeezed state of light is analyzed in the single-mode approximation. The parametric excitation of squeezed states for a quantum harmonic oscillator is considered.     

Majorana's stellar representation for the quantum geometric tensor of symmetric states

Majorana's stellar representation provides an intuitive picture in which quantum states in high-dimensional Hilbert space can be observed using the trajectory of Majorana stars. We consider the Majorana's stellar representation of the quantum geometric tensor for a spin state up to spin-3/2. The real and imaginary parts of the quantum geometric tensor, corresponding to the quantum metric tensor and Berry curvature, are therefore obtained in terms of the Majorana stars. Moreover, we work out the expressions of quantum geometric tensor for arbitrary spin in some important cases. Our results will benefit the comprehension of the quantum geometric tensor and provide interesting relations between the quantum geometric tensor and Majorana's stars.     

可积系统——求迹公式和h逆谱分析(英文)

研究了二维无关联四次振子系统,有理环面上积分 Hamiltonian运动方程给出了系统一系列周期轨道和经典物理量 ,使用半经典近似下的 Berry- Tabor求迹公式,得到了半经典的态密度.应用 Fourier变换分析了每条周期轨道对态密度的贡献,并与量子态密度的 Fourier变换结果比较证实了半经典求迹公式的有效性.Periodic orbits of two dimensional uncoupled quartic oscillator were calculated by inte grating Hamiltonian equations of motion on reasonable tori, and several classical quantities were also computed. Inserting them into Berry Tabor trace formula, a trace, i.e., the semiclassical density of states of the corresponding quantum system, was obtained. Finally, Fourier transform was adopted to verify the contribution of each periodic orbit. Good agreement between the semiclassical ...     

Geometric curvature and phase of the Rabi model

We study the geometric curvature and phase of the Rabi model. Under the rotating-wave approximation (RWA), we apply the gauge independent Berry curvature over a surface integral to calculate the Berry phase of the eigenstates for both single and two-qubit systems, which is found to be identical with the system of spin-1/2 particle in a magnetic field. We extend the idea to define a vacuum-induced geometric curvature when the system starts from an initial state with pure vacuum bosonic field. The induced geometric phase is related to the average photon number in a period which is possible to measure in the qubit–cavity system. We also calculate the geometric phase beyond the RWA and find an anomalous sudden change, which implies the breakdown of the adiabatic theorem and the Berry phases in an adiabatic cyclic evolution are ill-defined near the anti-crossing point in the spectrum.     

Emergent Newtonian dynamics and the geometric origin of mass

We consider a set of macroscopic (classical) degrees of freedom coupled to an arbitrary many-particle Hamiltonian system, quantum or classical. These degrees of freedom can represent positions of objects in space, their angles, shape distortions, magnetization, currents and so on. Expanding their dynamics near the adiabatic limit we find the emergent Newton’s second law (force is equal to the mass times acceleration) with an extra dissipative term. In systems with broken time reversal symmetry there is an additional Coriolis type force proportional to the Berry curvature. We give the microscopic definition of the mass tensor. The mass tensor is related to the non-equal time correlation functions in equilibrium and describes the dressing of the slow degree of freedom by virtual excitations in the system. In the classical (high-temperature) limit the mass tensor is given by the product of the inverse temperature and the Fubini–Study metric tensor determining the natural distance between the eigenstates of the Hamiltonian. For free particles this result reduces to the conventional definition of mass. This finding shows that any mass, at least in the classical limit, emerges from the distortions of the Hilbert space highlighting deep connections between any motion (not necessarily in space) and geometry. We illustrate our findings with four simple examples.     

Semiclassical diagonalization of quantum Hamiltonian and equations of motion with Berry phase corrections

It has been recently found that the equations of motion of several semiclassical systems must take into account terms arising from Berry phases contributions. Those terms are responsible for the spin Hall effect in semiconductor as well as the Magnus effect of light propagating in inhomogeneous media. Intensive ongoing research on this subject seems to indicate that a broad class of quantum systems may be affected by Berry phase terms. It is therefore important to find a general procedure allowing for the determination of semiclassical Hamiltonian with Berry Phase corrections. This article presents a general diagonalization method at order ħ for a large class of quantum Hamiltonians directly inducing Berry phase corrections. As a consequence, Berry phase terms on both coordinates and momentum operators naturally arise during the diagonalization procedure. This leads to new equations of motion for a wide class of semiclassical system. As physical applications we consider here a Dirac particle in an electromagnetic or static gravitational field, and the propagation of a Bloch electrons in an external electromagnetic field.