Non-adiabatic holonomy operators in classical and quantum completely integrable systems
G. Giachetta, L. Mangiarotti, G. Sardanashvily
给定一个完全可积系统，我们将经典和量子完整算子（广义Berry相因子）与参数流形上的不变环面上的任何联系起来，而不需要任何绝热近似.
Given a completely integrable system, we associate to any connection on its invariant tori fibred over a parameter manifold the classical and quantum holonomy operator (generalized Berry's phase factor), without any adiabatic approximation.
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Holonomy control operators in classical and quantum completely integrable Hamiltonian systems
完全可积的哈密顿系统看起来很有希望的可控性，因为他们的首次积分是稳定的内部发展下，人们可能希望找到一个扰动的哈密尔顿驱动的首次积分的意愿。直角坐标系对于这一目的是最方便的。相对于这些坐标，哈密顿量和首次积分仅依赖于作用量。我们引入了一个适当的扰动的哈密顿量的一个长期包含时间依赖的参数，使动作变量的演变只不过是一个位移沿曲线在参数空间。因此，可以通过适当选择参数函数来完全定义这种演化。有限能级量子系统的类完整可控性在量子计算中具有特殊的意义。本文给出了作用角变量下含时完全可积哈密顿系统的几何量子化。其哈密顿量和首次积分具有与时间无关的可数谱。构造了该可数能级量子系统的完整性控制算符。
An adiabatic change of parameters along a closed path may interchange the (quasi-)eigenenergies and eigenspaces of a closed quantum system. Such discrepancies induced by adiabatic cycles are refereed to as the exotic quantum holonomy, which is an extension of the geometric phase. "Small" adiabatic cycles induce no change on eigenspaces, whereas some "large" adiabatic cycles interchange eigenspaces. We explain the topological formulation for the eigenspace anholonomy, where the homotopy equivalence precisely distinguishes the larger cycles from smaller ones. An application to two level systems …
A topological formulation of the eigenspace anholonomy, where eigenspaces are interchanged by adiabatic cycles, is introduced. The anholonomy in two-level systems is identified with a disclination of the director (headless vector) of a Bloch vector, which characterizes eigenprojectors. The covering map structure behind the exotic quantum holonomy and the role of the homotopy classification of adiabatic cycles are elucidated. The extensions of this formulation to nonadiabatic cycles and N-level systems are outlined.
Quasi-holonomy in non-adiabatic quantum evolution
研究非绝热量子时间演化中沿复格拉斯曼流形回路子空间的准全纯性，通过分解薛定谔演化，获依赖动力学传播子的有效几何生成元，探讨多种分解方式及相关问题。
The correspondence between exotic quantum holonomy that occurs in families of Hermitian cycles, and exceptional points (EPs) for non-Hermitian quantum theory is examined in quantum kicked tops. Under a suitable condition, an explicit expressions of the adiabatic parameter dependencies of quasienergies and stationary states, which exhibit anholonomies, are obtained. It is also shown that the quantum kicked tops with the complexified adiabatic parameter have a higher order EP, which is broken into lower order EPs with the application of small perturbations. The stability of exotic holonomy again…
Mixed state non-Abelian holonomy for subsystems
动力学系统中的非阿贝尔完整性可能出现在能量简并态集的绝热输运中。我们研究这样一个holonomy结构的混合物的能量简并量子态。我们证明，这种结构有一个自然的解释，在标准的Wilczek-Zee holonomy与某一类的Hamilton耦合系统的一个附属。分析了利用离子阱进行完整量子计算的混合态完整性。
Adiabatic Berry Phase and Hannay Angle for Open Paths
我们通过定义一个广义规范位，其线积分给出了任意参数演化的相完整性，得到绝热Berry相。考虑到对于经典可积系统，如何获得开路Hannay角是一个很困难的问题，我们利用参数化相干态方法建立了开路Berry相位和Hannay角之间的联系.利用半经典波函数分析了开路Berry相位，得到开路Hannay角。进一步，通过用瞬时投影子的交换子及其微分表示绝热Berry相位，并利用算子的Wigner表示，得到分布函数及其微分之间的Poisson括号。这使我们能够讨论相位完整性的经典极限，它产生了开放路径的角度完整性。基于量子力学内积经典极限的思想，给出了汉内角的一个操作定义.本文还指出了开路Berry相位和Hannay角在波包复苏现象中的可能应用。
Geometric Phase for Non-Hermitian Hamiltonians and Its Holonomy Interpretation
对于任意可能包含例外点的非厄米矩阵哈密顿量H，我们构造了一个适当的参数空间M和M上的线丛L^n，使得初始哈密顿量的本征态的绝热几何相位与L^n的完整性相一致.我们研究的情况下，2 × 2矩阵哈密顿的细节，并表明，相反，在最近的一些出版物中提出的要求，几何相位所产生的环绕例外点一般是几何的，而不是拓扑性质。
An interplay of an exotic quantum holonomy and exceptional points is examined in one-dimensional Bose systems. The eigenenergy anholonomy, in which Hermitian adiabatic cycle induces nontrivial change in eigenenergies, can be interpreted as a manifestation of eigenenergy's Riemann surface structure, where the branch points are identified as the exceptional points which are degeneracy points in the complexified parameter space. It is also shown that the exceptional points are the divergent points of the non-Abelian gauge connection for the gauge theoretical formulation of the eigenspace anholono…
An adiabatic time evolution of a closed quantum system connects eigenspaces of initial and final Hermitian Hamiltonians for slowly driven systems, or, unitary Floquet operators for slowly modulated driven systems. We show that the connection of eigenspaces depends on a topological property of the adiabatic paths for given initial and final points. An example in slowly modulated periodically driven systems is shown. These analysis are based on the topological analysis of the exotic quantum holonomy in adiabatic closed paths.
We study the evolution of quantum eigenstates in the presence of level crossing under adiabatic cyclic change of environmental parameters. We find that exotic holonomies, indicated by exchange of the eigenstates after a single cyclic evolution, can arise from non-Abelian gauge potentials among non-degenerate levels. We illustrate our arguments with solvable two and three level models.
Using shortcuts to adiabaticity, we solve the time-dependent Schroedinger equation that is reduced to a classical nonlinear integrable equation. For a given time-dependent Hamiltonian, the counterdiabatic term is introduced to prevent nonadiabatic transitions. Using the fact that the equation for the dynamical invariant is equivalent to the Lax equation in nonlinear integrable systems, we obtain the counterdiabatic term exactly. The counterdiabatic term is available when the corresponding Lax pair exists and the solvable systems are classified in a unified and systematic way. Multisoliton pote…
A Unified Theory of Quantum Holonomies
量子系统的慢环境参数的周期性变化诱导量子完整性。相位全息学就是一个著名的例子。另一种是更奇特的一种，表现出特征值和特征空间完整性。我们介绍了一个理论公式，描述了相位和本征空间holonomies在平等的基础上。该理论的基石是一个有序基的规范连接，它在概念上不同于Mead-Truhlar-Berry连接及其Wilczek-Zee扩展。基于Fujikawa的形式，发展了本征空间完整性的规范不变处理。绝热量子holonomy的例子，包括奇异的光谱简并，显示。
Evolution Operator Can Always be Separated into the Product of Holonomy and Dynamic Operators
本研究证明量子系统演化算符总能分解为和乐与动力学算符的乘积，给出循环演化分解的矩阵表示及纯和乐条件，统一四类几何相位演化表示。
In this thesis we provide a uniform treatment of two non-adiabatic geometric phases for dynamical systems of mixed quantum states, namely those of Uhlmann and of Sjöqvist et al. We develop a holonomy theory for the latter which we also relate to the already existing theory for the former. This makes it clear what the similarities and differences between the two geometric phases are. We discuss and motivate constraints on the two phases. Furthermore, we discuss some topological properties of the holonomy of `real' quantum systems, and we introduce higher-order geometric phases for not necessari…
We present a formal geometric framework for the study of adiabatic quantum mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This framework generalizes earlier holonomy interpretations of the geometric phase to non-cyclic states appearing for non-Hermitian Hamiltonians. We start with an investigation of the space of non-degenerate operators on a finite-dimensional state space. We then show how the energy bands of a Hamiltonian family form a covering space. Likewise, we show that the eigenrays form a bundle, a generalization of a principal bundle, which admits a natural co…
Exotic quantum holonomy induced by degeneracy hidden in complex parameter space
在参数空间中，束缚态沿闭合回路的绝热变化不仅在态的相位上，而且在相应的本征空间和本征值上都能诱导出完整性。前者是著名的Berry相，而后者，即奇异的holonomy，是十年前发现的，它的起源还没有被理解。通过将该参数推广到复数中，揭示了奇异完整性与非厄米哈密顿量的简并性或例外点的对应关系。我们表明，这解释了所有已知的非平凡特性的异国情调的holonomies。
Frictional work and entropy production in integrable and non-integrable spin chains
本文研究了非可积自旋链中摩擦功与熵产的关系，发现其可由有效温度描述，并探讨了非可积性对功提取的影响。
A new simple proof of the adiabatic theorem is given in the finite dimensional case for nondegenerate as well as degenerate states. The explicitly integrable two level system is considered as an example. It is demonstrated that the error estimate given by the adiabatic theorem can not be improved.
Adiabatic quantum state preparation in integrable models
本文提出利用绝热算法在量子计算机上制备可积分模型的高能本征态，通过热力学 Bethe ansatz 显示算法深度与 qubit 数呈多项式关系，且在非相互作用 XY 自旋链中验证了该方法的效率。
The path dependence of adiabatic evolution in classical harmonic chains with clamps is examined. It is shown that cutting and joining a chain may braid adiabatic normal mode frequencies. Accordingly, different adiabatic paths with the same endpoints may transport a normal mode to a different one, and an adiabatic cycle pumps action variables, i.e., the adiabatic invariants of integrable classical systems. Another adiabatic pump for artificial edge modes induced by clamps is shown as an application. Extensions to completely integrable systems and quantum systems are outlined.
Geometric Phase and Chiral Anomaly in Path Integral Formulation
所有的几何相位，绝热和非绝热，制定在一个统一的方式在第二个量化的路径积分制定。薛定谔方程中固有的精确隐藏的局部对称性定义了完整性。所有的几何相位被证明是拓扑平凡的。的几何相位进行了简要的比较手征异常，这是自然制定的路径积分。
Geometric Phase in Eigenspace Evolution of Invariant and Adiabatic Action Operators
将几何相位理论推广到具有$N$ -倍退化的不变算子的本征空间的循环演化。相应的几何相位被解释为从格拉斯曼流形上的Stiefel U（N）-丛的万有联络继承的完整性。最重要的是，对于任意的初始状态，这个几何相位捕捉到了状态演化的固有几何特征。此外，还讨论了绝热作用量算符本征空间演化中的几何相位，并通过拉回U（N）-丛加以阐述。几个有趣的物理例子说明。
Non-adiabatic holonomic quantum gate in decoherence-free subspaces is of greatly practical importance due to its built-in fault tolerance, coherence stabilization virtues, and short run-time. Here we propose some compact schemes to implement two- and three-qubit controlled unitary quantum gates and Fredkin gate. For the controlled unitary quantum gates, the unitary operator acting on the target qubit is an arbitrary single-qubit gate operation. The controlled quantum gates can be directly implemented using non-adiabatic holonomy in decoherence-free subspaces and the required resource for the d…
The second quantized approach to geometric phases is reviewed. The second quantization generally induces a hidden local (time-dependent) gauge symmetry. This gauge symmetry defines the parallel transport and holonomy, and thus it controls all the known geometric phases, either adiabatic or non-adiabatic, in a unified manner. The transitional region from the adiabatic to non-adiabatic phases is thus analyzed in a quantitative way. It is then shown that the topology of the adiabatic Berry's phase is trivial in a precise sense and also the adiabatic phase is rather fragile against the non-adiabat…
It is shown that adiabatic cycles excite a quantum particle, which is confined in a one-dimensional region and is initially in an eigenstate. During the cycle, an infinitely sharp wall is applied and varied its strength and position. After the completion of the cycle, the state of the particle arrives another eigenstate. It is also shown that we may vary the final adiabatic state by choosing the parameters of the cycle. With a combination of these adiabatic cycles, we can connect an arbitrary pair of eigenstates. Hence, these cycles may be regarded as basis of the adiabatic excitations. A deta…
As a qubit usually has a limited lifetime, its manipulation should be as fast as possible, and thus non-adiabatic operation is more preferable. Moreover, as a qubit inevitably interacts with its surrounding environment, robust operations are of great significance. Here, we propose a scheme for quantum manipulation of the polariton qubit in circuit QED using non-adiabatic holonomy, which is inherently fast and robust. In particularly, the polariton qubit is shown to be robust against arbitrary low-frequency noise due to its near symmetric spectrum, which can also be convenient manipulated by ex…
Nonlinear optical realization of non-integrable phases accompanying quantum phase transitions
本文提出利用具有四方对称性的各向异性低双折射光纤中的非线性光学效应，通过绝热调节非线性磁化率使斯托克斯矢量积累与量子相变具有相同规范结构的非可积 Hannay 角，并讨论了纳米晶体沉积的实验实现方案。
Integrability as an attractor of adiabatic flows
本文利用量子几何张量研究自旋链绝热流，发现全局或边界破坏可积性时耦合空间最短路径均被可积区域吸引，并揭示弛豫各向异性发散及类似连续相变的普适可积—混沌转变。
Adiabatic driving and geometric phases in classical systems
本文在 Koopman-von Neumann 形式下研究经典可积系统的绝热驱动与几何相位，推导了相位显式形式及其与 Hannay 角的关系，并利用量子公式构建了生成绝热幺正流的经典绝热规范势。
Adiabatic driving and parallel transport for parameter-dependent Hamiltonians
本文利用Van Vleck-Primas微扰理论研究参数依赖哈密顿量特征向量的平行移动，定义了通过酉变换生成平行平移的非阿贝尔联络，并证明其和乐与Berry相位相关。
Temperature and integrability-breaking correspondence via adiabatic transformations
研究通过几何和绝热变换揭示温度与破坏积分性之间的对应关系，发现温度降低可使系统趋近于可积点，且局部可观测量的自相关函数表现出违反 ergodicity 的慢弛豫动力学。
Holonomy and Complementarity in Open Quantum Systems
本文研究开放量子系统中互补性与耗散的关系，通过准静态传输赋予几何解释，揭示了耗散如何影响几何热力学响应。
We construct a unified operator framework for quantum holonomies generated from bosonic systems. For a system whose Hamiltonian is bilinear in the creation and annihilation operators, we find a holonomy group determined only by a set of selected orthonormal modes obeying a stronger version of the adiabatic theorem. This photon-number independent description offers deeper insight as well as a computational advantage when compared to the standard formalism on geometric phases. In particular, a strong analogy between quantum holonomies and linear optical networks can be drawn. This relation provi…
Stabilizing the Kerr arbitrary cat states and holonomic universal control
本研究提出参量驱动KNR的新型工程哈密顿量，可稳定任意猫态、操控势阱生成碰撞态，首次探究其量子绝热相位，开发通用绝热量子计算协议，并给出超导电路实验方案，提升了玻色模式控制自由度。
Non-Abelian Geometric Quantum Memory with Atomic Ensemble
研究了一种基于原子系综的近三光子共振电磁感应透明（EIT）量子信息存储方案。每个四能级原子与两个经典控制场和一个量子探测场耦合。量子信息被动态地存储在相应的暗极化激元流形中。在我们的量子存储器中发现了一种内在的非平凡拓扑结构，这种结构是通过具有隐藏SU（3）动力学对称性的对称集体原子激发实现的。通过改变两个经典控制场的Rabi频率，可以将量子态恢复到非阿贝尔完整性，从而以纯粹的几何方式从最终态解码。
Continuous measurement-based holonomic quantum computation
该研究提出利用量子芝诺效应基于连续测量生成完整圈的方案，解析得出相关序列和时间，给出处理测量跃迁的方法，建立保持可纠错性的条件，还提及不满足纠错时用辅助量子比特使协议适用。
Topology, geometry, and gauge fields play key roles in quantum physics as exemplified by fundamental phenomena such as the Aharonov-Bohm effect, the integer quantum Hall effect, the spin Hall, and topological insulators. The concept of topological protection has also become a salient ingredient in many schemes for quantum information processing and fault-tolerant quantum computation. The physical properties of such systems crucially depend on the symmetry group of the underlying holonomy. We study here a laser-cooled gas of strontium atoms coupled to laser fields through a 4-level resonant tri…
Adiabatic Gauge Potential as a Tool for Detecting Chaos in Classical Systems
本研究将量子力学中的绝热量规势应用于经典FPUT与Toda模型，通过其时间方差区分不同混沌区域，关联混沌与扩散类型，提出混沌起始时间探测方法，验证了时间层级关系，为经典系统混沌检测提供了新工具。
We implement and characterize a numerical algorithm inspired by the $s$-source framework [Phys. Rev.~B 93, 045127 (2016)] for building a quantum many-body ground state wavefunction on a lattice of size $2L$ by applying adiabatic evolution to the corresponding ground state at size $L$, along with $L$ interleaved ancillae. The procedure can in principle be iterated to repeatedly double the size of the system. We implement the algorithm for several one dimensional spin model Hamiltonians, and find that the construction works particularly well when the gap is large and, interestingly, at scale inv…
Weyl's Relations, Integrable Matrix Models and Quantum Computation
基于有限维N的魏尔关系推广，展示海森堡交换关系在特定N-1维子空间中满足，并展示将算符投影到该子空间的线性映射。该设置用于构造参数依赖的可交换矩阵族，与表示量子可积模型的Type-1矩阵相关，并应用于量子计算中的Grover数据库搜索问题。
Double Occupancy Errors in Quantum Computing Operations: Corrections to Adiabaticity
在量子计算操作的背景下，我们研究了两个耦合量子点自旋量子比特的绝热动力学修正，每个量子点单独占据一个电子。Tunnel在操作结束时导致双重占用，并构成处理错误。我们用一个有效的两能级系统来模拟门操作，其中非绝热跃迁对应于双占有。该模型是可积的，具有三个独立的参数。我们确认的准确性Dykhne的公式，非微扰估计的过渡，并讨论其有效性的物理直观的条件。我们的半经典结果与精确时间演化的数值模拟非常吻合。类似的方法适用于不同背景下的两级系统。
Non-Abelian generalization of off-diagonal geometric phases
如果量子系统以非循环方式演化，则相应的几何相位或完整性可能无法完全定义。非对角几何相位已被开发来处理这种情况。在这里，我们将这些阶段推广到非阿贝尔的情况下，通过引入非对角的holonomies，涉及一个以上的子空间的基础希尔伯特空间的演变。提出了非对角完整性在绝热演化和干涉测量中的物理实现。
The choice of the electronic representation in on-the-fly quantum dynamics is crucial. The adiabatic representation is appealing since adiabatic states are readily available from quantum chemistry packages. The nuclear wavepackets are then expanded in a basis of Gaussian functions, which follow trajectories to explore the potential energy surfaces and approximate the potential using a local expansion of the adiabatic quantities. Nevertheless, the adiabatic representation is plagued with severe limitations when conical intersections are involved: the diagonal Born-Oppenheimer corrections (DBOCs…
Non-Adiabatic Quantum Optimization for Crossing Quantum Phase Transitions
针对有限时间内多体量子系统基态穿越量子相变时绝热性破坏引发的激发问题，受Kibble-Zurek机制启发分析现有最优控制程序的绝热性时间尺度，提出非绝热量子优化（NAQO）框架，利用Landau-Zener公式并考虑高激发态作用，在多种模型中性能优于现有方法。
A Landau-Zener formula for the Adiabatic Gauge Potential
本文研究 AGP 将非绝热跃迁概率由 LZ 公式抑制为零的机制，提出含普适前置因子的修正公式，并用复时间路径的 Aharonov-Bohm 相位解释，同时证明其适用于一类可积含时 Hamilton 量。
Adiabatic invariants are introduced and shown to provide an approximate second integral of motion for the non-integrable Dicke model, in the energy region where the system exhibits a regular dynamics. This low-energy region is always present and has been described both in a semiclassical and a full quantum analysis. Its Peres lattices exhibit that many observables vary smoothly with energy, along lines which beg for a formal description. It is shown how the adiabatic invariants provide a rationale to their presence in many cases. They are built employing the Born-Oppenheimer approximation, val…
Adiabatic Spectroscopy and a Variational Quantum Adiabatic Algorithm
针对哈密顿量基态制备问题，提出绝热谱学方法获取能谱信息，并设计变分量子绝热算法（VQAA）优化绝热路径，结合辅助协议直接评估基态重叠，在 $N=53$ 的 Ising 链上使总演化时间减少十倍，适用于含噪声量子设备。
Improving Performance of Quantum Heat Engines using modified Otto cycle
本文通过在 Otto 循环中引入特定 Hamiltonian 下的自由演化冲程，使系统进入低激发态，从而提升量子热机的吸热量、功输出与效率，并以可积与不可积模型验证了临界点附近的显著改善。
In the past decades, it was recognized that quantum chaos, which is essential for the emergence of statistical mechanics and thermodynamics, manifests itself in the effective description of the eigenstates of chaotic Hamiltonians through random matrix ensembles and the eigenstate thermalization hypothesis. Standard measures of chaos in quantum many-body systems are level statistics and the spectral form factor. In this work, we show that the norm of the adiabatic gauge potential, the generator of adiabatic deformations between eigenstates, serves as a much more sensitive measure of quantum cha…
Holonomic Quantum Computation
我们表明，广义Berry相位的概念，即，非阿贝尔完整性，可以用于实现量子计算。计算空间由流形$\cal M$参数化的哈密尔顿算子族的$n$ -折叠退化本征空间实现。$\cal M$的点表示控制场的经典配置，对于多体系统，表示子系统之间的耦合。控制$\cal M$中的绝热循环引起计算空间上的非平凡酉变换。对于一般系统，它表明，这种机制允许通用量子计算组成一个通用的循环对在$\cal M.$
Since its introduction by Hastings in [10], the technique of quasi-adiabatic continuation has become a central tool in the discussion and classification of ground state phases. It connects the ground states of self-adjoint Hamiltonians in the same phase by a unitary quasi-local transformation. This paper takes a step towards extending this result to non- self adjoint perturbations, though, for technical reason, we restrict ourselves here to weak perturbations of non-interacting spins. The extension to non-self adjoint perturbation is important for potential applications to Glauber dynamics (an…
We study adiabatic quantum quenches across a quantum multicritical point (MCP) using a quenching scheme that enables the system to hit the MCP along different paths. We show that the power-law scaling of the defect density with the rate of driving depends non-trivially on the path, i.e., the exponent varies continuously with the parameter $α$ that defines the path, up to a critical value $α= α_c$; on the other hand for $α\geq α_c$, the scaling exponent saturates to a constant value. We show that dynamically generated and {\it path($α$)-dependent} effective critical exponents associated with th…
Defining classical and quantum chaos through adiabatic transformations
通过绝热变换定义量子和经典混沌，利用保真度susceptibility度量混沌复杂性，并在自旋模型中验证其预测混沌出现的能力。
Quantum information scrambling in adiabatically-driven critical systems
本文将量子信息置乱概念扩展到绝热驱动的临界量子多体系统，以 Lipkin--Meshkov--Glick 和量子 Rabi 模型为例，揭示即使完美绝热演化也会置乱初始对称性破缺信息，并用量子相变、本征态参与数和 Loschmidt 回波等刻画其机制与不可逆性。
Driving a quantum system across quantum critical points leads to non-adiabatic excitations in the system. This in turn may adversely affect the functioning of a quantum machine which uses a quantum critical substance as its working medium. Here we propose a bath-engineered quantum engine (BEQE), in which we use the Kibble--Zurek mechanism and critical scaling laws to formulate a protocol for enhancing the performance of finite-time quantum engines operating close to quantum phase transitions. In the case of free fermionic systems, BEQE enables finite-time engines to outperform engines operatin…
Non-Abelian Geometric Phases in Triangular Structures And Universal SU(2) Control in Shape Space
本文研究了三角结构中非阿贝尔几何相位，利用形状理论推导了Wilczek-Zee连接，并展示了其受限holonomy群为SU(2)，实现单量子比特控制。同时提出了双量子比特操作及测量方法。
Geometric signatures of the onset of many-body ergodicity
本研究提出希尔伯特-基林度量，探测多体量子系统遍历区与可积区的边界，该边界由随系统尺寸始终最快增长的特征确定，结论经 Ising 和 PXP 模型数值研究证实。
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