David J Fernandez C, Nora Breton
The Aharonov-Anandan and Berry phases are determined for the cyclic motions of a non-relativistic charged spinless particle evolving in the superposition of the fields produced by a Penning trap and a rotating magnetic field. Discussion about the selection of the parameter's space and the relationship between the Berry phase and the symmetry of the binding potential is given.
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We show that the adiabatic approximation for nonselfadjoint hamiltonians seems to induce two non-equal expressions for the geometric phase. The first one is related to the spectral projector involved in the adiabatic theorem, the other one is the adiabatic limit of the nonadiabatic geometric phase. This apparent inconsistency is resolved by observing that the difference between the two expressions is compensated by a small deviation in the dynamical phases.
The adiabatic theorem for non-Hermitian quantum systems with real eigenvalues and the complex geometric phase
本文基于复几何相位、双正交系统泛函微积分和格隆沃尔不等式，证明了具有实特征值的可对角化的非厄米特系统中绝热定理仍成立，并验证了非厄米特系统中复贝里相位的定义。
A relativistic analogue of the quantum adiabatic approximation is developed for Klein-Gordon fields minimally coupled to electromagnetism, gravity and an arbitrary scalar potential. The corresponding adiabatic dynamical and geometrical phases are calculated. The method introduced in this paper avoids the use of an inner product on the space of solutions of the Klein-Gordon equation. Its practical advantages are demonstrated in the analysis of the relativistic Landau level problem and the rotating cosmic string.
We use the Rayleigh-Schrödinger perturbation theory to calculate the corrections to the adiabatic geometric phase due to a perturbation of the Hamiltonian. We show that these corrections are at least of second order in the perturbation parameter. As an application of our general results we address the problem of the adiabatic geometric phase for a one-dimensional particle which is confined to an infinite square well with moving walls.
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…
The geometric phase acquired by the vector states under an adiabatic evolution along a noncyclic path can be calculated correctly in any instantaneous basis of a Hamiltonian that varies in time due to a time-dependent classical field.
For a T-periodic non-Hermitian Hamiltonian H(t), we construct a class of adiabatic cyclic states of period T which are not eigenstates of the initial Hamiltonian H(0). We show that the corresponding adiabatic geometric phase angles are real and discuss their relationship with the conventional complex adiabatic geometric phase angles. We present a detailed calculation of the new adiabatic cyclic states and their geometric phases for a non-Hermitian analog of the spin 1/2 particle in a precessing magnetic field.
We concentrate on the geometric potential in the invariant perturbation theory of quantum adiabatic process which is presented in our recent papers. It is found out to be related to the geodesic curvature of the spherical curve in 2-dimension quantum systems. We also show that the geometric potential may affect adiabatic approximation remarkably.
In this paper we study the implementation of non-adiabatic geometrical quantum gates with in semiconductor quantum dots. Different quantum information enconding/manipulation schemes exploiting excitonic degrees of freedom are discussed. By means of the Aharanov-Anandan geometrical phase one can avoid the limitations of adiabatic schemes relying on adiabatic Berry phase; fast geometrical quantum gates can be in principle implemented
We examine the adiabatic dynamics of a quantum system coupled to a noisy classical control field. A stochastic phase shift is shown to arise in the off-diagonal elements of the system's density matrix which can cause decoherence. We derive the condition for onset of decoherence, and identify the noise properties that drive decoherence. We show how this decoherence mechansim causes: (1) a dephasing of the observable consequences of the adiabatic geometric phase; and (2) the loss of computational efficiency of the Shor algorithm when run on a sufficiently noisy geometric quantum computer.
We propose a non-adiabatic scheme for geometric quantum computation with trapped ions. By making use of the Aharonov-Anandan phase, the proposed scheme not only preserves the globally geometric nature in quantum computation, but also provides the advantage of non-adiabaticity that overcomes the problem of slow evolution in the existing adiabatic schemes. Moreover, the present scheme requires only two atomic levels in each ion, making it an appealing candidate for quantum computation.
We study a 2-qubit nuclear spin system for realizing an arbitrary geometric quantum phase gate by means of non-adiabatic operation. A single magnetic pulse with multi harmonic frequencies is applied to manipulate the quantum states of 2-qubit instantly. Using resonant transition approximation, the time dependent Hamiltonian of two nuclear spins can be solved analytically. The time evolution of the wave function is obtained without adiabatic approximation. The parameters of magnetic pulse, such as the frequency, amplitude, phase of each harmonic part as well as the time duration of the pulse, a…
In a recent Letter [Phys. Rev. Lett. {\bf 95}, 080502 (2005)], an interesting scheme was proposed to implement a type of conditional quantum phase gates with built-in fault-tolerant feature via adiabatic evolution of dark eigenstates. In this comment we elaborate the geometric nature of the gate scheme and clarify that it still belongs to a class of conventional geometric quantum computation.
We analyze the influence of a dissipative environment on geometric phases in a quantum system subject to non-adiabatic evolution. We find dissipative contributions to the acquired phase and modification of dephasing, considering the cases of weak short-correlated noise as well as of slow quasi-stationary noise. Motivated by recent experiments, we find the leading non-adiabatic corrections to the results, known for the adiabatic limit.
In this paper, we present a U(1)-invariant expansion theory of the adiabatic process. As its application, we propose and discuss new sufficient adiabatic approximation conditions. In the new conditions, we find a new invariant quantity referred as quantum geometric potential (QGP) contained in all time-dependent processes. Furthermore, we also give detailed discussion and analysis on the properties and effects of QGP.
The adiabatic theorem shows that the instantaneous eigenstate is a good approximation of the exact solution for a quantum system in adiabatic evolution. One may therefore expect that the geometric phase calculated by using the eigenstate should be also a good approximation of exact geometric phase. However, we find that the former phase may differ appreciably from the latter if the evolution time is large enough.
Sum rule for non-adiabatic geometric phases
该研究提出了非绝热几何相位的求和规则，揭示了其在量子计算中的限制作用。
The recent discovery of inconsistency (MS inconsistency) in the adiabatic approximation is discussed. In particular, the so-called, inconsistency in Berry phase is analyzed. On the contrary to some authors, we found that the MS inconsistency persisted and the so-called inconsistency in Berry phase does not exist.
We consider stimulated Raman adiabatic passage (STIRAP) processes in tripod systems and show how to generate purely geometric phase changes of the quantum states involved. The geometric phases are controlled by three laser fields where pulse shapes, relative field strength and phases can be controlled. We present a robust set of universal gates for quantum computing based on these geometric phases: a one-qubit phase gate, a Hadamard gate and a two-qubit phase gate.
Conditional geometric phase shift gate, which is fault tolerate to certain errors due to its geometric property, is made by NMR technique recently under adiabatic condition. By the adiabatic requirement, the result is inexact unless the Hamiltonian changes extremely slowly in the process. However, in quantum computation, everything has to be completed within the decoherence time. High running speed of every gate in quantum computation is demanded because the power of quantum computer can be exponentially proportional to the maximum number of logic gate operation that can be taken sequentially …
By analyzing an exactly solvable model in the second quantized formulation which allows a unified treatment of adiabatic and non-adiabatic geometric phases, it is shown that the topology of the adiabatic Berry's phase, which is characterized by the singularity associated with possible level crossing, is trivial in a precise sense. This topology of the geometric phase is quite different from the topology of the Aharonov-Bohm effect, where the topology is specified by the external local gauge field and it is exact for the slow as well as for the fast motion of the electron.
Quantum systems with adiabatic classical parameters are widely studied, e.g., in the modern holonomic quantum computation. We here provide complete geometric quantization of a Hamiltonian system with time-dependent parameters, without the adiabatic assumption. A Hamiltonian of such a system is affine in the temporal derivative of parameter functions. This leads to the geometric Berry factor phenomena.
Much long before the appearing time of the Comment by Cen, Li, and Yan,, the main issue addresed there by Cen et al had been resolved already. The information offered by the Comment is selective and misleading.
Steering a quantum harmonic oscillator state along cyclic trajectories leads to a path-dependent geometric phase. Here we describe an experiment observing this geometric phase in an electronic harmonic oscillator. We use a superconducting qubit as a non-linear probe of the phase, otherwise unobservable due to the linearity of the oscillator. Our results demonstrate that the geometric phase is, for a variety of cyclic trajectories, proportional to the area enclosed in the quadrature plane. At the transition to the non-adiabatic regime, we study corrections to the phase and dephasing of the qubi…
We introduce an adiabatic perturbation theory for quantum systems with degenerate energy spectra. This perturbative series enables one to rigorously establish conditions for the validity of the adiabatic theorem of quantum mechanics for degenerate systems. The same formalism can be used to find non-adiabatic corrections to the non-Abelian Wilczek-Zee geometric phase. These corrections are relevant to assess the validity of the practical implementation of the concept of fractional exchange statistics. We illustrate the formalism by exactly solving a time-dependent problem and comparing its solu…
We show that some composite pulses widely employed in NMR experiments are regarded as non-adiabatic geometric quantum gates with Aharanov-Anandan phases. Thus, we reveal the presence of a fundamental issue on quantum mechanics behind the traditional technique of the composite pulses. To examine the robustness of such composite pulses against fluctuations, we present a practical noise model in a two-level system. Then, we find that the composite pulses possesses purely geometrical nature even under a specific type of fluctuations.
We examine both quantum and classical versions of the problem of spin evolution in a slowly varying magnetic field. Main attention is given to the first- and second-order adiabatic corrections in the case of in-plane variations of the magnetic field. While the first-order correction relates to the adiabatic Berry phase and Coriolis-type lateral deflection of the spin, the second-order correction is shown to be responsible for the next-order geometric phase and in-plain deflection. A comparison between different approaches, including the exact (non-adiabatic) geometric phase, is presented.
The geometric phase associated with a many body ground state exhibits a signature of quantum phase transition. In this context, we have studied the behaviour of the geometric phase during a linear quench caused by a gradual turning off of the magnetic field interacting with a spin chain.
We investigate the geometric phase of an atom inside an adiabatic radio frequency (rf) potential created from a static magnetic field (B-field) and a time dependent rf field. The spatial motion of the atomic center of mass is shown to give rise to a geometric phase, or Berry's phase, to the adiabatically evolving atomic hyperfine spin along the local B-field. This phase is found to depend on both the static B-field along the semi-classical trajectory of the atomic center of mass and an ``effective magnetic field'' of the total B-field, including the oscillating rf field. Specific calculations …
The theory of geometric phase is generalized to a cyclic evolution of the eigenspace of an invariant operator with $N$-fold degeneracy. The corresponding geometric phase is interpreted as a holonomy inherited from the universal connection of a Stiefel U(N)-bundle over a Grassmann manifold. Most significantly, for an arbitrary initial state, this geometric phase captures the inherent geometric feature of the state evolution. Moreover, the geometric phase in the evolution of the eigenspace of an adiabatic action operator is also addressed, which is elaborated by a pullback U(N)-bundle. Several i…
We propose a theoretical framework that captures the geometric vector potential emerging from the non-adiabatic spin dynamics of itinerant carriers subject to arbitrary magnetic textures. Our approach results in a series of constraints on the geometric potential and the non-adiabatic geometric phase associated with it. These constraints play a decisive role when studying the geometric spin phase gathered by conducting electrons in ring interferometers under the action of in-plane magnetic textures, allowing a simple characterization of the topological transition recently reported by Saarikoski…
We show that a noncyclic phase of geometric origin has to be included in the approximate adiabatic wave function. The adiabatic noncyclic geometric phase for systems exhibiting a conical intersection as well as for an Aharonov-Bohm situation is worked out in detail. A spin-1/2 experiment to measure the adiabatic noncyclic geometric phase is discussed. We also analyze some misconceptions in the literature and textbooks concerning noncyclic geometric phases.
A geometric phase is found for a general quantum state that undergoes adiabatic evolution. For the case of eigenstates, it reduces to the original Berry's phase. Such a phase is applicable in both linear and nonlinear quantum systems. Furthermore, this new phase is related to Hannay's angles as we find that these angles, a classical concept, can arise naturally in quantum systems. The results are demonstrated with a two-level model.
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。
In [Phys. Rev. Lett. 95, 080502 (2005)], Zheng proposed a scheme for implementing a conditional phase shift via adiabatic passages. The author claims that the gate is "neither of dynamical nor geometric origin" on the grounds that the Hamiltonian does not follow a cyclic change. He further argues that "in comparison with the adiabatic geometric gates, the nontrivial cyclic loop is unnecessary, and thus the errors in obtaining the required solid angle are avoided, which makes this new kind of phase gates superior to the geometric gates." In this Comment, we point out that geometric operations, …
We study how the non-adiabatic effect causes the observable fluctuation in the "geometric phase" for a two-level system, which is defined as the experimentally measurable quantity in the adiabatic limit. From the Rabi's exact solution to this model, we give a reasonable explanation to the experimental discovery of phase fluctuation in the superconducting circuit system [P. J. Leek, \textit{et al}., Science \textbf{318}, 1889 (2007)], which seemed to be regarded as the conventional experimental error.
The conventional formulation of the non-adiabatic (Aharonov-Anandan) phase is based on the equivalence class $\{e^{iα(t)}ψ(t,\vec{x})\}$ which is not a symmetry of the Schrödinger equation. This equivalence class when understood as defining generalized rays in the Hilbert space is not generally consistent with the superposition principle in interference and polarization phenomena. The hidden local gauge symmetry, which arises from the arbitrariness of the choice of coordinates in the functional space, is then proposed as a basic gauge symmetry in the non-adiabatic phase. This re-formulation re…
We derive an elegant solution for a two-level system evolving adiabatically under the influence of a driving field with a time-dependent phase, which includes open system effects such as dephasing and spontaneous emission. This solution, which is obtained by working in the representation corresponding to the eigenstates of the time-dependent Hermitian Hamiltonian, enables the dynamic and geometric phases of the evolving density matrix to be separated and relatively easily calculated.
Multi-level spectral navigation with geometric diabatic-adiabatic control
本文提出了一种几何方法，通过在绝热和非绝热动力学之间插值，实现多级量子系统中高保真度的状态传输与优化。
We give a simple way to detect the geometric phase shift and the conditional geometric phase shift with Josephson junction system. Comparing with the previous work(Falcl G, Fazio R, Palma G.M., Siewert J and Verdal V, {\it Nature} {\bf 407}, 355(2000)), our scheme has two advantages. We use the non-adiabatic operation, thus the detection is less affected by the decoherence. Also, we take the time evolution on zero dynamic phase loop, we need not take any extra operation to cancel the dynamic phase.
We have developed an adiabatic Abelian geometric quantum computation strategy based on the non-degenerate energy eigenstates in (but not limited to) superconducting phase-qubit systems. The fidelity of the designed quantum gate was evaluated in the presence of simulated thermal fluctuation in superconducting phase qubits circuit and was found to be rather robust against the random errors. In addition, it was elucidated that the Berry phase in the designed adiabatic evolution may be detected directly via the quantum state tomography developed for superconducting qubits.
For adiabatic controls of quantum systems, the non-adiabatic transitions are reduced by increasing the operation time of processes. Perfect quantum adiabaticity usually requires the infinitely slow variation of control parameters. In this paper, we propose the dynamical quantum geometric tensor, as a metric in the control parameter space, to speed up quantum adiabatic processes and reach quantum adiabaticity in relatively short time. The optimal protocol to reach quantum adiabaticity is to vary the control parameter with a constant velocity along the geodesic path according to the metric. For …
In this work we study the geometrical and topological properties of non-equilibrium quantum systems driven by ac fields. We consider two tunnel coupled spin qubits driven by either spatially homogeneous or inhomogeneous ac fields. Our analysis is an extension of the classical model introduced by Berry with he addition of the spatial degree of freedom. We calculate the Berry and Aharonov-Anandan geometric phases, and demonstrate the influence of the different field parameters in the geometric properties. We also discuss the topological properties associated with the different driving regimes, a…
In this work Controlled phase shift gates are implemented on a qaudrupolar system, by using non-adiabatic geometric phases. A general procedure is given, for implementing controlled phase shift gates in an 'N' level system. The utility of such controlled phase shift gates, is demonstrated here by implementing 3-qubit Deutsch-Jozsa algorithm on a 7/2 quadrupolar nucleus oriented in a liquid crystal matrix.
Quantum technologies based on adiabatic techniques can be highly effective, but often at the cost of being very slow. Here we introduce a set of experimentally realistic, non-adiabatic protocols for spatial state preparation, which yield the same fidelity as their adiabatic counterparts, but on fast timescales. In particular, we consider a charged particle in a system of three tunnel-coupled quantum wells, where the presence of a magnetic field can induce a geometric phase during the tunnelling processes. We show that this leads to the appearance of complex tunnelling amplitudes and allows for…
Measurement protocol for non-adiabatic geometric phases of Floquet states
提出一种基于非绝热驱动相位绝热变化的协议，用于测量Floquet态的非绝热几何相位（AA相位），其实验要求与绝热相位测量相同。
The Non-Adiabatic Sub-Geometric Phase and Its Application on Quantum Transition
本研究将基于密度矩阵的绝热几何相位推广至非绝地亚几何相位，揭示其虚实部对量子跃迁的不同作用，并通过两类系统验证，补充了量子跃迁选择定则相关理论。
We examine the effect of spin-orbit coupling on geometric phases in hydrogenlike atoms exposed to a slowly varying magnetic field. The marginal geometric phases associated with the orbital angular momentum and the intrinsic spin fulfill a sum rule that explicitly relates them to the corresponding geometric phase of the whole system. The marginal geometric phases in the Zeeman and Paschen-Back limit are analyzed. We point out the existence of nodal points in the marginal phases that may be detected by topological means.
High-fidelity quantum operations are a key requirement for fault-tolerant quantum information processing. In electron spin resonance, manipulation of the quantum spin is usually achieved with time-dependent microwave fields. In contrast to the conventional dynamic approach, adiabatic geometric phase operations are expected to be less sensitive to certain kinds of noise and field inhomogeneities. Here, we investigate such phase gates applied to electron spins both through simulations and experiments, showing that the adiabatic geometric phase gate is indeed inherently robust against inhomogenei…
Geometric Quantum Mechanics is a novel and prospecting approach motivated by the belief that our world is ultimately geometrical. At the heart of that is a quantity called Quantum Geometric Tensor (or Fubini-Study metric), which is a complex tensor with the real part serving as the Riemannian metric that measures the `quantum distance', and the imaginary part being the Berry curvature. Following a physical introduction of the basic formalism, we illustrate its physical significance in both the adiabatic and non-adiabatic systems.
We design an adiabatic quantum algorithm for the counting problem, i.e., approximating the proportion, $α$, of the marked items in a given database. As the quantum system undergoes a designed cyclic adiabatic evolution, it acquires a Berry phase $2πα$. By estimating the Berry phase, we can approximate $α$, and solve the problem. For an error bound $ε$, the algorithm can solve the problem with cost of order $(\frac{1}ε)^{3/2}$, which is not as good as the optimal algorithm in the quantum circuit model, but better than the classical random algorithm. Moreover, since the Berry phase is a purely g…
Geometric-phase control of Krylov complexity in adiabatic dynamics
本文在缓慢演化自旋系统的瞬时本征态基下，研究了量子绝热动力学中Krylov复杂度与Berry相位的关联，发现初始Krylov基为叠加态时Krylov复杂度非零，单量子比特系统中其振荡频率与外场强度和Berry相位相关。
We study defects in adiabatic control of a quantum system caused by the entanglement of the system with its environment. Such defects can be assimilated to decoherence processes due to perturbative couplings between the system and the environment. To analyse these effects, we propose a geometric approach based on a field theory on the control manifold issued from the higher gauge theory associated with the $C^*$-geometric phases. We study a visualization method to analyse the defects of adiabatic control based on the drawing of the field strengths of the gauge theory. To illustrate the present…
Geometrical Amplitude factors in the the adiabatic evolution
本文在具有虚本征值的厄米哈密顿量中引入并重新推导了几何振幅概念，证明绝热循环演化会产生纯几何起源的附加振幅因子，并以广义谐振子为例加以验证。
We show that the definition of instantaneous eigenstate populations for a dynamical non-self-adjoint system is not obvious. The naive direct extension of the definition used for the self-adjoint case leads to inconsistencies; the resulting artifacts can induce a false inversion of population or a false adiabaticity. We show that the inconsistency can be avoided by introducing geometric phases in another possible definition of populations. An example is given which demonstrates both the anomalous effects and their removal by our approach.
Nonadiabatic geometric phases are only dependent on the evolution path of a quantum system but independent of the evolution details, and therefore quantum computation based on nonadiabatic geometric phases is robust against control errors. To realize nonadiabatic geometric quantum computation, it is necessary to ensure that the quantum system undergoes a cyclic evolution and the dynamical phases are removed from the total phases. To satisfy these conditions, the evolution paths in previous schemes are usually restricted to some special forms, e.g, orange-slice-shaped loops, which make the path…
We show that geometric phases may be generated in a quantum system subject to noise by adiabatic manipulations of the fluctuating fields, e.g., by variation of the system-environment coupling. For a two-state quantum system we express this phase in terms of the geometry of the path, traversed by the slowly varying direction and amplitude of the fluctuations. We discuss the origin of this phase and possibilities to separate it from the known environment-induced modification of the Berry phase.
We investigate the effects of systematic errors of the control parameters on single-qubit gates based on non-adiabatic non-Abelian geometric holonomies and those relying on purely dynamical evolution. It is explicitly shown that the systematic error in the Rabi frequency of the control fields affects these two kinds of gates in different ways. In the presence of this systematic error, the transformation produced by the non-adiabatic non-Abelian geometric gate is not unitary in the computational space, and the resulting gate infidelity is larger than that with the dynamical method. Our results …
We propose a method to design pulses in a resonant three-level system to enhance the robustness of non-adiabatic geometric gate operations. By optimizing the shape of the pulse envelope, we show that the gate operations are more robust against frequency detuning than they are with Gaussian and square pulses. Our method provides a way to design pulses that can be employed in a system where robustness against frequency variations or inhomogeneous broadening is required,and may be extended to ensure robustness against other physical imperfections such as intensity fluctuations and random noises.
Universal speeded-up adiabatic geometric quantum computation~(SAGQC) is studied in $Λ$-type three-level system with different coupling cases, i.e., time-dependent detuning, large detuning and one-photon resonance couplings, respectively. In these cases, the counterdiabatic driving method is used to speed up the universal quantum computation. These schemes in $Λ$-type three-level system are feasible in experiment because additional unaccessible ground-state-coupling is not needed. Only the shapes and phases of the initial adiabatic classical-field pulse are modified with the aid of effective tw…
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