Abelian and non-Abelian geometric phases in adiabatic open quantum systems
M. S. Sarandy, D. A. Lidar
备注：
v3: 10 pages, 2 figures. Substantially expanded version. Includes a proof of gauge invariance of the non-Abelian geometric phase, and an appendix on the left and right eigenvectors of the superoperator in the Jordan form
We introduce a self-consistent framework for the analysis of both Abelian and non-Abelian geometric phases associated with open quantum systems, undergoing cyclic adiabatic evolution. We derive a general expression for geometric phases, based on an adiabatic approximation developed within an inherently open-systems approach. This expression provides a natural generalization of the analogous one for closed quantum systems, and we prove that it satisfies all the properties one might expect of a good definition of a geometric phase, including gauge invariance. A striking consequence is the emergence of a finite time interval for the observation of geometric phases. The formalism is illustrated via the canonical example of a spin-1/2 particle in a time-dependent magnetic field. Remarkably, the geometric phase in this case is immune to dephasing and spontaneous emission in the renormalized Hamiltonian eigenstate basis. This result positively impacts holonomic quantum computing.
We give a gauge description of the adiabatic charge pumping in closed systems, both in Abelian and non-Abelian processes, and by means of asymptotic Wilson loops in a suitable parameter manifold. Our geometric formulation provides new insights into this issue, and a very simple algorithm for numerical computations. Indeed, as we show first, discretized Berry--Wilczeck--Zee holonomies are easy to implement. Finally, we study non-Abelian pumping in a solvable four-state model, already used in several contexts, to demonstrate the relevance of our approach.
We show in this Letter that the ground state degeneracy associated with the presence of non-Abelian anyons can be probed by using an adiabatic cooling process based on the non-Abelian entropy. In particular, we show that when the number of such anyons is increased adiabatically at sufficiently low temperatures, the non-Abelian liquid undergoes cooling, whereas heating occurs in the Abelian case. Estimates are provided for the cooling power produced by the non-Abelian anyon refrigerator, and its implementation in non-Abelian fractional quantum Hall liquids is discussed.
We implement a non-adiabatic universal set of holonomic quantum gates based on abelian holonomies using dynamical invariants, by Lie-algebraic methods. Unlike previous implementations, presented scheme does not rely on secondary methods such as double-loop or spin-echo and avoids associated experimental difficulties. It turns out that such gates exist purely in the non-adiabatic regime for these systems.
Qubits in topological quantum computation are built from non-Abelian anyons. Adiabatic braiding of anyons is exploited as topologically protected logical gate operations. Thus, the adiabaticity upon which the notion of quantum statistics is defined, plays a fundamental role in defining the non-Abelian anyons. We study the non-adiabatic effects in braidings of Ising-type anyons, namely Majorana fermions in topological superconductors, using the formalism of time-dependent Bogoliubov-de Gennes equations. Using this formalism, we consider non-adiabatic corrections to non-Abelian statistics from: …
Many quantum materials of interest, ex., bilayer graphene, possess a number of closely spaced but not fully degenerate bands near the Fermi level, where the coupling to the far detuned remote bands can induce Berry curvatures of the non-Abelian character in this active multiple-band manifold for transport effects. Under finite electric fields, non-adiabatic interband transition processes are expected to play significant roles in the associated Hall conduction. Here through an exemplified study on the valley Hall conduction in AB-stacked bilayer graphene, we show that the contribution arising f…
We present an efficient quantum algorithm for some independent set problems in graph theory, based on non-abelian adiabatic mixing. We illustrate the performance of our algorithm with analysis and numerical calculations for two different types of graphs, with the number of edges proportional to the number of vertices or its square. The theoretical advantages of our quantum algorithm over classical algorithms are discussed. Non-abelian adiabatic mixing can be a general technique to aid exploration in a landscape of near-degenerate ground states.
Non-Abelian geometric phases can be generated and detected in certain superconducting nanocircuits. Here we consider an example where the holonomies are related to the adiabatic charge dynamics of the Josephson network. We demonstrate that such a device can be applied both for adiabatic charge pumping and as an implementation of a quantum computer.
The common approach to topological quantum computation is to implement quantum gates by adiabatically moving non-Abelian anyons around each other. Here we present an alternative perspective based on the possibility of realizing the exchange (braiding) operators of anyons by adiabatically varying pairwise interactions between them rather than their positions. We analyze a system composed by four anyons whose couplings define a T-junction and we show that the braiding operator of two of them can be obtained through a particular adiabatic cycle in the space of the coupling parameters. We also dis…
We have investigated pumping in quantum dots from the perspective of non-Abelian (matrix) Berry phases by solving the time dependent Schr{ö}dinger equation exactly for adiabatic changes. Our results demonstrate that a pumped charge is related to the presence of a finite matrix Berry phase. When consecutive adiabatic cycles are performed the pumped charge of each cycle is different from the previous ones.
Demonstration of Majorana non-Abelian properties is a major challenge in the field of topological superconductivity. In this work, we propose a minimal device and protocol for testing non-Abelian properties using charge-transfer operations between a quantum dot and two Majorana bound states combined with reading the parity state using a second dot. We use an adiabatic perturbation theory to find fast adiabatic paths to perform operations and to account for nonadiabatic errors. We find the ideal parameter sweep and a region in parameter space which reduces the charge-transfer operation time 1-2…
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 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.
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.
Non-adiabatic non-Abelian geometric phase of spin-3/2 system in the rotating magnetic field is considered. Explicit expression for the corresponding effective non-Abelian gauge potential is obtained. This formula can be used for construction of quantum gates in quantum computations.
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 the adiabatic motion of ultra-cold, multi-level atoms in spatially varying laser fields can give rise to effective non-Abelian gauge fields if degenerate adiabatic eigenstates of the atom-laser interaction exist. A pair of such degenerate dark states emerges e.g. if laser fields couple three internal states of an atom to a fourth common one under pairwise two--photon-resonance conditions. For this so-called tripod scheme we derive general conditions for truly non-Abelian gauge potentials and discuss special examples. In particular we show that using orthogonal laser beams with orb…
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 possibility of realizing lattice analogs of fractional quantum Hall (FQH) states, so-called fractional Chern insulators (FCIs), in nearly flat topological (Chern) bands has attracted a lot of recent interest. Here, we make the connection between Abelian as well as non-Abelian FQH states and FCIs more precise. Using a gauge-fixed version of Qi's Wannier basis representation of a Chern band, we demonstrate that the interpolation between several FCI states, obtained by short-range lattice interactions in a spin-orbit-coupled kagome lattice model, and the corresponding continuum FQH states is …
Measurement protocol for non-adiabatic geometric phases of Floquet states
Quantum Advanced Research Center (QuARC), CSIC（量子先进研究中心（QuARC），西班牙国家研究委员会）；Instituto de Ciencia de Materiales de Madrid (ICMM), CSIC（马德里材料科学研究所（ICMM），西班牙国家研究委员会）
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 study the role of the quantum geometric tensor (QGT) in the evolution of quantum systems. We show that all its components play an important role on the extra phase acquired by a spinor and on the trajectory of an accelerated wavepacket in any realistic finite-duration experiment. While the adiabatic phase is determined by the Berry curvature (the imaginary part of the tensor), the non-adiabaticity is determined by the quantum metric (the real part of the tensor) and allows to determine corrections in the regimes where Landau-Zener approach is inapplicable. The particular case of a planar mi…
A connection is estabilished between the non-Abelian phases obtained via adiabatic driving and those acquired via a quantum Zeno dynamics induced by repeated projective measurements. In comparison to the adiabatic case, the Zeno dynamics is shown to be more flexible in tuning the system evolution, which paves the way to the implementation of unitary quantum gates and applications in quantum control.
The non-adiabatic geometric quantum computation (NGQC) has attracted a lot of attention for noise-resilient quantum control. However, previous implementations of NGQC require long evolution paths that make them more vulnerable to incoherent errors than their dynamical counterparts.In this work, we experimentally realize a universal short-path non-adiabatic geometric gate set (SPNGQC) with a 2-times shorter evolution path on a superconducting quantum processor. Characterizing with both quantum process tomography and randomized benchmarking methods, we report an average single-qubit gate fidelit…
We present a general unified approach for the study of quantum thermal machines, including both heat engines and refrigerators, operating under periodic adiabatic driving and in contact with thermal reservoirs kept at different temperatures. We show that many observables characterizing this operating mode and the performance of the machine are of geometric nature. Heat-work conversion mechanisms and dissipation of energy can be described, respectively, by the antisymmetric and symmetric components of a thermal geometric tensor defined in the space of time-dependent parameters generalized to in…
When a quantum system is driven adiabatically through a parametric cycle in a degenerate Hilbert space, the state would acquire a non-Abelian geometric phase, which is stable and forms the foundation for holonomic quantum computation (HQC). However, in the adiabatic limit, the environmental decoherence becomes a significant source of errors. Recently, various non-adiabatic HQC schemes have been proposed, but all at the price of increased sensitivity to control errors. Alternatively, there exist theoretical proposals for speeding up HQC by the technique of "shortcut to adiabaticity" (STA), but …
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.
We use the theory of dynamical invariants to yield a simple derivation of noncyclic analogues of the Abelian and non-Abelian geometric phases. This derivation relies only on the principle of gauge invariance and elucidates the existing definitions of the Abelian noncyclic geometric phase. We also discuss the adiabatic limit of the noncyclic geometric phase and compute the adiabatic non-Abelian noncyclic geometric phase for a spin 1 magnetic (or electric) quadrupole interacting with a precessing magnetic (electric) 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.
Topological phases of matter became a new standard to classify quantum systems in many cases, yet key quantities like the quantum geometric tensor providing local information about topological properties are still experimentally hard to access. In non-Abelian systems this accessibility to geometric properties can be even more restrictive due to the degeneracy of the states. We propose universal protocols to determine quantum geometric properties in non-Abelian systems. First, we show that for a weak resonant driving of the local parameters the coherent Rabi oscillations are related to the quan…
Geometric phases, which are ubiquitous in quantum mechanics, are commonly more than only scalar quantities. Indeed, often they are matrix-valued objects that are connected with non-Abelian geometries. Here we show how generalized, non-Abelian geometric phases can be realized using electromagnetic waves travelling through coupled photonic waveguide structures. The waveguides implement an effective Hamiltonian possessing a degenerate dark subspace, in which an adiabatic evolution can occur. The associated quantum metric induces the notion of a geodesic that defines the optimal adiabatic evolutio…
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…
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.
A practical quantum computer must be capable of performing high fidelity quantum gates on a set of quantum bits (qubits). In the presence of noise, the realization of such gates poses daunting challenges. Geometric phases, which possess intrinsic noise-tolerant features, hold the promise for performing robust quantum computation. In particular, quantum holonomies, i.e., non-Abelian geometric phases, naturally lead to universal quantum computation due to their non-commutativity. Although quantum gates based on adiabatic holonomies have already been proposed, the slow evolution eventually compro…
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.
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.
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.
In their recent paper, Yan-Xiong Du et al. [Phys. Rev. A 84, 034103 (2011)] claim to have found a non-Abelian adiabatic geometric phase associated with the energy eigenstates of a large-detuned $Λ$ three-level system. They further propose a test to detect the non-commutative feature of this geometric phase. On the contrary, we show that the non-Abelian geometric phase picked up by the energy eigenstates of a $Λ$ system is trivial in the adiabatic approximation, while, in the exact treatment of the time evolution, this phase is very small and cannot be separated from the non-Abelian dynamical p…
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
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 …
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…
The Kitaev honeycomb model is a system allowing for experimentally realisable quantum computation with topological protection of quantum information. Practical implementation of quantum information processing typically relies on adiabatic, i.e. slow dynamics. Here we show that the restriction to adiabatic dynamics can be overcome with optimal control theory, enabled by an extension of the fermionization of the Kitaev honeycomb model to the time-dependent case. Moreover we present a quantum control method that is applicable to large lattice models due to sub-exponential scaling.
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 adopt a geometric approach to describe the performance of adiabatic quantum machines, operating under slow time-dependent driving and in contact to two or more reservoirs with a temperature bias during all the cycle. We show that the problem of optimizing the power generation of a heat engine and the efficiency of both the heat engine and refrigerator operational modes is reduced to an isoperimetric problem with non-trivial underlying metrics and curvature. This corresponds to the maximization of the ratio between the area enclosed by a closed curve and its corresponding length. We illustra…
We study a quantum information storage scheme based on an atomic ensemble with near (also exact) three-photon resonance electromagnetically induced transparency (EIT). Each 4-level-atom is coupled to two classical control fields and a quantum probe field. Quantum information is adiabatically stored in the associated dark polariton manifold. An intrinsic non-trivial topological structure is discovered in our quantum memory implemented through the symmetric collective atomic excitations with a hidden SU(3) dynamical symmetry. By adiabatically changing the Rabi frequencies of two classical contro…
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.
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…
