The Geometric Phase in Quantum Systems

The discovery of the geometric phase is one of the most interesting and intriguing findings of the last few decades. It led to a deeper understanding of the concept of phase in quantum mechanics and motivated a surge of interest in fundamental quantum mechanical issues, disclosing unexpected applications in very diverse fields of physics. Although the key ideas underlying the existence of a purely geometrical phase had already been proposed in 1956 by Pancharatnam, it was Michael Berry who revived this issue 30 years later. The clarity of Berry's seminal paper, in 1984, was extraordinary. Research on the topic flourished at such a pace that it became difficult for non-experts to follow the many different theoretical ideas and experimental proposals which ensued. Diverse concepts in independent areas of mathematics, physics and chemistry were being applied, for what was (and can still be considered) a nascent arena for theory, experiments and technology. Although collections of papers by different authors appeared in the literature, sometimes with ample introductions, surprisingly, to the best of my knowledge, no specific and exhaustive book has ever been written on this subject. The Geometric Phase in Quantum Systems is the first thorough book on geometric phases and fills an important gap in the physical literature. Other books on the subject will undoubtedly follow. But it will take a fairly long time before other authors can cover that same variety of concepts in such a comprehensive manner. The book is enjoyable. The choice of topics presented is well balanced and appropriate. The appendices are well written, understandable and exhaustive - three rare qualities. I also find it praiseworthy that the authors decided to explicitly carry out most of the calculations, avoiding, as much as possible, the use of the joke `after a straightforward calculation, one finds...' This was one of the sentences I used to dislike most during my undergraduate studies. A student is inexperienced in such matters and needs to look at details. This book is addressed to graduate physics and chemistry students and was written thinking of students. However, I would recommend it also to young and mature physicists, even those who are already `into' the subject. It is a comprehensive work, jointly written by five researchers. After a simple introduction to the subject, the book gradually provides deeper concepts, more advanced theory and finally an interesting introduction and explanation of recent experiments. For its multidisciplinary features, this work could not have been written by one single author. The collaborative effort is undoubtedly one of its most interesting qualities. I would definitely recommend it to anyone who wants to learn more on the geometric phase, a topic that is both beautiful and intruiguing.

A systematic derivative expansion of the adiabatic phase

We show how geometric phases may be used to fully describe quantum systems, with or without gravity, by providing knowledge about the geometry and topology of its Hilbert space. We find a direct relation between geometric phases and von Neumann algebras. In particular, we show that a vanishing geometric phase implies the existence of a well-defined trace functional on the algebra. We discuss how this is realised within the AdS/CFT correspondence for the eternal black hole. On the other hand, a non-vanishing geometric phase indicates missing information for a local observer, associated to reference frames covering only parts of the quantum system considered. We illustrate this with several examples, ranging from a single spin in a magnetic field to Virasoro Berry phases and the geometric phase associated to the eternal black hole in AdS spacetime. For the latter, a non-vanishing geometric phase is tied to the presence of a centre in the associated von Neumann algebra.

The geometric (Berry-Pancharatnam) phase originates from the intrinsic geometry of the space of quantum states and can be observed in different situations, such as a cyclic evolution of a quantum system. Here, we utilize the geometric phase to obtain a surprising insight: It is possible to create nonlocal correlations in a fixed interferometer with independent photon inputs by varying the photons' internal states. In particular, we consider a cyclic interferometer that is fixed, i.e., that has no variable internal phase shifts or subsequent measurement settings. Instead, the measurement choices of the different parties correspond to the internal states of the input photons which influence the observed correlations via a collective $N$-photon geometric phase, constituting a different approach for the generation of nonlocality with respect to the usual paradigm. We observe a trade-off between the geometric phases and the visibility of the many-photon interference, impeding the generation of nonlocality. However, by making use of the dynamical quantum Zeno effect, we show that nonlocality can be created in the fixed cyclic interferometer using 12 (or more) independent photons.

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.

For pt.I see ibid., vol.25, p.1745 (1992). In the first paper the authors studied certain crucial mathematical properties of Abelian Berry's phases in two quantum systems with supersymmetrically related Hamiltonians. They follow up such investigation by extending their analysis to nonAbelian Berry's phases in this present study. They present the derivation of an explicit expression for the difference in the relevant nonAbelian Berry's connections. Also they have derived an expression for a connection one form, which contain the non-Abelian Berry's connection, and which is invariant in the two supersymmetrically related quantum systems. To illustrate their findings mentioned above, they take the example of a Hamiltonian expressible in spin quadrupole. In this example, with a particular choice of supersymmetric partner, they show that the two Berry's connection one forms of the two supersymmetrically related systems may be related by gauge transformation.

The Berry phase plays an important role in determining many physical properties of quantum systems. However, tuning the energy spectrum of a quantum system via Berry phase is comparatively rare because the Berry phase is usually a fixed constant. Here, we report the realization of an unusual valley-polarized energy spectra via continuously tunable Berry phases in Bernal-stacked bilayer graphene quantum dots. In our experiment, the Berry phase of electron orbital states is continuously tuned from about π to 2π by perpendicular magnetic fields. When the Berry phase equals π or 2π, the electron states in the two inequivalent valleys are energetically degenerate. By altering the Berry phase to noninteger multiples of π, large and continuously tunable valley-polarized energy spectra are realized. Our result reveals the Berry phase's essential role in valleytronics and the observed valley splitting, on the order of 10meV at a magnetic field of 1T, is about 100 times larger than Zeeman splitting for spin, shedding light on graphene-based valleytronics.

Since the appearance of Berry's seminal paper in 1984, geometric phases have been discovered in virtually all fields of physics. Here we address molecules and solids, and we limit our scope to the Berry's phases of the many-electron wavefunction. Many advances have occurred in very recent years relating to the theory of such phases and their observable consequences. After discussing the basic features of Berry's phases in a generic quantum system, we specialize to selected examples taken from molecular physics and condensed matter physics; in each of these cases, a Berry's phase of the electronic wavefunction leads to measurable effects.

In open quantum systems, we study the geometric phases acquired for a two-level atom coupled to a bath of fluctuating vacuum massless scalar fields due to linear acceleration and circular motion without and with a boundary. In free space, as we amplify acceleration, the geometric phase acquired purely due to linear acceleration case firstly is smaller than the circular acceleration case in the ultrarelativistic limit for the initial atomic state $\theta\in(0,\frac{\pi}{2})\cup(\frac{\pi}{2},\pi)$, then equals to the circular acceleration case in a certain acceleration, and finally, is larger than the circular acceleration case. The spontaneous transition rates show a similar feature. This result is different from the case of a bath of fluctuating vacuum electromagnetic fields that has been studied. Considering the initial atomic state $\theta \in(0,\pi)$, we find that the geometric phase acquired purely due to linear acceleration always equals to the circular acceleration case for the certain acceleration. The feature implies that, in a certain condition, one can simulate the case of the uniformly accelerated two-level atom by studying the properties of the two-level atom in circular motion. Adding a reflecting boundary, we observe that a larger value of a geometric phase can be obtained compared to the absence of a boundary. Besides, the geometric phase fluctuates along $z$, and the maximum of geometric phase is closer to the boundary for a larger acceleration. We also find that geometric phases can be acquired purely due to the linear acceleration case and circular acceleration case with $\theta \in(0,\pi)$ for a smaller $z$.

The geometric phase can be used as a fruitful venue of investigation to infer features of the quantum systems. Its application can reach new theoretical frontiers and imply innovative and challenging experimental proposals. Herein, we take advantage of the geometric phase to sense the corrections induced while a neutral particle travels at constant velocity in front of an imperfect sheet in quantum vacuum. As it is already known, two bodies in relative motion at constant velocity experience a quantum contactless dissipative force, known as quantum friction. This force has eluded experimental detection so far due to its small magnitude and short range. However, we give details of an innovative experiment designed to track traces of the quantum friction by measuring the velocity dependence of corrections to the geometric phase. We notice that the environmentally induced corrections can be decomposed in different contributions: corrections induced by the presence of the dielectric sheet and the motion of the particle in quantum vacuum. As the geometric phase accumulates over time, its correction becomes relevant at a relative short timescale, while the system still preserves purity. The experimentally viable scheme presented would be the first one in tracking traces of quantum friction through the study of decoherence effects on a NV center in diamond.

Aharonov and Anandan have recently reformulated and generalized Berry's phase by showing that a quantum system which evolves through a circuit C in projective Hilbert space acquires a geometrical phase \ensuremath{\beta}(C) related to the topology of the space and the geometry of the circuit. We present NMR interferometry experiments in a three-level system which demonstrate the Aharonov-Anandan phase and its topological invariance for different circuits.

The adiabatic control is a powerful technique for many practical applications in quantum state engineering, light-driven chemical reactions and geometrical quantum computations. This paper reveals a speed limit of nonadiabatic transition in a general time-dependent parametric quantum system that leads to an upper bound function which lays down an optimal criteria for the adiabatic controls. The upper bound function of transition rate between instantaneous eigenstates of a time-dependent system is determined by the power fluctuations of the system relative to the minimum gap between the instantaneous levels. In a parametric Hilbert space, the driving power corresponds to the quantum work done by the parametric force multiplying the parametric velocity along the parametric driving path. The general two-state time-dependent models are investigated as examples to calculate the bound functions in some general driving schemes with one and two driving parameters. The calculations show that the upper bound function provides a tighter real-time estimation of nonadiabatic transition and is closely dependent on the driving frequencies and the energy gap of the system. The deviations of the real phase from Berry phase on different closed paths are induced by the nonadiabatic transitions and can be efficiently controlled by the upper bound functions. When the upper bound is adiabatically controlled, the Berry phases of the electronic spin exhibit nonlinear step-like behaviors and it is closely related to topological structures of the complicated parametric paths on Bloch sphere.

Geometric phases induced in quantum evolutions have built‐in noise‐resilient characters, and thus can find applications in many robust quantum manipulation tasks. Here, a feasible and fast scheme for universal quantum computation on superconducting circuits with nonadiabatic non‐Abelian geometric phases is proposed, using resonant interaction of three‐level quantum system. In this scheme, arbitrary single‐qubit quantum gates can be implemented in a single‐loop scenario by shaping both the amplitudes and phases of the two driving microwave fields resonantly coupled to a transmon device. Moreover, nontrivial two‐qubit gates can also be realized with an auxiliary transmon simultaneously coupled to the two target transmons in an effective resonant way. In particular, this proposal can be compatible to various optimal control techniques, which further enhances the robustness of the quantum operations. Therefore, this proposal represents a promising way toward fault‐tolerant quantum computation on solid‐state quantum circuits.

In this paper a formalism for studying the dynamics of quantum systems coupled to classical spin environments is reviewed. The theory is based on generalized antisymmetric brackets and naturally predicts open-path off-diagonal geometric phases in the evolution of the density matrix. It is shown that such geometric phases must also be considered in the quantum-classical Liouville equation for a classical bath with canonical phase space coordinates; this occurs whenever the adiabatics basis is complex (as in the case of a magnetic field coupled to the quantum subsystem). When the quantum subsystem is weakly coupled to the spin environment, non-adiabatic transitions can be neglected and one can construct an effective non-Markovian computer simulation scheme for open quantum system dynamics in classical spin environments. In order to tackle this case, integration algorithms based on the symmetric Trotter factorization of the classical-like spin propagator are derived. Such algorithms are applied to a model comprising a quantum two-level system coupled to a single classical spin in an external magnetic field. Starting from an excited state, the population difference and the coherences of this two-state model are simulated in time while the dynamics of the classical spin is monitored in detail. It is the author's opinion that the numerical evidence provided in this paper is a first step toward developing the simulation of quantum dynamics in classical spin environments into an effective tool. In turn, the ability to simulate such a dynamics can have a positive impact on various fields, among which, for example, nano-science.

Broken gauge equivalence of Hamiltonians due to time-evolution and the Berry phase