The Geometric Phase in Quantum Systems

A systematic derivative expansion of the adiabatic phase

The Geometric Phase in Quantum Systems

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 anomalous Hall effect is usually associated with ferromagnets but a large anomalous Hall response can be found in topologically non-trivial half-Heusler antiferromagnets thanks to Berry phase effects associated with symmetry breaking. The quantum mechanical (Berry) phase of the electronic wavefunction plays a critical role in the anomalous1,2 and spin Hall effects3,4, including their quantized limits5,6,7. While progress has been made in understanding these effects in ferromagnets8, less is known in antiferromagnetic systems. Here we present a study of antiferromagnet GdPtBi, whose electronic structure is similar to that of the topologically non-trivial HgTe (refs 9,10,11), and where the Gd ions offer the possibility to tune the Berry phase via control of the spin texture. We show that this system supports an anomalous Hall angle ΘAH > 0.1, comparable to the largest observed in bulk ferromagnets12 and significantly larger than in other antiferromagnets13. Neutron scattering measurements and electronic structure calculations suggest that this effect originates from avoided crossing or Weyl points that develop near the Fermi level due to a breaking of combined time-reversal and lattice symmetries. Berry phase effects associated with such symmetry breaking have recently been explored in kagome networks14,15,16,17; our results extend this to half-Heusler systems with non-trivial band topology. The magnetic textures indicated here may also provide pathways towards realizing the topological insulating and semimetallic states9,10,11,18,19 predicted in this material class.

Geometric phases in quantum mechanics play an extraordinary role in broadening our understanding of fundamental significance of geometry in nature. One of the best known examples is the Berry phase [M.V. Berry (1984), Proc. Royal. Soc. London A, 392:45], which naturally emerges in quantum adiabatic evolution. So far the applicability and measurements of the Berry phase were mostly limited to systems of weakly interacting quasi-particles, where interference experiments are feasible. Here we show how one can go beyond this limitation and observe the Berry curvature, and hence the Berry phase, in generic systems as a nonadiabatic response of physical observables to the rate of change of an external parameter. These results can be interpreted as a dynamical quantum Hall effect in a parameter space. The conventional quantum Hall effect is a particular example of the general relation if one views the electric field as a rate of change of the vector potential. We illustrate our findings by analyzing the response of interacting spin chains to a rotating magnetic field. We observe the quantization of this response, which we term the rotational quantum Hall effect.

Bargmann invariants and null phase curves are known to be important ingredients in understanding the essential nature of the geometric phase in quantum mechanics. Null phase manifolds in quantum-mechanical ray spaces are submanifolds made up entirely of null phase curves, and so are equally important for geometric phase considerations. It is shown that the complete characterization of null phase manifolds involves both the Riemannian metric structure and the symplectic structure of ray space in equal measure, which thus brings together these two aspects in a natural manner.

The success of Berry phases in quantum mechanics stimulated the study of similar phenomena in other areas of physics, including the theory of living cell locomotion and motion of patterns in nonlinear media. More recently, geometric phases have been applied to systems operating in a strongly stochastic environment, such as molecular motors. We discuss such geometric effects in purely classical dissipative stochastic systems and their role in the theory of the stochastic pump effect (SPE).

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 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.

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.

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.

This document introduces the planned new search for the neutron Electric Dipole Moment at the Spallation Neutron Source at the Oak Ridge National Laboratory. A spin precession measurement is to be carried out using Ultracold neutrons diluted in a superfluid Helium bath at T = 0.5 K, where spin polarized 3He atoms act as detector of the neutron spin polarization. This manuscript describes some of the key aspects of the planned experiment with the contributions from Caltech to the development of the project. Techniques used in the design of magnet coils for Nuclear Magnetic Resonance were adapted to the geometry of the experiment. Described is an initial design approach using a pair of coils tuned to shield outer conductive elements from resistive heat loads, while inducing an oscillating field in the measurement volume. A small prototype was constructed to test the model of the field at room temperature. A large scale test of the high voltage system was carried out in a collaborative effort at the Los Alamos National Laboratory. The application and amplification of high voltage to polished steel electrodes immersed in a superfluid Helium bath was studied, as well as the electrical breakdown properties of the electrodes at low temperatures. A suite of Monte Carlo simulation software tools to model the interaction of neutrons, 3He atoms, and their spins with the experimental magnetic and electric fields was developed and implemented to further the study of expected systematic effects of the measurement, with particular focus on the false Electric Dipole Moment induced by a Geometric Phase akin to Berry’s phase. An analysis framework was developed and implemented using unbinned likelihood to fit the time modulated signal expected from the measurement data. A collaborative Monte Carlo data set was used to test the analysis methods.

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.