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Register a new userGeometric phase from Aharonov-Bohm to Pancharatnam-Berry and beyond
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Whenever a quantum system undergoes a cycle governed by a slow change of parameters, it acquires a phase factor: the geometric phase. Its most common formulations are known as the Aharonov-Bohm, Pancharatnam and Berry phases, but both prior and later manifestations exist. Though traditionally attributed to the foundations of quantum mechanics, the geometric phase has been generalized and became increasingly influential in many areas from condensed-matter physics and optics to high energy and particle physics and from fluid mechanics to gravity and cosmology. Interestingly, the geometric phase also offers unique opportunities for quantum information and computation. In this Review we first introduce the Aharonov-Bohm effect as an important realization of the geometric phase. Then we discuss in detail the broader meaning, consequences and realizations of the geometric phase emphasizing the most important mathematical methods and experimental techniques used in the study of geometric phase, in particular those related to recent works in optics and condensed-matter physics.
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The geometric phase acquired by an electron in a one-dimensional periodic lattice due to weak electric perturbation is found and referred to as the Pancharatnam-Zak phase. The underlying mathematical structure responsible for this phase is unveiled. As opposed to the well-known Zak phase, the Pancharatnam-Zak phase is a gauge invariant observable phase, and correctly characterizes the energy bands of the lattice. We demonstrate the gauge invariance of the Pancharatnam-Zak phase in two celebrated models displaying topological phases. A filled band generalization of this geometric phase is constructed and is observed to be sensitive to the Fermi-Dirac statistics of the band electrons. The measurement of the single-particle Pancharatnam-Zak phase in individual topological phases, as well as the statistical contribution in its many-particle generalization, should be accessible in various controlled quantum experiments.
We propose an experiment that would produce and measure a large Aharonov-Casher (A-C) phase in a solid-state system under macroscopic motion. A diamond crystal is mounted on a spinning disk in the presence of a uniform electric field. Internal magnetic states of a single NV defect, replacing interferometer trajectories, are coherently controlled by microwave pulses. The A-C phase shift is manifested as a relative phase, of up to 17 radians, between components of a superposition of magnetic substates, which is two orders of magnitude larger than that measured in any other atom-scale quantum system.
By example of the nonlinear Kerr-mode driven by a laser, we show that hysteresis phenomena in systems featuring a driven-dissipative phase transition (DPT) can be accurately described in terms of just two collective, dissipative Liouvillian eigenmodes. The key quantities are just two components of a nonabelian geometric connection, even though a single parameter is driven. This powerful geometric approach considerably simplifies the description of driven-dissipative phase transitions, extending the range of computationally accessible parameter regimes, and providing a new starting point for both experimental studies and analytical insights.
The working principle of ordinary refractive lenses can be explained in terms of the space-variant optical phase retardations they introduce, which reshape the optical wavefront curvature and hence affect the subsequent light propagation. These phases, in turn, are due to the varying optical path length seen by light at different transverse positions relative to the lens centre. A similar lensing behavior can however be obtained when the optical phases are introduced by an entirely different mechanism. Here, we consider the geometric phases that arise from the polarization transformations occurring in anisotropic optical media, named after Pancharatnam and Berry. The medium anisotropy axis is taken to be space-variant in the transverse plane and the resulting varying geometric phases give rise to the wavefront reshaping and lensing effect, which however depends also on the input polarization. We describe the realization and characterization of a cylindrical geometric-phase lens that is converging for a given input circular polarization state and diverging for the orthogonal one, which provides one of the simplest possible examples of optical element based on geometric phases. The demonstrated lens is flat and only few microns thick (not including the supporting substrates); moreover, its working wavelength can be tuned and the lensing can be switched on and off by the action of an external control electric field. Other kinds of lenses or more general phase elements inducing different wavefront distortions can be obtained by a similar approach. Besides their potential for optoelectronic technology, these devices offer good opportunities for introducing college-level students to an advanced topic of modern physics, such as the Berry phase, with the help of interesting optical demonstrations.
The interplay of $pi$-flux and lattice geometry can yield full localization of quantum dynamics in lattice systems, a striking interference phenomenon known as Aharonov-Bohm caging. At the level of the single-particle energy spectrum, this full-localization effect is attributed to the collapse of Bloch bands into a set of perfectly flat (dispersionless) bands. In such lattice models, the effects of inter-particle interactions generally lead to a breaking of the cages, and hence, to the spreading of the wavefunction over the lattice. Motivated by recent experimental realizations of analog Aharonov-Bohm cages for light, using coupled-waveguide arrays, we hereby demonstrate that caging always occurs in the presence of local nonlinearities. As a central result, we focus on special caged solutions, which are accompanied by a breathing motion of the field intensity, that we describe in terms of an effective two-mode model reminiscent of a bosonic Josephson junction. Moreover, we explore the quantum regime using small particle ensembles, and we observe quasi-caged collapse-revival dynamics with negligible leakage. The results stemming from this work open an interesting route towards the characterization of nonlinear dynamics in interacting flat band systems.
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