Hannay Angle Research Articles

Adiabatic driving, geometric phases, and the geometric tensor for classical states

We use the Hilbert space formulation of classical mechanics, known as the Koopman–von Neumann formalism, to study adiabatic driving, geometric phases, and the geometric tensor for classical states. In close relation to what happens to a quantum state, a classical Koopman–von Neumann eigenstate will acquire a geometric phase factor expiΦ after a closed variation of the parameters λ in its associated Hamiltonian. The explicit form of Φ is then derived for integrable systems, and its relation with the Hannay angle is shown. Additionally, we use quantum formulas to write an adiabatic gauge potential that generates adiabatic unitary flow between classical eigenstates, and we explicitly show the relationship between the potential and the classical geometric phase. We also define a classical analog of the geometric tensor, thus defining a Fubini–Study metric for classical states, and we use the singularities of the tensor to link the transition from Arnold–Liouville integrability to chaos with some of the mathematical formalism of quantum phase transitions. While the formulas and definitions we use originate in quantum mechanics, all the results found are purely classical, no classical or semiclassical limit is ever taken.

Heavy symmetric tops and the Hannay angle

The dynamics of a heavy symmetric top are studied in connection with the Hannay angle. When the top undergoes a steady precession due to gravity without nutation, the Hannay angle has a geometric nature such that it is identical to the solid angle subtended by the loop swept out by the symmetry axis of the top. Here, we show that the Hannay angle can also be described by the angle between two radial vectors on the disk of the top corresponding to the pure spinning motion and the coupled motion of spin and precession for one period of the precession. The geometric nature of the angle between the two radial vectors is verified by demonstrating, via parallel transport, that the magnitude of the angle is the same as that of the solid angle. In the presence of nutation, the path constructed by the symmetry axis is not closed, and the steady precession appears in the limit of infinite initial spin angular velocity. As a consequence, in an ideal situation of no friction, the Hannay angle as a pure geometric effect does not exist in the superposed motion of precession and nutation.

Sqrt{Toverline{T}}-deformed oscillator inspired by ModMax

Inspired by a recently proposed Duality and Conformal invariant modification of Maxwell theory (ModMax), we construct a one-parameter family of two-dimensional dynamical systems in classical mechanics that share many features with the ModMax theory. It consists of a couple of sqrt{Toverline{T}}-deformed oscillators that nevertheless preserves duality (q rightarrow p,p rightarrow -q) and depends on a continuous parameter gamma, as in the ModMax case. Despite its nonlinear features, the system is integrable. Remarkably, it can be interpreted as a pair of two coupled oscillators whose frequencies depend on some basic invariants that correspond to the duality symmetry and rotational symmetry. Based on the properties of the model, we can construct a nonlinear map dependent on gamma that maps the oscillator in 2D to the nonlinear one, but with parameter 2gamma. The reason behind the existence of such map can be revealed through a construction of two Lax pairs associated with the system. The dynamics also shows the phenomenon of energy transfer and we calculate a Hannay angle associated to geometric phases and holonomies.

Generalized Gauge Transformation with PT$PT$‐Symmetric Non‐Unitary Operator and Classical Correspondence of Non‐Hermitian Hamiltonian for a Periodically Driven System

AbstractThis paper demonstrates that the parity‐time ()‐symmetric non‐Hermitian Hamiltonian for a periodically driven system can be generated from a kernel Hamiltonian by a generalized gauge transformation. The kernel Hamiltonian is Hermitian and static, while the time‐dependent transformation operator has to be symmetric and non‐unitary in general. Biorthogonal sets of eigenstates appear necessarily as a consequence of the non‐Hermitian Hamiltonian. The wave functions and associated non‐adiabatic Berry phase for the nth eigenstate are obtained analytically. The classical version of the non‐Hermitian Hamiltonian becomes a complex function of canonical variables and time. The corresponding kernel Hamiltonian is derived with symmetric canonical‐variable transfer in the classical gauge transformation. Moreover, with the change of position‐momentum to angle‐action variables it is revealed that the non‐adiabatic Hannay's angle and Berry phase satisfy precisely the quantum‐classical correspondence, .

Non-Abelian quantum adiabatic dynamics and phase simulation with classical resonant oscillators

It is known that the dynamics and geometric phase of a quantum system could be mapped mathematically into a classical system of coupled oscillators without loss of physics. In this work, we show that this quantum-classical mapping can also be used to simulate non-Abelian dynamics and phase of the quantum system with energy degeneracy by interpreting the quantum degeneracy as the resonance between the eigenfrequencies of the classical oscillators. The quantum evolution and Wilczek-Zee phase in the degenerate subspace can also be described by the classical non-Abelian evolution and a non-Abelian geometric angle in the resonant subspace. By studying the adiabatic evolution in this classical non-Abelian case, our results can also be used to generalize the concept of averaging principle and Hannay's angle into the classical system with resonant eigenfrequencies. The quantum-classical mapping of a tripod-scheme Hamiltonian with two degenerate dark states is employed to illustrate the classical non-Abelian dynamics and geometric angles.

Emergence of a geometric phase shift in planar noncommutative quantum mechanics

Appearance of adiabatic geometric phase shift in the context of noncommutative quantum mechanics is studied using an exactly solvable model of 2D simple harmonic oscilator in Moyal plane, where momentum non-commutativity are also considered along with spatial noncommutativity. After finding a suitable Bopp shift, that bridges the noncommutative phase space operators with their effective commutative counterparts, having their dependence on the non-commutative parameters, we study the adiabatic evolution in the Heisenberg picture. An explicit expression for the geometric phase shift under adiabatic approximation is then found without using any perturbative technique. Lastly, this phase is found to be related to the Hannay angle of a classically analogous system, by studying the evolution of the coherent state of this system.

Berry-Hannay relation in nonlinear optomechanics

We address the quantum-classical comparison of phase measurements in optomechanics in the general framework of Berry phases for composite systems. While the relation between Berry phase and Hannay angle has been proven for a large set of quadratic Hamiltonians, such correspondence has not been shown so far in the case of non-linear interactions (e.g. when three or more operators are involved). Remarkably, considering the full optomechanical interaction we recover the aforementioned mathematical relation with the Hannay angle obtained from classical equations of motion. Our results link at a fundamental level previous proposals to measure decoherence, such as the one expressed by Marshall et al., with the no-go theorem shown by Armata et al., which provides boundaries to understand the quantum-to-classical transition in optomechanics.

Hannay Angles and Grassmannian Action—Angle Quantum States

Hannay Angles and Grassmannian Action—Angle Quantum States

Hannay angles in magnetic dynamics

We consider, within the framework developed by Hannay for classical integrable systems (Hannay, 1985), the geometric phases that occur in semi-classical magnetic dynamics. Such geometric phases are generically referred to as Hannay angles, and, in the context of magnetic dynamics, may arise as a result of both adiabatically-varying ellipticity and axis of magnetization precession. We elucidate both effects and their interplay for single-domain magnetic dynamics within a simple model with time-dependent anisotropies and external field. Subsequently, we consider spin waves and rederive, from our classical approach, some known results on what is commonly referred to as the magnon Berry phase. As an aside, these results are used to give an interpretation for geometric phases that occur in superfluids. Finally, we develop a Green’s function formalism for elliptical magnons. Within this formalism, we consider magnon transport in a mesoscopic ring and show how it is influenced by interference effects that are tuned by the Hannay angle that results from a varying ellipticity. Our results may inform the field of magnonics that seeks to utilize spin waves in applications.

Geometric Phases for Classical and Quantum Dynamics: Hannay Angle and Berry Phase for Loops on a Torus

In this paper we have considered closed trajectories of a particle on a two-torus where the loops are noncontractible (poloidal and toroidal loops and knots embedded on a regular torus). We have calculated Hannay angle and Berry phase for particle traversing such loops and knots when the torus itself is adiabatically revolving. Since noncontractible loops do not enclose any area Stokes theorem has to be applied with caution. In our computational scheme we have worked with line integrals directly thus avoiding Stokes theorem.

Finding the Hannay angle in dissipative oscillatory systems via conservative perturbation theory

Usage of a Hamiltonian perturbation theory for a nonconservative system is counterintuitive and, in general, a technical impossibility by definition. However, the time-independent dual Hamiltonian formalism for the nonconservative systems has opened the door for using various conservative perturbation theories for investigating the dynamics of such systems. Here we demonstrate that the Lie transform Hamiltonian perturbation theory can be adapted to find the perturbative solutions and the frequency corrections for the dissipative oscillatory systems. As a further application, we use the perturbation theory to analytically calculate the Hannay angle for the van der Pol oscillator's limit cycle trajectory when its parameters-the strength of the nonlinearity and the frequency of the linear part-evolve cyclically and adiabatically. For this van der Pol oscillator, we also numerically calculate the corresponding geometric phase and establish its equivalence with the Hannay angle.

Particle on a torus knot: Anholonomy and Hannay angle

The phenomenon of rotation of a vector under parallel transport along a closed path is known as anholonomy. In this paper, we have studied the anholonomy for noncontractible loops — closed paths in a curved surface that do not enclose any area and hence Stokes theorem is not directly applicable. Examples of such closed paths are poloidal and toroidal loops and knots on a torus. The present study is distinct from conventional results on anholonomy for closed paths on [Formula: see text] since in the latter case all closed paths are contractible or trivial cycles. We find that for some nontrivial cycles the anholonomy cancels out over the complete cycle. Next, we calculate Hannay angle for a particle traversing such noncontractible loops when the torus itself is revolving. Some new and interesting results are obtained especially for poloidal paths that is for paths that encircle the torus ring.

The Hannay angle, thirty years on

The Hannay angle, thirty years on

Hannay Angle: Yet Another Symmetry-Protected Topological Order Parameter in Classical Mechanics

Topological way of thinking now goes beyond conventional solid materials, and topological characterization of classical mechanical systems governed by Newton's equation of motion begins to attract much attention. To have a deeper insight on physical meaning of topological numbers in mechanical systems, we demonstrate the use of the Hannay angle, a classical counterpart of the Berry phase, as a symmetry protected topological order parameter. We first derive the Hannay angle using a canonical transformation that maps the Newton's equation to the Schr\"{o}dinger type equation. The Hannay angle is then used to characterize a simple spring-mass model topologically with a particular focus on the bulk-edge correspondence and new aspects of the symmetry in a classical system.

The magnetic monopole and the separation between fast and slow magnetic degrees of freedom

The Landau–Lifshitz–Gilbert (LLG) equation that describes the dynamics of a macroscopic magnetic moment finds its limit of validity at very short times. The reason for this limit is well understood in terms of separation of the characteristic time scales between slow degrees of freedom (the magnetization) and fast degrees of freedom. The fast degrees of freedom are introduced as the variation of the angular momentum responsible for the inertia. In order to study the effect of the fast degrees of freedom on the precession, we calculate the geometric phase of the magnetization (i.e. the Hannay angle) and the corresponding magnetic monopole. In the case of the pure precession (the slow manifold), a simple expression of the magnetic monopole is given as a function of the slowness parameter, i.e. as a function of the ratio of the slow over the fast characteristic times.

Coincidence of quantum-classical orbits for periodically driven two-dimensional anisotropic oscillator—Berry phase and Hannay angle

In this paper we study quantum–classical correspondence of a periodically driven two-dimensional (2D) anisotropic-oscillator. In terms of the time-dependent canonical transformation the stationary Lissajous orbits are found along with the Hannay’s angle. On the other hand, the stationary wave functions are derived analytically with the help of the generalized gauge transformation. The Berry phase in the original gauge is found to be times the nonadiabatic Hannay’s angle with the integer number n being the eigenstate index. By using the SU(2) coherent superposition of degenerate 2D-eigenstates for a fixed energy we construct the stationary wave function, the probability cloud of which is shown to coincide perfectly with the classical orbits.

Correspondences between quantum and classical orbits Berry phases and Hannay angles for harmonic oscillator system

On the basis of quantum-classical correspondence for two-dimensional anisotropic oscillator, we study quantum-classical correspondence for two-dimensional rotation and translation harmonic oscillator system from both quantum-classical orbits and geometric phases. Here, the two one-dimensional oscillators refer to a common harmonic oscillator and a rotation and translation harmonic oscillator. In terms of the generalized gauge transformation, we obtain the stationary Lissajous orbits and Hannay's angle. On the other hand, the eigenfunctions and Berry phases are derived analytically with the help of time-dependent gauge transformation. We may draw the conclusion that the nonadiabatic Berry phase in the original gauge is-n times the classical Hannay's angle, here n is the eigenfunction index. As a matter of fact, the quantum geometric phase and the classical Hannay's angle have the same nature according to Berry. Finally, by using the SU(2) coherent superposition of degenerate two-dimensional eigenfunctions for a fixed energy value, we construct the stationary wave functions and show that the spatial distribution of wave-function probability clouds is in excellent accordance with the classical orbits, indicating the exact quantum-classical correspondence. We also demonstrate the quantum-classical correspondences for the geometric phase-angle and the quantum-classical orbits in a unified form.

Berry phase and Hannay's angle of an electromagnetic mode system driven by harmonic field with Born-Oppenheimer approximation

In this paper, we investigate the Berry phase and Hannay's angle of an electromagnetic mode system driven by harmonic field with Born-Oppenheimer approximation and obtain their algebraic expressions by theoretical calculation. The semiclassical relation between Berry phase and Hannay's angle is discussed. We find that besides the usual connection term, the Berry phase of BO hybrid system contains a novel term brought forth by the coupling induced effective gauge potential. This quantum modification can be viewed as an effective Aharonov-Bohm effect. Moreover, a similar phenomenon is founded in the Hannay's angle of classical BO hybrid system, which indicates that the Berry phase and Hannay's angle possess the same relation as the usual one. Besides, our theory can also be used to generate Artificial gauge potentials for neutral atoms.

New method for calculating the Berry connection in atom—molecule systems

In the mean-field theory of atom—molecule systems, where the bosonic atoms combine to form molecules, there is no usual U(1) symmetry, which presents an apparent hurdle for calculating the Berry connection in these systems. We develop a perturbation expansion method of Hannay's angle suitable for calculating the Berry curvature in the atom—molecule systems. With this Berry curvature, the Berry connection can be computed naturally. We use a three-level atom—molecule system to illustrate our results. In particular, with this method, we compute the curvature for Hannay's angle analytically, and compare it to the Berry curvature obtained with the second-quantized model of the same system. An excellent agreement is found, indicating the validity of our method.

Berry phase and Hannay’s angle in a quantum-classical hybrid system

Berry phase, which had been discovered for more than two decades, provides us a very deep insight on the geometric structure of quantum mechanics. Its classical counterpart--Hannay's angle is defined if closed curves of action variables return to the same curves in phase space after a time evolution. In this paper, we study the Berry phase and Hannay's angle in a quantum-classical hybrid system under the Born-Oppenheimer approximation. By quantum-classical hybrid system, we denote a composite system consists of a quantum subsystem and a classical subsystem. The effects of subsystem-subsystem couplings on the Berry phase and Hannay's angle are explored. The results show that the Berry phase has been changed sharply by the couplings, whereas the couplings have small effect on the Hannay's angle.