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Wikipedia / Geometric phase

In classical and quantum mechanics, the geometric phase (also known as the Pancharatnam–Berry phase, Pancharatnam phase, or Berry phase) is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processeses, which results from the geometrical properties of the parameter space of the Hamiltonian. The phenomenon was independently discovered by S. Pancharatnam (1956) in classical optics and by H. C. Longuet-Higgins (1958) in molecular physics; it was later generalized by Michael Berry in (1984), and extended to nonadiabatic cycles by Y. Aharonov and J. Anandan (1987). Its analog in classical mechanics is the Hannay angle.

It can be seen in the conical intersection of potential energy surfaces and in the Aharonov–Bohm effect. The geometric phase around the conical intersection involving the ground electronic state of the C6H3F3+ molecular ion is discussed on pages 385–386 of the textbook by Bunker and Jensen. In the case of the Aharonov–Bohm effect, the adiabatic parameter is the magnetic field enclosed by two interference paths, and it is cyclic in the sense that these two paths form a loop. In the case of the conical intersection, the adiabatic parameters are the molecular coordinates. Apart from quantum mechanics, it arises in a variety of other wave systems, such as classical optics. As a rule of thumb, it can occur whenever there are at least two parameters characterizing a wave in the vicinity of some sort of singularity or hole in the topology; two parameters are required because either the set of nonsingular states will not be simply connected, or there will be a nonzero holonomy.

Waves are characterized by amplitude and phase, and may vary as a function of those parameters. The geometric phase occurs when both parameters are changed simultaneously but very slowly (adiabatically), and eventually brought back to the initial configuration. In quantum mechanics, this could involve rotations but also translations of particles, which are apparently undone at the end. One might expect that the waves in the system return to the initial state, as characterized by the amplitudes and phases (and accounting for the passage of time). However, if the parameter excursions correspond to a loop instead of a self-retracing back-and-forth variation, then it is possible that the initial and final states differ in their phases. This phase difference is the geometric phase, and its occurrence typically indicates that the system's parameter dependence is singular (its state is undefined) for some combination of parameters. To measure the geometric phase in a wave system, an interference experiment is required.

The Foucault pendulum is an example from classical mechanics that is sometimes used to illustrate the geometric phase. The geometric angle in an adiabatically driven harmonic oscillator is another simple example from classical mechanics.

Berry phase in quantum mechanics

In a quantum system at the n-th eigenstate, an adiabatic evolution of the Hamiltonian sees the system remain in the n-th eigenstate of the Hamiltonian, while also obtaining a phase factor. The phase obtained has a contribution from the state's time evolution and another from the variation of the eigenstate with the changing Hamiltonian. The second term corresponds to the Berry phase, and for non-cyclical variations of the Hamiltonian it can be made to vanish by a different choice of the phase associated with the eigenstates of the Hamiltonian at each point in the evolution.

However, if the variation is cyclical, the Berry phase cannot be cancelled; it is invariant and becomes an observable property of the system. By reviewing the proof of the adiabatic theorem given by Max Born and Vladimir Fock, in Zeitschrift für Physik 51, 165 (1928), we could characterize the whole change of the adiabatic process into a phase term. Under the adiabatic approximation, the coefficient of the n is Berry's phase with respect to parameter t. Changing the variable t can also be calculated by integrating the Berry curvature.

Examples of geometric phases

Foucault pendulum

In locations away from the equator, if a long and heavy pendulum is monitored over a period of hours, its plane of oscillation appears to change spontaneously as the Earth rotates. This is called a Foucault pendulum. An explanation in terms of geometric phases is given by Wilczek and Shapere:

To put it in different words, there are no inertial forces that could make the pendulum precess, so the precession (relative to the direction of motion of the path along which the pendulum is carried) is entirely due to the turning of this path. Thus the orientation of the pendulum undergoes parallel transport. For the original Foucault pendulum, the path is a circle of latitude, and by the Gauss–Bonnet theorem, the phase shift is given by the enclosed solid angle.

Derivation

Consider a vector around a closed loop, e.g. a spherical triangle, on the sphere: The angle by which it twists, , is proportional to the area inside the loop and will be considered below.

In a near-inertial frame moving in tandem with the Earth, but not sharing the rotation of the Earth about its own axis, the pendulum bob traces out a circular path during one sidereal day, i.e. a planar path since oscillation along the suspending rod is insignificant.

At the latitude of Paris, 48 degrees 51 minutes north, a full precession cycle takes just under 32 hours, so after one sidereal day, when the Earth is back in the same orientation as one sidereal day before, the oscillation plane has turned by just over 270 degrees. If the plane of swing was north–south at the outset, it is east–west one sidereal day later.

This also implies that there has been exchange of momentum; the Earth and the pendulum bob have exchanged momentum. The Earth is so much more massive than the pendulum bob that the Earth's change of momentum is unnoticeable. Nonetheless, since the pendulum bob's plane of swing has shifted, the conservation laws imply that an exchange must have occurred.

The precession of the oscillating plane can be demonstrated, by composing the infinitesimal rotations, that the precession rate is proportional to, in terms of the projection of the angular velocity of the Earth onto the normal are spherical angles. After 24 hours, the difference between initial and final orientations of the trace in the Earth frame is , which corresponds to the value given by the Gauss–Bonnet theorem is the Gaussian curvature). is also called the holonomy or geometric phase of the pendulum. When analyzing earthbound motions, the Earth frame is not an inertial frame, but rotates about the local vertical at an effective rate of radians per day. A simple method employing parallel transport within cones tangent to the Earth's surface can be used to describe the rotation angle of the swing plane of Foucault's pendulum.

From the perspective of an Earth-bound coordinate system (the measuring circle and spectator are Earth-bounded, also if terrain reaction to Coriolis force is not perceived by spectator when he moves), using a rectangular coordinate system with its axis pointing east and its axis pointing north, the precession of the pendulum is due to the Coriolis force (other fictitious forces as gravity and centrifugal force have no direct precession component). Consider a planar pendulum with constant natural frequency in the small angle approximation. There are two forces acting on the pendulum bob: the restoring force provided by gravity and the wire, and the Coriolis force (the centrifugal force, opposed to the gravitational restoring force, can be neglected). The Coriolis force at latitude where is the rotational frequency of the Earth, is the component of the Coriolis force in the direction, and is the component of the Coriolis force in the direction.

The restoring force, in the small-angle approximation

Using Newton's laws of motion

Switching to complex coordinates

To first order in

If time is measured in days, then and the pendulum rotates by an angle of during one day. The mathematics of these equations, i.e. derivation and so on, can be found in many textbooks on classical mechanics.

is practically trivial if one recalls a theorem of Pappus. enclosed by the cap is then given by

Polarized light in an optical fiber

A second example is linearly polarized light entering a single-mode optical fiber. Suppose the fiber traces out some path in space, and the light exits the fiber in the same direction as it entered. Then compare the initial and final polarizations. In semiclassical approximation the fiber functions as a waveguide, and the momentum of the light is at all times tangent to the fiber. The polarization can be thought of as an orientation perpendicular to the momentum. As the fiber traces out its path, the momentum vector of the light traces out a path on the sphere in momentum space. The path is closed, since initial and final directions of the light coincide, and the polarization is a vector tangent to the sphere. Going to momentum space is equivalent to taking the Gauss map. There are no forces that could make the polarization turn, just the constraint to remain tangent to the sphere. Thus the polarization undergoes parallel transport, and the phase shift is given by the enclosed solid angle (times the spin, which in case of light is1).

Stochastic pump effect

A stochastic pump is a classical stochastic system that responds with nonzero, on average, currents to periodic changes of parameters. The stochastic pump effect can be interpreted in terms of a geometric phase in evolution of the moment generating function of stochastic currents.

Spin

The geometric phase can be evaluated exactly for a spin- particle in a magnetic field.

Geometric phase defined on attractors

While Berry's formulation was originally defined for linear Hamiltonian systems, it was soon realized by Ning and Haken that similar geometric phase can be defined for entirely different systems such as nonlinear dissipative systems that possess certain cyclic attractors. They showed that such cyclic attractors exist in a class of nonlinear dissipative systems with certain symmetries. There are several important aspects of this generalization of Berry's phase: 1) Instead of the parameter space for the original Berry phase, this Ning-Haken generalization is defined in phase space; 2) Instead of the adiabatic evolution in quantum mechanical system, the evolution of the system in phase space needs not to be adiabatic. There is no restriction on the time scale of the temporal evolution; 3) Instead of a Hermitian system or non-hermitian system with linear damping, systems can be generally nonlinear and non-hermitian.

Exposure in molecular adiabatic potential surface intersections

There are several ways to compute the geometric phase in molecules within the Born–Oppenheimer is the adiabatic electronic wave function. The nonadiabatic coupling can be used to define a loop integral, analogous to a Wilson loop (1974) in field theory, developed independently for molecular framework by M.. The D-th adiabatic electronic state.

An alternative to the D then using only the j

Geometric phase and quantization of cyclotron motion

moves on a circular (cyclotron) orbit. is acceptable. Quantum-mechanically, only discrete energy levels (Landau levels for electrons in graphene. Although the derivation of these results is not difficult, there is an alternative way of deriving them, which offers in some respect better physical insight into the Landau level quantization. This alternative way is based on the semiclassical Bohr–Sommerfeld quantization picked up by the electron while it executes its (real-space) motion along the closed loop of the cyclotron orbit. of electrons in graphene.

See also

Notes

For simplicity, we consider electrons confined to a plane, such as 2DEG and magnetic field perpendicular to the plane.

is the Fermi velocity (of electrons in graphene).

Footnotes

Sources

Further reading