The Pancharatnam phase for displaced number states of the harmonic oscillator is discussed. In particular, it is examined how a single quantum oscillator, driven by a suitable transient external force, evolves form the initial eigenstate u m (x) to the final, displaced number state modified with a suitable phase factor. The significance of this, usually neglected, phase factor for the solution of the relevant time-dependent Schrodinger equation and for the geometric phase accumulated in the wavefunction during the time evolution of the system is examined. The general expression for the geometric phase for a noncyclic evolution from an initial displaced number state at time t 1 to the final state at time t 2 is derived and two applications, that of delta and harmonic forcing, are worked out. The special case of cyclic evolution is subsequently discussed and, in particular, the conditions leading to such an evolution are derived. The relationship to the classical notion of cyclic evolution is also examined in some detail and it is demonstrated that the general expression for the Pancharatnam phase reduces to the corresponding Berry phase.
The Pancharatnam phase for displaced number states of the harmonic oscillator is discussed. In particular, it is examined how a single quantum oscillator, driven by a suitable transient external force, evolves form the initial eigenstate u m (x) to the final, displaced number state modified with a suitable phase factor. The significance of this, usually neglected, phase factor for the solution of the relevant time-dependent Schrodinger equation and for the geometric phase accumulated in the wavefunction during the time evolution of the system is examined. The general expression for the geometric phase for a noncyclic evolution from an initial displaced number state at time t 1 to the final state at time t 2 is derived and two applications, that of delta and harmonic forcing, are worked out. The special case of cyclic evolution is subsequently discussed and, in particular, the conditions leading to such an evolution are derived. The relationship to the classical notion of cyclic evolution is also examined in some detail and it is demonstrated that the general expression for the Pancharatnam phase reduces to the corresponding Berry phase.
The Pancharatnam phase for ordinary and generalized squeezed states of the harmonic oscillator is discussed. In particular, the general expressions for the total and dynamical phases are obtained yielding, from their difference, the geometric phase for a nonadiabatic and noncyclic evolution from an initial generalized squeezed state at time t∣IH to the final state at a time t\ensuremath{''}. The special case of a two-photon interaction process, such as parametric amplification, is subsequently discussed. In particular, the conditions leading to cyclic evolution are examined and the corresponding Aharonov-Anandan phase is found from the general expression for the Pancharatnam phase.
The generalized time-dependent harmonic oscillator is studied. Though several approaches to the solution of this model have been available, yet a new approach is presented here, which is very suitable for the study of cyclic solutions and geometric phases. In this approach, finding the time evolution operator for the Schrödinger equation is reduced to solving an ordinary differential equation for a c-number vector which moves on a hyperboloid in a three-dimensional space. Cyclic solutions do not exist for all time intervals. A necessary and sufficient condition for the existence of cyclic solutions is given. There may exist some particular time interval in which all solutions with definite parity, or even all solutions are cyclic. Criteria for the appearance of such cases are given. The known relation that the nonadiabatic geometric phase for a cyclic solution is proportional to the classical Hannay angle is reestablished. However, this is valid only for special cyclic solutions. For more general ones, the nonadiabatic geometric phase may contain an extra term. Several cases with relatively simple Hamiltonians are solved and discussed in detail. Cyclic solutions exist in most cases. The pattern of the motion, say, finite or infinite, cannot be simply determined by the nature of the Hamiltonian (elliptic or hyperbolic, etc.). For a Hamiltonian with a definite nature, the motion can change from one pattern to another, that is, some kind of phase transition may occur, if some parameter in the Hamiltonian goes through some critical value.
In 1984 Berry addressed a quantum system undergoing a unitary and cyclic evolution under the action of a time-dependent Hamiltonian (M. V. Berry, 1984). The process was supposed to be adiabatic, meaning that the time scale of the system’s evolution was much shorter than the time scale of the changing Hamiltonian. Until Berry’s study, it was assumed that for a cyclic Hamiltonian the quantum state would acquire only so-called dynamical phases, deprived of physical meaning. Such phases could be eliminated by redefining the quantum state through a “gauge” transformation of the form |ψ〉 → eiα |ψ〉. However, Berry discovered that besides the dynamical, there was an additional phase that could not be “gauged away” and whose origin was geometric or topological. It depended on the path that |ψ〉 describes in the parameter space spanned by those parameters to which the Hamiltonian owed its time dependence. Berry’s discovery was the starting point for a great amount of investigations that brought to light topological aspects of both quantum and classical systems. Berry’s phase was soon recognized as a special case of more general phases that showed up when dealing with topological aspects of different systems. For example, the Aharonov-Bohm phase could be understood as a geometric phase. The rotation angle acquired by a parallel-transported vector after completing a closed loop in a gravitationally curved space-time region, is also a geometric, Berry-like phase. Another example is the precession of the plane of oscillation of a Foucault pendulum. Berry’s original formulation was directly applicable to the case of a spin-1/2 system evolving under the action of a slowly varying magnetic field that undergoes cyclic changes. A spin-1/2 system is a special case of a two-level system. Another instances are two-level atoms and polarized light, so that also in these cases we should expect to find geometric phases. In fact, the first experimental test of Berry’s phase was done using polarized, classical light (A. Tomita, 1986). Pancharatnam (S. Pancharatnam, 1956) anticipated Berry’s phase when he proposed, back in 1956, how to decide whether two polarization states are “in phase”. Pancharatnam’s prescription is an operational one, based upon observing whether the intensity of the interferogram formed by two polarized beams has maximal intensity. In that case, the two polarized beams are said to be “in phase”. Such a definition is analogous to the definition of distant parallelism in differential geometry. Polarized states can be subjected to different transformations which could be cyclic or not, adiabatic or not, unitary or not. The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects
The evolution of a quantum mechanical system under a nonadiabatic external perturbation at a time interval (0,T) is considered. It is shown that all the cyclic states of the system, mod Psi (T))=e i phi mod Psi (0)), are determined by the eigenvalues in (quasi-energies) and associated eigenvectors mod phi in (t)) of the Schrodinger operator S q (t)=H(t)-i( delta / delta t) acting on some Hilbert space. The set of linearly independent cyclic (quasi-energy) states possess some properties similar to the properties of the stationary states of a closed system. The Berry phases of the states associated with eigenvectors of the discrete spectrum of S q (t), which are single-valued functions of omega =2 pi /T, are supplied by the partial derivatives of the corresponding eigenvalues (quasi-energies) with respect to omega . The approach developed is illustrated by several applications to time-dependent systems: the system under an adiabatic perturbation, the forced harmonic oscillator, and the two-level system. Even in the case of a system with a time-independent Hamiltonian there exist nonstationary quasi-energy states that have nontrivial Berry phases ( beta /2 pi not an integer). The criterion of existence of such states is formulated in terms of the energies of the system, and the corresponding expression for the Berry phases is obtained. Some examples of nonstationary cyclic states of closed systems, including the coherent states of a force-free oscillator and the Wannier states of electrons in the parabolic band, are considered.
In a quantum system initially in the n-th eigenstate, an adiabatic evolution of the Hamiltonian ensures that the system remains in the corresponding instantaneous eigenstate while acquiring a phase factor. This phase has two components: one resulting from standard time evolution and another associated with the dependence of the eigenstate on the varying Hamiltonian, known as the Berry phase. In this work, we explore the concept of geometric amplitudes in the context of a Hermitian Hamiltonian with imaginary eigenvalues. We introduce the notion of geometric amplitude and provide a novel derivation of this concept. Our study reveals that a system undergoing cyclic evolution under adiabatic conditions acquires an additional amplitude factor of purely geometric origin. To illustrate this idea, we apply it to a concrete case: a generalized harmonic oscillator.
The generalized time-dependent harmonic oscillator is studied. Though several approaches to the solution of this model have been available, yet a new approach is presented here, which is very suitable for the study of cyclic solutions and geometric phases. In this approach, finding the time evolution operator for the Schr\\"odinger equation is reduced to solving an ordinary differential equation for a c-number vector which moves on a hyperboloid in a three-dimensional space. Cyclic solutions do not exist for all time intervals. A necessary and sufficient condition for the existence of cyclic solutions is given. There may exist some particular time interval in which all solutions with definite parity, or even all solutions, are cyclic. Criterions for the appearance of such cases are given. The known relation that the nonadiabatic geometric phase for a cyclic solution is proportional to the classical Hannay angle is reestablished. However, this is valid only for special cyclic solutions. For more general ones, the nonadiabatic geometric phase may contain an extra term. Several cases with relatively simple Hamiltonians are solved and discussed in detail. Cyclic solutions exist in most cases. The pattern of the motion, say, finite or infinite, can not be simply determined by the nature of the Hamiltonian (elliptic or hyperbolic, etc.). For a Hamiltonian with a definite nature, the motion can changes from one pattern to another, that is, some kind of phase transition may occur, if some parameter in the Hamiltonian goes through some critical value.