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Journal of Physics A Mathematical and Theoretical

Berry phase in arbitrary dimensions

Alagu Thilagam

Published Aug 12, 2010• 5 citations
Publisher

Abstract

We study the properties of the Berry phase in quantum systems where momentum and position operators satisfy the R -deformed Heisenberg relations and the noninteger dimension d and angular momentum ℓ appear as parameters. We focus on the dynamics of the generalized time-dependent harmonic oscillator which satisfies the Schrödinger wave mechanics formulated in arbitrary dimensions. We also examine quantum systems which can be expressed as linear combination of generators with time-dependent coefficients in arbitrary dimensions. We show the explicit dependence of the Berry phase on d and the parity ℓ of quantum systems which satisfy the R -deformed Heisenberg relations. Using an explicit relation of the Fubini–Study metric in fractional dimensions, we highlight a geometrical interpretation of d and ℓ of systems which are quantized in arbitrary dimensions.

DOI
10.1088/1751-8113/43/35/354004

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We study the properties of the Berry phase in quantum systems where momentum and position operators satisfy the R -deformed Heisenberg relations and the noninteger dimension d and angular momentum ℓ appear as parameters. We focus on the dynamics of the generalized time-dependent harmonic oscillator which satisfies the Schrödinger wave mechanics formulated in arbitrary dimensions. We also examine quantum systems which can be expressed as linear combination of generators with time-dependent coefficients in arbitrary dimensions. We show the explicit dependence of the Berry phase on d and the parity ℓ of quantum systems which satisfy the R -deformed Heisenberg relations. Using an explicit relation of the Fubini–Study metric in fractional dimensions, we highlight a geometrical interpretation of d and ℓ of systems which are quantized in arbitrary dimensions.
We formulate an algebraic approach to quantum mechanics in fractional dimensions in which the momentum and position operators P , Q satisfy the R -deformed Heisenberg relations, which depend on an operator ν. We find representations of P , Q in which the dimension d and angular momentum ℓ appear as parameters related to the eigenvalues of ν. We analyse the domain of P and find conditions which ensure that P is Hermitian. We investigate plane wave solutions and also free particle wavefunctions in fractional dimensions, and show that as a consequence of wavefunction continuity ℓ is quantized. The representations of P , Q also lead to the corresponding representations of paraboson operators which are used to solve the harmonic oscillator in dimension d , both algebraically and analytically. We demonstrate that the formalism extends also to time-dependent Hamiltonians by solving the time-dependent harmonic oscillator in any dimension d > 0 using the method of Lewis and Riesenfeld.
We construct representations of the Lie algebra using representations of the momentum and position operators satisfying the R -deformed Heisenberg relations, in which the fractional dimension d and angular momentum ℓ appear as parameters. The Bargmann index κ, which characterizes representations of the positive discrete series of , can take any positive value. We construct coherent states in fractional dimensions, in particular we extend the two well-known analytic representations of coherent states for , Perelomov and Barut–Girardello states, from dimension one to any dimension d . We generalize this construction to time-dependent coherent states by means of the symmetries of the quantum time-dependent harmonic oscillator in fractional dimensions. We investigate the uncertainty relations of the momentum and position operators with respect to these coherent states, and their dependence on the dimension.
We show the emergence of Berry phase in a forced harmonic oscillator system placed in the quantum space-time of Moyal type, where the time ‘t’ is also an operator. An effective commutative description of the system gives a time dependent generalised harmonic oscillator system with perturbation linear in position and momentum. The system is then diagonalised to get a generalised harmonic oscillator and then its adiabatic evolution over time-period T is studied in Heisenberg picture to compute the expression of geometric phase-shift.
In this paper, we study the generalized harmonic oscillator with arbitrary time-dependent mass and frequency subjected to a linear velocity-dependent frictional force from classical and quantum points of view. We obtain the solution of the classical equation of motion of this system for some particular cases and derive an equation of motion that describes three different systems. Furthermore, with the help of the quantum invariant method and using quadratic invariants we solve analytically and exactly the time-dependent Schrödinger equation for this system. Afterwards, we construct coherent states for the quantized system and employ them to investigate some of the system's quantum properties such as quantum fluctuations of the coordinate and the momentum as well as the corresponding uncertainty product. In addition, we derive the geometric, dynamical and Berry phases for this nonstationary system. Finally, we evaluate the dynamical and Berry phases for three special cases and surprisingly find identical expressions for the dynamical phase and the same formulae for the Berry's phase.
The most general time-dependence Hamiltonian for a harmonic oscillator is both linear and quadratic in the coordinate and the canonical momentum. It describes in general a harmonic oscillator with a time-dependent mass, a time-dependent friction (or antifriction) 'constant', and a time-dependent spring constant, acted upon by a time-dependent force. The energy, whose time derivative is the power, is in general different from the Hamiltonian. The generalized Berry phase for a given energy eigenstate has a state-dependent part, which vanishes if there is no damping, and an arbitrary state-independent part. If the Hamiltonian is identified as the energy, both the energy eigenvalues and the generalized Berry phase are different. In the adiabatic limit both approaches give the same total phase, which is the sum of the dynamical and Berry phases.