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arXiv:2110.04730 · Nuclear Physics B

Fingerprints of the quantum space-time in time-dependent quantum mechanics: An emergent geometric phase

Anwesha Chakraborty, Partha Nandi, Biswajit Chakraborty

Published Jan 29, 2022• 8 citations
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Abstract

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.

DOI
10.1016/j.nuclphysb.2022.115691

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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.
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Abstract In this paper we look for the existence of Berry phase in time dependent harmonic oscillators in noncommutative space. Two systems are considered in our study in the noncommutative framework. The first one is a system in which a scale invariant term is present in the Hamiltonian from the beginning, and the second one is a system in which a scale invariant term emerges due to a change of variables from the noncommutative to the commutative ones. We first compute the eigenstates of both the systems using the Lewis invariant approach. We then employ the Lewis invariant technique to obtain the geometric phase under adiabatic approximation. We also calculate the Berry phase explicitly by choosing appropriate forms of the time dependent parameters appearing in the Hamiltonian. Our analysis surprisingly reveals that a scale invariant time reversal symmetry breaking term may not always lead to a non trivial Berry phase.
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.
In a recent paper [Y. C. Ge and M. S. Child, Phys. Rev. Lett. 78, 2507 (1997)], by using a Gaussian wave function, Ge and Child presented a nonadiabatic relation between the quantum Berry phase and the classical Hannay angle for the time-dependent harmonic oscillator. In this paper, we present a perspective for this relation without the use of a trial wave function. In particular, an exact explicit formula for the cyclic evolution over the period T in the parameter space of action invariant is obtained; the $\ensuremath{-}(n+1/2)$ relation between the quantum geometric angle and the Hannay angle is rigorously established.
In this work we present a simple and elegant approach to study the adiabatic and nonadiabatic evolution of a generalized damped harmonic oscillator which is described by the generalized Caldirola–Kanai Hamiltonian, in both classical and quantum contexts. Based on time-dependent dynamical invariants, we find that the geometric phase acquired when the damped oscillator evolves adiabatically in time provides a direct connection between the classical Hannay’s angle and the quantum Berry’s phase. In addition, we solve the time-dependent Schrödinger equation for this system and calculate various quantum properties of the damped generalized harmonic one, such as coherent states, expectation values of the position and momentum operators, their quantum fluctuations and the associated uncertainty product.