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arXiv:hep-th/0204116 · Physical Review A

Time-dependent driven anharmonic oscillator and adiabaticity

Hyeong-Chan Kim, Jae Hyung Yee

Published Sep 27, 2002• 5 citations
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Abstract

We use the Liouville--von Neumann approach to study the dynamics and the adiabaticity of a time-dependent driven anharmonic oscillator as an example of nonequilibrium quantum dynamics. We show that the adiabaticity is minimally broken in the sense that a Gaussian wave packet at the past infinity evolves to coherent states; however slowly the potential changes, its coherence factor is of the order of the coupling. We also show that the dynamics are governed by an equation of motion similar to the Kepler motion, which satisfies angular momentum conservation.

DOI
10.1103/physreva.66.032117

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We use the Liouville--von Neumann approach to study the dynamics and the adiabaticity of a time-dependent driven anharmonic oscillator as an example of nonequilibrium quantum dynamics. We show that the adiabaticity is minimally broken in the sense that a Gaussian wave packet at the past infinity evolves to coherent states; however slowly the potential changes, its coherence factor is of the order of the coupling. We also show that the dynamics are governed by an equation of motion similar to the Kepler motion, which satisfies angular momentum conservation.
The dynamics of an initially excital Gaussian wave packet (harmonic oscillator coherent state, HOCS) in a general one-dimensional anharmonic system is studied analytically with the aid of a second quantization formalism for times shorter than a classical oscillation period. The initial state population probability PQ(t), its classical counterpart PC(t), and the overlap probability PQC(t) between a time evolved wave packet and a classically moving Gaussian with stationary shape are calculated in order to analyze the spreading of the wave packet. It was found that, in general, PQ(t) and PC(t) agree very well if the potential energy function has no strongly changing low order derivatives. This agreement is held even if the wave packet spreads drastically within the time scale considered as can be demonstrated by analyzing PQC(t). The analytical expressions for PQ(t), PC(t), and PQC(t) are applied to a dissociative third-order oscillator and a strongly anharmonic fourth-order oscillator. While for a harmonic system, the initial decay time of PQ(t) and PC(t) (which are identical in this case) are only dependent on the energy expectation value 〈H〉 with respect to the initial HOCS, differences up to an order of magnitude occur for HOCS with the same energy in the model systems.
Unique set of coherent states for the anharmonic oscillator is obtained by requiring i. under the quantum mechanical time evolution a coherent state evolves into another, governed by trajectory in the classical phase space (of a related hamiltonian); ii. the resolution of identity involves exactly the classical phase space measure.The rules are invariant under unitary transformations of the quantum theory and canonical transformations of the classical theory. The states are almost, but not quite, minimal uncertainty wave packets. The construction can be generalized to quantum versions of integrable classical theories.
In this paper, the analytic solution of the time-dependent Schrödinger equation is obtained for the wave packet in two-dimensional oscillator potential. The quantum dynamics of the wave packet is investigated based on this analytic solution. To our knowledge, this is the first time we solve, analytically and exactly this kind of time-dependent Schrödinger equation in a two-dimensional system, in which the Gaussian parameters satisfy the coupled nonlinear differential equations. The coherent states and their rotations of the system are discussed in detail. We find also that this analytic solution includes four kinds of modes of the evolutions for the wave packets: rigid, rotational, vibrational states and a combination of the rotation and vibration without spreading.
Quantum dynamical properties of a general time-dependent coupled oscillator are investigated based on the theory of two-dimensional (2D) dynamical invariants. The quantum dynamical invariant of the system satisfies the Liouville–von Neumann equation and it coincides with its classical counterpart. The mathematical formula of this invariant involves a cross term which couples the two oscillators mutually. However, we show that, by introducing two pairs of annihilation and creation operators, it is possible to uncouple the original invariant operator so that it becomes the one that describes two independent subsystems. The eigenvalue problem of this decoupled quantum invariant can be solved by using a unitary transformation approach. Through this procedure, we eventually obtain the eigenfunctions of the invariant operator and the wave functions of the system in the Fock state. The wave functions that we have developed are necessary in studying the basic quantum characteristics of the system. In order to show the validity of our theory, we apply our consequences to the derivation of the fluctuations of canonical variables and the uncertainty products for a particular 2D oscillatory system whose masses are exponentially increasing.
An adiabatic invariant, which is a conserved quantity, is useful for studying quantum and classical properties of dynamical systems. Adiabatic invariants for time-dependent superconducting qubit-oscillator systems and resonators are investigated using the Liouville-von Neumann equation. At first, we derive an invariant for a simple superconducting qubit-oscillator through the introduction of its reduced Hamiltonian. Afterwards, an adiabatic invariant for a nanomechanical resonator linearly interfaced with a superconducting circuit, via a coupling with a time-dependent strength, is evaluated using the technique of unitary transformation. The accuracy of conservation for such invariant quantities is represented in detail. Based on the results of our developments in this paper, perturbation theory is applicable to the research of quantum characteristics of more complicated qubit systems that are described by a time-dependent Hamiltonian involving nonlinear terms.
We construct a quantum Markovian master equation for a driven system coupled to a thermal bath. The derivation utilizes an explicit solution of the propagator of the driven system. This enables the validity of the master equation to be extended beyond the adiabatic limit. The nonadiabatic master equation (NAME) is derived employing the weak system-bath coupling limit. The NAME is valid when a separation of timescales exists between the bath dynamics and the external driving. In contrast to the adiabatic master equation, the NAME leads to coupled equations of motion for the population and coherence. We employ the NAME to solve the example of an open driven time-dependent harmonic oscillator. For the harmonic oscillator the NAME predicts the emergence of coherence associated with the dissipation term. As a result of the nonadiabatic driving the thermalization rate is suppressed. The solution is compared with both numerical calculations and the adiabatic master equation.