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Register a new userSum Rule of Quantum Uncertainties: Coupled Harmonic Oscillator System with Time-Dependent Parameters
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Uncertainties $(Delta x)^2$ and $(Delta p)^2$ are analytically derived in an $N$-coupled harmonic oscillator system when spring and coupling constants are arbitrarily time-dependent and each oscillator is in an arbitrary excited state. When $N = 2$, those uncertainties are shown as just arithmetic average of uncertainties of two single harmonic oscillators. We call this property as sum rule of quantum uncertainty. However, this arithmetic average property is not generally maintained when $N geq 3$, but it is recovered in $N$-coupled oscillator systems if and only if $(N-1)$ quantum numbers are equal. The generalization of our results to a more general quantum system is briefly discussed.
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The dynamics of mixedness and entanglement is examined by solving the time-dependent Schr{o}dinger equation for three coupled harmonic oscillator system with arbitrary time-dependent frequency and coupling constants parameters. We assume that part of oscillators is inaccessible and remaining oscillators accessible. We compute the dynamics of entanglement between inaccessible and accessible oscillators. In order to show the dynamics pictorially we introduce three quenched models. In the quenched models both mixedness and entanglement exhibit oscillatory behavior in time with multi-frequencies. It is shown that the mixedness for the case of one inaccessible oscillator is larger than that for the case of two inaccessible oscillators in the most time interval. Contrary to the mixedness entanglement for the case of one inaccessible oscillator is smaller than that for the case of two inaccessible oscillators in the most time interval.
Using Schwinger Variational Principle we solve the problem of quantum harmonic oscillator with time dependent frequency. Here, we do not take the usual approach which implicitly assumes an adiabatic behavior for the frequency. Instead, we propose a new solution where the frequency only needs continuity in its first derivative or to have a finite set of removable discontinuities.
Using operator ordering techniques based on BCH-like relations of the su(1,1) Lie algebra and a time-splitting approach,we present an alternative method of solving the dynamics of a time-dependent quantum harmonic oscillator for any initial state. We find an iterative analytical solution given by simple recurrence relations that are very well suited for numerical calculations. We use our solution to reproduce and analyse some results from literature in order to prove the usefulness of the method and, based on these references, we discuss efficiency in squeezing, when comparing the parametric resonance modulation and the Janszky-Adam scheme.
In the context of the de Broglie-Bohm pilot wave theory, numerical simulations for simple systems have shown that states that are initially out of quantum equilibrium - thus violating the Born rule - usually relax over time to the expected $|psi|^2$ distribution on a coarse-grained level. We analyze the relaxation of nonequilibrium initial distributions for a system of coupled one-dimensional harmonic oscillators in which the coupling depends explicitly on time through numerical simulations, focusing in the influence of different parameters such as the number of modes, the coarse-graining length and the coupling constant. We show that in general the system studied here tends to equilibrium, but the relaxation can be retarded depending on the values of the parameters, particularly to the one related to the strength of the interaction. Possible implications on the detection of relic nonequilibrium systems are discussed.
The solution of the Feinberg-Horodecki (FH) equation for a time-dependent mass (TDM) harmonic oscillator quantum system is studied. A certain interaction is applied to a mass to provide a particular spectrum of stationary energies. The related spectrum of the harmonic oscillator potential acting on the TDM oscillators is found. We apply the time version of the asymptotic iteration method (AIM) to calculate analytical expressions of the TDM stationary state energies and their wave functions. It is shown that the obtained solutions reduce to those of simple harmonic oscillator as the time-dependent of the mass reduces to
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