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Chinese Physics Letters

Time Evolution of a Harmonic Chain with Fixed Boundary Conditions

H. Lu, Jing-Dong Bao

Published Apr 1, 2011• 3 citations
Publisher

Abstract

We investigate the time evolution of a coupled harmonic-oscillator chain under two boundary conditions: two ends fixed and one end fixed. The dynamics of the coupled chain and the steady variances of the coordinates are explicitly analyzed by the entire Hamiltonian using a diagonalization approach. Our result shows the desirable symmetry for the case with two ends fixed. In particular, a Langevin simulation technique is proposed to sample the harmonic chain across the entire equilibrium distribution.

DOI
10.1088/0256-307x/28/4/040505

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We investigate the time evolution of a coupled harmonic-oscillator chain under two boundary conditions: two ends fixed and one end fixed. The dynamics of the coupled chain and the steady variances of the coordinates are explicitly analyzed by the entire Hamiltonian using a diagonalization approach. Our result shows the desirable symmetry for the case with two ends fixed. In particular, a Langevin simulation technique is proposed to sample the harmonic chain across the entire equilibrium distribution.
We generalize the classical time evolution problem of a harmonic-oscillator chain with periodic and fixed-end boundary conditions to the quantum mechanical case. It is shown explicitly that the high-temperature limits of our quantum mechanical treatment give the known classical results. We also show that the finite diffusion constant for the periodic chain arises from the zero-frequency normal mode which represents the rotational mode of the periodic chain, and the diffusion constant vanishes as tempperature approaches zero. The zero-frequency normal mode is also responsible for the divergence of the mean square displacement of a tagged oscillator. On the other hand, the absence of the zero-frequency normal mode in a fixed-end chain gives rise to zero diffusion constant and finite mean square displacement. The finiteness of diffusion constant and the divergence of mean square displacement in this model are responsible for the symmetry of the system.
Starting from quantum Langevin equations for operators we study thermal properties of a one-dimensional harmonic chain to whose ends independent heat baths are attached. In this paper, we mainly discuss the thermal equilibrium state that the chain eventually approaches if the heat baths are at equal temperatures. In the classical limit, this state is determined by the Gibbs ensemble of the free chain, whereas in the quantal case, this is only true if the strength of coupling between chain and heat baths is made infinitely small. We find that corrections for finite coupling strength are appreciable only in boundary layers near both ends of the chain. The thickness of the boundary layers depends only on the temperature and not on the damping constant. Outside these boundary layers we find an analogy between thermal properties of the chain and a discrete random walk.
The problem of the quantum description of the damping of a 1D chain of coupled oscillators with periodic boundary conditions in the Markovian regime is solved. The equations of motion of the normal modes are derived, and the evolution in time of their mean values and variances is explicitly studied. Furthermore, we investigate the effect of the diffusion and the friction on the dynamics of the chain. It is shown that when the off-diagonal elements of the diffusion and friction tensors are taken into account, the most notable contribution of these processes is observed in the mode in which all the oscillators undergo an in-phase vibration.
We describe a quantum system consisting of a one-dimensional linear chain of n identical harmonic oscillators coupled by a nearest neighbor interaction. Two boundary conditions are taken into account: periodic boundary conditions (where the nth oscillator is coupled back to the first oscillator) and fixed wall boundary conditions (where the first oscillator and the nth oscillator are coupled to a fixed wall). The two systems are characterized by their Hamiltonian. For their quantization, we treat these systems as Wigner quantum systems (WQSs), allowing more solutions than just the canonical quantization solution. In this WQS approach, one is led to certain algebraic relations for operators (which are linear combinations of position and momentum operators) that should satisfy triple relations involving commutators and anti-commutators. These triple relations have a solution in terms of the Lie superalgebra gl(1|n). We study a particular class of gl(1|n) representations V(p), the so-called ladder representations. For these representations, we determine the spectrum of the Hamiltonian and of the position operators (for both types of boundary conditions). Furthermore, we compute the eigenvectors of the position operators in terms of stationary states. This leads to explicit expressions for position probabilities of the n oscillators in the chain. An analysis of the plots of such position probability distributions gives rise to some interesting observations. In particular, the physical behavior of the system as a WQS is very much in agreement with what one would expect from the classical case, except that all physical quantities (energy, position, and momentum of each oscillator) have a finite spectrum.
In the following, we study the dissipative time evolution of a quantum chain consisting of three coupled harmonic oscillators, the first and third of which weakly interact quadratically with two independent thermal baths in equilibrium at different temperatures. Due to the quadratic form of the total Hamiltonian, the uni- tary dynamics of the compound system is formally analytically solvable and defines a one-parameter group of Gaussian maps which enables us to solve the exact dynamics of the chain numerically. Following the Gorini-Kossakowski-Sudarshan-Lindblad approach to open quantum systems, one can perform the rotating-wave approximation with respect to the interacting or noninteracting chain Hamiltonian and respectively derive the so-called global and local master equations. The solutions of the ensuing different master equations can then be compared with the exact one, possibly sorting out the two approaches in correspondence to different timescales of the system. We derive the steady states of the open chain quantum dynamics in the two approaches and show that the behavior of fidelity between them versus interoscillator coupling depends on the two bath temperatures, revealing the existence of a temperature-dependent critical interoscillator coupling strength that determines the domain of validity of each approach. When the newly found coupling is less than this critical value, the local approach outperforms the global approach, whereas for larger interoscillator coupling, the global approach is a better approximation of the exact evolution. This critical value of interoscillator coupling depends on the two bath temperatures, which then play a crucial role in deciding the best possible approximating open dynamics.
In the following, we study the dissipative time-evolution of a quantum chain consisting of three coupled harmonic oscillators, the first and third of which weakly interact quadratically with two independent thermal baths in equilibrium at different temperatures. Due to the quadratic form of the total Hamiltonian, the unitary dynamics of the compound system is formally analytically solvable and defines a one-parameter group of Gaussian maps which enables us to solve the exact dynamics of the chain numerically. Following the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) approach to open quantum systems, one can perform the rotating wave approximation with respect to the interacting, or non-interacting chain Hamiltonian and respectively derive the so-called global and local master equations. The solutions of the ensuing different master equations can then be compared with the exact one, possibly sorting out the two approaches in correspondence to different time-scales of the system. We derive the steady states of the open chain quantum dynamics in the two approaches and show that the behaviour of fidelity between them versus inter-oscillator coupling depends on the two bath temperatures, revealing the existence of a temperature-dependent critical inter-oscillator coupling strength that determines the domain of validity of each approach. When the newly found coupling is less than this critical value, the local approach outperforms the global approach, whereas for larger inter-oscillator coupling, the global approach is a better approximation of the exact evolution. This critical value of inter-oscillator coupling depends on the two bath temperatures, which then play a crucial role in deciding the best possible approximating open dynamics.