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Journal of Physics A Mathematical and Theoretical

Non-equilibrium quantum dynamics of lattice vibrations

Yamen Hamdouni, F. Benrachi

Published Aug 12, 2013• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1751-8113/46/35/355303

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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.
Quantum diffusion equations featuring explicitly time-dependent transport coefficients are derived from generalized non-Markovian Langevin equations. Generalized fluctuation-dissipation relations and analytic expressions for calculating the friction and diffusion coefficients in nuclear processes are obtained. The asymptotic behavior of the transport coefficients and correlation functions for a damped harmonic oscillator that is linearly coupled in momentum to a heat bath is studied. The coupling to a heat bath in momentum is responsible for the appearance of the diffusion coefficient in coordinate. The problem of regression of correlations in quantum dissipative systems is analyzed.
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.
This thesis explores the dynamics of linearly coupled quantum and classical oscillators interacting with environments. In the quantum regime, such dynamics is described in terms of the Liouvillian superoperator, which leads to non-unitary time evolution of the system, specifically of the density matrix. In particular, when the correlation time of the environment is significantly shorter than the characteristic timescale of the system, the dynamics are classified as Markovian. Therefore, the equation of motion for the density matrix that fulfils this property is known as the Lindblad master equation. Systems of coupled cavities and waveguides in the presence of gain and loss can be described by this equation and, more importantly, by non-Hermitian Hamiltonians. In particular, when two coupled cavities exhibit balance between gain and loss, the system is PT -symmetric. However, for certain values in the parameter space, this symmetry is broken and the non-Hermitian Hamiltonian is no longer diagonalizable. Such values are denoted as exceptional points. In the case of large arrays of coupled quantum oscillators, the dynamics description becomes more challenging since the dimensions of the density matrix grow exponentially. To address this, we study the dynamics of these arrays via the quantum state diffusion approach. In such approach, the Lindblad master equation is unraveled by a stochastic Schrödinger equation. Then, we decouple this equation by diagonalizing the non-Hermitian Hamiltonian that describes the system. Such diagonalization leads to a transformation of a system of coupled oscillators into a decoupled one, offering a great reduction in the basis size. For systems of classical oscillators with dissipative nearest-neighbor coupling, the dynamics leads to synchronized motion. We study how an array of such oscillators reaches the synchronized state. This behavior is found when the system is dissipation free, which implies that all the eigenvectors of the dynamical matrix are damped, except one. To measure the degree of synchronization, we introduce the inverse of participation ratio, whose maximum value is associated with a delocalized state and, therefore, fully synchronized motion. For larger arrays of oscillators, we examine the case where the frequency fluctuations of each oscillator are obtained from a Gaussian distribution with zero mean and a given standard deviation. Then, we derive a scaling law for the largest value of the mentioned standard deviation that still allows the oscillators fully synchronize their motion.:1 Introduction 2 Open quantum systems 3 Dynamics of two coupled cavities in the presence of losses 4 Unraveling of Lindblad equation for system of oscillators 5 Synchronization of dissipatively coupled harmonic oscillators 6 Conclusions and outlook A Derivation of equations of motion for the amplitudes B Quantum state diffusion unraveling for N decoupled oscillators C Model for disspative coupling D Perturbation theory for disorder in the frequencies References
We study the perturbative response of a complex quantum system on time changes of an external parameter XXX. The driven dynamics is treated in adiabatic basis of the system's Hamiltonian H^[X]\hat{H}[X]H^[X]. Within a random matrix approach we obtained non--Markovian Fokker--Planck equation for the occupancy of given adiabatic state. We observed normal diffusion regime of the driven quantum dynamics at quite small values of the memory time defined by the time scales of the XXX--correlations and energy--distribution of the coupling matrix elements (∂H^/∂X)nm(\partial \hat{H}/\partial X)_{nm}(∂H^/∂X)nm​. Here the normal energy diffusion was found to drop out with the width of the matrix elements' energy--distribution and the diffusion may be significantly suppressed with the decrease of the correlations between the matrix elements. In the opposite limit of relatively large memory times we obtained ballistic regime of the dynamics.
We analyze a system coupled to a bath of independent harmonic oscillators. We transform the bath in chain structure by solving an inverse eigenvalue problem. We solve the equations of motion for the collective variables defined by this transformation, and we derive the exact dynamics for a harmonic oscillator in terms of the microscopic motion of the environmental modes. We compare this approach to the well-known generalized Langevin equation and we show that our dynamics satisfies this equation.
The exact stochastic decomposition of non-Markovian dissipative quantum dynamics is combined with the time-dependent semiclassical initial value formalism. It is shown that even in the challenging regime of moderate friction and low temperatures, where non-Markovian effects are substantial, this approach allows for the accurate description of dissipative dynamics in anharmonic potentials over many oscillation periods until thermalization is reached. The problem of convergence of the stochastic average at long times, which plagues full quantum mechanical implementations, is avoided through a joint sampling of the stochastic noise and the semiclassical phase-space distribution.