Master Equation Solvers      

Master equation solvers solve for the time evolution of the density matrix/Wigner function of a quantum system. These codes are very memory intensive since, for a single degree of freedom, the memory requirements go as L^(2n) where n is the space dimensionality and L the number of points per dimension. In the case n=1, we have run system sizes as large as 8K X 8K. The basic method of solution is a split-operator spectral technique. We have implemented algorithms that offer second and fourth order accuracy in time. Code performance is controlled typically by the performance achieved by the FFT implementation in the spectral method. The Wigner representation is often advantageous since it allows the use of nonsymmetric grids to optimize accuracy.

This is an F90/MPI code for parallel integration of a set of 4(N^2) strongly-coupled PDE's, where N represents a grid resolution and was typically 2048 or 4096. The individual PDE's represent the elements of a position-dependent density operator for an ensemble of two-level atoms driven by a time-dependent electromagnetic field. In order to deal with an extreme separation of timescales between the center-of-mass dynamics and the atomic internal-state dynamics, the code uses two nested loops for the integration. The inner loop propagates the internal dynamics only, using a fine timestep and a second-order Runge-Kutta integrator. The outer loop propagates the center-of-mass state with a course timestep, using the FFT split-operator technique. Running on 64 nodes and with N=2048, the code could perform (125 fine)x(2800 coarse) timesteps in about 4 hours of wall-clock time. At this resolution, execution time is so far dominated by integration of the internal-state dynamics. We believe that this component of the code can be significantly improved via cache optimizations, and the execution time should ultimately be dominated by the 2D FFT's. Improved versions of the base code are under development.

The purpose of the code is to simulate experiments being performed at the University of Texas (Austin) and at the University of Auckland, on dynamical localization in the quantum delta-kicked rotor. Previous simulation codes have not explicitly treated the atomic internal-state dynamics, and so were incapable of directly addressing questions about the role of atomic spontaneous emission. In FY 98 we had focused on validating the F90/MPI code by comparing its output to those of a proven HPF code without internal dynamics, in a parameter regime where spontaneous emission can justifiably be neglected. We now plan to use it for detailed studies of the effects of decoherence on dynamical localization in atom-optical systems.

This is a C++/MPI code for parallel integration of a set of N^2 strongly coupled, highly nonlinear ODE's with multiplicative white noise. The set of SDE's represent a measurement-based Stochastic Master Equation for single-atom cavity quantum electrodynamics (QED), and this code for integrating such an equation is the first its kind. In order to keep the integration accurate and stable, it was necessary to use a semi-implicit Milstein integration algorithm and a timestep on the order of one thousandth the largest term in the equation. Unlike most stochastic integration scenarios, we had to achieve strong convergence in order to produce accurate individual trajectories (as opposed to accurate reproduction of statistical moments).

The purpose of the code is to simulate the conditional evolution of an open quantum system, under conditions of continuous but partial observation. A rigorous mathematical basis for deriving the equations of motion has only recently been established, and experiments in quantum optics are just beginning to be able to investigate this type of physical system. Our intention is to use this code for the development of schemes for quantum feedback control of open quantum systems, which will provide crucial guidance for initial experimental work in this area. During the past year we have validated the code by comparison with predictions based on analytic simplifications of the full equations of motion. Execution time is so far limited by cache problems; a new version is under development.

Code development during FY 00 has succeeded in producing a combined version of the two codes described above. We can now simulate cavity-QED systems with quantized atomic center-of-mass motion and partial continuous observation, yielding numerical results that may be directly compared with experimental data.

The scientific goals for production runs will be to study (this has been submitted as part of a separate NERSC project in FY01):

Decoherence in the quantum delta-kicked rotor and other systems

Conditional dynamics in single-atom optical bistability

Quantum feedback control of atomic motion in an optical cavity

A common theme for all of this work is that of studying the relationships among decoherence, measurement, information, and entropy in open quantum systems. The long-term motivation for this work is to clarify the mechanisms for entropy production in nonequilibrium quantum statistical mechanics, especially with regards to the subtle interplay between decoherence and dynamical complexity.

Salman Habib / LANL / habib@lanl.gov / August 00