Numerical data for "The Timescales of Quantum Breaking"

This is the numerical data that belongs to the paper

M. Michel, S. Zell, The Timescales of Quantum Breaking, arXiv:2306.09410.

The code used to produce this numerical data can be reviewed in the QuantumBreaking-TimeScales repository on github.

This code relies on the computer program TimeEvolver, which was presented in

M. Michel, S. Zell, TimeEvolver: A Program for Time Evolution With Improved Error Bound, Comput. Phys. Commun. 277 (2022) 108374, arXiv:2205.15346.

In the following we outline the procedure to generate the data and plots presented in this paper. All equation numbers refer to the latest version of arXiv:2306.09410.

In this work we studied the dynamics of a multi-mode quantum system defined by eq. (3.6) and how the timescale of evolution depends on its defining parameters \(N, Q, C_m\) and \(\lambda\). To extract a quantitative scaling, all but one of these are kept fixed while the remaining one is varied over a certain interval. For each choice of parameters, the exact time evolution is computed numerically using the TimeEvolver software package.

We consider three parameter regimes, each with a corresponding folder, namely underCritical (defined by eq. (4.4)), overCriticalFewSpecies (defined by eq. (4.13)) and overCriticalManySpecies (defined by eq. (4.19)). Every folder is divided into subfolders, where the name of the subfolder indicates the parameter that is varied. (As a concrete example, in undercritical/Cm, all parameters except for Cm are set by eq. (4.4) and the system is studied for different values of Cm.) In some cases, an additional parameter deviates from the values as set by eqs. (4.4), (4.13) or (4.19). This is indicated with an underscore followed by the name of the parameter and its value. In addition to that, there are also subfolders named realTimePlots containing data illustrating the real time evolution in different dynamical phases.

Apart from the three parameter regimes described above, we also study the system for the parameter choice of eq. (4.25) in the folder interpolating. Finally, we also investigate a different system defined by eq. (2.35) in the folder PPM.

Beside the raw data in the form of h5 files each folder contains a Mathematica-notebook which is responsible for data analysis and producing the plots presented in the paper. These have specific adaptions for each parameter and are named accordingly.

To extract a dynamic timescale we prepare the system in a specific initial state (given by eq. (2.10)) and
track the time it takes for the system to move "sufficiently far" away from it. In order to quantify this, we evaluate in each simulation the maximal deviation of the \(n_0\)-mode from its initial value (see eq. (4.5)). Then we determine the minimal relative excursion over all runs in the scan range and after multiplying by a factor of 0.8, this sets the threshold at which we extract a time duration for each run (see eq. (4.6).

Finally, we fit these values with various fit functions to determine quantitatively how the timescale of evolution depend on the original parameters \(N, Q, C_m\) and \(\lambda\).

We refer to the paper for a comprehensive description as well as further details.