Codes used in:

A. Balaz, I. Vidanovic, A. Bogojevic, A. Belic, and A. Pelster

"Fast Converging Path Integrals for Time-Dependent Potentials I: Recursive Calculation of Short-Time Expansion of the Propagator"

*J. Stat. Mech.* (2011) P03004

arXiv e-print: 0912.2743

A. Balaz, I. Vidanovic, A. Bogojevic, A. Belic, and A. Pelster

"Fast Converging Path Integrals for Time-Dependent Potentials II: Generalization to Many-body Systems and Real-Time Formalism"

*J. Stat. Mech.* (2011) P03005

arXiv e-print: 1011.5185

We calculate the short-time expansion of the propagator for a general quantum system with many degrees of freedom in a time-dependent potential to orders that have not yet been accessible before. Based on the earlier developed approach, the propagator is expressed in terms of a discretized effective potential, for which we derive and analytically solve a set of efficient recursion relations. Such a discretized effective potential can be used for solving a plethora of non-equilibrium many-body quantum problems within the exact diagonalization (arXiv:0911.5145 and arXiv:0911.5154) or Path-Integral Monte-Carlo approach, or to set up various analytic approximation techniques to study dynamic properties of quantum systems in time-dependent potentials. The analytically derived results are numerically verified by treating several simple models in both imaginary and real time.

The following Mathematica and C codes were written for this paper:
Calculation of imaginary-time single-particle one-dimensional effective action for a general time-dependent potential:

Mathematica notebook and p=10 result
Calculation of real-time single-particle one-dimensional effective action for a general time-dependent potential:

Mathematica notebook and p=10 result

Calculation of imaginary-time single-particle one-dimensional effective action for a forced harmonic oscillator:

Mathematica notebook and p=20 result, Serial SPEEDUP C code for p=8

Calculation of real-time single-particle one-dimensional effective action for a forced harmonic oscillator:

Mathematica notebook and p=20 result
Calculation of imaginary-time single-particle one-dimensional effective action for a time-dependent harmonic oscillator:

Mathematica notebook and p=20 result, Serial SPEEDUP C code for p=8
Calculation of real-time single-particle one-dimensional effective action for a time-dependent harmonic oscillator:

Mathematica notebook and p=20 result
Calculation of imaginary-time single-particle one-dimensional effective action for a time-dependent pure quartic oscillator:

Mathematica notebook and p=20 result, Serial SPEEDUP C code for p=7
Calculation of real-time single-particle one-dimensional effective action for a time-dependent pure quartic oscillator:

Mathematica notebook and p=20 result