Demos.OscillatorChain.TDSE

Wavepacket dynamics of chains of oscillators

For the case of homogeneous chains of harmonic oscillators, we study the quantum-mechanical propagation of a vibrational excitation initially localized at the central site of the chain. These simulations shall serve here as a testing ground for the various propagation schemes implemented in WavePacket and/or in WaveTrain.

Quantum simulations

For the example of a cyclic homogeneous hexamer (N=6) we initialize our propagations with one particle (at|near the center) in a coherent state (displacement of 20 dimensionless units) and all other particles in their respective vibrational ground states (zero displacements).

The conventional grid-based WavePacket software package using a Gauss-Hermite grid with 8 points for each degree of freedom and a second order Strang splitting scheme to propagate in (real) time takes about half an hour of CPU time.

• Animated wavepacket
• Input data file
• Logfile output

The tensor-train based WaveTrain software package using second quantization with 8 basis functions for each degree of freedom and a fourth order symmetrized Euler scheme to propagate in (real) time takes more than one hour of CPU time.

• Animated simulation
• Input script
• Output file

Classical simulations

For the quantum dynamics of the chains of oscillators investigated here, reference data for assessing our conventional or tensor train-based results can be easily generated on the basis of the quantum-classical correspondence. According to the Ehrenfest theorem, the quantum-mechanical expectation values of positions and momenta coincide with results from classical trajectories, as long as the vibrational Hamiltonian is a polynomial of order not higher than two. This is indeed the case for our vibrational model Hamiltonian.

• Animated simulation
• Input script
• Output file

Comparison

In the three simulations shown above, the mean values of positions and momenta of the six particles, as well as the corresponding uncertainties (quantum simulations only) are identical to three or four digits. Of course, it would be straightforward to reach higher precision by choosing larger basis sets. In general, it can be expected that the vibrationally displaced states require larger basis sets than those used for our calculations of the vibrational ground states.

Note also the behavior of the total energies: In our classical simulations of the cyclic hexamer, the initial displacement of one particle by 20 dimensionless units corresponds to an energy of exactly 6E-4 which is well conserved during our 50 time steps. In contrast, the quantum simulations of the same hexamer yield a total energy of 5.796E-3 which is also well conserved during our 50 time steps. The difference of 5.196E-3 is essentially attributed to the zero point energy which was found to be 5.032E-3, see our calculations of the vibrational ground states.

Scaling behavior

While the conventional, grid-based numerical schemes implemented in WavePacket were successful in a propagation for N=6 for the examples given above, it should be mentioned, however, that on a standard PC one cannot go beyond N=7 or N=8 due to the exponential growth of the computational effort.

This is in marked contrast to the tensor-train based techniques implemented in WaveTrain which can handle chains comprising tens of sites, see our publication on coupled excitons and phonons. Even though the simulations for N=6 took more than twice as long as with WavePacket, it is emphasized that, by employing efficient low-rank tensor-train decompositions, both the required storage and the CPU time are found to scale much more favorably, thus mitigating the curse of dimensionality.