Time-Dependent Two-Particle Reduced Density Matrix Theory for Atoms in Strong Laser Pulses

1 Institute for Theoretical Physics, Vienna University of Technology, Vienna, Austria , EU 2 Photon Science Center, Graduate School of Engineering, The University of Tokyo, Japan

Fabian Lackner1, Iva Březinová1, Stefan Donsa1, Takeshi Sato² , Kenichi L. Ishikawa², and Joachim Burgdörfer1

N-representable 2-RDMs belong to at least one wavefunction

Theoretical background

Abstract

References:

[1] F. Lackner et al. Phys. Rev. A 91, 023412 (2015)

[2] Koji Yasuda and Hiroshi Nakatsuji, Phys. Rev. A 56, 2648 (1997)

[3] David A. Mazziotti, Phys. Rev. A 60, 3618 (1999)

[4] David.A. Mazziotti.,Phys. Rev. E 65, 026704 (2002)

Acknowledgments: This work has been supported by the FWF DK Solids4fun, FWF SFB ViCoM, FWF SFB Next Lite and WWTF project MA14-002. Calculations have been performed on the Vienna Scientific Cluster.

Conclusions and Outlook Propagation of the 2-RDM is an efficient alternative to expensive MCTDHF calculations The accuracy is high enough to reproduce high-harmonic spectra Our method can be applied to other systems: Hubbard model, ultra-cold atoms,… Future studies: ionization probabilities

N-representability

Purification

The electronic response of atoms to strong laser pulses is governed by the Schrödinger equation

Contact: fabian.lackner@tuwien.ac.at

For the theoretical description of dynamical many-body systems two main approaches are available to date: multi-configurational methods such as MCTDHF, and time-dependent density functional theory (TDDFT). While MCTDHF is in principle exact but computationally very demanding, TDDFT suffers from unknown xc-functionals. We propose an accurate and efficient alternative based on the propagation of the 2-RDM [1]. We apply our method to high-harmonic generation of atomic targets in strong laser fields. We obtain excellent agreement with MCTDHF benchmark calculations.

Equation of motion for the 2-RDM

The time evolution of RDMs is given by the Bogoliubov-Born-Gree-Kirkwood-Yvon hierarchy

With the Hamiltonian:

At present there is no reconstruction that conserves N-representability automatically

Closing the equation requires reconstruction of the 3-RDM

High harmonic generation of Beryllium

Commutator: Propagation of pairs

Collision operator: Interaction between pair and surrounding particles

Reconstruction

Reduced density matrices are the equal time limit of many-body propagators:

Contraction Consistency

The intensity of the high-harmonic radiation is given by Lamor’s formula

Hartree-Fock

A direct solution is not feasible for

Instead of we propagate the 2-RDM:

The 2-RDM contains sufficient information to calculate all two-particle observable e.g. the energy

and the 1-RDM

The reconstruction has to be contraction consistent

in order to ensure energy conservation, spin conservation and consistency with the equation of motion (EOM) for the 1-RDM

Orbital expansion

For numerical efficiency we expand the 2-RDM in time-dependent orbitals

Analogous to the diagrammatic expansion of many- body propagators the 3-RDMs can be expanded in

connected diagrams unconnected diagrams

Depending on the treatment of the three-particle cumulant several reconstructions exist

Valdemoro:

Nakastuji-Yasuda [2]:

Second-order approximation by evaluating the simplest connected diagram for

Mazziotti [3]:

Reducing the four-particle cumulant relation to the three particle subspace gives the implicit equation

None of the proposed reconstruction functionals is contraction consistent

Contraction consistency can be enforced for arbitrary reconstruction functionals using the unitary decomposition of the 3-RDM

Through a more sophisticated ansatz [1] we force the kernel component to have zero diagonal and off-diagonal contractions of all individual spin-blocks

The orthogonal component is obtained by making the ansatz

The reconstruction of the orthogonal complement of the 3-RDM signifificantly improves the accuracy and ensures spin and energy conservation

Results

Necessary conditions are known in the form of positivity conditions [4]

We use the unitary decomposition of the 2-RDM component with negative eigenvalues

And purify the full 2-RDM according to

The reconstruction error for various different reconstruction functionals

governed by EOM

with

and solving the linear set of equations for