Exponential propagators for the Schrödinger equation with a time-dependent potential.

We consider the numerical integration of the Schrödinger equation with a time-dependent Hamiltonian given as the sum of the kinetic energy and a time-dependent potential. Commutator-free (CF) propagators are exponential propagators that have shown to be highly efficient for general time-dependent Hamiltonians. We propose new CF propagators that are tailored for Hamiltonians of the said structure, showing a considerably improved performance. We obtain new fourth- and sixth-order CF propagators as well as a novel sixth-order propagator that incorporates a double commutator that only depends on coordinates, so this term can be considered as cost-free. The algorithms require the computation of the action of exponentials on a vector similar to the well-known exponential midpoint propagator, and this is carried out using the Lanczos method. We illustrate the performance of the new methods on several numerical examples.

[1]  Sergio Blanes,et al.  Symplectic integrators for the matrix Hill equation , 2017, J. Comput. Appl. Math..

[2]  Alexander Ostermann,et al.  Magnus integrators on multicore CPUs and GPUs , 2017, Comput. Phys. Commun..

[3]  M. Feit,et al.  Solution of the Schrödinger equation by a spectral method , 1982 .

[4]  Sverker Holmgren,et al.  Accurate time propagation for the Schrodinger equation with an explicitly time-dependent Hamiltonian. , 2008, The Journal of chemical physics.

[5]  M. Thalhammer,et al.  On the convergence of splitting methods for linear evolutionary Schrödinger equations involving an unbounded potential , 2009 .

[6]  Mechthild Thalhammer,et al.  A fourth-order commutator-free exponential integrator for nonautonomous differential equations , 2006, SIAM J. Numer. Anal..

[7]  J. M. Sanz-Serna,et al.  Classical numerical integrators for wave‐packet dynamics , 1996 .

[8]  Sergio Blanes,et al.  Time-Average on the Numerical Integration of Nonautonomous Differential Equations , 2018, SIAM J. Numer. Anal..

[9]  Arieh Iserles,et al.  Efficient methods for linear Schrödinger equation in the semiclassical regime with time-dependent potential , 2016, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[10]  P.-V. Koseleff.  Formal Calculus for Lie Methods in Hamiltonian Mechanics (Translation) , 1994 .

[11]  Sergio Blanes,et al.  Splitting methods for the time-dependent Schrödinger equation , 2000 .

[12]  R. Folk,et al.  Construction of high-order force-gradient algorithms for integration of motion in classical and quantum systems. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  G. Quispel,et al.  Splitting methods , 2002, Acta Numerica.

[14]  Siu A. Chin,et al.  Symplectic integrators from composite operator factorizations , 1997 .

[15]  D. Manolopoulos,et al.  Symplectic integrators tailored to the time‐dependent Schrödinger equation , 1996 .

[16]  T. Carrington,et al.  Using preconditioned adaptive step size Runge-Kutta methods for solving the time-dependent Schrödinger equation. , 2004, The Journal of chemical physics.

[17]  Mechthild Thalhammer,et al.  High-Order Exponential Operator Splitting Methods for Time-Dependent Schrödinger Equations , 2008, SIAM J. Numer. Anal..

[18]  Ronnie Kosloff,et al.  The solution of the time dependent Schrödinger equation by the (t,t’) method: The use of global polynomial propagators for time dependent Hamiltonians , 1994 .

[19]  W. Magnus.  On the exponential solution of differential equations for a linear operator , 1954 .

[20]  T. Park,et al.  Unitary quantum time evolution by iterative Lanczos reduction , 1986 .

[21]  C. Lubich.  From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis , 2008 .

[22]  Mechthild Thalhammer,et al.  Convergence Analysis of High-Order Time-Splitting Pseudospectral Methods for Nonlinear Schrödinger Equations , 2012, SIAM J. Numer. Anal..

[23]  S. Blanes,et al.  Symplectic time-average propagators for the Schrödinger equation with a time-dependent Hamiltonian. , 2017, The Journal of chemical physics.

[24]  H. Munthe-Kaas,et al.  Computations in a free Lie algebra , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[25]  S. Blanes,et al.  The Magnus expansion and some of its applications , 2008, 0810.5488.

[26]  J. M. Sanz-Serna,et al.  Numerical Hamiltonian Problems , 1994 .

[27]  S. Blanes,et al.  Fourth-and sixth-order commutator-free Magnus integrators for linear and non-linear dynamical systems , 2006 .

[28]  Angel Rubio,et al.  Propagators for the time-dependent Kohn-Sham equations. , 2004, The Journal of chemical physics.

[29]  Arieh Iserles,et al.  Magnus-Lanczos Methods with Simplified Commutators for the Schrödinger Equation with a Time-Dependent Potential , 2018, SIAM J. Numer. Anal..

[30]  S. Gray,et al.  Classical Hamiltonian structures in wave packet dynamics , 1994 .

[31]  Holger Fehske,et al.  High-order commutator-free exponential time-propagation of driven quantum systems , 2011, J. Comput. Phys..

[32]  Arieh Iserles,et al.  Effective Approximation for the Semiclassical Schrödinger Equation , 2014, Foundations of Computational Mathematics.

[33]  R. K. Preston,et al.  Quantum versus classical dynamics in the treatment of multiple photon excitation of the anharmonic oscillator , 1977 .

[34]  Y. Saad.  Analysis of some Krylov subspace approximations to the matrix exponential operator , 1992 .

[35]  Fernando Casas,et al.  An efficient algorithm based on splitting for the time integration of the Schrödinger equation , 2015, J. Comput. Phys..