Mamadou Ndong

Automatic computer procedure for generating exact and analytical kinetic energy operators based on the polyspherical approach

We develop a new general code to automatically derive exact analytical kinetic energy operators in terms of polyspherical coordinates. Computer procedures based on symbolic calculations are implemented. Sets of orthogonal or non-orthogonal vectors are used to parametrize the molecular systems in space. For each set of vectors, and whatever the size of the system, the exact analytical kinetic energy operator (including the overall rotation and the Coriolis coupling) can be derived by the program. The correctness of the implementation is tested for different sets of vectors and for several systems of various sizes.

Optimal control simulation of the Deutsch-Jozsa algorithm in a two-dimensional double well coupled to an environment.

The quantum Deutsch-Jozsa algorithm is implemented by using vibrational modes of a two-dimensional double well. The laser fields realizing the different gates (NOT, CNOT, and HADAMARD) on the two-qubit space are computed by the multitarget optimal control theory. The stability of the performance index is checked by coupling the system to an environment. Firstly, the two-dimensional subspace is coupled to a small number Nb of oscillators in order to simulate intramolecular vibrational energy redistribution. The complete (2+Nb)D problem is solved by the coupled harmonic adiabatic channel method which allows including coupled modes up to Nb=5. Secondly, the computational subspace is coupled to a continuous bath of oscillators in order to simulate a confined environment expected to be favorable to achieve molecular computing, for instance, molecules confined in matrices or in a fullerene. The spectral density of the bath is approximated by an Ohmic law with a cutoff for some hundreds of cm(-1). The time scale of the bath dynamics (of the order of 10 fs) is then smaller than the relaxation time and the controlled dynamics (2 ps) so that Markovian dissipative dynamics is used.

Laser control in a bifurcating region

We present a complete analysis of the laser control of a model molecular system using both optimal control theory and adiabatic techniques. This molecule has a particular potential energy surface with a bifurcating region connecting three potential wells which allows a variety of processes such as isomerization, tunneling, or implementation of quantum gates on one or two qubits. The parameters of the model have been chosen so as to reproduce the main features of H{sub 3}CO which is a molecule benchmark for such dynamics. We show the feasibility of different processes and we investigate their robustness against variations of laser field. We discuss the conditions under which each method of control gives the best results. We also point out the relation between optimal control theory and local control.

Automatic computer procedure for generating exact and analytical kinetic energy operators based on the polyspherical approach: General formulation and removal of singularities

We present new techniques for an automatic computation of the kinetic energy operator in analytical form. These techniques are based on the use of the polyspherical approach and are extended to take into account Cartesian coordinates as well. An automatic procedure is developed where analytical expressions are obtained by symbolic calculations. This procedure is a full generalization of the one presented in Ndong et al., [J. Chem. Phys. 136, 034107 (2012)]. The correctness of the new implementation is analyzed by comparison with results obtained from the TNUM program. We give several illustrations that could be useful for users of the code. In particular, we discuss some cyclic compounds which are important in photochemistry. Among others, we show that choosing a well-adapted parameterization and decomposition into subsystems can allow one to avoid singularities in the kinetic energy operator. We also discuss a relation between polyspherical and Z-matrix coordinates: this comparison could be helpful for building an interface between the new code and a quantum chemistry package.

A Chebychev propagator for inhomogeneous Schrödinger equations

A propagation scheme for time-dependent inhomogeneous Schrödinger equations is presented. Such equations occur in time dependent optimal control theory and in reactive scattering. A formal solution based on a polynomial expansion of the inhomogeneous term is derived. It is subjected to an approximation in terms of Chebychev polynomials. Different variants for the inhomogeneous propagator are demonstrated and applied to two examples from optimal control theory. Convergence behavior and numerical efficiency are analyzed.

A Chebychev propagator with iterative time ordering for explicitly time-dependent Hamiltonians

A propagation method for time-dependent Schrodinger equations with an explicitly time-dependent Hamiltonian is developed where time ordering is achieved iteratively. The explicit time dependence of the time-dependent Schrodinger equation is rewritten as an inhomogeneous term. At each step of the iteration, the resulting inhomogeneous Schrodinger equation is solved with the Chebychev propagation scheme presented in the work of M. Ndong et al. [J. Chem. Phys. 130, 124108 (2009)]. The iteratively time-ordering Chebychev propagator is shown to be robust, efficient, and accurate and compares very favorably with all other available propagation schemes.

NOT gate in a cis-trans photoisomerization model

We numerically study the implementation of a NOT gate by laser pulses in a model molecular system presenting two electronic surfaces coupled by nonadiabatic interactions. The two states of the bit are the fundamental states of the cis-trans isomers of the molecule. The gate is classical in the sense that it involves a one-qubit flip so that the encoding of the outputs is based on population analysis which does not take the phases into account. This gate can also be viewed as a double photoswitch process with the property that the same electric field controls the two isomerizations. As an example, we consider one-dimensional cuts in a model of the retinal in rhodopsin already proposed in the literature. The laser pulses are computed by the multitarget optimal control theory with chirped pulses as trial fields. Very high fidelities are obtained. We also examine the stability of the control when the system is coupled to a bath of oscillators modeled by an ohmic spectral density. The bath correlation time scale being smaller than the pulse duration, the dynamics is carried out in the Markovian approximation.

Robust coherent superposition of states by single-shot shaped pulse

We adapt a single-shot shaped pulse technique to produce robust coherent superpositions of quantum states with a high fidelity of control. We derive simple pulses of low areas for the corresponding Rabi frequency which are robust with respect to pulse area imperfections. Such features of robustness, high-fidelity, and low Rabi frequency area are crucial steps towards the experimental implementation of scalable quantum gates.