TY - JOUR

T1 - Evolution of an N -level system via automated vectorization of the Liouville equations and application to optically controlled polarization rotation

AU - Shahriar, M. S.

AU - Wang, Ye

AU - Krishnamurthy, Subramanian

AU - Tu, Y.

AU - Pati, G. S.

AU - Tseng, S.

N1 - Funding Information:
This work was supported by AFOSR [grant number FA9550-10-01-0228], [grant number FA9550-09-01-0652]; NASA [grant number NNX09AU90A]; NSF [grant number 0630388]; the DARPA ZOE program [grant number W31P4Q-09-1-0014].

PY - 2014/2/23

Y1 - 2014/2/23

N2 - The Liouville equation governing the evolution of the density matrix for an atomic/molecular system is expressed in terms of a commutator between the density matrix and the Hamiltonian, along with terms that account for decay and redistribution. To find solutions of this equation, it is convenient first to reformulate the Liouville equation by defining a vector corresponding to the elements of the density operator, and determining the corresponding time-evolution matrix. For a system of N energy levels, the size of the evolution matrix is N2×N2. When N is very large, evaluating the elements of these matrices becomes very cumbersome. We describe a novel algorithm that can produce the evolution matrix in an automated fashion for an arbitrary value of N. As a non-trivial example, we apply this algorithm to a 15-level atomic system used for producing optically controlled polarization rotation. We also point out how such a code can be extended for use in an atomic system with arbitrary number of energy levels.

AB - The Liouville equation governing the evolution of the density matrix for an atomic/molecular system is expressed in terms of a commutator between the density matrix and the Hamiltonian, along with terms that account for decay and redistribution. To find solutions of this equation, it is convenient first to reformulate the Liouville equation by defining a vector corresponding to the elements of the density operator, and determining the corresponding time-evolution matrix. For a system of N energy levels, the size of the evolution matrix is N2×N2. When N is very large, evaluating the elements of these matrices becomes very cumbersome. We describe a novel algorithm that can produce the evolution matrix in an automated fashion for an arbitrary value of N. As a non-trivial example, we apply this algorithm to a 15-level atomic system used for producing optically controlled polarization rotation. We also point out how such a code can be extended for use in an atomic system with arbitrary number of energy levels.

KW - multi-level coherent process

KW - novel computational algorithm

KW - optically controlled birefringence

UR - http://www.scopus.com/inward/record.url?scp=84900632291&partnerID=8YFLogxK

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U2 - 10.1080/09500340.2013.865806

DO - 10.1080/09500340.2013.865806

M3 - Article

AN - SCOPUS:84900632291

SN - 0950-0340

VL - 61

SP - 351

EP - 367

JO - Journal of Modern Optics

JF - Journal of Modern Optics

IS - 4

ER -