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

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.

[1]  Maira Amezcua,et al.  Quantum Optics , 2012 .

[2]  Prospects for Doppler cooling of three-electronic-level molecules , 2010, 1012.3696.

[3]  J. Lough,et al.  Quantum optic techniques for diagnostics of a gradient magnetometer atom interferometer , 2009 .

[4]  M. Shahriar,et al.  Demonstration of a continuously guided atomic interferometer by single-zone optical excitation , 2003, quant-ph/0311163.

[5]  D. Steck.  Rubidium 85 D Line Data , 2008 .

[6]  F. Shimizu.  On Laser Cooling , 2001 .

[7]  Harold Metcalf,et al.  Laser Cooling and Trapping , 1999, Peking University-World Scientific Advanced Physics Series.

[8]  M. S. Shahriar,et al.  Dark-state-based three-element vector model for the stimulated Raman interaction , 1997 .

[9]  E. Arimondo.  Coherent Population Trapping in Laser Spectroscopy , 1996 .

[10]  Bruce W. Shore,et al.  Theory of Coherent Atomic Excitation , 1991 .

[11]  Chu,et al.  Atomic interferometry using stimulated Raman transitions. , 1991, Physical review letters.

[12]  Bruce W. Shore,et al.  The Theory of Coherent Atomic Excitation , 1991 .

[13]  Wineland,et al.  Quantum Zeno effect. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[14]  Shaoul Ezekiel,et al.  Ac Stark shifts in a two-zone Raman interaction , 1989 .

[15]  Peter L. Knight,et al.  Population trapping and dispersion in a three-level system , 1982 .

[16]  C. R. Stroud,et al.  Coherent trapping of atomic populations. , 1978, Optics letters.

[17]  Ian J. Spalding,et al.  Laser physics , 1977, Nature.

[18]  G. Orriols,et al.  Nonabsorbing atomic coherences by coherent two-photon transitions in a three-level optical pumping , 1976 .

[19]  C. R. Stroud,et al.  Double optical resonance , 1976 .