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Physical Review A

Analytic results for the population dynamics of a driven dipolar molecular system

Wen-Jing Zhang, Kang Jin, Luling Jin, Xiao-Tao Xie

Published Apr 22, 2016• 10 citations
Publisher

Abstract

We theoretically investigate the dipolar molecular system driven by monochromatic periodic, linear, parabolic, and ${\mathrm{sech}}^{2}$ forms external fields, respectively. The two-level Hamiltonian model with nonzero diagonal dipole matrix elements is adopted to describe the population dynamics of the driven dipolar molecule, and the corresponding exact solutions are presented in terms of the confluent Heun equations without the generalized rotating-wave approximation. The analytic solutions derived here are valid in the whole parameter space.

DOI
10.1103/physreva.93.043840

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We theoretically investigate the dipolar molecular system driven by monochromatic periodic, linear, parabolic, and sech2{\mathrm{sech}}^{2}sech2 forms external fields, respectively. The two-level Hamiltonian model with nonzero diagonal dipole matrix elements is adopted to describe the population dynamics of the driven dipolar molecule, and the corresponding exact solutions are presented in terms of the confluent Heun equations without the generalized rotating-wave approximation. The analytic solutions derived here are valid in the whole parameter space.
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We investigate population dynamics in NNN-level systems driven beyond the linear regime by a strong external field, which couples to the system through an operator with nonzero diagonal elements. As concrete example we consider the case of dipolar molecular systems. We identify limiting cases of the Hamiltonian leading to wave functions that can be written in terms of ordinary exponentials, and focus on the limits of slowly and rapidly varying fields of arbitrary strength. For rapidly varying fields we prove for arbitrary NNN that the population dynamics is independent of the sign of the projection of the field onto the dipole coupling. In the opposite limit of slowly varying fields the population of the target level is optimized by a dipole resonance condition. As a result population transfer is maximized for one sign of the field and suppressed for the other one, so that a switch based on flopping the field polarization can be devised. For significant sign dependence the resonance linewidth with respect to the field strength is small. In the intermediate regime of moderate field variation, the integral of lowest order in the coupling can be rewritten as a sum of terms resembling the two limiting cases, plus correction terms for N>2N>2N>2, so that a less pronounced sign-dependence still exists.
We theoretically investigate the exact solutions for generalized parity–time( <?CDATA PT{\mathscr{P}}{\mathscr{T}}PT?> P T )-reversal-symmetric Rabi models driven by external fields with monochromatic periodic, linear, and parabolic forms, respectively. The corresponding exact solutions are presented in terms of the confluent Heun equations without any approximation. In principle, the analytic solutions derived here are valid in the whole parameter space. Such a kind of study may offer potential coherent control schemes of the <?CDATA PT{\mathscr{P}}{\mathscr{T}}PT?> P T -symmetric two-level systems.
We investigate and establish an analytic connection between the exact solutions describing the dynamics of a two-level system driven by periodic external fields, focusing on the cases of linear driving and the so-called rotating-wave approximation, or circular driving. In both cases, the exact solutions can be obtained by mapping the Schrodinger equation onto Heun equations: the confluent Heun equation for linear driving and the Heun equation for the rotating-wave case. In particular, we demonstrate a direct analytic connection between the exact solutions for linear driving and those for the rotating-wave case. This result is obtained by analyzing local solutions expressed in terms of hypergeometric functions, which, in the case of the confluent Heun equation, can be derived by considering path-multiplicative Floquet solutions involving a bilateral series. This series leads to two continued-fraction expansions that can be perturbatively solved by imposing a suitable consistency condition. The connection between the linear-driving and rotating-wave solutions is established through a perturbative procedure that allows us to recover not only the rotating-wave approximation itself, but also the correct Stark and Bloch-Siegert shifts, as well as the so-called high-frequency approximation.
The dynamics of the density matrix of the two-state system under the influence of time-dependent deterministic and fluctuating forces is investigated. The exact formal solutions for the diagonal and off-diagonal elements are derived and transformed into coupled nonconvolutive master equations and integral relations. Thereby, the evolution of any observable relevant to the two-state system may be described. Solutions of the dynamical equations in analytic form are presented both for high and low temperatures. An iterative numerical method is put forward in which the familiar noninteracting-blip approximation is systematically improved by taking into account all bath-induced nearest-neighbor interblip correlations. The expanded treatment is found to be indispensable in studies of the dynamics of the off-diagonal elements at low temperatures. Finally, we discuss selection rules, as well as possibilities of control of tunneling, which occur in the presence of monochromatic driving.