Time-dependent treatment of a general three-level system

Both unitary evolution and the effects of dissipation and decoherence for a general three-level system are of wide interest in quantum optics, molecular physics, and elsewhere. A previous paper presented a technique for solving the time-dependent operator equations involved but under certain restrictive conditions. We now extend our results to a general three-level system with arbitrary time-dependent Hamiltonians and Lindblad operators. Analytical handling of the SU(3) algebra of the eight operators involved leaves behind a set of coupled first-order differential equations for classical functions. Solution of this set gives a complete solution of the quantum problem, without having to invoke rotating-wave or other approximations. Numerical illustrations for multiphoton couplings and quantum control are given.

Figure 1

Diagonal and off-diagonal elements of the density matrix and entropy $S$ for a two-level system described by Eqs. (5, 9). With $ϵ\left(t\right)=A\cos \left(\omega t\right)$, $2J=B\cos (\Omega t+\delta )$, results are shown in the left-hand panels for an initial pure state 1, and $A=45$, $B=6$, $\omega =1$, $\Omega =0$, $\delta =0$, $\Gamma =0.3$. These results coincide with those of Ref. 13. The right-hand panels show for comparison the results of a simplified model for decoherence in Ref. 6.

Figure 2

Populations of a three-level system described by Eq. (2), starting with initial state 1. Time on the horizontal axis is in units of $1∕Re{\Delta }_1$. The parameters are $\mid G_1\mid =\mid G_2\mid =0.5$ and (a) ${\Delta }_1=0.5-0.01i$, ${\Delta }_2=0.5+0.01i$, (b) ${\Delta }_1=-{\Delta }_2=5-i$. The intermediate level 2, shown by a thick line, remains essentially at zero for all time in (b). The results coincide with those of Ref. 1.

Figure 3

Populations of a three-level system described by Eq. (4), starting with initial state 1. Time on the horizontal axis is in units of $\sigma$. Thick and thin lines describe levels 2 and 3, respectively, and a dashed line the population of level 1. Values of parameters described in the text are (a) $\alpha =2$, $\gamma =0$, $\delta =0$, (b) $\alpha =2$, $\gamma =0$, $\delta =0.866$, (c) $\alpha =5$, $\gamma =0$, $\delta =0$, (d) $\alpha =5$, $\gamma =0.5$, $\delta =0$.

Figure 4

Populations of a three-level system described by Eq. (4), starting with initial state 1. Time on the horizontal axis is in units of $\sigma$. Values of parameters described in the text are $A_2=A_1=2.5$, $t_1=12$, $t_2=t_1-\sigma$, and (a) $\sigma =8$, (b) $\sigma =3$, (c) $\sigma =1.5$, (d) $\sigma =0.9$, all with $\gamma =0$ and $\Delta =0$, (e) $\sigma =3$, $\gamma =0.15$, $\Delta =0$, (f) $\sigma =3$, $\gamma =0$, $\Delta =1.5$.