1. Introduction
A two-level quantum system is an important model in quantum mechanics. In the field of laser–atom interaction, the two-level system is also crucial to understanding the main physical phenomena [1]. With regard to quantum control, a single or several delayed frequency chirped laser pulses are used [2–5]. Recently, Golubev and Kuleff presented a simple approach to arbitrarily control the population of a two-level system by using analytical single resonant laser pulses [2]. Csehi presented an analytical electric field expression to simultaneously control the population and phase dynamics of two-level quantum systems. The field works well in the two-level atom model [3]. It is believed that the two-level model is valid if the two levels are resonant with the external laser pulse and all other levels are highly detuned. There are no quantitative studies on the validity of the two-level atom model, apart from the V-type three-level model [6]. In the V-type three-level atom model, a third state couples to the ground state, but the effect of coupling to the excited state was not studied.
2. Model and calculation
In the present work, the population transfer of sodium is explored in the analytical laser pulse of Golubev and Kuleff in the two-level, three-level and many-level models. The many-level model can be considered as the real sodium atom. In the three-level model, the effect of a third state coupling to the excited state can be studied by changing the energy level difference and electric dipole moment between the third state and the excited state.
| i | (i)Two-level sodium (3s–3p) |
The Hamiltonian of two-level sodium interacting with an external classical monochromatic field has the form
$\begin{eqnarray}H={\varepsilon }_{1}\left|1\right\rangle \left\langle 1\right|+{\varepsilon }_{2}\left|2\right\rangle \left\langle 2\right|+\mu E(t)(\left|1\right\rangle \left\langle 2\right|+\left|2\right\rangle \left\langle 1\right|),\end{eqnarray}$
where $\left|1\right\rangle ,\,\left|2\right\rangle $ and ${\varepsilon }_{1},{\varepsilon }_{2}$ are the two eigenstates and eigenenergies, respectively, and $\mu $ is the dipole momentum.The electric field $E(t)$ has the form
$\begin{eqnarray}\begin{array}{l}E(t)=\displaystyle \frac{1}{\mu }\displaystyle \frac{\alpha ({a}_{f}-{a}_{i}){{\rm{e}}}^{\alpha t}}{(1+{{\rm{e}}}^{\alpha t})\sqrt{(1-{a}_{i}+(1-{a}_{f}){{\rm{e}}}^{\alpha t})({a}_{i}+{a}_{f}{{\rm{e}}}^{\alpha t})}}\\ \,\,\times \,\sin ({\omega }_{0}t+\varphi ),\end{array}\end{eqnarray}$
with the control function $f(t)=\tfrac{{a}_{i}{{\rm{e}}}^{-\alpha t}+{a}_{f}}{1+{{\rm{e}}}^{-\alpha t}}.$For the 3s $\to $ 3p transition of atomic sodium, $\mu $ is 2.50 a.u. The parameters in the electric field are set as $\alpha =0.01,\varphi =0,\,{\omega }_{0}=0.077\,{\rm{a}}.{\rm{u}}{\rm{.}},\,{a}_{i}=1,\,{a}_{f}=0.$
By directly solving the time-dependent Schrödinger equation, the time evolution of the population is obtained, as shown in figure 1. The pulses can completely transform the population from 3s to 3p, exactly according to the control function.
| i | (ii)Three-level sodium (3s–3p–4s) |
Sodium is taken as a three-level system (3s–3p–4s). The Hamiltonian of this system interacting with electric field has the form
$\begin{eqnarray}\begin{array}{l}H={\varepsilon }_{1}\left|1\right\rangle \left\langle 1\right|+{\varepsilon }_{2}\left|2\right\rangle \left\langle 2\right|+{\varepsilon }_{3}\left|3\right\rangle \left\langle 3\right|+{\mu }_{1}E(t)\left(\left|1\right\rangle \left\langle 2\right|\right.\\ \,\,+\left.\left|2\right\rangle \left\langle 1\right|\right)+{\mu }_{2}E(t)(\left|2\right\rangle \left\langle 3\right|+\left|3\right\rangle \left\langle 2\right|).\end{array}\end{eqnarray}$
By numerically solving the time-dependent Schrödinger equation, one gets the time evolution of the population, as shown in figure 2.
To study the effect of coupling to the excited state on the two-level quantum system, three-level sodium can be taken as a simple model of a three-level ladder system. Setting ${\mu }_{2}=a{\mu }_{1}$ and ${\omega }_{23}={\varepsilon }_{3}-{\varepsilon }_{2}=b{\omega }_{12}=b({\varepsilon }_{2}-{\varepsilon }_{1}),$ changing $a,b$ from 0 to 10, one gets the parameter dependence of the population, shown in figure 3.
Figure 1. Time evolution of populations (bottom) of the 3s state and 3p state of sodium atoms and the analytical laser pulse (top) obtained using the parameters $\alpha =0.01,\,\varphi =0,\,{\omega }_{0}=0.077\,{\rm{a}}.{\rm{u}}{\rm{.}},\,{a}_{i}=1,\,{a}_{f}=0.$ |
Figure 2. Time evolution of populations of three-level sodium atoms in the same analytical laser pulse as in figure 1. |
From figure 3, when ${\omega }_{23}\geqslant {\omega }_{12}$ the impact of the third state can be omitted, as common sense would suggest. The most important influence of the third state happens in the vicinity of ${\omega }_{23}\approx {\omega }_{12}.$ When ${\mu }_{2}\approx 1.5{\mu }_{1},$ the population of the 4s state reaches its maximum value of about 80%. The third state has the effect of stealing population from the second state. When ${\omega }_{23}$ becomes bigger, most populations remain in the 3s state. The 4s state couples strongly with the 3p state, preventing the population from increasing.
| i | (iii)Many-level sodium |
Figure 3. The parameter dependence of 3s state, 3p state and 4s state populations. Red means that the population is approximately equal to 1 and blue means the population is approximately equal to 0. |
For this case, real sodium is like a many-level system (3s, 4s, 5s, 3p, 4p, 5p, 3d, 4d, 5d, 4f, 5f). The time-dependent multilevel approach is used to solve the problem of sodium interacting with the electric field [7]. The Hamiltonian is
$\begin{eqnarray}H=-\displaystyle \frac{1}{2}{{\rm{\nabla }}}^{2}+V(r)+zE(t).\end{eqnarray}$
Because one is concentrating on the valence electron, the potential $V(r)$ is the model potential in [8]. The form of the model potential is
$\begin{eqnarray}V=-\displaystyle \frac{1}{r}[\tilde{Z}+(Z-\tilde{Z})\exp (-{a}_{1}r)+{a}_{2}r\exp (-{a}_{3}r)].\end{eqnarray}$
By solving the field-free Hamiltonian, the energy level of sodium is obtained. After some operations on the time-dependent Schrödinger equation, one obtains the following set of coupled equations: ${\dot{a}}_{k}(t)=-{\rm{i}}E(t){\sum }_{j}{a}_{j}(t){\unicode{x0003C}z\unicode{x0003E}}_{kj}{{\rm{e}}}^{-{\rm{i}}({E}_{j}-{E}_{k})t}.$
By solving the coupled equations, the population evolution can be found from ${P}_{j}(t)={\left|{a}_{j}(t)\right|}^{2},$ which shown in figure 4. In the many-level approximation, the population cannot all go to the 3p state, and the 3p state population only reaches 90%.
Figure 4. Time evolution of populations of many-level sodium in the same analytical laser pulse as in figure 1. |
3. Conclusion
In this work, the population transfer of sodium in analytical laser pulses is explored in three approximations. It is easily found that the population can reach the final state at 100% exactly according to the control function. But for real sodium, i.e. the many-level situation, the 3p population only reaches 90% after the pulses. This suggests that the analytical electric field proposed by Golubev and Kuleff is a good controlling pulse for the population transfer of a two-level atom, but for real experiments it needs to be examined more carefully. For the three-level ladder system the impact of the third state mainly arises in the vicinity of the resonance frequency. This effect can be divided into two categories. The first is the stealing of population from the second state when the dipole moment is nearly equal to the dipole moment of the resonance levels. The second is strongly coupling the second state and preventing population transfer from the lowest state when the dipole moment is much greater than the dipole moment of the resonance levels.