Pulse engineering for population control under dephasing and dissipation

We apply reverse engineering to find electromagnetic pulses that allow for the control of populations in quantum systems under dephasing and thermal noises. In particular, we discuss two-level systems given their importance in the description of several molecular systems as well as quantum computing. Such an investigation naturally finds applications in a multitude of physical situations involving the control of quantum systems. We present an analytical description of the pulse which solves a constrained dynamics where the initial and final populations are fixed a priori. This constrained dynamics is sometimes impossible, and we precisely spot the conditions for that. One of our results is the presentation of analytical conditions for the establishment of steady states for finite coherence in the presence of noise. This might naturally find applications in quantum memories.

Figure 1

A two-level system representing, for instance, a coupled donor-acceptor system interacts simultaneously with the environment and an externally controlled electromagnetic pulse $E\left(t\right)$. The purpose of the pulse is to induce a predetermined change of population. The populations are indicated by the filling of the bars corresponding to the ground $|g\rangle$ and excited $|e\rangle$ states.

Figure 2

Top panel: Laser pulse as in Eq. (9). Middle panel: Ground-state population ${\rho }_{gg}\left(t\right)$ is shown in blue (solid), excited-state population ${\rho }_{ee}\left(t\right)$ is shown in red (dot-dashed), and the control function $f\left(t\right)$ is shown in black (dashed). Lower panel: The absolute value of the coherence $|{\rho }_{eg}\left(t\right)|=|{\rho }_{ge}\left(t\right)|$ is shown in red (solid) and the function $h\left(t\right)$ is shown in black (dashed). For all plots, $a_i=0.8$, $a_f=0.3$, $\mathrm{\Gamma }=0$, and $\gamma ={10}^{-3}\text{a.u.}$

Figure 3

Imaginary part of the function $h\left(t\right)$ as function of time and the target final population $a_f$ for two choices of $\gamma$. The frames in red (dashed) show the accessible region for $a_f$.

Figure 4

Accessible range when $\alpha$ is varied. We consider the pure dephasing case with $\gamma ={10}^{-3}$ and $\mathrm{\Gamma }=0$. For each $\alpha$ we also display the pulse width $\delta t$. The plots on the right illustrate the breaking of the RWA used to build the pulse. One can see that the time evolution does not closely follow the desired time dependence given by the functions $f\left(t\right)$ and $h\left(t\right)$.

Figure 5

Population inversion under dephasing as a function of frequency for our protocol (horizontal dotted black line) and the models Landau-Zener (blue-dashed line), Rozen-Zener (purple-solid line), Allen-Eberly (yellow-dotted-dashed line), and Denkov-Kunik (green-dotted line) solved in [23]. Expressions and parameters used here for these models are those on Table1 and Fig. 1 of [23]. Different rates $\omega /\gamma$ are fixed in each plot, and $T$ is a fixed typical time also defined in [23]. For the curve corresponding to our case, $\gamma /\alpha ={10}^{-1}$ in all plots.

Figure 6

Top panel: Laser pulse obtained using Eq. (9). Middle panel: Ground-state population ${\rho }_{gg}\left(t\right)$ is shown in blue, excited-state population ${\rho }_{ee}\left(t\right)$ is shown in red (dot-dashed), and the control $f\left(t\right)$ is shown in black (dashed). Lower panel: The absolute value of the coherences $|{\rho }_{eg}\left(t\right)|=|{\rho }_{ge}\left(t\right)|$ are shown in red, and the function $h\left(t\right)$ is shown in black (dashed). For all plots, $a_i=0.8$, $a_f=0.3$, $\mathrm{\Gamma }={10}^{-4}\mathrm{a}.\mathrm{u}.$, $\gamma =0$, $\overline{n}=0$.

Figure 7

Panel (a): Imaginary part of $h\left(t\right)$ as a function of time and $a_f$ for $\mathrm{\Gamma }={10}^{-4}\mathrm{a}.\mathrm{u}.$, $\gamma =0$, and $\overline{n}=0$. Panel (b): Imaginary part of $h\left(t\right)$ as a function of time and $a_f$ for $\mathrm{\Gamma }={10}^{-3}\mathrm{a}.\mathrm{u}.$, $\gamma =0$, and $\overline{n}=0$. Panel (c): Imaginary part of $h\left(t\right)$ as a function of time and $a_f$ for $\mathrm{\Gamma }={10}^{-4}\mathrm{a}.\mathrm{u}.$, $\gamma ={10}^{-2}\mathrm{a}.\mathrm{u}.$, and $\overline{n}=0$. Panel (d): Imaginary part of $h\left(t\right)$ as a function of time and $\overline{n}$. Here we kept $a_f=0.4$. The frames in red (dashed) show the accessible regions. For all panels $a_i=0.8$.

Figure 8

Top panel: Laser pulse as Eq. (9). Middle panel: Ground-state population ${\rho }_{gg}\left(t\right)$ is shown in blue, excited-state population ${\rho }_{ee}\left(t\right)$ is shown in red (dot-dashed), and the control $f\left(t\right)$ is shown in black (dashed). Lower panel: The absolute value of the coherence $|{\rho }_{eg}\left(t\right)|=|{\rho }_{ge}\left(t\right)|$ is shown in red and the function $h\left(t\right)$ is shown in black (dashed). The dotted horizontal line shows the steady-state value of coherence $h_{\infty }$ obtained from Eq. (10). For all plots, $a_i=0.8$, $a_f=0.6$, $\mathrm{\Gamma }={10}^{-4}\mathrm{a}.\mathrm{u}.$, $\gamma ={10}^{-3}\mathrm{a}.\mathrm{u}.$, $\overline{n}=0.3$.

Figure 9

Plots of the coherence and populations against time for different magnitudes of errors in frequency, i.e., different values of ${\sigma }_{\omega }$. The dashed lines show the ideal evolution while the solid lines correspond to the averaged dynamics over 1000 realizations. Each realization corresponds to a randomly chosen frequency from a Gaussian distribution centered at $\omega =2\times {10}^{-2}\mathrm{a}.\mathrm{u}.$ with dispersion ${\sigma }_{\omega }$ shown in each plot. Top panels: Pure dephasing case with $\gamma ={10}^{-3}\mathrm{a}.\mathrm{u}.$ and $\mathrm{\Gamma }=0$. Bottom panels: Dissipative case with $\gamma ={10}^{-3}\mathrm{a}.\mathrm{u}.$, $\mathrm{\Gamma }={10}^{-4}\mathrm{a}.\mathrm{u}.$, and $\overline{n}=0.3$.