How to Make Distinct Dynamical Systems Appear Spectrally Identical

We show that a laser pulse can always be found that induces a desired optical response from an arbitrary dynamical system. As illustrations, driving fields are computed to induce the same optical response from a variety of distinct systems (open and closed, quantum and classical). As a result, the observed induced dipolar spectra without detailed information on the driving field are not sufficient to characterize atomic and molecular systems. The formulation may also be applied to design materials with specified optical characteristics. These findings reveal unexplored flexibilities of nonlinear optics.

Figure 1

The control fields $E_a(t)$, $E_b(t)$, $E_c(t)$, $E_d(t)$, and $E_e(t)$ induce the same nonlinear optical response $Y(t)$ on the following: closed quantum systems [(a) and (b)], (c) an open quantum system, (d) a closed classical system, and (e) an open classical system. In (a) and (b), the system is a hydrogen atom initially prepared in the ground and first excited states, respectively. $E_b(t)$, $E_d(t)$, and $E_e(t)$ are scaled for comparison with the first field $E(t)$ that is applied to a model of an argon atom to produce the induced dipolar spectra $Y(t)$, which is used for tracking in the remaining cases. Compare with Fig. 2 for the importance of small differences in the control fields, which are not apparent in the time domain.

Figure 2

The control fields (a–e) $E_a(t)$, $E_b(t)$, $E_c(t)$, $E_d(t)$, and $E_e(t)$ and the target IDS $Y(t)$ from Fig. 1 in frequency domain. $\mathrm{abs}\mathcal{F}[·]$ denotes the absolute value of the Fourier transform. The ${\omega }_0$ is the carrier frequency of $E(t)$ in Fig. 1. The spectrum of the tracking fields (f) $E_a(t)$, $E_c(t)$, $E_d(t)$, and $E_e(t)$ from Fig. 1.