Assuring robustness to noise in optimal quantum control experiments

Closed-loop optimal quantum control experiments operate in the inherent presence of laser noise. In many applications, attaining high quality results [i.e., a high signal-to-noise $(S∕N)$ ratio for the optimized objective] is as important as producing a high control yield. Enhancement of the $S∕N$ ratio will typically be in competition with the mean signal, however, the latter competition can be balanced by biasing the optimization experiments towards higher mean yields while retaining a good $S∕N$ ratio. Other strategies can also direct the optimization to reduce the standard deviation of the statistical signal distribution. The ability to enhance the $S∕N$ ratio through an optimized choice of the control is demonstrated for two condensed phase model systems: second harmonic generation in a nonlinear optical crystal and stimulated emission pumping in a dye solution.

Figure 1

Experimental quantum control apparatus for simultaneously optimizing the mean signal $\mu$ and maximizing the signal-to-noise ratio $\mu ∕\sigma$, where $\sigma$ is the signal standard deviation. In any particular application the dashed box on the left would contain the laboratory sample placed under control. In the present work illustrating the concept, the laser pulses are modulated in a liquid-crystal pulse shaper and focused for two different experiments (BS: beam splitter). In one arm, second harmonic generation (SHG) is created in a BBO crystal. The other arm contains a flow cell with Coumarin 152 dissolved in ethanol. Population transfer is monitored with stimulated emission pumping (SEP), by overlapping the shaped pump pulse spatially and temporally delayed (DS: delay stage) with a weak probe pulse in the interaction region of the flow cell. The probe pulse consists of a white light continuum (WLC) created by focusing a part of the short pulse into a sapphire plate. The SEP signal is spectrally filtered around 480 nm using a bandpass filter with $\Delta \lambda =10\mathrm{nm}$. For both SEP and SHG experiments, the distribution formed by 1000 individual laser pulse signals is detected using a 1.2-ns rise time photodiode (PD), a Boxcar integrator, and an A/D converter.

Figure 2

(a) Absorption and fluorescence emission of Coumarin 152 dissolved in ethanol. The gray bars indicate the spectral position of the two-photon excitation at 400 nm and the stimulated emission at 480 nm. (b) Schematic of the population transfer in Coumarin 152. Two photon excitation at the center of the molecular absorption using a 800-nm shaped pulse creates a wave packet in the excited $S_1$ state which is detected by stimulated emission of a weak WLC pulse bandpass filtered around 480 nm.

Figure 3

SHG $S∕N$ performance: Comparison between one-parameter variations and full multiparameter optimization. The dashed line in plots (b)–(e) is the reference $S∕N$ output for a transform-limited pulse taken once during each generation. (a) SHG $S∕N$ dependence upon the laser power of a transform-limited pulse. (b) SHG $S∕N$ dependence upon the pulse length expressed in terms of positive and negative chirp (shaded circles). (c) SHG $S∕N$ performance (black circles) during learning control using the fitness function $F_{\mathrm{SHG}}^{\left(1\right)}$ in Eq. (1) with full multiparameter phase and amplitude modulation. (d) Mean SHG signal ${\mu }_{\mathrm{SHG}}$ of the fittest individual during the multiparameter phase and amplitude optimization. (e) SHG standard deviation ${\sigma }_{\mathrm{SHG}}$ of the fittest individual during the multiparameter phase and amplitude optimization. (f) SHG signal histograms for 1000 pulses of the fittest pulse-shape during the 8th, the 24th, and the 51st generation of the optimization.

Figure 4

Cost-benefit dependence of SHG mean signal ${\mu }_{\mathrm{SHG}}$ and $S∕N$ optimization results using $F_{\mathrm{SHG}}^{\left(2\right)}$ in Eq. (2). The results are normalized to that of the transform-limited pulse corresponding to the filled gray point at 1.0 on both axes. The black dots represent results of individual optimization experiments, where $\beta$ in Eq. (2) was increased in order to obtain higher ${\mu }_{\mathrm{SHG}}$. This enhancement of ${\mu }_{\mathrm{SHG}}$ by a factor of 300% in going from $\beta =0$ to $\beta =10$ was achieved at only a modest reduction of 28% in the $S∕N$ ratio. The gray dots represent results for transform-limited pulses at increasing power upon going to the right in the figure. Comparison with the latter dots shows that introducing the fitness function in Eq. (2) biases the optimal control outcome benefits towards significantly improving the mean signal ${\mu }_{\mathrm{SHG}}$ while maintaining a good $S∕N$ ratio, as evident by the divergence in the two data tracks proceeding to the left in the figure.

Figure 5

Correlation of SHG $S∕N$ ratio and standard deviation ${\sigma }_{\mathrm{SHG}}$ obtained by the best pulse for each generation, shown for three different optimization experiments using the three fitness functions $F_{\mathrm{SHG}}^{\left(1\right)}$, $F_{\mathrm{SHG}}^{\left(2\right)}$ and $F_{\mathrm{SHG}}^{\left(3\right)}$ [Eqs. (1, 2, 3)]. The black circles contain bias towards smaller standard deviation ${\sigma }_{\mathrm{SHG}}$, using $F_{\mathrm{SHG}}^{\left(3\right)}$. The open circles correspond to no bias using $F_{\mathrm{SHG}}^{\left(1\right)}$. The gray circles correspond to bias towards higher mean yield ${\mu }_{\mathrm{SHG}}$, using $F^{\left(2\right)}$. By suitably choosing the fitness function broad regions of the $S∕N-{\sigma }_{\mathrm{SHG}}$ plane can be accessed.

Figure 6

Robust control pulse characteristics found for differently biased SHG $S∕N$ optimizations. The left column shows the temporal pulse intensities (solid line) and phases (dashed line) while the right column shows the spectral attenuation. The robust pulse forms without bias using Eq. (1) are shown in (d) and (i). Going up in the figure corresponds to increasing ${\mu }_{\mathrm{SHG}}$ bias with Eq. (2) by varying $\beta$. The bottom rows (e) and (j) correspond to control pulse bias for reducing ${\sigma }_{\mathrm{SHG}}$ with Eq. (3) applying $\gamma =5$.

Figure 7

(a) SEP pump-probe plots for ${\lambda }_{\text{probe}}=480\mathrm{nm}$ and ${\lambda }_{\text{probe}}=520\mathrm{nm}$. For each time delay, the signals of 1500 shots were averaged. (b): Dependence of the SEP mean amplification and the SEP $S∕N$ ratio on the laser intensity of a transform-limited pulse normalized to the intensity of the reference pulse. The signal at the right is recorded at ${\lambda }_{\text{probe}}=480\mathrm{nm}$.

Figure 8

Comparison of different pump-probe SEP signals created by three different pump pulses. For each plot the SEP signal is detected at 480 nm with $\Delta \lambda =10\mathrm{nm}$ and averaged over 200 pulses. (a) Transform-limited pulse, (b) control pulse found with $F_{\mathrm{SEP}}^{\left(1\right)}$ in Eq. (1), and (c) control pulse found with $F_{\mathrm{SEP}}^{\left(2\right)}$ Eq. (2). In (b) and (c) the goal was to seek an optimized pulse at a delay time of $\Delta t=3\mathrm{ps}$, although the improved quality of the signal is evident at all times.

Figure 9

(a) Cost-benefit dependence of SEP $S∕N$ optimizations using different weighting factors $\beta$ in $F_{\mathrm{SEP}}^{\left(2\right)}$ of Eq. (2). The black dots represent results of individual optimization experiments where $\beta$ was increased in order to bias the search for higher mean signals ${\mu }_{\mathrm{SEP}}$. The mean signal was increased by a factor of 2 in going from $\beta =0$ to $\beta =10$ while maintaining the $S∕N$ ratio at the same value. The small gray dots represent results for transform-limited pulses of different intensities, with all of the results normalized to the value of the most intense transform-limited pulse, shown as the gray filled dot. (b) Histograms from the SEP $S∕N$ optimization with $\beta =10$ of the pulse-to-pulse distribution of the fittest pulse form in generations 8, 45, and 80, respectively.

Figure 10

Temporal pulse intensities and phases, (a)–(c), and associated spectral attenuations, (d)–(f), of SEP $S∕N$ controls, obtained with fitness function $F_{\mathrm{SEP}}^{\left(1\right)}$ (a) and (d), as well as fitness function $F_{\mathrm{SEP}}^{\left(2\right)}$ for $\beta =3$, (b) and (e), and $\beta =10$, (c) and (f).