Robustness of controlled quantum dynamics

The control of multilevel quantum systems is sensitive to implementation errors in the control field and uncertainties associated with system Hamiltonian parameters. A small variation in the control field spectrum or the system Hamiltonian can cause an otherwise optimal field to deviate from controlling desired quantum state transitions and reaching a particular objective. An accurate analysis of robustness is thus essential in understanding and achieving model-based quantum control, such as in the control of chemical reactions based on ab initio or experimental estimates of the molecular Hamiltonian. In this paper, theoretical foundations for quantum control robustness analysis are presented from both a distributional perspective—in terms of moments of the transition amplitude, interferences, and transition probability—and a worst-case perspective. Based on this theory, analytical expressions and a computationally efficient method for determining the robustness of coherently controlled quantum dynamics are derived. The robustness analysis reveals that there generally exists a set of control pathways that are more resistant to destructive interferences in the presence of control field and system parameter uncertainty. These robust pathways interfere and combine to yield a relatively accurate transition amplitude and high transition probability when uncertainty is present.

Figure 1

Top: Calculation of the first moment of $A^{\alpha }\left(E\left[A^{\alpha }\right]\right)$. The log plot shows that for an amplitude mode with a Gaussian distribution, the moment of amplitude with increasing power becomes exponentially larger relative to its nominal value. Bottom: The bar plot shows the Dyson terms involved in ${\varepsilon }_1$ at nominal and expected case $[\sigma \left(A_k\right)$=0.3]. It suggests that terms of higher orders, which may not be negligible under the nominal condition, will become significant in a noisy environment.

Figure 2

Plot of expected population transfer (top) and interference (bottom). The data suggests that control field implementation uncertainty increases destructive interference and, in turn, reduces the moment of transition probability.

Figure 3

Bar plot of interferences between different pathways involved in the control field dynamics of ${\varepsilon }_1$ from Table 2 in the nominal (top) and expected (bottom) case $[\sigma \left(A_k\right)=0.3]$. As seen from the plot, Gaussian uncertainty increases the destructive interferences between transitions.

Figure 4

Plot of nominal and expected transition amplitude $[\sigma \left(A_k\right)=0.3]$ associated with ${\varepsilon }_1$ from Table 2 decomposed in terms of each Dyson term.

Figure 5

Plot of expected population transfer (left) and interference (right) for uncertainty distributions that differ across amplitude modes. The $x$ axis shows increasing standard deviation of the second amplitude mode, while the rest of the amplitude modes are fixed at 0.3. The data suggest that certain pathways are more resistant to uncertainty, leading to higher expected transition probability and reduced destructive interferences.

Figure 6

Plot of the temporal field of ${\varepsilon }_1$ and ${\varepsilon }_8$ as defined in Table 2 (left) and their corresponding population transfer trajectory (right) in the nominal and expected case $[\sigma \left(A_1\right)=\sigma \left(A_2\right)=0.45$ and $\sigma \left(A_2\right)=0.65]$.

Figure 7

Left: Plot of the expectation value of transition probability vs increasing amplitude strength. The figure shows that under nominal condition, better control is achieved as stronger fields are optimized (circle). In the presence of amplitude uncertainty $\left(\sigma =0.2\right)$, however, there is a particular field strength corresponding to a robust point which is not necessarily optimal under the nominal condition (square). Right: The bar plot shows the significant number of Dyson terms involved in the dynamics of two control fields with different field strength. It indicates that the more robust field involves fewer significant Dyson terms and, hence, amplitude pathways relative to the nonrobust field.