Investigating constrained quantum control through a kinematic-to-dynamic-variable transformation

A search for the variables that control a quantum system's dynamics occurs over a landscape, defined as the target objective as a function of the variables. Prior studies show that upon satisfaction of three specific assumptions, the topology of the landscape is free of suboptimal traps that could prematurely halt the search for an optimal control. One key assumption is free access to all necessary control variables; however, in practice, the controls are always limited in some fashion which may result in constraint-induced traps on the landscape. This paper aims to introduce the means to systematically explore the nature of constrained controls that yield suboptimal outcomes. The procedure utilizes kinematic controls, which comprise a simple set of time-independent variables, and then performs a landscape topology-preserving transformation into corresponding dynamic controls. The equivalent landscape topology of these two formulations permits the study of a family of dynamic controls that reflect constrained control landscape behavior. In particular, constrained dynamic controls are identified as isolated points on the landscape or as suboptimal level sets. The wide range of such dynamic controls indicates the richness and complexity of constraint-induced features on the landscape.

Figure 1

A schematic representation of the transformation from a constrained kinematic control to a family of dynamic controls. A particular kinematic control matrix $A$ (with two controls $K_1,K_2$ shown, though typically a larger number of $A$ matrix elements are admitted as control variables) obtained along a constrained state-to-state landscape ascent can be mapped to a variety of dynamic controls through the minimization of Eq. (7). Any point on the kinematic landscape, as well as the entire trajectory (represented by the bold black curve), can be transformed into a family of dynamic controls, including a number $L_F$ of distinct fields $\varepsilon \left(t\right)$ and $L_H$ Hamiltonian structures $\{H_0,\mu \}$.

Figure 2

A schematic representing the transformation of trapped kinematic controls, lying on a suboptimal level set, to associated dynamic controls. Two connected kinematic control matrices $A_1^{\mathrm{trap}}$ and $A_2^{\mathrm{trap}}$ with control variables $K_1,K_2$ are shown that yield the same suboptimal $P_{i\to f}\left(A_1^{\mathrm{trap}}\right)=P_{i\to f}\left(A_2^{\mathrm{trap}}\right)<1.0$. The kinematic $\to$ dynamic transformation procedure explored in the present work will show that a family of related dynamic controls can be obtained at any point along the level set to fully reflect the kinematic control topological landscape features. In this instance, $A_1^{\mathrm{trap}}$ can be mapped to a number $L_F$ of corresponding dynamic fields ${\varepsilon }_1^l\left(t\right),l=1,...,L_F$; likewise, $A_2^{\mathrm{trap}}$ can be mapped to $L_F^{\prime }$ fields ${\varepsilon }_2^{l^{\prime }},l^{\prime }=1,...,L_F^{\prime }$. An analogous statement applies to a mapping to Hamiltonian structure variables $\{H_0,\mu \}$.

Figure 3

Constrained kinematic optimization of $P_{1\to 4}$ starting from the initial ($N=4$)-dimensional system in Eq. (23) and terminating at Eq. (24). The constrained d-morph procedure introduced in Sec. 2a was used to increase $P_{1\to 4}$ while keeping $C$ [cf. Eq. (11)] fixed at 4.2475 to $\sim {10}^{-6}$. The final $A$ matrix corresponds to an isolated trap at $P_{1\to 4}=0.7745$. All quantities are dimensionless.

Figure 4

Dynamic fields identified through the kinematic $\to$ dynamic control variable transformation procedure in Sec. 3, corresponding to the constrained kinematic $P_{1\to 4}$ landscape ascent in Fig. 3. The fields yielding the initial $P_{1\to 4}=0.0443$ (dashed curve) and final $P_{1\to 4}=0.7745$ (solid curve) are shown. No constraints were placed on the field during the mapping procedure. All quantities are dimensionless.

Figure 5

Dynamic fields identified through the kinematic $\to$ dynamic control variable transformation procedure in Sec. 3, corresponding to the constrained kinematic $P_{1\to 4}$ landscape ascent in Fig. 3. The fields yielding the initial $P_{1\to 4}=0.0443$ (dashed curve) and final $P_{1\to 4}=0.7745$ (solid curve) are shown. During the mapping procedure, a Gaussian envelope was imposed on the fields, enforcing a structure distinct from that seen for the fields in Fig. 4. All units are dimensionless.

Figure 6

(a) $H_0$ and (b) $\mu$ matrix elements identified through the kinematic $\to$ dynamic control variable transformation procedure in Sec. 3, corresponding to the same constrained kinematic $P_{1\to 4}$ landscape ascent in Fig. 3. The fixed reference field used throughout the transformation procedure is shown in (c). The matrix element ${\mu }_{14}$ [index 7 on the abscissa of (b)] changes most dramatically, which reflects the coupling directly to the target transition $|1\rangle \to |4\rangle$. All quantities are dimensionless.

Figure 7

(a) Increase in $P_{1\to 6}$ for an ($N=6$)-dimensional system which is constrained by an insufficient number of control variables: 12 matrix elements of $A$ are held fixed at randomly generated initial values [see Eq. (25)] and the remaining 9 elements are treated as control variables. The changes in the admitted controls during the landscape ascent are shown in (b). A trap is encountered at $P_{1\to 6}=0.4113$. All quantities are dimensionless.

Figure 8

Results from a search for a transition dipole matrix $\mu$ that reflects the kinematic trajectory shown in Fig. 7. (a) The initial and final values for the elements of the ($N=6$)-dimensional $\mu$ matrix; $H_0$ was held fixed during the kinematic $\to$ dynamic mapping and its diagonal elements are listed in row 2 of Table 1. (b) The power spectrum of the applied field [shown in (d)] with the system's natural transition frequencies arising from $H_0$ overlaid. Importantly, the frequency corresponding to the “direct” $|1\rangle \to |6\rangle$ transition ($\sim$17) lies well outside the spectral region of the field, while other indirect transitions show better overlap with the field's frequency components. (c) The population dynamics for all $P_{1\to j},j=1,...,6$, transitions. In addition to $|1\rangle \to |6\rangle$, the $|1\rangle \to |2\rangle$ and $|1\rangle \to |5\rangle$ transitions noticeably contribute. These results indicate that various intermediate transitions, aside from the direct $|1\rangle \to |6\rangle$ transition, play important roles. All quantities are dimensionless.

Figure 9

Projected Hessian eigenvalues before and after the constraint parameter morphing procedure discussed in Sec. 2b is performed. The constraint parameters forming $C$ in Eq. (11) are the elements of $B$ and the scalar $\gamma$. The eigenvalues morph from circles to squares. Importantly, the initial eigenvalues show a single zero member, which corresponds to a trivial solution arising from the imposition of the (single) constraint $C$. The next smallest eigenvalue is of order ${10}^{-1}$. The constraint parameter morphing procedure was halted when an eigenvalue on the order of ${10}^{-3}$ was found. A subsequent level set traversal was then performed utilizing the newly formed projected Hessian (nontrivial) null space. All quantities are dimensionless.

Figure 10

Dynamic fields reflecting a kinematic trap level set traversal following the morphing of the topology of an initially isolated trap. Only the fields corresponding to the level set's initial and final kinematic controls are shown. In this instance, a Gaussian envelope was imposed on the fields during the kinematic $\to$ dynamic control transformation. All quantities are dimensionless.