General tracking control of arbitrary N-level quantum systems using piecewise time-independent potentials

Abstract

Here we propose a tracking quantum control protocol for arbitrary N-level systems. The goal is to make the expected value of an observable ${\mathcal O}$ to follow a predetermined trajectory S(t). For so, we drive the quantum state $|\varPsi (t) \rangle$ evolution through an external potential V which depends on $M_V$ tunable parameters (e.g., the amplitude and phase (thus $M_V = 2$) of a laser field in the dipolar condition). At instants $t_n$, these parameters can be rapidly switched to specific values and then kept constant during time intervals $\Delta t$. The method determines which sets of parameters values can result in $\langle \varPsi (t) | {\mathcal O} |\varPsi (t) \rangle = S(t)$. It is numerically robust (no intrinsic divergences) and relatively fast since we need to solve only nonlinear algebraic (instead of a system of coupled nonlinear differential) equations to obtain the parameters at the successive $\Delta t$’s. For a given S(t), the required minimum $M_V = M_{\min }$ ‘degrees of freedom’ of V attaining the control is a good figure of merit of the problem difficulty. For instance, the control cannot be unconditionally realizable if $M_{\min } > 2$ and V is due to a laser field (the usual context in real applications). As it is discussed and exemplified, in these cases a possible procedure is to relax the control in certain problematic (but short) time intervals. Finally, when existing the approach can systematically access distinct possible solutions, thereby allowing a relatively simple way to search for the best implementation conditions. Illustrations for 3-, 4-, and 5-level systems and some comparisons with calculations in the literature are presented.

Appendices

Appendix 1: An equivalent system to Eq. (5), the existence of a $M_{\min }$, and a numerical test for the controllability of a given QC task

With the help of Eq. (2), it is easy to show that Eq. (5) is fully equivalent to

$$\begin{aligned}&{\mathcal U}^{(\varepsilon )} \, C^{({\varepsilon })} = \tilde{C}^{({\varepsilon })} \ \ \text{ or } \text{ likewise } \ \ \, {{\mathcal U}^{(\varepsilon )}}^{\dagger } \, \tilde{C}^{({\varepsilon })} = C^{({\varepsilon })}, \end{aligned}$$

(15)

$$\begin{aligned}&\text{ with } \ \ {\mathcal U}^{(\varepsilon )} = \exp [-i \tilde{\tau } H^{(\varepsilon )}/\hbar ]. \end{aligned}$$

(16)

To each unitary ${\mathcal U}^{(\varepsilon )}$ satisfying to Eq. (15), it must correspond an unique $H^{(\varepsilon )}$ [109]. In practice, we do not use Eqs. (15)–(16) because they are numerically more cumbersome to solve than Eq. (5). Indeed, the mentioned interrelation between the elements of $H^{(\varepsilon )}$ (denoted by $I_H$) implies into interdependence (denoted by $I_{{\mathcal U}}$) between the ${\mathcal U}^{(\varepsilon )}$ matrix elements. But how $I_H$ implies in $I_{{\mathcal U}}$ is generally difficult to determine,³ so hard to handle in Eqs. (15)–(16). However, the structure of Eqs. (15)–(16) is twofold helpful: to prove the existence of a $M_{min}$ and to implement the numerical test mentioned in Sect. 2.3.

A basic linear algebra result⁴ is the following. For a ${\mathcal U}^{(\varepsilon )}$ solving Eq. (15), it must be possible to write both $\tilde{C}^{({\varepsilon })}$ and $C^{({\varepsilon })}$ ⁵ as linear combinations of the rows of, respectively, ${\mathcal U}^{(\varepsilon )}$ and ${{\mathcal U}^{(\varepsilon )}}^{\dagger }$. Furthermore, since ${\mathcal U}^{(\varepsilon )}$ obeys Eq. (16), its matrix structure is obviously dictated by the specific characteristics of $H^{(\varepsilon )}$.

To establish the existence of a $M_{min}$ for Eq. (5), we start arguing that always there are solutions if V is totally arbitrary. Assume for $H^{(\varepsilon )}$ independent $v_{n m}$’s and $\varphi _{n m}$’s in Eq. (1) (a situation barely found in practice, but useful for our reasoning here), thus $M_V = N^2$. Reciprocally, the ${\mathcal U}^{(\varepsilon )}_{n m}$’s from Eq. (16) can all be taken as independent from each other, and the number of unknown real variables specifying ${\mathcal U}^{(\varepsilon )}$ is then $N^2$. From the converse of the linear algebra property above, these $N^2$ variable suffice to solve Eq. (15). Moreover, they can lead to many possible solutions (this fact is reinforced considering that often we have many choices for the $\tilde{C}^{({\varepsilon })}$’s, cf. Eq. (3)). Hence, for each ${\mathcal U}^{(\varepsilon )}$ matching Eq. (15), we get a distinct $H^{(\varepsilon )}$ by numerically calculating $\ln [{\mathcal U}^{(\varepsilon )}]$ [110] (any unitary matrix has a well-defined logarithm [109]). So, the one step control is always possible in this case.

Now, for a certain QC (totally specified by $C^{({\varepsilon })}$ and $\tilde{C}^{({\varepsilon })}$), one may depart from the previous ‘ideal’ situation of $N^2$ parameters. One could systematically decrease $M_V$ by considering more and more restrictive $I_{{\mathcal U}}$—assuming specific functional relations between the ${\mathcal U}^{(\varepsilon )}_{n m}$’s, which set the number of free variables—and check for the existence of solutions for Eq. (15). This might be a complicated procedure. However, the point here is not its actual implementation. Instead, to realize that from this process we conceivably can find $M_{\min } \le N^2$ for ${\mathcal U}$. Since there is a one-to-one association $I_H \leftrightarrow I_{{\mathcal U}}$, such $M_{\min }$ also corresponds to the minimum number of parameters for $H^{(\varepsilon )}$. Lastly, given the equivalence between Eqs. (15)–(16) and Eq. (5), for each $C^{({\varepsilon })}$ and $\tilde{C}^{({\varepsilon })}$, the same $M_{\min }$ also unconditionally solves Eq. (5).

Finally, we can employ a simple numerical test to evaluate whether the considered number of parameters $M_V$ (say, $M_V = 2$) is adequate for the control. Suppose we are having difficulties to find solutions for Eq. (5). So, for a specific S and a tentative form for $H^{({\varepsilon })}$, at each control step we can generate distinct $\tilde{C}^{({\varepsilon })}$’s and reasonable (e.g., experimentally feasible) values for the control parameters, creating thus a grid in the parameters space. Each point in this grid corresponds to a ‘probe’ $H^{(\varepsilon )}$. So, from Eq. (15) (with ${\mathcal U}^{(\varepsilon )}$ coming directly from $H^{(\varepsilon )}$ through Eq. (16)), we verify whether these probe Hamiltonians can lead, as we sweep the grid, to proper solutions. This kind of analysis is also used in some other methods [50, 54]. But here, since for each proposed $H^{(\varepsilon )}$ Eq. (15) becomes a straightforward linear system of equations, the procedure is computationally very fast. Hence, we can test relatively large grids. These checks, of course, are not mathematical formal demonstrations that a certain QC is or is not generally achievable for a specified $M_V$. Nevertheless, they can indicate whether typically the given $M_V$ is enough for the control task. When the best parameter intervals are determined (once the adequate $M_V$ is found), we can come back to Eq. (5) for the detailed calculations of $H^{(\varepsilon )}$.

Appendix 2: Parameterization for the 1D harmonic oscillator

Using atomic units (so mass and $\hbar$ are set to 1), Ref. [58] considers a 1D harmonic oscillator (HO) Hamiltonian $H_0 = -(1/2) \, d^2/dx^2 + \omega ^2 x^2/2$ interacting with a dipolar laser field potential $V = - \epsilon \, \mu$, with $\epsilon = \epsilon (t)$ and the dipole function given by $\mu = x^2/2 - \gamma \, x$. For $| k \rangle$ and $e_k = (k+1/2) \, \omega$, respectively, the eigenstate and eigenenergy k ($= 0, 1, 2, \ldots$) of $H_0$, we have the matrix elements (for $\delta _{k l}$ the Kronecker’s delta)

$$\begin{aligned} \langle k | V | l \rangle= & {} - \epsilon \, \left\langle k \left| \frac{x^2}{2} - \gamma \, x \right| l \right\rangle = \epsilon \, c_{k l}, \nonumber \\ \langle k | H_0 | l \rangle= & {} \left( k +\frac{1}{2}\right) \, \omega \, \delta _{k l}. \end{aligned}$$

(17)

From basic quantum results for the HO (e.g., [111]), for any $k,l = 0,1,2,\ldots$, we get

$$\begin{aligned} c_{k l}= & {} - \left\{ \frac{1}{4 \omega } \Big [ (2 k + 1) \delta _{k l}\right. \nonumber \\&\left. +\,\sqrt{(k + 2)(k + 1)} \delta _{k l-2} + \sqrt{k (k - 1)} \delta _{k l+2} \Big ]\right. \nonumber \\&\left. -\,\frac{\gamma }{\sqrt{2 \omega }} \Big [ \sqrt{k} \delta _{k l+1} + \sqrt{k+1} \delta _{k l-1} \Big ] \right\} . \end{aligned}$$

(18)

So, the parameterization in Eq. (1) for no phases for $\epsilon (t)$ ($\epsilon$ real) yields $\varphi _{k+1 \, l+1} = 0$ and $v_{k+1 \, l+1} = \epsilon \, c_{k l}$ (recall that in our method notation, the indices n, m of $\varphi _{n m}$ and $v_{n m}$ range over $1, 2, \ldots , N$). Hence, $v_{n n} \ne 0$ and all the $v_{n m}$’s are proportional to each other and to an unique control parameter $\epsilon$.

In the basis $\{ |k \rangle \}$, the matrix elements of the observable ${\mathcal O} = x$ read

$$\begin{aligned} \langle k | x | l \rangle = \frac{1}{\sqrt{2 \omega }} \Big [ \sqrt{k} \delta _{k l+1} + \sqrt{k+1} \delta _{k l-1} \Big ]. \end{aligned}$$

(19)

Hence, by taking the first N levels ($k=0,1,\ldots ,N-1$) of the OH, we can construct matrices of distinct sizes for the observable x. By numerically diagonalizing them, we find the following corresponding sets (with $\omega = 4$)

$$\begin{aligned} N=3 :&\{ o_1 = -o_3, o_2 = 0, o_3 = +0.612 \}, \nonumber \\ N=4 :&\{ o_1 = -o_4, o_2 = -o_3, o_3 = +0.262, o_4 = +0.825 \},\nonumber \\ N=5 :&\{ o_1 = -o_5, o_2 = -o_4, o_3 = 0, o_4 = +0.479, \nonumber \\&o_5 = +1.010 \}. \end{aligned}$$

(20)

The target trajectory S(t) for this problem in Eq. (13) ($t \ge 0$) has numerical minimum and maximum of -0.2031 and +1. Since we must choose a set of observable eigenvalues able to span the full range of values of S, from Eq. (20) we see that we need to work at least with 5 levels of the OH to achieve the QC.