B. O. Volkov, A. N. Pechen, “Higher-order traps for some strongly degenerate quantum control systems”, Russian Math. Surveys, 78:2 (2023), 390

Control of quantum systems attracts high interest due to fundamental reasons and applications to quantum technologies [1]. Controlled dynamics of a $N$-level closed quantum system is described by Schrödinger’s equation

A question important for quantum control is to establish, for a controlled system, whether or not the objective has a trapping behaviour [2]. An $n$th-order trap for the objective functional $J_O$, where $n\geqslant2$, is a control $f_0\in \mathfrak{H}^0$ such that (a) $f_0$ is not a point of global maximum of $J_O$ and (b) the Taylor expansion of the objective functional at the point $f_0$ has the form

Proof. For such $\rho_0$ and $O$ the complete controllabillity of the quantum system $(H_0,V)$ implies that the control $f_0\equiv0$ is not a point of global extremum of $J_O$ for $T\geqslant T_{\min}$ [6]. Without loss of generality we can assume that $\lambda_N=0$ [6]. Let $V_t:=e^{itH_0}Ve^{-itH_0}$ and let $A^n_{lk}\colon \mathfrak{H}^0 \to\mathbb{C}$ be the form of order $n$ defined by

$$ \begin{equation*} A^n_{lk}\langle f\rangle:=\int_0^Tdt_1\int_0^{t_1}dt_2\dotsb \int_0^{t_{n-1}}dt_n\,f(t_1)\cdots f(t_n)\langle l|V_{t_1}\dotsb V_{t_n}|k\rangle. \end{equation*} \notag $$

Let $A^0_{lk}=\delta_{lk}$ ($\delta_{lk}$ is the Kronecker symbol). By direct calculations, one can obtain a formula for the Fréchet differential of order $n$ of the objective functional $J_O$ at $f_0$:

$$ \begin{equation} \frac 1{n!}J_O^{(n)}(f_0)(f,\dots,f)=\sum_{j=0}^n\,\sum_{l=1}^{N-1} (-1)^{n-j}i^{n} \lambda_l A^j_{lN}\langle f \rangle\overline{A^{n-j}_{lN}\langle f \rangle}. \end{equation} \tag{1} $$

For $n=1$ we get that $J'_O(f_0)=0$. Since $\langle l|V|N\rangle=0$ for $l\neq N-1$, we have

$$ \begin{equation*} \frac 1{2!}J''_O(f_0)(f,f)=\lambda_{N-1}|A^1_{(N-1)N}\langle f\rangle|^2=\lambda_{N-1}v^2_{N-1}\biggl(\,\int_0^Tf(t)\,dt\biggr)^2. \end{equation*} \notag $$

We introduce the space $\mathfrak{H}^{1}=\biggl\{f \in \mathfrak{H}^0\colon \displaystyle\int_0^Tf(t)\,dt=0\biggr\}$. Then $J''_O(f_0)(f,f)<0$ for $f\in \mathfrak{H}^0 \setminus\mathfrak{H}^1$ and $J''_O(f_0)(f,f)=0$ for $f\in \mathfrak{H}^1$. Note that the following holds true for the quantum system $(H_0,V)$. If $1<l$ and $n\leqslant N-1$, then $\langle l|V_{s_n}\cdots V_{s_1}|N\rangle=\langle l|V^n|N\rangle$, and so the form $A^n_{lN}\langle f\rangle=\dfrac{\langle l|V^n|N\rangle}{n!}\biggl(\,\displaystyle\int_0^Tf(t)\,dt\biggr)^{n}$ vanishes on $\mathfrak{H}^{1}$. Also, if $n<N-1$, then $\langle 1|V_{s_n}\cdots V_{s_1}|N\rangle=0$, and so $A^n_{1N}=0$. Then it follows from (1) that $J_O^{(n)}(f_0)(f,\dots,f)=0$ $J^{(n)}_O(f_0)(f,\ldots,f)=0$ for $3\leqslant n \leqslant 2N-3$ and $f\in \mathfrak{H}^1$. Moreover,

$$ \begin{equation*} \begin{aligned} \, &\frac{1}{(2N-2)!}J^{(2N-2)}_O(f_0)(f,\dots,f)= \lambda_1|A_{1N}^{N-1}\langle f \rangle|^2 \\ &\qquad\qquad\qquad=\lambda_1\biggl|\int_{[0,T]^{N-1}}K(t_1,\dots,t_{N-1}) f(t_1)\cdots f(t_{N-1})\,dt_1\cdots dt_{N-1}\biggr|^2\geqslant 0 \end{aligned} \end{equation*} \notag $$

for $f\in\mathfrak{H}^1$, where

$$ \begin{equation*} K(t_1,t_2,\dots,t_{N-1})=\frac1{(N-1)!}v_{1}v_{2}\cdots v_{N-1}e^{i(a-b)\max(t_1,\dots,t_{N-1})}. \end{equation*} \notag $$

Thus the control $f_0\equiv 0$ is a trap of the $(2N-3)$d order. $\Box$

The authors are grateful to S. A. Kuznetsov for pointing out to them the proof of controllability in [9].

Citation in format AMSBIB

\Bibitem{VolPec23}

\by B.~O.~Volkov, A.~N.~Pechen

\paper Higher-order traps for some strongly degenerate quantum control systems

\jour Russian Math. Surveys

\yr 2023

\vol 78

\issue 2

\pages 390--392

\mathnet{http://mi.mathnet.ru//eng/rm10069}

\crossref{https://doi.org/10.4213/rm10069e}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4653853}

\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2023RuMaS..78..390V}

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