# Note about wide potentials in control of Schrodinger dynamics

Consider the dimensionless Schrodinger initial value problem (IVP)
$$
i\partial_t\psi+\frac{1}{2}\nabla^2\psi-V(x,u(t))\psi=0, \ \ \
\psi(x,0)=\psi_0(x)\in L^{2}\left(\mathbb{C}\right)
$$
with normalization condition $\|\psi_0\|_{L^2(\mathbb{C})}=1.$ We formulate a common design problem of finding the time-dependent parameter $u$ that evolves an initial state $\psi_0$ through the Schrodinger equation into some desired state $\psi_d$ as follows.

Define an admissible class of controls $\mathcal{U}$ such that $u\in H^1([0,T])$ with fixed initial and terminal conditions $u(0)$ and $u(T)$. We solve
$$
\inf_{u\in \mathcal{U}}{J}=\frac{1}{2}\inf_{u\in \mathcal{U}} \left [J^{\rm infidelity}(u) + J^{\rm regular}(u)\right]
$$
subject to the above IVP where
$J^{\rm infidelity}(u)= 1-\left|{\left\langle\psi_d(x),\psi(x,T)\right\rangle}\right|^2_{L^2(\mathbb{C}^3)}$ and $J^{\rm regular}(u)=\gamma\int_0^T|\dot{u}|^2 dt.$ The infidelity penalizes mismatches in the realized state $\psi(x,T)$ with the desired state $\psi_d$ and has a minimum of 0. Sparber, and others, proved that this optimal control problem is well-posed upon adding the Tikhonov regularization $J^{\rm regular}.$

Now consider the following setup in terms of the initial state, the final state, and the form of the potential
$$
V(x,u(t),v(t))=-\frac{u(t)}{2}\mathrm{sech}^2(v(t)x),
$$
$$
\psi_0(x)=-\frac{1}{\sqrt{2}}\mathrm{sech}(x),\quad \psi_d(x)=-\frac{3}{2\sqrt{3}}\mathrm{sech}^2(x),
$$

The task is now to find two control parameters $u$ and $v$ with boundary values consistent with $\psi_0$ and $\psi_d$ being eigenfunctions of the time-independent Schrodinger equation. Solving the control problem numerically for different values of the regularization parameter $\gamma$ reveals some interesting features.
For $\gamma=10^{-4}$ we have

and for $\gamma=0$ we have

with outputs that are very close in the sense of $L^2(\mathbb{C})$

The regularized potential is certainly desirable in terms of manufacturability since it is not as wide as the unregularized potential. To try to understand the sensitive dependence of the control problem on the Tikhonov regularization parameter $\gamma$ we make a few modifications to the problem and then pursue a dimensionality reduction via Galerkin's method.

We set the potential $V=\frac{1}{2}x^2u$ with control boundary values $u(0)=1,\ u(T)=100.$ To discuss the initial and desired states, let's first derive the reduced system.
We use a Galerkin expansion of the form
$$
\psi(x,t)=\sum_{j=0}^{N}c_n(t)\varphi_j(x;u(t)),
$$
where each of the basis functions $\varphi_j(x;u(t))$ is an instantaneously normalized eigenfunction of the time-independent linear Schrodinger equation. The basis states are the well-known Hermite function generated by the following Rodrigues formula

$$
\varphi_n(x;u)= (-1)^n\frac{\pi^{-1/4}}{\sqrt{2^nu^{n/4}\,n!}}u^{1/8}e^{\frac{u^{1/4}x^2}{2}}\partial_x^ne^{-u^{1/4}x^2}
$$

Keeping only the first three terms ($N=2$), and discarding the single odd mode because of the symmetry of the even quadratic potential, we find a finite-dimensional Hamiltonian system for the time-dependent coefficients (relabeling the even terms )
$$
i\dot{c}_j=\partial_{c^{\dagger}_j}\mathcal{H},\qquad i\dot{c}^{\dagger}_j=-\partial_{c_j}\mathcal{H},\qquad j=0,1
$$
where the Hamiltonian is given by
$$
\mathcal{H}=\frac{\sqrt{u}}{2} \left(\left|{c_0}\right|^2 +5 \left|{c_1}\right|^2\right)-\frac{\dot{u}}{2 \sqrt{2} u}\Im\left\{c_0c^{\dagger}_1\right\}.
$$

The initial condition for $c_0$ and $c_1$ is such that they are a fixed-point of the system at $t=0$ and we choose a desired condition $c_d$ which is a fixed point at $t=T$. It is possible, through a sequence of canonical transformations and making use of the conserved mass of the system, to rewrite the Hamiltonian as the one corresponding to simple harmonic motion
$$
\mathcal{H}(q,p;u)=\sqrt{u}\left(q^2+p^2+\frac{1}{2}\right)-
\frac{p \dot{u}}{8 u}\sqrt{4-2p^2-2q^2}
$$

Now we investigate a finite-dimensional Hamiltonian control problem. Since the fixed point of the Hamiltonian corresponds to its minimum value, the optimal control problem is now
$$
\min_{u\in \mathcal{U}}{J}=\min_{u\in \mathcal{U}}\left\{\mathcal{H}(q,p;u)\bigg|_{t=T}+\frac{\gamma}{2}\int_0^T\dot{u}^2dt\right\},
$$
subject to the above Hamiltonian system.

We now solve this control problem numerically. To make a visual comparison with a suboptimal control, these are the Schrodinger dynamics for a linear function $u$ which interpolates between $u_0$ and $u_T$:
The dotted white line indicates the end of the control time, after which, the control is kept at a constant value of $u_T.$

We solve the Galerkin-reduced control problem numerically, and use the control to solve the Schrodinger IVP to see if the reduced control is effective. Indeed, the Galerkin-reduced strategy finds effective controls. We make a comparison among different values of $\gamma$. The false-color plots are for the absolute values square of the wavefunction corresponding to the computed optimal control. For $\gamma=10^{-4}$ we have:
For $\gamma=5*10^{-5}$ we have:
For $\gamma=10^{-5}$ we have:
We see that an interesting phenomenon develops around $t=0.5$ as we reduce the Tikhonov parameter $\gamma$. Zooming in, we see

The goal is to find estimates on the objective depending on the Tikhonov regularization to figure out a priori which Tikhonov parameter finds the best balance between smoothness and optimality at the level of the ODE control problem. Interestingly, $\gamma=5*10^{-5}$ not only acheives this, but it also happens to give the best result at the level of the PDE control problem.

Everything below here is incorrect:
Our goal now is to try to understand this phenomenon. Using Pontryagin's framework, the optimality conditions for the states, controls, and for the (adjoint) costates $\lambda$ and $\mu$ are
$$
\dot{q}=2p\sqrt{u},\ \ \ \ \ \ \ \ \ \ \dot{p}=-2q\sqrt{u},
$$
$$
\dot{\lambda}=\frac{\dot{u}}{\sqrt{u}}q+\mu\sqrt{u},\ \ \ \ \ \ \dot{\mu}=\frac{\dot{u}}{\sqrt{u}}p-\lambda\sqrt{u}
$$
$$
\gamma\ddot{u}=u^{-1/2}\left(qu-\lambda p-\frac{1}{4}\right)
$$
with $q(0)=q_0,\ p(0)=p_0$ consistent with the fixed point of the Hamiltonian $\mathcal{H}$, $\lambda(T)=\mu(T)=0$ and $u(0)=u_0,\ u(T)=u_T$.

It's not clear how to make progress. Even when $\gamma=0,$ the last optimality conditions reads
$$
u_*=\frac{4\lambda+1}{4q},
$$