# Applications of Bilinear Control Theory in Nonlinear Spectroscopy

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Here, I wish to motivate bilinear system identification techniques for 3rd order spectroscopy.

The von Neumann equation describes time evolution of a density matrix $$\rho = \sum_i p_i {\left| \psi_i \right>}{\left< \psi_i \right|}$$ where $$\{{\left| \psi_i \right>}\}$$ spans the Hilbert space:

${\frac{\partial \rho(t)}{\partial t}} = -\frac{i}{\hbar}[H(t),\rho(t)]$

Integrating:

$\rho(t) = \rho(0) + \frac{-i}{\hbar}\int_{0}^{t}{dt_1[H(t_1),\rho(t_1)]}$

One can solve by repeatedly inserting the above equation into itself:

$\rho(t) = \rho(0) + \frac{-i}{\hbar}\int_{0}^{t}{dt_1[H(t_1),\rho(0)] + \left(\frac{-i}{\hbar}\right)^2 \int_{0}^{t}{dt_1 \int_{0}^{t_1}{dt_2[H(t_1),[H(t_2),\rho(t_2)]]}}}$

And so on:

$\rho(t) = \rho(0) + \sum_{n=1}^{\infty}{\left(\frac{-i}{\hbar}\right)^n \int_{0}^{t}{dt_1 \int_{0}^{t_1}{dt_2 ... \int_{0}^{t_{n-1}}{dt_n [H(t_1),[H(t_2),...[H(t_n),\rho(t_n)]...]]}}}}$

This series is traditionally defined as

$\rho(t) = \sum_{n=0}^{\infty}{\rho_n(t)} = \rho(0) + \sum_{n=1}^{\infty}{\rho_n(t)}$

where $$\rho_0(t) = \rho(0)$$ and $$\rho_n(t) = \int_{0}^{t}{dt_1 ... \int_{0}^{t_{n-1}}{dt_n [H(t_1),...[H(t_n),\rho(t_n)]...]}}$$.

Re-inspecting the above equation, I find a recursive relationship for the nth order perturbation using linearity of the commutator:

$\rho_n(t) = \frac{-i}{\hbar}\int_{0}^{t}{d\tau[H(\tau),\rho_{n-1}(\tau)]}$

By differentiating both sides, I derive an equation of motion for the nth order perturbation:

${\frac{\partial \rho_n(t)}{\partial t}} = \frac{-i}{\hbar}[H(t),\rho_{n-1}(t)]$

In nonlinear spectroscopy, polarization is related to the perturbation of the density matrix as follows (see Mukamel):

$P_n(t) = {\left< \mu \rho_n(t) \right>}$

I wish to use a general form to relate the polarization measurement $$P_3(t)$$ with nth order perturbations of the density matrix. In superoperator form (where the elements of $$n\times n$$ density matrices concatenate to create $$1\times n^2$$ vectors), let us define $${\mathbf{x}}$$ as a concatenation of $$\rho_n, 0\le n\le 3$$:

${\mathbf{x}}(t) = \left( \begin{array}{cccc} \rho_0(t) \\ \rho_1(t) \\ \rho_2(t) \\ \rho_3(t) \\ \end{array} \right)$

The third-order polarization measurement $$P_3(t)$$ is therefore

$P_3(t) = \left(\begin{array}{c} 0 \\ 0 \\ 0 \\ F \\ \end{array}\right) \cdot {\mathbf{x}}(t)$

such that applying $$F$$ (in operator form) behaves as follows: $$F \cdot = {\left< \mu \cdot \right>}$$. The $$1\times 4n^2$$ vector $${\mathbf{x}}(t)$$ is governed by the time evolution equation

${\frac{\partial {\mathbf{x}}(t)}{\partial t}} = \left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ M(t) & 0 & 0 & 0 \\ 0 & M(t) & 0 & 0 \\ 0 & 0 & M(t) & 0 \\ \end{array} \right) {\mathbf{x}}(t)$

where $${\mathbf{x}}(0) = \left( \begin{array}{cccc} \rho(0) \\ 0 \\ 0 \\ 0 \\ \end{array} \right)$$ and applying $$M(t)$$ (in operator form) behaves as follows: $$M(t) \cdot = \frac{-i}{\hbar}[H(t),\cdot]$$.

This is a general form to describe polarization measurement $$P_3(t)$$ and perturbations of the density matrix.

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Next, I investigate rudimentary linear control theory.

Standard state-space representation of a linear system has this form: $\dot{{\mathbf{x}}}(t) = A(t) {\mathbf{x}}(t) + B(t) {\mathbf{u}}(t)$ ${\mathbf{y}}(t) = C(t) {\mathbf{x}}(t) + D(t) {\mathbf{u}}(t)$

If the system is LTI (linear and time-invariant), then matrices $$A,B,C,D$$ will be time-independent. Our hope is to formulate the nonlinear spectroscopy problem in a form suitable for control theory analysis.

Our observation $$P_3(t)$$ can be interpreted as $${\mathbf{y}}(t)$$ as shown above. In particular, $$C(t) = C = \begin{matrix}(0 & 0 & 0 & F)\end{matrix}$$ and $$D(t) = {\mathbf{0}}$$. Can the time dependence within $$H(t)$$ be modeled as input $${\mathbf{u}}(t)$$ to the system? In general, $$H(t) = H_0 + E(t) \cdot \mu$$.

Let us define superoperators $$M_0, M_1$$ such that $$M(t) = A_0 + E(t) \cdot M_1$$. By linearity of the commutator, applying $$M_0$$ (in operator form) can behave as $$M_0 \cdot = \frac{-i}{\hbar}[H_0,\cdot]$$, and applying $$M_1$$ (in operator form) can behave as $$M_1 \cdot = \frac{-i}{\hbar}[\mu,\cdot]$$.

Now, all matrices are time-independent, but there remains coupling between $${\mathbf{x}}(t)$$ and $$E(t)$$. In particular, we have the evolution equation:

$\dot{{\mathbf{x}}}(t) = \left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ M_0 & 0 & 0 & 0 \\ 0 & M_0 & 0 & 0 \\ 0 & 0 & M_0 & 0 \\ \end{array} \right) {\mathbf{x}}(t) + \left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ M_1 & 0 & 0 & 0 \\ 0 & M_1 & 0 & 0 \\ 0 & 0 & M_1 & 0 \\ \end{array} \right) {\mathbf{x}}(t) E(t)$

Bilinear system theory may be well-equipped to handle this coupling.

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In general, 3rd order spectroscopy experiments employ four-wave mixing: at time $$t = t_0$$, an electric pulse excites a molecule in its ground state. Two more pulses fire at the molecule (at time $$t = t_1, t_2$$), adjusting the molecular state. A last pulse (at $$t = t_3$$) is designed such that the molecule is forced back into its ground state, emitting some measured energy. This last pulse is the “measurement”.

In a classical controls context, the input $${\mathbf{u}}(t)$$ models the electric field controls $$E(t)$$. $$E(t)$$ is pulse-like at $$t = 0, t_1, t_2$$, and $$0$$ everywhere else. (Since the last pulse constitutes a measurement, there are effectively no inputs after time $$t = t_2$$).

The system evolution, then, after time $$t = t_2$$, is linear with equations as given below: $\dot{{\mathbf{x}}}(t) = A{\mathbf{x}}(t) = \left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ M_0 & 0 & 0 & 0 \\ 0 & M_0 & 0 & 0 \\ 0 & 0 & M_0 & 0 \\ \end{array} \right) {\mathbf{x}}(t)$ ${\mathbf{y}}(t) =C{\mathbf{x}}(t) = \begin{matrix}(0 & 0 & 0 & F)\end{matrix} {\mathbf{x}}(t)$

For convenience, let us define time $$t = t_2 = 0$$. In general, we may not know $$M_0$$ nor $$x(0)$$. We can use classical subspace identification techniques to approximate $$M_0$$. Then, we will use some methods from bilinear control theory to approximate $$M_1$$.

The closed-form solution to this linear system is given below: ${\mathbf{x}}(t) = e^{At}{\mathbf{x}}(0)$

Here, $$e^A$$ is shorthand for the series expansion of $$e^x$$:

$e^A = I + A + A^2 / 2 + A^3 / 6 + ...$

In this formulation, $$A$$ is nilpotent, so $$e^A$$ has a finite number of terms. In particular:

$e^{At} = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ M_0 t & 1 & 0 & 0 \\ M_0^2 t^2/2 & M_0 t & 1 & 0 \\ M_0^3 t^3/6 & M_0^2 t^2/2 & M_0 t & 1 \\ \end{array} \right)$

Since $${\mathbf{y}}(t) = \begin{matrix}(0 & 0 & 0 & F)\end{matrix} {\mathbf{x}}(t)$$, we can find a closed-form solution for $${\mathbf{y}}(t)$$:

${\mathbf{y}}(t) = F\begin{matrix}(M_0^3 t^3/6 & M_0^2 t^2/2 & M_0 t & 1)\end{matrix} {\mathbf{x}}(0)$

Expanding $${\mathbf{x}}(0)$$ in terms of its perturbation components, the above equation simplifies:

${\mathbf{y}}(t) = F [M_0^3 t^3 \rho_0(0)/6 + M_0^2 t^2 \rho_1(0)/2 + M_0 t \rho_2(0) + \rho_3(0)]$

Thus, for a particular $$E(t)$$, the output polarization maps to a cubic function of time (after the third pulse). System identification techniques to identify $$M_0$$ and $$\{\rho_i(0)|1\le i\le 3\}$$ could prove useful. (Here, $$\rho_0$$ is a constant, so it should still represent the ground state of the molecule, as it did before $$E(t)$$ was applied.)