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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.)

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