14.5: Nonlinear response functions

If the internal variable of the system does not respond linearly to the application of the external agent, the principles that we discussed in prior sections can be extended to include nonlinear terms, as long as the interaction remains perturbative. That is, we can include the expansion in higher orders of \(f(t)\) in our description of \(\overline {A(t)} \) in eq. (14.1.04), which will ultimately be implemented using higher order perturbation theory expansions.

To illustrate how this reasoning works, let’s look at the second-order term in eq. (14.1.04):

\[\delta \overline {A(t)} ^{\left( 2 \right)} = \int dt_{1}\int dt_{2} \, R^{\left( 2 \right)} ( t;t_{1},t_{2} ) f_{1}( t_1 )f_{2}( t_2 ) \label{eq15.5.01}\]

We can write the Hamiltonian as \(H=H_0-f_1(t_1)\hat{A}-f_2(t_2)\hat{A}\). Equation (\ref{eq15.5.01}) is meant to account for changes to \(A\) caused by two external forces, \(f_1\) and \(f_2\), applied at times \(t_1\) and \(t_2\) on a system that was initially at equilibrium. These forces can be two distinct agents acting on the same variable, or by setting \(f_1=f_2\) we can simply be talking about a quadratic dependence of \(f\) on \(A\). Again, the time-dependence of either \(f_n(t)\) can be arbitrarily complex, and we will integrate over the entire history of the application of two external forces up to the time of observation \(t\) (Figure 14.5.1).

The second-order response function is formulated to enforce causality through a time ordering that requires (1) that all forces must be applied before a response is observed and (2) that the application of \(f_1\) must precede \(f_2\). That is, \(t \geqslant t_{2} \geqslant t_{1}\), which we enforce with the time-ordering of the integration limits. We will call the system stationary so that we are only concerned with the time intervals between consecutive interaction times. We expect that eq. (\ref{eq15.5.01}) will take the form

\[\delta \overline {A(t)} ^{\left( 2 \right)} = \int_{ - \infty }^{t} dt_{2} \int_{ - \infty }^{t_{2}} dt_{1} \, R^{\left( 2 \right)} ( t - t_{2},t_{2} - t_{1}) f_{1}( t_1 ) f_{2}( t_2 ) \label{eq15.5.02}\]

We define the intervals between adjacent interactions

\[\begin{aligned}
& \tau _{1} = t_{2} - t_{1} \\
& \tau _{2} = t - t_{2} \\
\end{aligned} \label{eq15.5.03}\]

Switching integration variables from \(t_1\) and \(t_2\) to \(\tau_1\) and \(\tau_2\), we find

\[\delta \overline {A(t)} ^{\left( 2 \right)} = \int_0^{\infty } d\tau _{1} \int_0^{\infty } d\tau _{2} \, \goodbreak R^{\left( 2 \right)}\left( \tau _{1},\tau _{2} \right) f_{1} ( t - \tau _{1} - \tau _{2} ) f_{2} ( t - \tau _{2} ) \label{eq15.5.04}\]

In this expression, the time arguments for \(f_1\) and \(f_2\) ensure the proper time ordering and that \(\tau_1\) preceeds \(\tau_2\).

We can follow most of the same reasoning as in Sec. 14.3, keeping terms to second order, to derive the quantum mechanical form of the second-order (nonlinear) response function. At this point, there are enough integrals and time-ordered operators to worry about in the derivation that it becomes easier to use a density matrix approach. We will again evaluate eq. (14.3.19), making use of the series expansion of the interaction picture density matrix. Taking the system to be initially at equilibrium, the expansion to second order is

\[\begin{aligned}
\rho_I \nobreak (t) & =\rho_{\text{eq}}-\frac{\mathrm{i}}{\hbar} \int_{-\infty}^t d t_1\left[V_I(t_1), \rho_{\text{eq}}\right]+ \goodbreak \left(\frac{-\mathrm{i}}{\hbar}\right)^2 \int_{-\infty}^t d t_2 \int_{-\infty}^{t_2} d t_1 \goodbreak \left[V_I(t_2),\left[V_I(t_1), \rho_{\text{eq}}\right]\right]+ \goodbreak \ldots \\
& =\rho_I^{(0)}+\rho_I^{(1)}+\rho_I^{(2)}+\ldots
\end{aligned} \label{eq15.5.05}\]

The linear response function is obtained from \(\rho _I^{\left( 1 \right)}\), while the second-order corrections to the changes in \(A(t)\), denoted \(\overline {A^{\left( 2 \right)}(t)} \), come from inserting \(\rho _I^{\left( 2 \right)}\) into eq. (14.3.19):

\[\begin{aligned}
\overline{A^{(2)}(t)} & =\operatorname{Tr}\left(A_I(t) \rho_I^{(2)}(t)\right) \\
& =\operatorname{Tr}\left\{A_I(t)\left(-\frac{\mathrm{i}}{\hbar}\right)^2 \int_{-\infty}^t d t_2 \int_{-\infty}^{t_2} d \nobreak t_1 \goodbreak \left[A_I(t_2),\left[A_I(t_1), \rho_{\text{eq}}\right]\right] \goodbreak \left(-f_2 (t_2) \right) \left( -f_1 (t_1) \right) \right\} \\
& =\left(\frac{\mathrm{i}}{\hbar}\right)^2 \int_{-\infty}^t d t_2 \int_{-\infty}^{t_2} d\nobreak t_1 \goodbreak \operatorname{Tr} \left(A_I(t)\left[A_I(t_2),\left[A_I(t_1), \rho_{\text{eq}}\right]\right]\right) \goodbreak f_2(t_2) f_1(t_1)
\end{aligned} \label{eq15.5.06}\]

Switching time variables to the time intervals in eq. (\ref{eq15.5.03}), we find:

\[\overline{A^{(2)}(t)}=\left(\frac{\mathrm{i}}{\hbar}\right)^2 \int_0^{\infty} d \tau_2 \int_0^{\infty} d \nobreak \tau_1 \goodbreak \operatorname{Tr}\left(A_I(t)\left[A_I(\tau_2),\left[A_I(\tau_1), \rho_{\text{eq}}\right]\right]\right) \goodbreak f_2\left(t-\tau_2\right) f_1\left(t-\tau_2-\tau_1\right) \label{eq15.5.07}\]

Comparing this result with eq. (\ref{eq15.5.04}), we can see that the second order response function is given by the nested commutator products within the trace. Rewriting the nested commutator products as

\[A_{I}(t)\left[ A_{I}(\tau_2),\left[ A_{I}(\tau_1),\rho _{\text{eq}} \right] \right] = \left[ \left[ A_{I}(\tau_2),A_{I}(\tau_1) \right],A_{I}\left( 0 \right) \right]\rho _{\text{eq}} \label{eq15.5.08}\]

we find that the response function is

\[R^{\left( 2 \right)} ( \tau _{1},\tau _{2} ) = \left( \frac{\mathrm{i}}{\hbar } \right)^{2}\operatorname{Tr}\left( \left[ \left[ A_{I}(\tau_2),A_{I}(\tau_1) \right],A_{I} (0) \right]\rho _{\text{eq}} \right) \label{eq15.5.09}\]

Equivalently, we can write

\[R^{\left( 2 \right)} ( \tau _{1},\tau _{2} ) = \left( \frac{\mathrm{i}}{\hbar } \right)^{2}\left\langle \left[ \left[ A_{I}(\tau_2),A_{I}(\tau_1) \right],A_{I} ( 0 ) \right] \right\rangle \label{eq15.5.10}\]

If one expands the nested commutator in eq. (\ref{eq15.5.10}), one finds that there are four separate correlation functions that vary by the time-ordering of the operators. For Hermitian operators, only two of these correlation functions are independent, and the other two are their complex conjugates. We will discuss the interpretation of nonlinear response functions and define nonlinear susceptibilities when we take up nonlinear spectroscopy.

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