5.3: The Density Matrix in the Interaction Picture
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For the case in which we wish to describe a material Hamiltonian \(H_0\) under the influence of an external potential \(V(t)\),
\[H (t) = H _ {0} + V (t) \label{4.21}\]
we can also formulate the density operator in the interaction picture, \(\rho_I\). From our original definition of the interaction picture wavefunctions
\[| \psi _ {I} \rangle = U _ {0}^{\dagger} | \psi _ {S} \rangle \label{4.22}\]
We obtain \(\rho_I\) as
\[\rho _ {I} = U _ {0}^{\dagger} \rho _ {S} U _ {0} \label{4.23}\]
Similar to the discussion of the density operator in the Schrödinger equation, above, the equation of motion in the interaction picture is
\[\dfrac {\partial \rho _ {I}} {\partial t} = - \dfrac {i} {\hbar} \left[ V _ {I} (t) , \rho _ {I} (t) \right] \label{4.24}\]
where, as before, \(V _ {I} = U _ {0}^{\dagger} V U _ {0}\).
Equation \ref{4.24} can be integrated to obtain
\[\rho _ {I} (t) = \rho _ {I} \left( t _ {0} \right) - \dfrac {i} {\hbar} \int _ {t _ {0}}^{t} d t^{\prime} \left[ V _ {I} \left( t^{\prime} \right) , \rho _ {I} \left( t^{\prime} \right) \right] \label{4.25}\]
Repeated substitution of \(\rho _ {I} (t)\) into itself in this expression gives a perturbation series expansion
\[.\begin{align} \rho _ {I} (t) &= \rho _ {0} - \dfrac {i} {\hbar} \int _ {t _ {0}}^{t} d t _ {2} \left[ V _ {I} \left( t _ {1} \right) , \rho _ {0} \right] \\[4pt] & + \left( - \dfrac {i} {\hbar} \right) \int _ {t _ {0}}^{t} d t _ {2} \int _ {t _ {0}}^{t _ {2}} d t _ {1} \left[ V _ {I} \left( t _ {2} \right) , \left[ V _ {I} \left( t _ {1} \right) , \rho _ {0} \right] \right] + \cdots \\[4pt] & + \left( - \dfrac {i} {\hbar} \right)^{n} \int _ {t _ {0}}^{t} d t _ {n} \int _ {t _ {0}}^{t _ {n}} d t _ {n - 1} \\[4pt] & + \cdots \label{4.26}\\[4pt] &= \rho^{( 0 )} + \rho^{( 1 )} + \rho^{( 2 )} + \cdots + \rho^{( n )} + \cdots \label{4.27} \end{align}\]
Here \(\rho _ {0} = \rho \left( t _ {0} \right)\) and \(\rho^{( n )}\) is the nth-order expansion of the density matrix. This perturbative expansion will play an important role later in the description of nonlinear spectroscopy. An nth order expansion term will be proportional to the observed polarization in an nth order nonlinear spectroscopy, and the commutators observed in Equation \ref{4.26} are proportional to nonlinear response functions. Similar to Equation \ref{4.15}, Equation \ref{4.26} can also be expressed as
\[\rho _ {I} (t) = U _ {0} \rho _ {I} ( 0 ) U _ {0}^{\dagger} \label{4.28}\]
This is the solution to the Liouville equation in the interaction picture. In describing the time-evolution of the density matrix, particularly when describing relaxation processes later, it is useful to use a superoperator notation to simplify the expressions above. The Liouville equation can be written in shorthand in terms of the Liovillian superoperator \(\hat {\hat {\mathcal {L}}}\)
\[\dfrac {\partial \hat {\rho} _ {I}} {\partial t} = \dfrac {- i} {\hbar} \hat {\mathcal {L}} \hat {\rho} _ {l} \label{4.29}\]
where \(\hat {\hat {\mathcal {L}}}\) is defined in the Schrödinger picture as
\[\hat {\hat {L}} \hat {A} \equiv [ H , \hat {A} ] \label{4.30}\]
Similarly, the time propagation described by Equation \ref{4.28} can also be written in terms of a superoperator \(\hat {\boldsymbol {\hat {G}}}\), the time-propagator, as
\[\rho _ {I} (t) = \hat {\hat {G}} (t) \rho _ {I} ( 0 ) \label{4.31}\]
\(\hat {\boldsymbol {\hat {G}}}\) is defined in the interaction picture as
\[\hat {\hat {G}} \hat {A} _ {I} \equiv U _ {0} \hat {A} _ {I} U _ {0}^{\dagger} \label{4.32}\]
Given the eigenstates of \(H_0\), the propagation for a particular density matrix element is
\[ \begin{align} \hat {G} (t) \rho _ {a b} & = e^{- i H _ {d} t h} | a \rangle \langle b | e^{iH_0 t \hbar} \\[4pt] &= e^{- i \omega _ {\omega} t} | a \rangle \langle b | \end{align} \label{4.33}\]
Using the Liouville space time-propagator, the evolution of the density matrix to arbitrary order in Equation \ref{4.26} can be written as
\[\rho _ {I}^{( n )} = \left( - \dfrac {i} {\hbar} \right)^{n} \int _ {t _ {0}}^{t} d t _ {n} \int _ {t _ {0}}^{t _ {n}} d t _ {n - 1} \ldots \int _ {t _ {0}}^{t _ {2}} d t _ {1} \hat {G} \left( t - t _ {n} \right) V \left( t _ {n} \right) \hat {G} \left( t _ {n} - t _ {n - 1} \right) V \left( t _ {n - 1} \right) \cdots \hat {G} \left( t _ {2} - t _ {1} \right) V \left( t _ {1} \right) \rho _ {0} \label{4.34}\]