18.4: Redfield Theory and Master Equation

Redfield Equation

To describe the exchange of amplitude between system states induced by the bath, we will want to evaluate the matrix elements of the reduced density matrix in the system eigenstates (\(a,b,c,d\)). To begin, we use eq. (18.4.18) to write the time-dependent matrix elements as \[\begin{aligned}
\dot{\sigma}_{a b}^I(t)=-\sum_{c, d} \frac{1}{\hbar^2} \int_0^{\infty} d \tau & \left[\hat{A}_{a c}(t) \hat{A}_{c d}(t-\tau) \sigma_{c b}^I(t)-\hat{A}_{a c}(t-\tau) \sigma_{c d}^I (t) \hat{A}_{c b}(t)\right] C_{\beta \beta}(\tau) \\
-&\left[\hat{A}_{a c}(t) \sigma_{c d}^I(t) \hat{A}_{c b}( t - \tau) -\sigma \vphantom{j}_{a c}^I (t) \hat{A}_{c d}(t-\tau) \hat{A}_{c b}(t)\right] C \vphantom{|}_{\beta \beta}^*(\tau)
\end{aligned}\]

Now, let's convert the time-dependence from expression in terms of the interaction picture into a Schrodinger representation, using \[\begin{aligned}
\left\langle a \left| \hat{A}(t) \right| b \right\rangle & =e^{\mathrm{i} \omega_{a b}t} A_{a b} \\[4pt]
\left\langle a\left|\sigma^I\right| b\right\rangle & =e^{\mathrm{i} \omega_{a b} t} \sigma_{a b} \\
\dot{\sigma}_{a b}^I & =\frac{\partial}{\partial t}\left\langle a\left|\sigma^I\right| b\right\rangle
\end{aligned}\]

To see how this turns out, consider the first term in eq. (18.4.24):

\[\dot{\sigma}_{a b}^I(t)=-\sum_{c, d} \frac{1}{\hbar^2} \int_0^{\infty} \! d \tau \, \hat{A}_{a c}(t) \hat{A}_{c d}(t-\tau) \sigma_{c b}^I(t) C_{\beta \beta}(\tau)\]

Returning to the Schrödinger picture we find \[\dot{\sigma}_{a b}(t) e^{\mathrm{i} \omega_{a b} \tau}+\mathrm{i} \omega_{a b} e^{\mathrm{i} \omega_{a b} \tau} \sigma_{a b}
= -\sum_{c, d} \frac{1}{\hbar^2} \hat{A}_{a c} \, \hat{A}_{c d} \, \sigma_{d b}(t) \, e^{\mathrm{i} \omega_{a c} t+\mathrm{i} \omega_{c d} t+\mathrm{i} \omega_\omega t} \ \int_0^{\infty} \! d \tau \, e^{-\mathrm{i} \omega_{c a} \tau} C_{\beta \beta}(\tau)\]

Defining the Fourier-Laplace transform of the bath correlation function:

\[\tilde{C}_{\beta \beta}(\omega)=\int_0^{\infty} d \tau \, e^{\mathrm{i} \omega \tau} C_{\beta \beta}(\tau)\]

We have

\[\dot{\sigma}_{a b}(t)=-\mathrm{i} \omega_{a b} \sigma_{a b}-\sum_{c, d} \frac{1}{\hbar^2} \hat{A}_{a c} \hat{A}_{c d} \sigma_{c b}(t) \tilde{C}_{\beta \beta}\left(\omega_{d c}\right)\]

Here the spectral representation of the bath correlation function is being evaluated at the energy gap between system states \( \omega_{d c} \). So the evolution of coherences and populations in the system states is governed by their interactions with other system states, governed by the matrix elements, and this is modified depending on the fluctuations of the bath at different system state energy gaps. In this manner, eq. (18.4.24) becomes

\[\begin{align}
\dot{\sigma}_{a b}(t) =& \begin{aligned} \Space{5px}{3.56ex}{2ex}\\
\, -\mathrm{i} \omega_{a b} \sigma_{a b}(t)-\frac{1}{\hbar^2} \sum_{c, d}& \left[\hat{A}_{a c} \hat{A}_{c d} \sigma_{c b}(t) \tilde{C}_{\beta \beta}\left(\omega_{c c}\right)-\hat{A}_{a c} \hat{A}_{d b} \sigma_{c d}(t) \tilde{C}_{\beta \beta}\left(\omega_{c a}\right)\right. \\
&\left. \, -\hat{A}_{a c} \hat{A}_{c b} \sigma_{c d}(t) \tilde{C}_{\beta \beta}^*\left(-\omega_{c b}\right)+\hat{A}_{c d} \hat{A}_{c b} \sigma_{a c}(t) \tilde{C}_{\beta \beta}^*\left(-\omega_{c d}\right)\right] \end{aligned}\\[5pt]
=&-\mathrm{i} \omega_{a b} \sigma_{a b}(t)-\sum_{c, d}\left[\Gamma_{a c, c d}^{+} \sigma_{d b}(t)-\Gamma_{a c, d b}^{-} \sigma_{c d}(t)-\Gamma_{a c, d b}^{+} \sigma_{c d}(t)+\Gamma_{c d, d b}^{-} \sigma_{a c}(t)\right] \end{align}\]

We see that the off-diagonal reduced density matrix elements—the coherences—oscillate with at a frequency given by the energy splitting between the states, but the amplitude of these states are modified, rising and falling at rates given by the constants \({\rm{\Gamma }}_{ab,cd}^\pm \). The rate constants are defined through:

\[\begin{align}
\Gamma _{ab,cd}^ + &= \frac{1}{\hbar ^2} A_{ab} A_{cd} \ \tilde C_{\beta \beta } ( \omega _{dc} ) \\[3pt]
\Gamma _{ab,cd}^ - &= \frac{1}{\hbar ^2} A_{ab} A_{cd} \ \tilde C_{\beta \beta} ( \omega _{ba}) \end{align}\]

The coupled differential equations in eq. (18.4.31) express the relaxation dynamics of the system states almost entirely in terms of the system Hamiltonian. The influence of the bath only enters through the bath correlation function.

The common alternate way of writing eq. (18.4.31) is in terms of the relaxation super-operator, \(\mathcal{R}\) \[\dot \sigma _{ab} (t) = - \mathrm{i} \omega _{ab} \sigma _{ab} - \sum \limits_{c,d} \mathcal{R}_{ab,cd} \, \sigma _{cd} (t) \]

or in the interaction picture \[ \dot \sigma_{ab}^I (t) = \sum\limits_{c,d} \sigma _{cd}^I (t) \, \mathcal{R}_{ab,cd} \; e^{\mathrm{i} ({E_a} - {E_b} - {E_c} + {E_d}) t / \hbar } \]

Equation (18.4.34), the reduced density matrix equation of motion for a Markovian bath, is known as the Redfield equation. It describes the irreversible and oscillatory components of the amplitude in the \(| a \rangle \langle b | \) coherence as features resulting from dissipation to the bath and feeding from other states. \(\mathcal{R}\) describes the rates of change of the diagonal and off-diagonal elements of \(\sigma \vphantom{j}_I\) and is expressed as: \[ \mathcal{R}_{ab,cd} = \delta _{db} \sum\limits_k \left[ \Gamma _{ak,kc}^+ - \Gamma _{db,ad}^+ - \Gamma _{db,ad}^- \right] + \delta _{ac} \sum\limits_k \Gamma _{dk,kb}^- \]

where \(k\) refers to a system state.

The derivation described above can be performed without assuming a form to the system-bath interaction potential as we did in eq. (18.4.13). If so, one can write the relaxation operator in terms of a correlation function for the system-bath interaction \[\begin{align}
\Gamma _{ab,cd}^ + &= \frac{1}{\hbar ^2} \int_0^\infty \! d \tau \left\langle V_{ab} (\tau) V_{cd} (0) \vphantom{X^2} \right\rangle \! \! _{_{B}} e^{ - \mathrm{i} \omega _{cd} \tau } \\
\Gamma _{ab,cd}^ - &= \frac{1}{\hbar ^2} \int_0^\infty \! d \tau \left\langle V_{ab} (0) V_{cd} (\tau) \vphantom{X^2} \right\rangle \! \! _{_{B}} e^{ - \mathrm{i} \omega _{ab} \tau }\end{align}\]

The tetradic notation for the Redfield relaxation operator accounts for all possible dynamic evolution processes within a multistate system. In particular we identify four classes of relaxation processes, depending on the number of states involved:

• \(aa,aa\): Population Relaxation
  • Rate of loss of the population in \(a\)
• \(ab,ab\): Coherence Relaxation or Dephasing
  • Damping of the coherence \(ab\)
• \(aa,bb\): Population Transfer
  • Rate of transfer of population from state \(b\) to state \(a\)
• \(ab,cd\): Coherence Transfer
  • Rate at which amplitude in an oscillating superposition between two states (\(c\) and \(d\)) couples to form oscillating amplitude between two other states (\(a\) and \(b\))

The origin and meaning of these terms will be discussed below.

Secular Approximation

From eq. (18.4.35) we note that the largest changes in matrix elements of result from a resonance condition:

\[\begin{gathered}
\exp \left[ {\mathrm{i} \left( {E_a} - {E_b} - {E_c} + {E_d} \right)t/\hbar } \vphantom{X^2} \right] \approx 1 \\
{E_a} - {E_b} - {E_c} + {E_d} \approx 0
\end{gathered}\]

which is satisfied for three possible cases:

\[\begin{aligned}
a = c \;\, ; \;\, b = d & \implies & \mathcal{R}_{ab,ab}\\[2pt]
a = b\;\, ; \;\, c = d & \implies & \mathcal{R}_{aa,cc}\\[2pt]
a = b = c = d & \implies & \mathcal{R}_{aa,aa}
\end{aligned}\]

In evaluating relaxation rates, often only these secular terms are retained. Whether this approximation is valid must be considered on a case-by-case basis and depends on the nature of the system eigenvalues and the bath correlation function.

Population Relaxation and the Master Equation

To understand the information in the relaxation operator and the classification of relaxation processes, let’s first consider the relaxation of the diagonal elements of the reduced density matrix. Using the secular approximation,

\[\dot \sigma _{aa} (t) = - \sum\limits_b \mathcal{R}_{aa,bb} \, \sigma _{bb} (t) \]

Considering first the case that \(a \ne b\), eq. (18.4.40) gives the relaxation operator as

\[\mathcal{R}_{aa,bb} = - \Gamma _{ba,ab}^+ - \Gamma _{ba,ab}^- \]

Recognizing that \({\Gamma ^ + }\) and \({\Gamma ^ - }\) are Hermitian conjugates,

\[\begin{aligned}
\mathcal{R}_{aa,bb} &= - \frac{1}{\hbar ^2} \left| A_{ab} \vphantom{X^2} \right|^2 \int_0^\infty \! d\tau \, C_{\beta \beta } (\tau) \, e^{ - \mathrm{i} \omega _{ba} \tau } \; + \;c.c.\\[3pt]
&= - \frac{1}{\hbar ^2} \int_0^\infty \! d\tau \left\langle V_{ba}( \tau ) V_{ab} (0) \vphantom{X^2} \right\rangle \! \! _{_B} e^{ - \mathrm{i} \omega _{ab} \tau } \; + \;c.c.
\end{aligned}\]

So \(\mathcal{R}_{aa,bb}\) is a real valued quantity. However, since \(\left\langle {V_{ba}( \tau ) V_{ab}(0)} \right\rangle = \left\langle V_{ba}(0) V_{ab} ( - \tau ) \right\rangle \), \[ \mathcal{R}_{aa,bb} = - \frac{1}{\hbar ^2} \int_{ - \infty }^{ + \infty } \! dt {\left\langle V_{ba}( \tau ) V_{ab}(0) \vphantom{X^2} \right\rangle \! \! _{_B}} e^{\mathrm{i} \omega _{ba} \tau }\]

So we see that the relaxation tensor results in the same rate of population relaxation between states \(a\) and \(b\) as we previously derived from Fermi’s Golden rule:

\[\mathcal{R}_{aa,bb} = - w_{ab} \qquad \quad \left( a \ne b \right)\]

For the case that \(a = b\), eq. (18.4.40) gives the relaxation operator as

\[\begin{aligned}
\mathcal{R}_{aa,aa} &= - \left(\Gamma_{aa,aa}^ + + \Gamma_{aa,aa}^ - \right) + \sum\limits_k \left( \Gamma_{ak,ka}^ + + \Gamma_{ak,ka}^ - \right) \\
&= \sum\limits_{k \, \ne \, a} \left( \Gamma_{ak,ka}^ + + \Gamma_{ak,ka}^ - \right) \end{aligned}\]

The relaxation accounts for the dissipation induced by bath interactions with all states of the system (last term), but with the influence of self-relaxation (first term) removed. The net result is that we are left with the net rate of relaxation from \(a\) to all other system states (\(a \ne k\))

\[\mathcal{R}_{aa,aa} = \sum\limits_{k \, \ne \, a} w_{ka} \]

This term \(\mathcal{R}_{aa,aa}\) is also referred to as the inverse of \(T_1\), the population lifetime of the \(a\) state.

The combination of these observations shows that the diagonal elements of the reduced density matrix follow a master equation that describes the net gain and loss of population in a particular state

\[\dot \sigma _{aa} (t) = \sum\limits_{b \, \ne \, a} w_{ab} \sigma _{bb} (t) - \sum\limits_{k \, \ne \, a} w_{ka} \sigma _{aa} (t) \]

Coherence Relaxation

Now let’s consider the relaxation of the off-diagonal elements of the reduced density matrix. It is instructive to limit ourselves at first to one term in the relaxation operator, so that we write the equation of motion as

\[\dot \sigma _{ab} \left( t \right) = - \mathrm{i} \omega _{ab} \sigma _{ab} (t) - \mathcal{R}_{ab,ab} \, \sigma _{ab} (t) + \cdots \]

The relaxation operator gives

\[\begin{aligned}
\mathcal{R}_{ab,ab} &= - \left( \Gamma _{aa,bb}^ + + \Gamma _{aa,bb}^ - \right) + \sum\limits_k \left( \Gamma _{ak,ka}^ + + \Gamma _{bk,kb}^ - \right) \\
& = - \left( \Gamma _{aa,bb}^ + + \Gamma _{aa,bb}^ - - \Gamma _{\vphantom{b} aa,aa}^ + - \Gamma _{bb,bb}^ - \right) +
\left( \sum\limits_{k \, \ne \, a} \Gamma _{ak,ka}^ + + \sum\limits_{k \, \ne \, b}\Gamma _{bk,kb}^ - \right)
\end{aligned}\]

In the second step, we have separated the sum into two terms, one involving relaxation constants for the two coherent states, and the second involving all other states. The latter term looks very similar to the relaxation rates in eq. (18.4.46). In fact, if we factor out the imaginary parts of these terms and add them as a correction to the frequency in eq. (18.4.49), \(\omega _{ab} \to \left( \omega _{ab} + \mathop{\rm Im} [\text{term} \ 2 ] \right) \), then the remaining expression is directly related to the population lifetimes of the \(a\) and \(b\) states:

\[\begin{aligned}
{\mathop{\rm Re}\nolimits} \left( \sum\limits_{k \, \ne \, a} \Gamma _{ak,ka}^ + + \sum\limits_{k \, \ne \, b} \Gamma _{bk,kb}^ - \right) &= \frac{1}{2}\sum\limits_{k \, \ne \, b} w_{kb} - \frac{1}{2}\sum\limits_{k \, \ne \, a} w_{ka} \\
&= \frac{1}{2}\left( \frac{1}{ T_{1,a}} + \frac{1}{T_{1,b}} \right)
\end{aligned}\]

This term accounts for the decay of the coherence as a sum of the rates of relaxation of the \(a\) and \(b\) states.

The meaning of the first term on the right hand side of eq. (18.4.51) is a little less obvious. If we write out the four contributing relaxation factors explicitly using the system-bath correlation functions in eqs. (18.4.37) and (18.4.38), the real part can be written as

\[\begin{aligned}
\mathop{\rm Re} \left( \Gamma _{aa,bb}^+ + \Gamma _{aa,bb}^- - \Gamma _{aa,aa \vphantom{b}}^+ - \Gamma _{bb,bb}^- \right)
&= \int\limits_0^\infty d\tau \left\langle \left[ V_{bb}(\tau) - V_{aa}(\tau) \vphantom{X^2} \right]\left[ V_{bb} (0) - V_{aa}(0) \vphantom{X^2} \right] \vphantom{\sum X^2} \! \right\rangle_B \\
&\equiv \int\limits_0^\infty d\tau \left\langle \Delta V(\tau)\Delta V(0) \vphantom{X^2}\right\rangle _{_B} \end{aligned}\]

In essence, this term involves an integral over a correlation function that describes variations in the \(a\)-\(b\) energy gap, which varies as a result of its interactions with the bath. So this term, in essence, accounts for the fluctuations of the energy gap that we previously treated with stochastic models. Of course, in the current case, we have made a Markovian bath assumption, so the fluctuations are treated as rapid and only assigned an interaction strength \( \Gamma\) which is related to the linewidth. In an identical manner to the fast modulation limit of the stochastic model we see that the relaxation rate is related to the square of the amplitude of modulation, multiplied by the correlation time for the bath:

\[\int\limits_0^\infty \! d\tau \left\langle \Delta V(\tau) \Delta V(0) \vphantom{X^2} \right\rangle \! \! _{_B} = \left\langle \Delta {V^2} \right\rangle \tau _c \equiv \Gamma \]

As earlier this is how the pure dephasing contribution to the Lorentzian lineshape is defined. It is also assigned a time scale \(T_2^* = \Gamma ^{ - 1}\).

To summarize, we see that the relaxation of coherences has contributions from pure-dephasing and from the lifetime of the states involved. Explicitly, the equation of motion in eq. (18.4.49) can be re-written

\[\dot{\sigma}_{a b}(t) = -\mathrm{i} \undertilde\omega \vphantom{\omega}_{a b} \, \sigma_{a b}(t) - \frac{1}{T_2} \sigma_{a b}(t)\]

where the dephasing time is

\[\frac{1}{T_2} = \frac{1}{T_2^*} + \frac{1}{2} \left( \frac{1}{T_{1,a}} + \frac{1}{T_{1,b}} \right)\]

and the frequency has been corrected as a result interactions with the bath with the (small) imaginary contributions to to \(\mathcal{R}_{ab,ab}\):

\[ \undertilde\omega \vphantom{x}_{a b} =\omega_{a b}+\operatorname{Im}\left[\mathcal{R}_{a b, a b}\right]\]