2022.04.09 — two level system driven with noise
last update: 2023.01.20
in this post I want to collect some results / formulas concerning population transfer between two quantum states.
dynamical equations
we assume that the hamiltonian $h$ governing the system dynamics is composed of a time independent "unperturbed" part $h_o$
and a time dependent "perturbation" $h'$, i.e.
\begin{equation}
h = h_o + h'
\end{equation}
let $\ket{-}$ and $\ket{+}$ be respectively the low and high energy eigenstates of $h_o$, i.e.
\begin{equation}
h_o = \Delta \epsilon \sigma_z
\end{equation}
where
\begin{equation}
\sigma_z = \op{+}{+} - \op{-}{-}
\end{equation}
is the $z$ pauli matrix, so that the energy separation between $\ket{+}$ and $\ket{-}$ is
\begin{equation}
\bra{+} h_o \ket{+} - \bra{-} h_o \ket{-} = 2 \Delta \epsilon
\end{equation}
the perturbation $h'$ has the form
\begin{equation}
h' = \epsilon(t) \sigma_x
\end{equation}
where
\begin{equation}
\sigma_x = \op{+}{-} + \op{-}{+}
\end{equation}
is the $x$ pauli matrix, and
\begin{equation}
\epsilon(t) = \epsilon_o \cos{\phi(t)}
\end{equation}
where $\epsilon_o$ is some constant and
\begin{equation}
\phi(t) = \phi_o + \int_{t_o}^\infty dt' \omega(t')
\end{equation}
where $\phi_o$ is the phase of the perturbation at $t=t_o$ and
\begin{equation}
\omega(t) = \omega_o + \Delta \omega + \delta \omega(t)
\end{equation}
where
\begin{equation}
\hbar \omega_o = 2 \Delta \epsilon
\label{omegao}
\end{equation}
and $\Delta \omega$ is some fixed "detuning",
while $\delta \omega(t)$ is a randomly fluctuating phase with the following statistical properties
\begin{align}
\mathbb{E} \left[ \delta \omega(t) \right] & = 0 \\
\mathbb{E} \left[ \delta \omega(t_1) \delta \omega(t_2) \right] & = 2 \gamma \delta(t_1 - t_2)
\end{align}
where $\mathbb{E}\left[ \cdot \right]$ denotes a (classical) expectation value, $\gamma$ is a constant and $\delta(t)$ is the dirac delta function.
by the way, if we want to, we can write the total hamiltonian $h$ in the following, somewhat more "geometric" form
\begin{equation}
h = \bm{h} \cdot \bm{\sigma}
\end{equation}
where
\begin{equation}
\bm{h} = \left( h_x,h_y,h_z \right) = \left( \epsilon(t),0,\Delta \epsilon \right)
\end{equation}
and
\begin{equation}
\bm{\sigma} = \left( \sigma_x,\sigma_y,\sigma_z \right)
\end{equation}
given a state $\ket{\psi(t)}$, we can define amplitudes $c_\pm(t)$ by the rule
\begin{equation}
c_\pm (t) = \ip{\pm}{\psi(t)} e^{\pm i \frac{\phi(t)}{2}}
\end{equation}
if $\ket{\psi (t)}$ satisfies the schrodinger equation,
i.e. $\frac{d}{dt} \ket{\psi(t)} = \left(i \hbar\right)^{-1} h \ket{\psi(t)}$,
then the $c_\pm(t)$ satisfy the equation:
\begin{equation}
\begin{split}
\frac{d}{dt} c_\pm(t)
& =
\ip{\pm}{\dot{\psi}} e^{\pm i \frac{\phi(t)}{2}} + \ip{\pm}{\psi(t)} \frac{d}{dt} e^{\pm i \frac{\phi(t)}{2}}
\\
& = \bra{\pm} \left[ \left(i \hbar\right)^{-1} h \pm \frac{i}{2} \frac{d}{dt} \phi(t) \right] \ket{\psi} e^{\pm i \frac{\phi(t)}{2}}
\\
& = \bra{\pm} \left[ \left(i \hbar\right)^{-1} h + \frac{i}{2} \frac{d}{dt} \phi(t) \sigma_z \right] \ket{\psi} e^{\pm i \frac{\phi(t)}{2}}
\end{split}
\label{cpm}
\end{equation}
inspecting the bracketed expression, we find
\begin{equation}
\begin{split}
&
\left(i \hbar\right)^{-1} h + \frac{i}{2} \frac{d}{dt} \phi(t) \sigma_z
\\
= &
\left(i \hbar\right)^{-1} \left( h_o + h' \right) + \frac{i}{2} \frac{d}{dt} \phi(t) \sigma_z
\\
= &
\left(i \hbar\right)^{-1} \left( \Delta \epsilon \sigma_z + \epsilon(t) \sigma_x \right) + \frac{i}{2} \frac{d}{dt} \phi(t) \sigma_z
\\
= &
\left(i \hbar\right)^{-1} \left( \Delta \epsilon \sigma_z + \epsilon(t) \sigma_x \right)
+
\frac{i}{2} \left( \omega_o + \Delta \omega + \delta \omega(t) \right) \sigma_z
\\
= &
i \left(
\left(
\frac{\omega_o}{2} - \frac{\Delta \epsilon}{\hbar}
+
\frac{1}{2} \left(\Delta \omega + \delta \omega(t)\right)
\right) \sigma_z - \frac{\epsilon(t)}{\hbar} \sigma_x
\right)
\\
(!) = &
i \left(
\frac{1}{2} \left(\Delta \omega + \delta \omega(t)\right) \sigma_z - \frac{\epsilon(t)}{\hbar} \sigma_x
\right)
\end{split}
\end{equation}
where in the step marked (!) we apply equation \eqref{omegao}. inserting this expression back into equation \eqref{cpm}, we get
\begin{equation}
\begin{split}
\frac{d}{dt} c_\pm(t)
& =
\bra{\pm} \left[
i \left(
\frac{1}{2} \left(
\Delta \omega + \delta \omega(t)
\right) \sigma_z - \frac{\epsilon(t)}{\hbar} \sigma_x
\right)
\right] \ket{\psi} e^{\pm i \frac{\phi(t)}{2}}
\\
& =
\left( i \hbar \right)^{-1} \left(
\mp \frac{1}{2} \hbar \left(
\Delta \omega + \delta \omega(t)
\right) c_\pm(t)
+ \left[
\bra{\pm} \epsilon(t) \sigma_x \ket{\psi(t)} e^{\pm i \frac{\phi(t)}{2}}
\right]
\right)
\end{split}
\label{hp}
\end{equation}
zooming in now on the bracketed expression:
\begin{equation}
\begin{split}
&
\bra{\pm} \epsilon(t) \sigma_x \ket{\psi(t)} e^{\pm i \frac{\phi(t)}{2}}
\\
= &
\bra{\mp} \epsilon_o \cos \phi(t) \ket{\psi(t)} e^{\pm i \frac{\phi(t)}{2}}
\\
= &
\bra{\mp} \epsilon_o \cos \phi(t) \ket{\psi(t)} e^{\pm i \frac{\phi(t)}{2}}
\\
= &
\bra{\mp} \epsilon_o
\frac{1}{2}
\left(
e^{i\phi(t)} + e^{-i\phi(t)}
\right)
\ket{\psi(t)} e^{\pm i \frac{\phi(t)}{2}}
\\
(!)
= &
\frac{1}{2}
\bra{\mp} \epsilon_o
\ket{\psi(t)} e^{\mp i \frac{\phi(t)}{2}}
\\
= &
\frac{1}{2} \epsilon_o c_\mp(t)
\end{split}
\label{cmp}
\end{equation}
where in the step marked (!) we apply the rotating wave approximation, valid for $\omega_o \gg \Delta \omega, \delta \omega, \gamma, \frac{\epsilon_o}{\hbar}$.
substituting \eqref{cmp} into \eqref{hp}, we arrive at
\begin{equation}
\dot{c}_\pm = (i \hbar)^{-1} \left( \mp \frac{1}{2} \hbar \left( \Delta \omega + \delta \omega \right)c_\pm + \frac{1}{2} \epsilon c_\mp \right)
\end{equation}
this equation can be recast in the following form of a "modified" schrodinger equation:
\begin{equation}
\ket{\dot{\tilde{\psi}}} = \left( i \hbar \right)^{-1} \tilde h \ket{\tilde{\psi}}
\end{equation}
where
\begin{equation}
\ket{\tilde{\psi}} = e^{i \frac{\phi(t)}{2} \sigma_z} \ket{\psi}
\label{hpp}
\end{equation}
and
\begin{equation}
\tilde{h} = \bm{\tilde{h}} \cdot \bm{\sigma}
\end{equation}
where
\begin{equation}
\bm{\tilde{h}}
= \left( \tilde{h}_x, \tilde{h}_y, \tilde{h}_z \right)
= \left(\frac{1}{2} \epsilon_o, 0, -\frac{1}{2} \hbar \left( \Delta \omega + \delta \omega(t) \right) \right)
\end{equation}
the time evolution of the expectation values $\Sigma_i = \bra{\tilde{\psi}} \sigma_i \ket{\tilde{\psi}}$ of the spin operators $\sigma_i$ for a state $\ket{\tilde{\psi}}$ satisfying \eqref{hpp} is given by
\begin{equation}
\begin{split}
\frac{d}{dt} {\Sigma}_i(t)
& =
\frac{d}{dt} \bra{\tilde{\psi}} \sigma_i \ket{\tilde{\psi}}
\\
& =
\bra{\tilde{\psi}}
\left[
(i \hbar)^{-1}
\comm{\tilde{h}}{\sigma_i}
\right]
\ket{\tilde{\psi}}
\\
& =
\bra{\tilde{\psi}}
\left[
(i \hbar)^{-1}
\sum_{j=1}^3 \tilde{h}_j
\comm{\sigma_j}{\sigma_i}
\right]
\ket{\tilde{\psi}}
\\
(!) & =
(i \hbar)^{-1}
\sum_{j=1}^3 \tilde{h}_j
\left( \sum_{k=1}^3 2 i \epsilon_{jik} \bra{\tilde{\psi}} \sigma_k \ket{\tilde{\psi}} \right)
\\
& =
(i \hbar)^{-1}
\sum_{j=1}^3 \tilde{h}_j
\left( \sum_{k=1}^3 2 i \epsilon_{jik} \Sigma_k \right)
\\
& =
\sum_{j,k=1}^3 \epsilon_{ikj} \Sigma_k \frac{2\tilde{h}_j}{\hbar}
\\
& =
\left( \bm{\Sigma} \times \bm{\Omega} \right)_i
\end{split}
\label{rabi}
\end{equation}
i.e.
\begin{equation}
\dot{\bm{\Sigma}} = \bm{\Sigma} \times \bm{\Omega}
\label{rabivec}
\end{equation}
where
\begin{equation}
\bm{\Sigma} = \left( \Sigma_x, \Sigma_y, \Sigma_z \right)
\end{equation}
and
\begin{equation}
\begin{split}
\bm{\Omega}
& =
\left( \Omega_x, \Omega_y, \Omega_z \right)
\\
& =
\frac{2 \bm{\tilde{h}}}{\hbar}
\\
& =
\left( \tilde{\omega}, 0, -\Delta \omega - \delta \omega(t) \right)
\end{split}
\end{equation}
where
\begin{equation}
\tilde{\omega} = \frac{\epsilon_o}{\hbar}
\end{equation}
note that back in equation \eqref{rabi} we apply in the step marked (!) the commutation relations for the paul spin operators.
it is worth asking of what relevance is equation \eqref{rabivec},
since it gives the time evolution of a state evolving under the influence of $\tilde{h}$, while our system evolves under $h$.
if we are only concerned in populations $P_\pm(t) = \left| \ip{\pm}{\psi} \right|^2$ (as opposed to coherences), however, we find that we are ok:
\begin{equation}
\begin{split}
P_\pm(t)
& =
\left| \ip{\pm}{\psi} \right|^2
\\
& =
\left|
\bra{\pm}\left(e^{-i \frac{\phi(t)}{2} \sigma_z} e^{+i \frac{\phi(t)}{2} \sigma_z}\right) \ket{\psi}
\right|^2
\\
& =
\left|
\left( e^{+i \frac{\phi(t)}{2} \sigma_z} \ket{\pm} \right)^\dagger \left( e^{+i \frac{\phi(t)}{2} \sigma_z} \ket{\psi} \right)
\right|^2
\\
& =
\left|
\left(
e^{
\pm i \frac{
\phi(t)
}{
2
}
}
\ket{\pm}
\right)^\dagger
\ket{
\tilde{\psi}
}
\right|^2
\\
& =
\left|
e^{\mp i \frac{\phi(t)}{2}} \ip{\pm}{\tilde{\psi}}
\right|^2
\\
& =
\left|
\ip{\pm}{\tilde{\psi}}
\right|^2
\\
& =
\ip{\tilde{\psi}}{\pm}\ip{\pm}{\tilde{\psi}}
\\
& =
\frac{1}{2} \left(
\ip{\tilde{\psi}}{\pm}\ip{\pm}{\tilde{\psi}} + \ip{\tilde{\psi}}{\mp}\ip{\mp}{\tilde{\psi}}
\right) +
\frac{1}{2} \left(
\ip{\tilde{\psi}}{\pm}\ip{\pm}{\tilde{\psi}} - \ip{\tilde{\psi}}{\mp}\ip{\mp}{\tilde{\psi}}
\right)
\\
& =
\frac{1}{2} \left(
P_\pm + P_\mp
\right) +
\frac{1}{2} \left(
\pm \Sigma_z
\right)
\\
& =
\frac{1}{2} \left(
1 \pm \Sigma_z
\right)
\end{split}
\end{equation}
an expression for the dynamics of the classical expectation value
$\bm{\tilde{\Sigma}} = \mathbb{E} \left[ \bm{\Sigma} \right]$ of the spins $\bm{\Sigma}$ over the distribution of phase noises $\delta \omega(t)$
can be determined using results summarized in appendix b of agarwal's "quantum statistical theory of optical-resonance phenomena in fluctuating laser fields"
(link).
for a dyamical system of the form
\begin{equation}
\dot{\bm{x}} = M \bm{x} + F(t) \bm{x}
\label{ag}
\end{equation}
where $M$ is a time-independent operator and $F(t)$ is a stochastic operator satisfying
\begin{align}
\mathbb{E} \left[
F(t)
\right]
& =
0\,,
\label{age1}
\\
\mathbb{E} \left[
F_{ij}(t_1) F_{kl}(t_2)
\right]
& =
2 Q_{ijkl} \delta(t_1 - t_2)
\label{age2}
\end{align}
for some $Q_{ijkl}$, the dynamics of the expectation value $\mathbb{E} \left[\bm{x}\right] = \bm{y}$ satisfies the equation
\begin{equation}
\dot{\bm{y}} = M \bm{y} + Q \bm{y}
\end{equation}
where
\begin{equation}
Q_{ij} = \sum_{k} Q_{ikkj}
\end{equation}
defining
\begin{align}
\bm{\Omega_o} & = \left(\tilde{\omega},0,-\Delta \omega\right) \\
\bm{\delta \Omega}(t) & = \left(0,0,-\delta \omega(t)\right) \\
\end{align}
so that
\begin{equation}
\bm{\Omega} = \bm{\Omega_o} + \bm{\delta \Omega}(t)
\end{equation}
and making the identifications
\begin{align}
\bm{x} & \to \bm{\Sigma} \, , \\
M & \to \cdot \times \bm{\Omega_o} \, , \\
F(t) & \to \cdot \times \bm{\delta \Omega}(t) \, , \\
\end{align}
we find that conditions \eqref{ag}, \eqref{age1}, and \eqref{age2} are satisfied and that
\begin{equation}
Q \to - \gamma \left( 1 - \bm{\hat{z}} \bm{\hat{z}}^\intercal \right)
\end{equation}
with
\begin{equation}
\bm{\hat{z}} = \left(0,0,1\right)
\end{equation}
so we arrive finally at the following dynamical equation for $\bm{\tilde{\Sigma}}$
\begin{equation}
\dot{\bm{\tilde{\Sigma}}} = \bm{\tilde{\Sigma}} \times \bm{\Omega_o} - \Gamma \bm{\tilde{\Sigma}}
\label{dyn}
\end{equation}
where
\begin{equation}
\Gamma = \gamma \left( 1 - \bm{\hat{z}} \bm{\hat{z}}^\intercal \right)
\end{equation}
solution
together $\left[ \cdot \times \bm{\Omega_o} - \Gamma \right]$ form a linear operator, which we will call $A$, so that
\begin{equation}
\dot{\bm{\tilde{\Sigma}}} = A \bm{\tilde{\Sigma}}
\end{equation}
as a linear operator, $A$ possesses 3 eigenvectors $\bm{u}_i,\,i=1,2,3$ with associated eigenvalues $\lambda_i.$
if the system is initiatilized in a spin state $\bm{\tilde{\Sigma}}_o$, then after a time interval $T$ the spin state $\bm{\tilde{\Sigma}}'$ will be
\begin{equation}
\bm{\tilde{\Sigma}}' =
\sum_{j=1}^3
\bm{u}_j
\left(
\bm{\hat{x}}_j \cdot S^{-1} \bm{\tilde{\Sigma}}_o
\right)
e^{\lambda_j T}
\end{equation}
where
\begin{equation}
S_{ij} = \bm{\hat{x}}_i \cdot \bm{u}_j
\end{equation}
where
\begin{equation}
\bm{\hat{x}}_1 = \bm{\hat{x}},\,\, \bm{\hat{x}}_2 = \bm{\hat{y}},\,\, \bm{\hat{x}}_3 = \bm{\hat{z}}
\end{equation}
we are interested in
the $z$ component $\bm{\hat{z}} \cdot \bm{\tilde{\Sigma}}'$ of the final spin state
starting from an initial state of $\bm{\tilde{\Sigma}}_o = -\bm{\hat{z}}$.
this is given by
\begin{equation}
\bm{\hat{z}} \cdot \bm{\tilde{\Sigma}}' =
-
\sum_{j=1}^3
\left( \bm{\hat{z}} \cdot \bm{u}_j \right)
\left(
\bm{\hat{x}}_j \cdot S^{-1} \bm{\hat{z}}
\right)
e^{\lambda_j T}
\end{equation}
noise source
suppose the phase / frequency noise is supplied from a source $\delta \omega(t)$ with rms value $\delta \omega_o$ and a correlation time $\tau$.
the correlation function $\mathbb{E} \left[ \delta \omega(t_1) \delta \omega(t_2) \right]$ of such a noise should be
\begin{equation}
\mathbb{E} \left[ \delta \omega(t_1) \delta \omega(t_2) \right] = \delta \omega_o^2 e^{-\frac{|t_1 - t_2|}{\tau}}
\end{equation}
if the correlation time $\tau$ is much shorter than all the other relevant time scales
(i.e.
$\tilde{\omega}^{-1}$,
$\Delta \omega^{-1}$,
the interaction time $T$),
then we can reasonably make the substitution
\begin{equation}
\frac{e^{-\frac{|t_1 - t_2|}{\tau}}}{2 \tau} \to \delta(t_1 - t_2)
\end{equation}
so that for such a noise source we can make the identification
\begin{equation}
\gamma \to \delta \omega_o^2 \tau
\end{equation}
rabi frequency
the parameter $\tilde{\omega}$ corresponds to the rabi (radial) frequency,
as can be seen by letting $\gamma = \Delta \omega = 0$
and inspecting the dynamics of equation \eqref{dyn} for an initial $\bm{\tilde{\Sigma}}(t_o) = \bm{\hat{z}}$.
the spin will trace out a circular motion along a circle of unit radius with a velocity — and thus angular velocity — of
$|\dot{\bm{\tilde{\Sigma}}}(t_o)| = \tilde{\omega}$.
according to hilborn's "einstein coefficients, cross sections, f values, dipole moments, and all that"
(link),
the rabi frequency is related to the electric field amplitude $E$ of the perturbing radiation
and the dipole moment $\mu_{+-}$ of the transition by the relation
\begin{equation}
\tilde{\omega} = \frac{\mu_{+-}E}{\hbar}
\end{equation}
the cycle-averaged intensity $I$ of a plane electromagnetic wave with electric field amplitude $E$ is given by
\begin{equation}
I = \frac{1}{2} c \epsilon_o E^2\,,
\end{equation}
while from chadwick et al.'s "quantum state specific reactant preparation in a molecular beam by rapid adiabatic passage"
(link) we find the transition moment $\mu_{+-}$
to be related to the transition's einstein coefficient $A_{+-}$ by the relationship
\begin{equation}
\mu_{+-} = c\left(j_-, m_- ; 1, 0 | j_+ m_+ \right) \sqrt{\frac{3 \varepsilon_o h c^3}{2 \omega_o^3} A_{+-}}
\end{equation}
where
-
$\varepsilon_o$ is the permittivity of free space
(not to be confused with the strength $\epsilon_o$ of the perturbation $h'$)
-
$c\left(j_1,m_1;j_2,m_2|j_3,m_3\right)$ is the clebsch-gordan coefficient,
-
$j_{\pm}$, $m_{\pm}$ are the $j$ and $m$ quantum numbers of the upper and lower states,
-
$h = 2 \pi \hbar$ is planck's constant
we can rearrange terms and obtain the following intuitively-appealing expression for the rabi frequency $\bar{f} = \frac{\bar{\omega}}{2 \pi}$:
\begin{equation}
\bar{f} = \sqrt{A_{+-}A_\gamma}
\end{equation}
where
\begin{equation}
A_\gamma = \frac{3}{2} \times \frac{1}{\pi^4} \times \frac{I \times \pi \left(\frac{\lambda}{2}\right)^2}{E_\gamma}
\end{equation}
where $\lambda = \frac{c}{2 \pi \omega_o}$ is the (vacuum) wavelength of the radiation
and $E_\gamma = \hbar \omega_o$ is the energy of the photons making up the radiation.
so we can think of $A_\gamma$ roughly speaking as the rate at which photons pass through a circle of diameter equal to the wavelength.
the rate $\bar{f}$ at which radiation stimulates rabi cycling is then seen to be the geometric mean of $A_{+-}$, the spontaneous emission rate, and
the rate at which photons strike the molecule.