mirror of
https://github.com/vale981/notes_io_loop
synced 2025-03-04 17:11:40 -05:00
finish TeXing the transmission
This commit is contained in:
parent
3cf8e35a6e
commit
4452ccd21e
1 changed files with 193 additions and 9 deletions
|
|
@ -951,6 +951,8 @@ as can be ascertained from \cref{fig:delta_choice}.
|
|||
\end{figure}
|
||||
|
||||
|
||||
|
||||
|
||||
\subsection{Effects of Asymmetric Damping}
|
||||
\label{sec:effects-asymm-damp}
|
||||
|
||||
|
|
@ -982,7 +984,7 @@ The diagonalizing transformation is then
|
|||
\eu^{\iu ϕ} & \eu^{\iu ϕ}
|
||||
\end{pmatrix}
|
||||
& T^{-1}_{\mp;i,0} &\triangleq
|
||||
\frac{1}{\sqrt{2}\sqrt{1-Δ^{2}}}
|
||||
\frac{1}{\sqrt{2}\cos(ψ)}
|
||||
\begin{pmatrix}
|
||||
-1 & \eu^{-\iu (ϕ-ψ)}\\
|
||||
1 & \eu^{-\iu (ϕ+ψ)}
|
||||
|
|
@ -1007,7 +1009,7 @@ With this, we have for \(m=\pm,-N,-N+1,\ldots,-1,1,2,\ldots,N\)
|
|||
\begin{equation}
|
||||
\label{eq:94}
|
||||
\begin{aligned}
|
||||
ω^{0}_{m\neq \pm} &= ω_{n}^{B} & ω^{0}_{\pm} &= \pm δ \sqrt{1-Δ^{2}}\\
|
||||
ω^{0}_{m\neq \pm} &= ω_{n}^{B} & ω^{0}_{\pm} &= \pm δ \cos(ψ)\\
|
||||
η^{0}_{m\neq\pm} &= η_{B} = η_S+2δΔ & η^{0}_{\pm} &= n_{S} + δΔ =
|
||||
\frac{η_{S}+ η_{B}}{2}.
|
||||
\end{aligned}
|
||||
|
|
@ -1017,16 +1019,90 @@ Next, we compute the transformation of the interaction
|
|||
\begin{equation}
|
||||
\label{eq:96}
|
||||
\begin{aligned}
|
||||
V_{\pm,n\not\in \{+,-\}} &= \frac{J_{0,n}}{\sqrt{2}\sqrt{1-Δ^{2}}}\eu^{-\iu(ϕ\pm ψ)} &
|
||||
V_{\pm,n\not\in \{+,-\}} &= \frac{J_{0,n}}{\sqrt{2}\cos(ψ)}\eu^{-\iu(ϕ\pm ψ)} &
|
||||
V_{n\not\in \{+,-\},\pm}&= \frac{J_{n,0}}{\sqrt{2}}\eu^{\iu ϕ} & V_{n\neq \pm,
|
||||
m\neq \pm} &=
|
||||
J_{nm} \\
|
||||
V_{+,-}&=\frac{J_{00}}{2\sqrt{1-Δ^{2}}}\eu^{-\iu ψ} & V_{-,+} &=\frac{J_{00}}{2\sqrt{1-Δ^{2}}}\eu^{\iu ψ}.
|
||||
V_{+,-}&=\frac{J_{00}}{2\cos(ψ)}\eu^{-\iu ψ} & V_{-,+} &=\frac{J_{00}}{2\cos(ψ)}\eu^{\iu ψ}.
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
Evidently, the matrix \(T^{-1}VT\) is not hermitian except in the
|
||||
limit \(Δ\to 0\).
|
||||
|
||||
|
||||
\subsection{Rotating-Wave Interaction}
|
||||
\label{sec:rotat-wave-inter}
|
||||
The modulation term in \cref{eq:89} can be written as
|
||||
\begin{equation}
|
||||
\label{eq:103}
|
||||
V_{nm}(t) = \hat{V}_{nm} f(t),
|
||||
\end{equation}
|
||||
where \(f(t)\) is proportional to the voltage applied to the
|
||||
EOM. Note that \(V_{nm}\) is already expressed in the basis of the
|
||||
\(c_{m}\) (see \cref{eq:96}).
|
||||
|
||||
Let us now assume that the voltage modulation takes the form of
|
||||
\begin{equation}
|
||||
\label{eq:104}
|
||||
f(t)=∑_{j}\frac{\hat{f}_{j}}{2} \sin\qty[\pqty{\hat{ω}_{j}+\hat{δ}_{j}}t
|
||||
+ \varphi_{j}] = -\iu ∑_{j}\hat{f}_{j}\bqty{\eu^{\iu
|
||||
\pqty{\hat{ω}_{j}+\hat{δ}_{j}}t}\eu^{\iu \varphi} - \eu^{-\iu
|
||||
\pqty{\hat{ω}_{j}+\hat{δ}_{j}}t}\eu^{-\iu \varphi_{j}}}.
|
||||
\end{equation}
|
||||
|
||||
Transforming into the rotating frame of the \(\tilde{c}_{m}\) (see
|
||||
\cref{eq:66}), we have
|
||||
\begin{equation}
|
||||
\label{eq:105}
|
||||
\tilde{V}_{mn}=-\iu ∑_{j}\hat{V}_{mn}\hat{f}_{i}\bqty{\eu^{i \pqty{\hat{ω_{j}} +\hat{δ}_{j} -
|
||||
\pqty{ω^{0}_{n}-ω^{0}_{m}}}t} \eu^{\iu \varphi_{j}}
|
||||
- \eu^{-i \pqty{\hat{ω_{j}} +
|
||||
\hat{δ}_{j}+\pqty{ω^{0}_{n}-ω^{0}_{m}}}t} \eu^{-\iu \varphi_{j}}}.
|
||||
\end{equation}
|
||||
|
||||
As discussed in \cref{sec:choice-hybridization}, we want to couple the
|
||||
\(+\) and \(m\neq \pm\). Comparing with \cref{eq:12}, we set
|
||||
\begin{align}
|
||||
\label{eq:107}
|
||||
\hat{ω_{j}}&=ω^{0}_{j} - ω^{0}_{+} = - \pqty{ω^{0}_{-j} - ω^{0}_{-}}& \hat{δ}_{j} = ε_{+}-ε_{j}
|
||||
\end{align}
|
||||
where \(0<j\leq N\) and obtain from \cref{eq:105} by neglecting all
|
||||
counter-rotating terms
|
||||
\begin{equation}
|
||||
\label{eq:108}
|
||||
\begin{aligned}
|
||||
\tilde{V}_{+,j} &= -\iu \hat{f}_{j} \hat{V}_{+,j} \eu^{i\varphi_{j}}
|
||||
\eu^{\iu \pqty{ε_{+}-ε_{j}}t}
|
||||
& \tilde{V}_{j,+} &= \iu \hat{f}_{j} \hat{V}_{j,+}
|
||||
\eu^{-\iu\varphi_{j}} \eu^{\iu
|
||||
\pqty{ε_{j}-ε_{+}}t}\\
|
||||
\tilde{V}_{-j,-} &= -\iu \hat{f}_{j} \hat{V}_{-j,-} \eu^{i\varphi_{j}}
|
||||
\eu^{\iu \pqty{ε_{+}-ε_{j}}t}
|
||||
& \tilde{V}_{-,-j} &= \iu \hat{f}_{j} \hat{V}_{-,-j}
|
||||
\eu^{-\iu\varphi_{j}} \eu^{\iu
|
||||
\pqty{ε_{j}-ε_{+}}t}.
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
Note, that \cref{eq:108} implies \(ε_{-} = -ε_{+}\) and
|
||||
\(ε_{-j} = -ε_{j}\).
|
||||
|
||||
Comparing \cref{eq:108} with \cref{eq:96} and identifying
|
||||
\(J_{mn}=f(t)\hat{J}_{mn}\) we get
|
||||
\begin{align}
|
||||
\label{eq:113}
|
||||
\begin{aligned}
|
||||
V^{0}_{+,j} &= -\iu \hat{f}_{j}
|
||||
\frac{\hat{J}_{0,j}}{\sqrt{2}\cos(ψ)}
|
||||
\eu^{-i\pqty{ϕ+ ψ-\varphi_{j}}}
|
||||
& V^{0}_{j,+} &= \iu \hat{f}_{j} \frac{\hat{J}_{0,j}}{\sqrt{2}}
|
||||
\eu^{i\pqty{ϕ-\varphi_{j}}}\\
|
||||
V^{0}_{-j,-} &= -\iu \hat{f}_{j} \frac{\hat{J}_{-j,0}}{\sqrt{2}}
|
||||
\eu^{i\pqty{ϕ+\varphi_{j}}}
|
||||
& V^{0}_{-,-j} &= \iu \hat{f}_{j} \frac{\hat{J}_{0,-j}}{\sqrt{2}\cos(ψ)}
|
||||
\eu^{-i\pqty{ϕ- ψ+\varphi_{j}}}.
|
||||
\end{aligned}
|
||||
\end{align}
|
||||
|
||||
\subsection{Steady-state Transmission on the Small Ring}
|
||||
\label{sec:steadyst-transm}
|
||||
|
||||
|
|
@ -1048,17 +1124,17 @@ Using \cref{sec:effects-asymm-damp}, we identify
|
|||
\eu^{\iu m
|
||||
ψ}\pqty{δ_{m,+}-δ_{m,-}}\\
|
||||
&& U^{-1}_{m,β}&\to U^{-1}_{m,0} =
|
||||
T^{-1}_{m;S,0}=\frac{1}{\sqrt{2}\sqrt{1-Δ^{2}}}\pqty{δ_{m,+}-δ_{m,-}}
|
||||
T^{-1}_{m;S,0}=\frac{1}{\sqrt{2}\cos(ψ)}\pqty{δ_{m,+}-δ_{m,-}}
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
which yields using \cref{eq:88}
|
||||
\begin{equation}
|
||||
\label{eq:98}
|
||||
η_{m} = \frac{\abs{κ} πn_{B}}{2c\sqrt{1-Δ^{2}}} \pqty{δ_{m,+}
|
||||
η_{m} = \frac{\abs{κ} πn_{B}}{2c\cos(ψ)} \pqty{δ_{m,+}
|
||||
\eu^{\iu ψ} + δ_{m,-}\eu^{-\iu ψ}}.
|
||||
\end{equation}
|
||||
|
||||
To obtain the \(ω_{γ}\) and \(O_{m,γ}\) we have to diagonalize
|
||||
To obtain the \(ω_{γ}\), \(λ_{γ}\) and \(O_{m,γ}\) we have to diagonalize
|
||||
\begin{equation}
|
||||
\label{eq:99}
|
||||
V^{0}_{n,m} + \pqty{ε_{m} -\iu \pqty{η^{0}_{m} + η_{m}} δ_{nm}}.
|
||||
|
|
@ -1066,7 +1142,7 @@ To obtain the \(ω_{γ}\) and \(O_{m,γ}\) we have to diagonalize
|
|||
As
|
||||
\begin{equation}
|
||||
\label{eq:100}
|
||||
\Im η_{\pm}= \pm \frac{\abs{κ} πn_{B}}{2c\sqrt{1-Δ^{2}}}Δ,
|
||||
\Im η_{\pm}= \pm \frac{\abs{κ} πn_{B}}{2c\cos(ψ)}Δ,
|
||||
\end{equation}
|
||||
we obtain a correction to the on-site energies \(ε_{\pm}=\pm ε_{A}\),
|
||||
whereas
|
||||
|
|
@ -1074,7 +1150,115 @@ whereas
|
|||
\label{eq:101}
|
||||
\Re η_{\pm} = \frac{\abs{κ} πn_{B}}{2c}
|
||||
\end{equation}
|
||||
is independent of \(Δ\) which is consistent with the symmetry of \(η^{0}_{\pm}\).
|
||||
is independent of \(Δ\) which is consistent with the symmetry of
|
||||
\(η^{0}_{\pm}\). Note however that \cref{eq:100} is of very small
|
||||
magnitude as it is the product of two quantities that are small
|
||||
compared to \(Ω_{B}\) and \(1\) respectively.
|
||||
|
||||
|
||||
In light of \cref{eq:113} we may choose\footnote{The right hand sides
|
||||
denote quantities from the previous set of notes.}
|
||||
\begin{gather}
|
||||
\label{eq:114}
|
||||
ε_{+} = ε_{A}- \Im η_{+}= ε_{A} - \frac{\abs{κ}
|
||||
πn_{B}}{2c\cos(ψ)}Δ\\
|
||||
\begin{aligned}
|
||||
ε_{j} &= ω_{j}, & \varphi_{j} &= 0, & \hat{f}_{i}&= \frac{\sqrt{2}}{\hat{J}_{0,i}}η_{j}
|
||||
\end{aligned}
|
||||
\end{gather}
|
||||
so that \(V^{0}_{n,m}\) most closely resembles the target
|
||||
Hamiltonian. The drive phases \(\varphi_{j}\) can also be set to
|
||||
\(\varphi_{j}=ϕ + \frac{π}{2}\), to remove the phase in the
|
||||
interaction if it is known. The phase in the interaction does not
|
||||
influence the observable \(ρ_{A}\). However it does influence the
|
||||
interference with a reference light beam.
|
||||
|
||||
To calculate the susceptibility (see \cref{eq:87}), we evaluate
|
||||
\begin{align}
|
||||
\label{eq:116}
|
||||
U_{γ} &= ∑_{σ=\pm} \eu^{\iu \tilde{ω}_{σ} t}O^{-1}_{γσ}
|
||||
T^{-1}_{σ;S0} = ∑_{σ=\pm} \frac{σ
|
||||
}{\sqrt{2}\cos(ψ)} \eu^{\iu \pqty{ω_{σ}^{0}-ε_{σ}} t}O^{-1}_{γσ}\\
|
||||
\bar{U}_{γ}&= ∑_{σ=\pm} \eu^{-\iu \pqty{ω_{σ}^{0}-ε_{σ}}
|
||||
t}O_{σγ}T_{S0;σ} = ∑_{σ=\pm} \frac{σ \eu^{\iu σ ψ}}{\sqrt{2}}\eu^{-\iu \pqty{ω_{σ}^{0}-ε_{σ}}t}O_{σγ}
|
||||
\end{align}
|
||||
with
|
||||
\begin{equation}
|
||||
\label{eq:117}
|
||||
\tilde{ω}_{\pm} \equiv ω_{σ}^{0}-ε_{σ} = ω_{S} \pm δ\cos(ψ) - ε_{A}
|
||||
+ \frac{\abs{κ}
|
||||
πn_{B}}{2c\cos(ψ)}Δ.
|
||||
\end{equation}
|
||||
|
||||
Finally we arrive at
|
||||
\begin{equation}
|
||||
\label{eq:118}
|
||||
\begin{aligned}
|
||||
χ(t,s) &= χ_{0}Θ(s)
|
||||
∑_{γ,σ,σ'}\eu^{{-\iu\bqty{\tilde{ω}_{σ}t -
|
||||
\tilde{ω}_{σ'}s + ω_{γ}(t-s)} -
|
||||
λ_{γ}(t-s)}}O_{σ,{γ}}O^{-1}_{γ,σ'}\frac{σ σ' \eu^{\iu σ
|
||||
ψ}}{\cos(ψ)}\\
|
||||
&= θ(s) χ_{1}(t-s) + χ_{2}(t,s).
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
where
|
||||
\begin{equation}
|
||||
\label{eq:119}
|
||||
χ_{0}=\frac{\abs{κ}π n_{B}}{2c}.
|
||||
\end{equation}
|
||||
The stationary and non-stationary susceptibilities work out to be
|
||||
\begin{align}
|
||||
\label{eq:121}
|
||||
χ_{1}(t) &= χ_{0}
|
||||
∑_{γ,σ}\eu^{{-\bqty{\iu\pqty{\tilde{ω}_{σ} + ω_{γ}} +
|
||||
λ_{γ}}t}}O_{σ,{γ}}O^{-1}_{γ,σ'}\frac{\eu^{\iu σ
|
||||
ψ}}{\cos(ψ)}\\
|
||||
χ_{2}(t,s) &= -χ_{0} ∑_{γσ} \eu^{-\bqty{\iu\pqty{\tilde{ω}_{σ} +
|
||||
ω_{γ}} + λ_{γ}}t} \eu^{\bqty{\iu\pqty{\tilde{ω}_{\bar{σ}} +
|
||||
ω_{γ}} + λ_{γ}}s} O_{σ,{γ}}O^{-1}_{γ,\bar{σ}},
|
||||
\end{align}
|
||||
where \(\bar{σ}=-σ\). As \(V^{0}_{mn}\) decomposes into two blocks,
|
||||
where modes with \(n=-,-1,-2,\dots,-N\) don't couple to modes with
|
||||
\(n=+,1,2,\dots,N\) so that we obtain two non-overlapping sets of
|
||||
eigenstates, we have
|
||||
\begin{equation}
|
||||
\label{eq:122}
|
||||
O_{σ,γ}=0 \wedge O^{-1}_{\bar{σ},γ} =0\; \forall σ=\pm\implies χ_{2}(t,s) = 0
|
||||
\end{equation}
|
||||
and the non-stationary contribution to the susceptibility vanishes.
|
||||
|
||||
We can now proceed to calculate the response of the system to a
|
||||
coherent input beam with frequency \(ω\) in the limit of
|
||||
\(t\gg λ_{γ}\)
|
||||
\begin{equation}
|
||||
\label{eq:123}
|
||||
∫_{0}^{t}\eu^{-\iu ω s} χ(t-s)\dd{s} = \eu^{\iu ω t} χ_{0} ∑_{σγ}\frac{O_{σ,{γ}}O^{-1}_{γ,σ'}}{\iu\pqty{\tilde{ω}_{σ} + ω_{γ}-ω} +
|
||||
λ_{γ}}\frac{\eu^{\iu σψ}}{\cos(ψ)}\equiv \eu^{-\iu ω t}T_{S}(ω).
|
||||
\end{equation}
|
||||
|
||||
Both the magnetic and the electric field are proportional to
|
||||
\(\Im b_{\outputf}\) so that the absolute value of the pointing
|
||||
vector, e.g. the intensity, averaged over the oscillation period of
|
||||
the input light becomes
|
||||
\begin{equation}
|
||||
\label{eq:128}
|
||||
\bar{I} = I_{0}\pqty{1-2\Re{T_{S}(ω)} + \abs{T_{S}(ω)}^{2}} \approx
|
||||
I_{0}\pqty{1-2\Re{T_{S}(ω)}} ,
|
||||
\end{equation}
|
||||
where we have used \(b_{\inputf}=b_{0}\eu^{-\iu ωt}\).
|
||||
|
||||
For \(Δ=0\) we have
|
||||
|
||||
\begin{equation}
|
||||
\label{eq:130}
|
||||
\Re{T_{S}(ω)} = χ_{0} ∑_{σγ}\frac{λ_{γ}\abs{O_{σ,γ}}^{2}}{\pqty{\tilde{ω}_{σ} + ω_{γ}-ω}^{2} +
|
||||
λ_{γ}^{2}},
|
||||
\end{equation}
|
||||
whereas \(Δ\neq 0\) will very slightly shift the peaks and influence
|
||||
the peak heights. We also see, that we only have a good signal on the
|
||||
states that have some overlap with the small loop. We therefore have
|
||||
to attach an additional transmission line to the big loop.
|
||||
|
||||
\newpage
|
||||
\printbibliography{}
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue