mirror of
https://github.com/vale981/notes_io_loop
synced 2025-03-04 09:01:41 -05:00
1502 lines
60 KiB
TeX
1502 lines
60 KiB
TeX
\documentclass[fontsize=11pt,paper=a4,open=any,
|
||
twoside=no,toc=listof,toc=bibliography,headings=optiontohead,
|
||
captions=nooneline,captions=tableabove,english,DIV=12,numbers=noenddot,final,parskip=false,
|
||
headinclude=true,footinclude=false,BCOR=0mm]{scrartcl}
|
||
\pdfvariable suppressoptionalinfo 512\relax
|
||
\synctex=1
|
||
|
||
\author{Valentin Boettcher}
|
||
\usepackage{hirostyle}
|
||
\usepackage{hiromacros}
|
||
\addbibresource{references.bib}
|
||
|
||
\title{Input-Output Theory for Modulated Optical Fibre Resonators}
|
||
\date{2023}
|
||
\graphicspath{{graphics}}
|
||
|
||
\newcommand{\inputf}[0]{\ensuremath{\mathrm{in}}}
|
||
\newcommand{\outputf}[0]{\ensuremath{\mathrm{out}}}
|
||
\usetikzlibrary{math}
|
||
% \usetikzlibrary{external}
|
||
% \tikzexternalize[prefix=tikz/]
|
||
\usepackage{pgfplots}
|
||
|
||
\begin{document}
|
||
\maketitle
|
||
\tableofcontents
|
||
|
||
\section{Microscopic Derivation}
|
||
\label{sec:micr-deriv}
|
||
The setup we are describing consists of a general driven photonic
|
||
system \(A\) and a transmission line \(B\). The \(A\) system is
|
||
considered to have the Hamiltonian
|
||
\begin{equation}
|
||
\label{eq:1}
|
||
H_{A}=H_{0}+V(t) = ∑_{j,β;i,α} \pqty{H_{0}}_{i,α;j,β}a_{j,β}^†a_{i,α}= ∑_{m} ω^{0}_{m} c_{m}^†c_{m} + V(t),
|
||
\end{equation}
|
||
where \(\comm{a_{i,α}}{a^†_{j,β}}=δ_{ij}δ_{αβ}\). We assume that the
|
||
system \(A\) consists of several distinct resonators/cavities indexed
|
||
by the first index on the \(a^†\), who each have their own lengths
|
||
\(L_{A,i}\) and eigen-momenta \(k_{i,α} = 2πα/L_{A,i}\) with
|
||
\(α\in\ZZ\).
|
||
|
||
The eigenmodes of the system \(c_{m}\) are linear combinations of the
|
||
bare modes in the photonic system where we have
|
||
\begin{equation}
|
||
\label{eq:43}
|
||
c_{m} = ∑_{i,α} T^\ast_{i,α;m}a_{i,α},
|
||
\end{equation}
|
||
where \(T_{i,α;m}\) is the matrix whose rows are the normalized
|
||
eigenvectors of the matrix \(\pqty{H_{0}}_{i,α;j,β}\).
|
||
|
||
|
||
We designate the bare modes of the
|
||
EM field that are actually in contact with the transmission line as
|
||
the modes with subsystem index \(i=i_{0}\) which is suppressed for
|
||
clarity in all expressions concerning that subsystem. We find modes
|
||
\(a_{β}\) for the electric field in the subsystem in contact with the
|
||
transmission line
|
||
\begin{equation}
|
||
\label{eq:4}
|
||
E_{A}(x,t)= \iu \sqrt{\frac{\hbar}{2ε_{0}n_{A}^{2} L_{A,\perp}^{2} L_{A}}} ∑_{β}
|
||
\sqrt{ω_{k_β}} \pqty{a_{β}(t)
|
||
\eu^{\iu k_{β} x } - a_{β}^†(t) \eu^{-\iu k_{β} x}},
|
||
\end{equation}
|
||
where \(L_{A,\perp}\) is a length scale that can be interpreted as the
|
||
diameter of the transmission line~\cite{Jacobs} and \(L_{A}\) is the
|
||
length of the cavity/resonator that hosts the electric field. The
|
||
modes have wave numbers \(k_{β} = 2πβ/L_{A}\) for \(β \in \ZZ\) and
|
||
frequencies \(ω_{k_β} = c \abs{k_{β}}/n_{A}\), where \(n_{A}\) is the
|
||
refractive index inside the cavity. For simplicity we set \(\hbar
|
||
= 1\) such that energy is measured in units of frequency.
|
||
|
||
\subsection{Transformations and Rotating Frames}
|
||
\label{sec:rotating-frames}
|
||
|
||
The bare \(a_{β}\) modes are linear combinations of the \(c_{m}\) and
|
||
can be related through
|
||
\begin{equation}
|
||
\label{eq:5}
|
||
a_{β} = ∑_{m} U_{βm} c_{m},
|
||
\end{equation}
|
||
where \(U_{βm} = T_{i_{0},β;m}\) is a not necessarily square matrix
|
||
that obeys the unitarity relation \(U U^† = \id\). Transitioning into
|
||
a rotating frame with respect to \(H_{0}\) and employing the rotating
|
||
wave approximation removes all but the slowest-oscillating rotating
|
||
terms from the interaction
|
||
\begin{multline}
|
||
\label{eq:12}
|
||
h_{m}(t) = c_{m}(t)\eu^{\iu \pqty{ω^{0}_{m}-ε_{m}}t} \equiv c_{m}(t)\eu^{\iu \tilde{ω}^{0}_{m}t}
|
||
\\\implies H_{A} \to \tilde{H}_{A}=
|
||
∑_{mn}\pqty{V_{mn}(t) + ε_{m}δ_{mn}} \eu^{\iu t (ω^{0}_{n}-ω^{0}_{m})}\eu^{-\iu
|
||
(ε_{n}-ε_{m})t}
|
||
{c}_{m}^†{c}_{n} \approx ∑_{mn}\pqty{V^{0}_{mn}+ ε_{m}δ_{mn}} {h}_{m}^†{h}_{n},
|
||
\end{multline}
|
||
where
|
||
\(\abs{ε_{m}}\ll\abs{ω^{0}_{m}}\)\(\abs{ε_{m}-ε_{n}}\ll\abs{ω^{0}_{m}
|
||
- ω^{0}_{n}}\) are the detunings of the drive with respect to the
|
||
energy levels of \(H_{0}\). \emph{This constitutes our target
|
||
Hamiltonian which we can control through the modulation of
|
||
\(V(t)\)}
|
||
\begin{equation}
|
||
\label{eq:109}
|
||
H^{T}_{mn}= {V^{0}_{mn} + δ_{mn}{ε_{m}-\iu η_{m}}}.
|
||
\end{equation}
|
||
|
||
Due to the coupling to the transmission line we will find that the
|
||
equation of motion for the \(h_{m}\) becomes non-unitary with a
|
||
damping term
|
||
\begin{equation}
|
||
\label{eq:33}
|
||
\iu \dot{h}_{m} = ∑_{n}\bqty{V^{0}_{mn} + δ_{mn}{ε_{m}-\iu η_{m}}}h_{n}.
|
||
\end{equation}
|
||
|
||
We can subsequently find a (non-unitary) transformation that diagonalizes
|
||
the RWA interaction
|
||
\begin{equation}
|
||
\label{eq:30}
|
||
∑_{mn}\pqty{O^{-1}}_{γm}\bqty{V^{0}_{mn} + δ_{mn}\pqty{ε_{m}-\iu
|
||
η_{m}}}O_{nγ'} = \pqty{ω_{γ}-\iu λ_{γ}} δ_{γ,γ'}.
|
||
\end{equation}
|
||
For \(η_{m}=0\) the columns of \(O\) are the normalized eigenvectors
|
||
of \(V_{mn}^{0}=\mel{m}{V^{0}}{n}\). So if \(\ket{ψ_{j}}\) is an
|
||
eigenvector of \(V\), then \(\braket{i}{ψ_{j}} = O_{ij}\)
|
||
\footnote{This is just a reminder for Valentin who can't seem to keep
|
||
this in his head.}. For finite \(η_{m}\) we will find that the
|
||
eigenvalues will feature an imaginary part and \(O_{mγ}\) is no longer
|
||
unitary, except for the case where all \(η_{m}\) are the same. This
|
||
situation occurs if there are other dominating sources of loss such
|
||
that the coupling to the transmission line is not a factor or if we
|
||
couple to a sufficiently narrow range of modes so that the variation
|
||
of damping rates becomes negligible.
|
||
|
||
Transforming the \(h_{m}\) according to
|
||
\begin{equation}
|
||
\label{eq:13}
|
||
d_{γ} = ∑_{n}O^{-1}_{γn} h_{n} = ∑_{n}O^{-1}_{γn} \eu^{\iu
|
||
\tilde{ω}^{0}_{n} t}{c}_{n} \implies \iu \dot{d}_{γ} = ω_{γ}d_{γ}
|
||
\end{equation}
|
||
leaves us with a very simple equation of motion.
|
||
|
||
|
||
In summary, the bare modes of the resonators are denoted by
|
||
\(a_{j,α}\) where \(j\) refers to the resonator and \(α\) labels the
|
||
mode within that resonator. The eigenmodes \(c_{m}\) of the coupled
|
||
oscillators obeying \(H_{0}\) are related to the bare modes
|
||
\(α_{j,α}\) by \cref{eq:43}. The frame in which the equations of
|
||
motion for the modes reflects the target Hamiltonian
|
||
\(H^{T}\) are reached through a transformation into a
|
||
rotating frame in \cref{eq:12} leading to the \(h_{n}\) modes. The
|
||
resulting equations of motion for the \(h_{n}\) can be decoupled by
|
||
the transformation \(O_{mγ}\) giving the eigenmodes of the target
|
||
Hamiltonian \(d_{γ}\) including damping.
|
||
|
||
It is important to keep in mind that the actual observables are the
|
||
\(α_{β} = α_{i_{0},β}\) which couple to the transmission line, wheras
|
||
the modes in the rotating frame \(h_{n}\) correspond to the amplitudes
|
||
that evolve according to the target Hamiltonian.
|
||
|
||
Let us list the relation between the \(a\), \(c\), \(h\) and \(d\) operators
|
||
for later reference
|
||
\begin{equation}
|
||
\label{eq:67}
|
||
\begin{array}{@{}l| c c c c@{}}
|
||
& α_{β} & c_{m} & h_{n} & d_{γ}\\
|
||
\midrule
|
||
a_{β} & 1 & U_{βm}\equiv T_{i_{0},β;m} & U_{βm}\eu^{-\iu \tilde{ω}^0_mt} & ∑_{m}U_{βm}O_{mγ}\eu^{-\iu \tilde{ω}^0_mt} \\
|
||
c_{m} & U^{-1}_{mβ} = U^\ast_{βm} & 1 & δ_{mn}\eu^{-\iu \tilde{ω}^0_mt} & O_{mγ}\eu^{-\iu \tilde{ω}^0_nt} \\
|
||
h_{n} & U^\ast_{βn}\eu^{\iu \tilde{ω}^0_nt} & δ_{nm}\eu^{\iu \tilde{ω}^0_mt} & 1 & O_{nγ} \\
|
||
d_{γ} & ∑_{m} U^\ast_{βm} O^{-1}_{γm} \eu^{\iu \tilde{ω}^0_mt} & O^{-1}_{γn} \eu^{\iu \tilde{ω}^0_mt} & O^{-1}_{γm} & 1. \\
|
||
\end{array}
|
||
\end{equation}
|
||
The quantity \(x_{i}\) in each row is obtained from the quantity
|
||
\(y_{j}\) heading each row through the transformation \(A_{ij}(t)\) in
|
||
each cell by \(x_{i} = ∑_{j}A_{ij}(t)y_{j}\).
|
||
|
||
\subsection{Coupling to the Transmission Line}
|
||
\label{sec:coupl-transm-line}
|
||
|
||
The transmission line is considered to only
|
||
have one polarization direction and one dimension of
|
||
propagation, so that the vector potential is effectively scalar and we
|
||
have
|
||
\begin{equation}
|
||
\label{eq:2}
|
||
E_{B}(x, t) = \iu\sqrt{\frac{\hbar}{2ε_{0}n_{B}^{2}
|
||
(2π)^{3}L_{B,\perp}^{2}}} ∫{\sqrt{ω^{B}_{k}}} \pqty{b_{k}(t)
|
||
\eu^{\iu k x } - b_{k}^†(t) \eu^{-\iu k x}}\dd{k},
|
||
\end{equation}
|
||
with \(\comm{b_{k}}{b_{q}^†}=δ(k-q)\), \(ω^{B}_{k} = c \abs{k}/n_{B}\)
|
||
with \(n_{B}\) being the refractive index of the fibre and
|
||
\(L_{B,\perp}\) being the perpendicular length scale as discussed
|
||
above. Note that the \(b_{k}\) here have dimensions of \(\sqrt{[L]}\)
|
||
as opposed to \(\sqrt{[t]}\), as is the usual convention in
|
||
input-output theory. If a stochastic theory is desired, the latter
|
||
convention is preferrable and can be obtained through substituting
|
||
\(k\to \pm ω/c n_{B}\) and rescaling
|
||
\(b_{k}\to b_{k}/ \sqrt{c n_{B}^{-1}}\).
|
||
|
||
An interaction between the transmission line and the
|
||
system \(A\) roughly inspired by coupled mode theory is
|
||
\begin{equation}
|
||
\label{eq:3}
|
||
H_{I} = g_{0} ∫ E_{A,+}(x,t)E_{B,-}(x,t) f(x) \dd{x} + \hc,
|
||
\end{equation}
|
||
where the subscripts \(\pm\) denote positive or negative frequency
|
||
portions of the fields and \(f(x)\) is a dimensionless weighting
|
||
function with compact support \([-Δx/2, Δx/2]\) whose maximum is
|
||
unity. Coupling only the positive/negative parts simplifies the
|
||
calculations and is consistent with the later application of the
|
||
rotating wave approximation. A possible phase shift between the fields
|
||
has been absorbed into the definition of the creation and annihilation
|
||
operators.
|
||
|
||
Expanding the fields in \cref{eq:3} we obtain
|
||
\begin{equation}
|
||
\label{eq:6}
|
||
H_{I} = {g_{0}} \frac{\hbar Δx}{2 ε_{0}n_{A}n_{B} (2π)^{3}
|
||
L_{A,\perp}L_{B,\perp}\sqrt{L_{A}}} ∑_{β}∫
|
||
\sqrt{ω^{B}_{k}ω_{k_{β}}}\,\tilde{f}(k-k_{β})\, b^†_{k}
|
||
a_{β} \dd{k} + \hc
|
||
\end{equation}
|
||
The Fourier transform of the weighting function
|
||
\begin{equation}
|
||
\label{eq:7}
|
||
\tilde{f}(k) = \frac{1}{Δx} ∫f(x)\eu^{-\iu k x} \dd{x}
|
||
\end{equation}
|
||
controls how ``far'' the interaction reaches in \(k\)-space. In the
|
||
extreme case \(Δx\to 0\) every \(b_{k}\) couples to every \(a_{β}\),
|
||
whereas for \(Δx\to ∞\) only modes with matching wave-numbers
|
||
couple. As the \(b_{k}\) will contain both the coherent drive with a
|
||
laser and the output field amplitudes it is desirable to have this
|
||
coupling to be as local in \(k\)-space as possible for targeted
|
||
control and precise readout. In the limit of weak coupling between
|
||
transmission line and system, which we will assume in a short while,
|
||
the rotating wave approximation will ensure that our result won't
|
||
depend significantly on the choice of \(f\).
|
||
|
||
The coupling constant \(g_{0}\) in \cref{eq:6} has the dimensions of
|
||
\([L]^{2}\times [ε_{0}]\). We define a new coupling constant that has
|
||
units of energy as
|
||
\begin{equation}
|
||
\label{eq:8}
|
||
g_{0} = g\frac{n_{A}n_{B}ε_{0} L_{A,\perp}L_{B,\perp} 2(2π)^{3}}{\hbar ω_{0}},
|
||
\end{equation}
|
||
where \(ω_{0}\) is a typical frequency\footnote{For example, the
|
||
frequency of the drive laser.}.
|
||
Using this, \cref{eq:6} becomes
|
||
\begin{equation}
|
||
\label{eq:9}
|
||
\begin{aligned}
|
||
H_{I} &= \frac{gΔx}{
|
||
\sqrt{L_{A}}} ∑_{β}∫
|
||
G_{β}(k) b^†_{k}
|
||
a_{β} \dd{k} + \hc
|
||
&G_{β}(k) &= \frac{\sqrt{ω^{B}_{k}ω_{k_β}}}{ω_{0}} \tilde{f}(k-k_{β}).
|
||
\end{aligned}
|
||
\end{equation}
|
||
We note that for
|
||
a \(ω_{k_{β}}= ω_{0} + δω\) with \(δω \ll ω_{0}\) the coupling factor
|
||
\(G_{β}(k)\) only depends on the difference \(k-k_{β}\). By defining
|
||
\begin{equation}
|
||
\label{eq:11}
|
||
\mathcal{O(k)} = \frac{Δx}{\sqrt{L_{A}}} ∑_{β} G_β(k)a_{β} =
|
||
\frac{Δx}{\sqrt{L_{A}}} ∑_{β,m} G_β(k)U_{β,m}c_{m}
|
||
\end{equation}
|
||
the interaction takes on the more familiar form
|
||
\begin{equation}
|
||
\label{eq:14}
|
||
H_{I} = {g} ∫
|
||
b^†_{k} \mathcal{O}(k)
|
||
\dd{k} + \hc
|
||
\end{equation}
|
||
|
||
Changing variables from \(k\) to\footnote{This is a bit
|
||
unconventional.} \(ω^{B}_{k}=k c / n_{B}\) in
|
||
\cref{eq:9} we obtain
|
||
\begin{equation}
|
||
\label{eq:17}
|
||
H_{I} = \frac{gΔx}{\sqrt{L_{A}}} ∑_{β}∫_{-∞}^{∞}
|
||
G'_{β}(ω)f^†_{ω}
|
||
{a_{β}} \dd{ω} + \hc,
|
||
\end{equation}
|
||
where \(f_{ω}=\sqrt{\frac{n_{B}}{c}}b_{\frac{ω n_{B}}{c}}\) with
|
||
\(\comm{f_{ω}}{f_{ω'}^†}=δ(ω-ω')\) and
|
||
\(G'_{β}(ω)=G_{β}\pqty{\frac{ω n_{B}}{c}}\).
|
||
|
||
|
||
\subsection{Rotating Wave and First Markov Approximation}
|
||
\label{sec:rotating-wave-first}
|
||
Following the route taken in \cite{Jacobs}, the next step would be to
|
||
transition into a rotating frame so that
|
||
\(\tilde{H}_{A}=\tilde{H}_{B}=0\) and apply the rotating wave
|
||
approximation. Here, the rotating terms that would occur have the
|
||
frequencies of the form \(ω^{0}_m + ω_{γ}\) which are not guaranteed
|
||
to be spaced sufficiently far apart for the RWA to
|
||
apply\footnote{Consider, for example the SSH model where the
|
||
\(k\)-space density can be arbitrarily high depending on the length
|
||
of the chain.}. We therefore work in the frame of the
|
||
\(h_{m}\) and \(\tilde{f}_{ω} = f_{ω}\eu^{\iu \abs{ω}t}\) to
|
||
obtain
|
||
\begin{equation}
|
||
\label{eq:10}
|
||
\tilde{H}_{I}= \frac{gΔx}{
|
||
\sqrt{L_{A}}} ∑_{β,m}∫
|
||
G'_{β}(ω) \eu^{-\iu
|
||
(\tilde{ω}^{0}_{m}-\abs{ω}) t}
|
||
U_{β,m} \tilde{f}_k^†h_{m} \dd{ω} + \hc.
|
||
\end{equation}
|
||
|
||
|
||
\begin{figure}[H]
|
||
\centering
|
||
{\fontsize{8pt}{1em}
|
||
\input{graphics/rwa_illustr.pdf_tex}}
|
||
\caption{\label{fig:rwa_illustr} In the rotating wave approximation
|
||
the bare frequencies of the resonator only couple to the
|
||
transmission line in frequency sub-intervals
|
||
\([ω_{m}-λ_{m}, ω_{m}+λ_{m}]\). A second effect that comes into
|
||
play is the geometrically induced coupling amplitude
|
||
\(\tilde{G'}_{m}(ω)\), which is visualized around \(ω_{m}\) under
|
||
the assumption that all \(G'_{β}\) that enter into
|
||
\(∑_{β}U_{βm}G'_{β}\) share a similar profile which is a valid
|
||
assumption in the applications considered \cref{sec:appl-non-mark}.}
|
||
\end{figure}
|
||
For \(g \ll \tilde{ω}_{m}^{0}\) each \(h_{m}\) in \cref{eq:10} only
|
||
interacts with non-overlapping sub-intervals
|
||
\([\tilde{ω}^{0}_{m}-λ_{m}, \tilde{ω}^{0}_{m}+λ_{m}]\) of the transmission frequency axis
|
||
(rotating wave approximation) with \(g\ll λ_{m} \ll \tilde{ω}_{m}^{0}\). This
|
||
situation is illustrated in \cref{fig:rwa_illustr}. Also, the coupling
|
||
amplitude \(G_{β}(ω)\) is local in frequency space and can assist the
|
||
RWA depending on the choice of parameters and how close the
|
||
\(\tilde{ω}^{0}_{m}\) are to the \(ω_{k_{β}}\), i.e. how local in
|
||
frequency space \(U_{βm}\) is. We obtain
|
||
\begin{equation}
|
||
\label{eq:16}
|
||
\tilde{H}_{I}\approx \frac{gΔx}{
|
||
\sqrt{L_{A}}} ∑_{β,m}∫_{\tilde{ω}^{0}_{m}-λ_{m}}^{\tilde{ω}^{0}_{m}+λ_{m}}
|
||
\eu^{-\iu
|
||
(\tilde{ω}^{0}_{m}-\abs{ω}) t}
|
||
U_{β,m} \pqty{G'_{β}(ω) \tilde{f}_{ω}^† + G'_{β}(-ω)
|
||
\tilde{f}_{-ω}^†}h_{m} \dd{ω} + \hc
|
||
\end{equation}
|
||
For any finite \(Δx\) and
|
||
\(\tilde{ω}_{0}^{m},ω_{k_{β}}\gg \frac{2πc}{Δx n_{A}}\) we can assume
|
||
\begin{equation}
|
||
\label{eq:44}
|
||
G'_{β}\pqty{-\sgn(β) ω}\approx 0
|
||
\end{equation}
|
||
in \cref{eq:16}.
|
||
|
||
As each \({h}_{m}\) is now interacting with non-overlapping
|
||
transmission-line field modes, we can introduce a separate field for
|
||
each \({h}_{m}\) that commutes with all other fields and extend
|
||
the integration bounds to infinity again\footnote{This is called the
|
||
``First Markov Approximation'' in \refcite{Gardiner1985}.}.
|
||
Care has to be taken to maintain consistency with \cref{eq:44},
|
||
\begin{equation}
|
||
\label{eq:16}
|
||
\tilde{H}_{I}= \frac{gΔx}{
|
||
\sqrt{L_{A}}} ∑_{β,m}∫_{0}^{∞}
|
||
\eu^{-\iu
|
||
(\tilde{ω}^{0}_{m}-\abs{ω}) t}
|
||
U_{β,m} G'_{β}(\sgn({β})ω) \tilde{f}^{m,†}_{\sgn({β})ω}{h}_{m} \dd{ω} + \hc
|
||
\end{equation}
|
||
which becomes\footnote{A lot of discussion for a simple result :).}
|
||
\begin{equation}
|
||
\label{eq:18}
|
||
H_{I}= ∑_{m}∫_{-∞}^{∞}
|
||
\tilde{G}_{m}(k) {b}^{m,†}_{k}{c}_{m} \dd{k}
|
||
\end{equation}
|
||
upon transitioning out of the rotating frame with
|
||
\begin{equation}
|
||
\label{eq:15}
|
||
\tilde{G}_{m}(k) =
|
||
\frac{gΔx}{
|
||
\sqrt{L_{A}}} ∑_{β\gtrless 0}U_{β,m} G_{β}(k)δ_{\sgn(β),\sgn(k)}
|
||
\end{equation}
|
||
|
||
The equation of motion
|
||
for the transmission line modes become
|
||
\begin{gather}
|
||
\iu\dot{b}^{m}_{k} = ω^{B}_{k} b_{k}^{m,†} +
|
||
\tilde{G}_{m}(k) \eu^{-\iu \tilde{ω}^{0}_{m}t}h_{m}\\
|
||
\label{eq:19}
|
||
\implies b^{m}_{k}(t) = b^{m}_{k}(0) \eu^{-\iu ω_{k}^{B}t} -\iu
|
||
\tilde{G}_{m}(k) ∫_{0}^{t}\eu^{-\iu
|
||
ω_{k}^{B}(t-s)} \eu^{-\iu ω_{k}^{B}s}h_{m}(s)\dd{s}.
|
||
\end{gather}
|
||
The equation of motion for \(h_{m}\) is
|
||
\begin{equation}
|
||
\label{eq:21}
|
||
\iu\dot{h}_{m} = ∑_{n}\bqty{V^{0}_{mn} + ε_{m}δ_{nm}}{h}_n +
|
||
\underbrace{\eu^{\iu \tilde{ω}_{m}^{0}t}∫_{-∞}^{∞}\tilde{G}^\ast_{m}(k)
|
||
b_{k}^{m}(t)\dd{k}}_{\equiv I}.
|
||
\end{equation}
|
||
Further inspection of the rightmost term in \cref{eq:21} yields
|
||
\begin{equation}
|
||
\label{eq:22}
|
||
\begin{aligned}
|
||
I &= \eu^{\iu \tilde{ω}_{m}^{0}t} ∫_{-∞}^{∞}\tilde{G}^\ast_{m}(k)
|
||
b_{k}^{m}(t)\dd{k} \\
|
||
&= ∫_{-∞}^{∞}\tilde{G}^\ast_{m}(k)
|
||
b_{k}^{m}(0)\eu^{-\iu ({ω}^{B}_{k} - \tilde{ω}^{0}_{m})t}\dd{k} -\iu ∫_{0}^{t}∫_{-∞}^{∞}\abs{\tilde{G}_{m}(k)}^{2}
|
||
{h}_{m}(s)\eu^{-\iu {ω}^{B}_{k}(t-s)} \eu^{\iu
|
||
\tilde{ω}^{0}_{m}(t-s)}\dd{k}\dd{s}\\
|
||
&=II + III.
|
||
\end{aligned}
|
||
\end{equation}
|
||
As the RWA limits the interaction of each \(h_{m}\) to an narrow
|
||
frequency/momentum band, we assume that \(\tilde{G}_{m}\) is
|
||
approximately constant close to the resonance frequencies \([\tilde{ω}^{0}_{m}-λ_{m}, \tilde{ω}^{0}_{m}+λ_{m}]\)
|
||
\begin{equation}
|
||
\label{eq:23}
|
||
\begin{aligned}
|
||
\tilde{G}_{m}(k) &\approx
|
||
δ_{m}\tilde{G}_{m}\pqty{\sgn(k) ω_{m}^{0}\frac{n_{B}}{c n_{A}}} =
|
||
δ_{m}\frac{gΔx}{\sqrt{L_{A}}}∑_{β}U_{βm}G_{β}\pqty{\sgn(k) ω_{m}^{0}\frac{n_{B}}{c
|
||
n_{A}}} δ_{\sgn(β),\sgn(k)} \\
|
||
&\equiv ∑_{β}g^{0}_{β}\sqrt{ω^{B}_{k}} U_{βm}δ_{\sgn(β),\sgn(k)} \equiv g_{m, \sgn(k)}\sqrt{ω^{B}_{k}}
|
||
\end{aligned}
|
||
\end{equation}
|
||
in the interval (see \cref{eq:16}) where \(δ_{m}\) is a possible
|
||
scaling factor to better approximate \(\tilde{G}_{m}(k)\) as a
|
||
constant in \cref{eq:16}.
|
||
|
||
Using this in \(I\), we obtain
|
||
\begin{equation}
|
||
\label{eq:24}
|
||
\begin{aligned}
|
||
II &= {\eu^{\iu ω_{m}^{0}t}} \bqty{g_{m,+}^\ast ∫_{0}^{∞}\sqrt{ω^{B}_{k}}b^{m}_{k}(0)\eu^{-\iu
|
||
ω^{B}_{k}t}\dd{k} + g_{m,-}^\ast∫_{-∞}^{0}\sqrt{ω^{B}_{k}}b^{m}_{k}(0)\eu^{-\iu
|
||
ω^{B}_{k}t}\dd{k}}\\
|
||
&\equiv {\eu^{\iu ω_{m}^{0}t}}\pqty{
|
||
g_{m,+}^\ast b_{\inputf,+}^{m}(t) + g_{m,-}^\ast b_{\inputf,-}^{m}(t)},
|
||
\end{aligned}
|
||
\end{equation}
|
||
where \(b_{\inputf,+(-)}^{m}(t)\) is identified as the
|
||
right(left)-moving input field and is proportional to the annihilation
|
||
part of the electric field. The second part of \cref{eq:22} becomes
|
||
\begin{equation}
|
||
\label{eq:25}
|
||
III= -\iu ∫_{0}^{t}\eu^{\iu \tilde{ω}^{0}_{m}(t-s)}{h}_{m}(s)
|
||
\bqty{ \abs{g_{m,+}}^{2} ∫_{0}^{∞}ω^{B}_{k}\eu^{-\iu ω^{B}_{k}(t-s)} \dd{k} + \abs{g_{m,-}}^{2} ∫_{-∞}^{0}ω^{B}_{k}\eu^{-\iu ω^{B}_{k}(t-s)} \dd{k}}\dd{s}.
|
||
\end{equation}
|
||
Now we use the identity
|
||
\begin{equation}
|
||
\label{eq:26}
|
||
∫_{0}^{∞}ω^{B}_{k}\eu^{-\iu ω^{B}_{k}(t-s)} \dd{k} = -\iu ∂_{s}\frac{n_{B}}{c}
|
||
\bqty{\mathcal{P}\frac{-i}{t-s} + π δ(t-s)},
|
||
\end{equation}
|
||
but neglect the principal value, as it leads only to rapidly
|
||
oscillating terms that are inconsistent with the RWA, to obtain
|
||
\begin{equation}
|
||
\label{eq:27}
|
||
\begin{aligned}
|
||
III&= -2\iu η_{m}∫_{0}^{t}\eu^{\iu
|
||
\tilde{ω}^{0}_m(t-s)}\bqty{\pqty{\tilde{ω}^{0}_{m}+\iu ∂_{s}}\tilde{c}_{m}(s)}
|
||
δ(t-s)\dd{s}\\
|
||
&= -\iu η_{m} h_{m}(t) + \frac{η_{m}}{\tilde{ω}^{0}_{m}}
|
||
\dot{h}_{m}(t) \overset{η_{m}\ll \tilde{ω}_{m}^{0},\, V\ll H_{0}}{\approx}-\iu η_{m} h_{m}(t),
|
||
\end{aligned}
|
||
\end{equation}
|
||
where the factor \(1/2\) in the last equality stems from the fact that
|
||
we only use half of the delta function and
|
||
\begin{equation}
|
||
\label{eq:45}
|
||
η_{m}\equiv π\frac{\tilde{ω}^{0}_{m} n_{B}}{c}\bqty{\abs{g_{m,-}}^{2}+\abs{g_{m,+}}^{2}}.
|
||
\end{equation}
|
||
We have also neglected the correction to the eigen-energy
|
||
\(η_{m}/\tilde{ω}^{0}_{m}\) for the same reason we neglected the
|
||
principal value.
|
||
|
||
Note that \cref{eq:45} is an incoherent sum of the couplings to the
|
||
right moving and left moving fields in the transmission line.
|
||
Altogether we arrive at
|
||
\begin{equation}
|
||
\label{eq:28}
|
||
\iu\dot{h}_{m} = {∑_{n}\underbrace{\bqty{V^{0}_{mn} +
|
||
\pqty{ε_{m}-\iu η_{m}}δ_{nm}}}_{\equiv H^{T}_{mn}}{h}_n +
|
||
{\eu^{\iu \tilde{ω}_{m}^{0}t}}
|
||
∑_{σ=\pm}g_{m,σ}^\ast b_{\inputf,σ}^{m}(t)} .
|
||
\end{equation}
|
||
The usual situation is that \(b^{m}_{\inputf, -} = 0\) and we can
|
||
restrict ourselves to the coupling to the right-moving input field.
|
||
|
||
\subsection{Input-Output Relation and further Simplifications}
|
||
\label{sec:input-outp-relat}
|
||
Integrating \cref{eq:19} over all \(k\) yields
|
||
\begin{equation}
|
||
\label{eq:29}
|
||
\begin{aligned}
|
||
{b_{\outputf}^{m}(x,t)} &\equiv
|
||
∫ \sqrt{ω^{B}_{k}} b_{k}^{m}(t) \eu^{\iu k
|
||
t}\dd{k}\\
|
||
&= b_{\inputf}^{m}(x, t) -\iu
|
||
g_{m,\sgn(x)}\frac{\tilde{ω}^{0}_{m}π n_{B}}{c}
|
||
{h}_{m}(τ(x,t))\eu^{-i \tilde{ω}^{0}_{m}τ(x,t)}Θ(τ(x,t)),
|
||
\end{aligned}
|
||
\end{equation}
|
||
which is the input-output relation with the retarded time
|
||
\begin{equation}
|
||
\label{eq:20}
|
||
τ(x,t)=t - \frac{\abs{x}n_{B}}{c}.
|
||
\end{equation}
|
||
The coupling constant accounts for the direction of propagation and
|
||
the time argument is properly retarded. We defined
|
||
\begin{equation}
|
||
\label{eq:48}
|
||
b_{\inputf}^{m}(x,t) = ∫\sqrt{ω^{B}_{k}} b_{k}^{m}(0)\eu^{\iu \pqty{kx -
|
||
ω_{k}^{B}t}}\dd{k}
|
||
\end{equation}
|
||
used that
|
||
\begin{equation}
|
||
\label{eq:42}
|
||
∫_{0}^{∞}ω^{B}_{k}\eu^{-\iu ω^{B}_{k}(t-s)}\eu^{\pm\iu k x} \dd{k} =
|
||
-\iu ∂_{s}\frac{n_{B}}{c}
|
||
\bqty{\mathcal{P}\frac{-i}{t-s \pm \frac{x n_{B}}{c}} + π δ\pqty{t-s\mp
|
||
\frac{x n_{B}}{c}}},
|
||
\end{equation}
|
||
along with similar arguments to the above.
|
||
|
||
The case of \(x=0\) is recovered by defining
|
||
\begin{equation}
|
||
\label{eq:47}
|
||
\lim_{x\to0} g_{m,\sgn(x)=0} = \frac{1}{2} \pqty{g_{m,+} + g_{m,-}},
|
||
\end{equation}
|
||
which amounts to taking half of each delta function in
|
||
\cref{eq:42}. It shall be noted, that it is physical to assume
|
||
\(x>0\), as we necessarily measure outside the fibre-coupler between
|
||
transmission line and resonator. By neglecting the \(k\)-dependence of
|
||
the coupling in \cref{eq:23} through invocation of the RWA we have
|
||
effectively ignored length \(Δx\), but to maintain consistency with
|
||
\cref{eq:44} we should assume it to be finite.
|
||
We can also neglect the retardation if \(x / v_{g}\) is
|
||
much smaller than a typical timescale we're interested in.
|
||
|
||
|
||
To integrate \cref{eq:28}, we
|
||
first diagonalize \(V^{0}_{mn} + δ_{mn}\pqty{ε_{m}-i η_{m}}\)
|
||
\begin{equation}
|
||
\label{eq:65}
|
||
V^{0}_{mn} + δ_{mn}\pqty{ε_{m}-i η_{m}} \to ∑_{γγ'}
|
||
δ_{γγ'}(ω_{γ}-\iu λ_{γ})
|
||
\end{equation}
|
||
using the transformation \(O_{nγ}\) (see \cref{sec:rotating-frames})
|
||
and find
|
||
\begin{equation}
|
||
\label{eq:32}
|
||
\dot{d}_{γ} = ∑_{m}O^{-1}_{γm}\dot{h}_{m} =
|
||
-\iu\bqty{\pqty{ω_{γ} - \iu λ_{γ}}d_{γ} +
|
||
∑_{σ=\pm}∑_{m}\pqty{O^{-1}}_{γm}{g_{m,σ}^\ast } \eu^{\iu \tilde{ω}_{m}^{0}t}
|
||
b_{\inputf,σ}^{m}(t)}.
|
||
\end{equation}
|
||
|
||
We now introduce some additional simplifications beginning with
|
||
equating all input fields \(b_{\inputf}^{m}\). This is allowed, as we
|
||
will transition to the classical picture later, where the commutation
|
||
relations do not matter. We also assume that we're working in a region
|
||
in \(m\) space, where the \(\sqrt{ω_{0}}g_{β}^{0}\approx \sqrt{κ}\) and
|
||
\(\tilde{ω}^{0}_{m}\approx{ω_{0}}\), where \(ω_{0}\) is a typical
|
||
frequency in the input field, can be assumed to be approximately
|
||
constant. With these considerations in mind we can simplify
|
||
\cref{eq:45,eq:32} to
|
||
\begin{equation}
|
||
\label{eq:64}
|
||
η_{m}=\abs{κ}\frac{πn_{B}}{c}∑_{σ=\pm,β,β'}U_{βm}U^\ast_{β'm}δ_{\sgn(β),σ} δ_{\sgn(β'),σ}
|
||
\end{equation}
|
||
and
|
||
\begin{gather}
|
||
\label{eq:34}
|
||
\dot{d}_{γ} =
|
||
-\iu\bqty{\pqty{ω_{γ}-\iu λ_{γ}}d_{γ} + \sqrt{κ^\ast} ∑_{σ=\pm}
|
||
U^{σ}_{γ}(t) {b_{\inputf}(t)}}\\
|
||
U^{σ}_{γ}(t) = ∑_{m,β} δ_{\sgn({β}),σ}U^\ast_{βm}\pqty{O^{-1}}_{γm}\eu^{\iu \tilde{ω}_{m}^{0}t}.
|
||
\end{gather}
|
||
|
||
These simplifications still capture the essence of the physics, as
|
||
demonstrated in the current long-range SSH experiment.
|
||
|
||
We can now proceed to integrate \cref{eq:34} to obtain
|
||
\begin{equation}
|
||
\label{eq:36}
|
||
d_{γ}(t)= d_{γ}(0) \eu^{-\pqty{\iu ω_{γ} + {λ}_{γ}}t} -
|
||
\frac{i}{\sqrt{κ}} Σ_{σ=\pm} ∫_{0}^{t}χ_{γ}(t-s) U^{σ}_{γ}(s)
|
||
{b_{\inputf,σ}(t)}\dd{s}
|
||
\end{equation}
|
||
with
|
||
\begin{equation}
|
||
\label{eq:37}
|
||
χ_{γ}(t) = \abs{κ} \eu^{-\pqty{\iu ω_{γ} + λ_{γ}}t}.
|
||
\end{equation}
|
||
|
||
When constructing the total output field, we have to remember how the
|
||
separate fields \(b_{\outputf,m}\) came about. We assumed that each
|
||
\(c_{m}\) only interacted with a finite range of modes (see
|
||
\cref{eq:16}) in the transmission line and then just extended the
|
||
resulting sub-fields back to full independent fields for
|
||
simplicity. Now, we have to perform the reverse process, which amounts
|
||
to summing together all system (resonator) contributions in
|
||
\cref{eq:34} as these only excite the sub-fields and we can safely
|
||
glue them back together. To be consistent, we have to sum together the
|
||
finite ranges of the input fields which amounts to having \emph{one}
|
||
whole copy of the input field.
|
||
This leads us to
|
||
\begin{equation}
|
||
\label{eq:38}
|
||
{b_{\outputf}(x,t)} \equiv
|
||
b_{\inputf}(x, t) -i θ(τ(x,t)) \frac{\sqrt{κ} πn_{B}}{c}
|
||
∑_{γ}\bqty{U^{\sgn(x)}_{γ}\pqty{τ(x,t)}}^\ast d_{γ}(τ(x,t))
|
||
\end{equation}
|
||
|
||
Transitioning to expectation values and using \(\ev{d_{γ}(0)}=0\) we
|
||
find
|
||
\begin{equation}
|
||
\label{eq:39}
|
||
\ev{{b_{\outputf}(x,t)}} =
|
||
\ev{b_{\inputf}(x,t)} - ∑_{σ=\pm}∫_{0}^{τ(x,t)}χ_{\sgn(x),σ}(τ(x,t),s) \ev{b_{\inputf,σ}(s)} \dd{s}
|
||
\end{equation}
|
||
with the time non-local susceptibility for the left and right moving
|
||
input fields
|
||
\begin{equation}
|
||
\label{eq:40}
|
||
χ_{δ,σ}(t,s) = \frac{π n_{B}}{c}Θ(t) ∑_{γ}\pqty{U^{δ}_{γ}(t)}^\astχ_{γ}(t-s)U^{σ}_{γ}(s).
|
||
\end{equation}
|
||
|
||
For an input field with no left-moving components and a measurement
|
||
position \(x>0\) we have
|
||
\begin{equation}
|
||
\label{eq:31}
|
||
\ev{{b_{\outputf}(x>0,t)}} =
|
||
\ev{b_{\inputf}(x,t)} -∫_{0}^{τ(x,t)}χ_{++}(τ(x,t),s) \ev{b_{\inputf}(s)} \dd{s}.
|
||
\end{equation}
|
||
with \(b_{\inputf}(s) = b_{\inputf,+}(s) + b_{\inputf,-}(s) =
|
||
b_{\inputf,+}(s)\).
|
||
|
||
\subsection{Langevin-Equations for Lossy Oscillators}
|
||
\label{sec:lang-equat-lossy}
|
||
|
||
In the above we have assumed that \(H_{0}\) is hermitian. This,
|
||
however, ceases to be the case when we assume some a-priori
|
||
phenomenological decay in the bare components of the system and we
|
||
cannot write \(H_{0}=H^{H}_{0} - \iu η \id\) with \(H^{H}_{0}\)
|
||
hermitian. To retain consistency, the decay rates have to be
|
||
introduced on the level of the equations of motion of the mode
|
||
operators \(a_{i,α}\) after deriving them from the hermitian
|
||
Hamiltonian. The equations of motion can then still be decoupled by
|
||
diagonalizing the non-hermitian that includes the phenomenological
|
||
decay.
|
||
|
||
We find\footnote{Assuming that the non-hermiticity is small enough for
|
||
the matrix to remain diagonalizable.}
|
||
\begin{equation}
|
||
\label{eq:53}
|
||
∑_{iα;jβ}\pqty{T^{-1}}_{m;i,α} \pqty{H_{0}}_{i,α;j,β}T_{j,β;n} =
|
||
\pqty{ω_{m}^{0}-\iu η_{m}^{0}}δ_{nm},
|
||
\end{equation}
|
||
where \(T\) is the matrix whose rows are the eigenvectors of
|
||
\(H_{0}\). Note that \(T\) is not unitary anymore. For notational
|
||
convenience we will write \(T^{-1}_{m;i,α}\) instead of
|
||
\(\pqty{T^{-1}}_{m;i,α}\) and use explicit fractions if we want to
|
||
express the multiplicative inverse. The mode operators transform as
|
||
\begin{equation}
|
||
\label{eq:60}
|
||
c_{m} = ∑_{i,α} T^{-1}_{m;i,α}a_{i,α},
|
||
\end{equation}
|
||
which are \emph{not} to be identified with bosons anymore, as the
|
||
non-unitarity of \(T\) breaks the bosonic commutation
|
||
relations. Again, we express the modes that are in contact with the
|
||
transmission line as \(a_{α}=a_{i_{0},α}\) and find
|
||
\begin{equation}
|
||
\label{eq:69}
|
||
α_{α} = ∑_{α} T_{i_{0},α;m}c_{m} \equiv ∑_{α}U_{αm} c_{m}.
|
||
\end{equation}
|
||
For convenience we define
|
||
\begin{equation}
|
||
\label{eq:70}
|
||
U^{-1}_{mα}\equiv T^{-1}_{m;i_{0}α}.
|
||
\end{equation}
|
||
|
||
The modulation term \(V\) transforms as
|
||
\begin{equation}
|
||
\label{eq:74}
|
||
V_{mn}=∑_{iα;jβ}\pqty{T^{-1}}_{m;i,α} V_{i,α;j,β}T_{j,β;n},
|
||
\end{equation}
|
||
and is no longer hermitian.
|
||
|
||
We start by writing down the equations of motion for the original
|
||
modes, assuming \(H_{0}\) to be hermitian, introduce the non-hermitian
|
||
terms and express everything in terms of the \(c_{m}\) using
|
||
\(T\). Subsequently, we change into a rotating frame
|
||
\begin{equation}
|
||
\label{eq:66}
|
||
h_{m} = c_{m}\eu^{\iu \tilde{ω}^{0}_{m}t},
|
||
\end{equation}
|
||
rotating away only the unitary evolution. Applying the rotating wave
|
||
and first Markov approximations works out precisely as in
|
||
\cref{sec:rotating-wave-first}.
|
||
|
||
|
||
To account for non-unitarity we have to make the following
|
||
replacements along the way
|
||
\begin{align}
|
||
\label{eq:68}
|
||
\tilde{G}_{m}(k) &\rightarrow \tilde{G}_{m}(k)= \frac{gΔx}{\sqrt{L_{A}}} ∑_{β} U_{βm}
|
||
G_{β}(k) δ_{\sgn(β),\sgn(k)}\\
|
||
\tilde{G}^\ast_{m}(k) &\rightarrow\tilde{G}^{-1}_{m}(k) = \frac{g^\astΔx}{\sqrt{L_{A}}} ∑_{β} U^{-1}_{mβ}
|
||
G^\ast_{β}(k) δ_{\sgn(β),\sgn(k)}\\
|
||
g_{m,σ}&\rightarrow g_{m,σ}=∑_{β}g^{0}_{β} U_{βm}δ_{\sgn(β),σ}\\
|
||
g^\ast_{m,σ}&\rightarrow \bar{g}_{m,σ}=∑_{β}\pqty{g^{0}_{β}}^\ast U^{-1}_{mβ}δ_{\sgn(β),σ}\\
|
||
\end{align}
|
||
which gives us
|
||
\begin{align}
|
||
\label{eq:72}
|
||
η_{m}=\frac{\tilde{ω}_{m}^{0}π n_{B}}{c} ∑_{σ} g_{mσ}\bar{g}_{mσ},
|
||
\end{align}
|
||
which might have an imaginary part.
|
||
|
||
This leaves us with
|
||
\begin{equation}
|
||
\label{eq:73}
|
||
\iu\dot{h}_{m} = {∑_{n}\Bqty{V^{0}_{mn} + \bqty{ε_{m}-\iu \pqty{η_{m} +
|
||
η_{m}^{0}}}δ_{nm}}{h}_n +
|
||
{\eu^{\iu \tilde{ω}_{m}^{0}t}}
|
||
∑_{σ=\pm}\bar{g}_{m,σ} b_{\inputf,σ}^{m}(t)},
|
||
\end{equation}
|
||
where the \(η_{m}\) might shift the energies \(ε_{m}\)
|
||
slightly.
|
||
|
||
Diagonalizing
|
||
\begin{equation}
|
||
\label{eq:77}
|
||
∑_{mn}O^{-1}_{γ'm}\bqty{∑_{n}\Bqty{V^{0}_{mn}+ \bqty{ε_{m}-\iu
|
||
\pqty{η_{m}^{0}+η_{m}}}δ_{nm}}h_{m}}O_{nγ} = \pqty{ω_{γ}-\iu λ_{γ}}δ_{γ,γ'}
|
||
\end{equation}
|
||
and defining
|
||
\begin{equation}
|
||
\label{eq:78}
|
||
d_{γ} = ∑_{n}O^{-1}_{γn}h_{n} \implies h_{n}=∑_{γ}O_{nγ}d_{γ}
|
||
\end{equation}
|
||
will give us the equivalent of \cref{eq:32}
|
||
\begin{equation}
|
||
\label{eq:80}
|
||
\dot{d}_{γ}=-\iu\pqty{ω_{γ}+\sqrt{κ^\ast}∑_{σ=\pm}U^{σ}_{γ}{b_{\inputf,σ}}}d_{γ}
|
||
- λ_{γ}d_{γ}
|
||
\end{equation}
|
||
where we have set \(\sqrt{ω_{0}}g_{β}^{0}=\sqrt{κ}\) and defined
|
||
\begin{equation}
|
||
\label{eq:82}
|
||
U^{σ}_{γ} =
|
||
∑_{mβ}\eu^{\iu\tilde{ω}^0_mt}O^{-1}_{γm}U^{-1}_{mβ}δ_{\sgn(β),σ}.
|
||
\end{equation}
|
||
This also simplifies \cref{eq:64} to
|
||
\begin{equation}
|
||
\label{eq:88}
|
||
η_{m}=\abs{κ}\frac{πn_{B}}{c}∑_{σ=\pm,β,β'}U_{βm}U^{-1}_{mβ}δ_{\sgn(β),σ} δ_{\sgn(β'),σ}.
|
||
\end{equation}
|
||
|
||
Further defining
|
||
\begin{align}
|
||
\label{eq:83}
|
||
\bar{U}^{σ}_{γ}&=∑_{mβ}\eu^{-\iu\tilde{ω}^0_mt}O_{mγ}U_{βm}δ_{\sgn(β),σ}\qq{and}&
|
||
χ_{γ}&=\abs{κ}\eu^{-\pqty{\iu ω_{γ}+λ_{γ}}t},
|
||
\end{align}
|
||
we obtain
|
||
\begin{equation}
|
||
\label{eq:86}
|
||
\ev{{b_{\outputf}(x,t)}} =
|
||
\ev{b_{\inputf}(x,t)} - ∑_{σ=\pm}∫_{0}^{τ(x,t)}χ_{\sgn(x),σ}(τ(x,t),s) \ev{b_{\inputf,σ}(s)} \dd{s}
|
||
\end{equation}
|
||
with the time non-local susceptibility for the left and right moving
|
||
input fields
|
||
\begin{equation}
|
||
\label{eq:87}
|
||
χ_{δ,σ}(t,s) = \frac{π n_{B}}{c}Θ(t) ∑_{γ}\bar{U}^{δ}_{γ}(t)χ_{γ}(t-s)U^{σ}_{γ}(s).
|
||
\end{equation}
|
||
|
||
These equations are essentially the same as \cref{eq:39,eq:40},
|
||
accounting for the non-unitary transformations and the apriori decay
|
||
rates when diagonalizing the equations of motion for the \(h_{m}\).
|
||
|
||
For completeness, we give the equivalent of \cref{eq:67} for the non-unitary case
|
||
\begin{equation}
|
||
\label{eq:35}
|
||
\begin{array}{@{}l| c c c c@{}}
|
||
& α_{β} & c_{m} & h_{n} & d_{γ}\\
|
||
\midrule
|
||
a_{β} & 1 & U_{βm}\equiv T_{i_{0},β;m} & U_{βm}\eu^{-\iu \tilde{ω}^0_mt} & ∑_{m}U_{βm}O_{mγ}\eu^{-\iu \tilde{ω}^0_mt} \\
|
||
c_{m} & U^{-1}_{mβ}\equiv T^{-1}_{m;i_{0},β} & 1 & δ_{mn}\eu^{-\iu \tilde{ω}^0_mt} & O_{mγ}\eu^{-\iu \tilde{ω}^0_nt} \\
|
||
h_{n} & U^{-1}_{nβ}\eu^{\iu \tilde{ω}^0_nt} & δ_{nm}\eu^{\iu \tilde{ω}^0_mt} & 1 & O_{nγ} \\
|
||
d_{γ} & ∑_{m} U^{-1}_{mβ} O^{-1}_{γm} \eu^{\iu \tilde{ω}^0_mt} & O^{-1}_{γn} \eu^{\iu \tilde{ω}^0_mt} & O^{-1}_{γm} & 1 \\
|
||
\end{array}.
|
||
\end{equation}
|
||
|
||
\section{Application to the Non-Markovian Quantum Walk}
|
||
\label{sec:appl-non-mark}
|
||
The experimental setup for implementing the non-Markovian quantum walk
|
||
discussed in~\cite{Ricottone2020,Kitagawa2010} is illustrated in
|
||
\cref{fig:schematic}. The abstract system introduced in
|
||
\cref{sec:micr-deriv} is replaced by a small \(S\) and a large \(B\)
|
||
fibre loop with lengths \(L_{B}\gg L_{S}\). The resonant modes of the
|
||
loops do have the free spectral ranges
|
||
\begin{equation}
|
||
\label{eq:41}
|
||
\begin{aligned}
|
||
Ω_{B} &= \frac{2πc}{n_{B}} & Ω_{S} &= \frac{2πc}{L_{S}},
|
||
\end{aligned}
|
||
\end{equation}
|
||
where \(n\) is the refractive index of the respective fibres.
|
||
|
||
Attaching the transmission line to the smaller loop has the advantage
|
||
that we only excite and detect what we will later identify as the
|
||
\(A\) level of the non-Markovian quantum walk in momentum space.
|
||
|
||
We assume that the loops share some common eigenfrequency
|
||
\begin{equation}
|
||
\label{eq:46}
|
||
ω_{s} = n_{0}^{S}Ω_{S} = n_{0}^{B}Ω_{B}\iff \frac{n_{0}^{B}}{L_{B}}
|
||
= \frac{n_{0}^{S}}{L_{S}}
|
||
\end{equation}
|
||
with \(n,m\gg 1\) and \(ω_{s}\) close to the frequency of the
|
||
laser. We choose to index the eigenfrequencies of the loops relative
|
||
to \(ω_{s}\) so that
|
||
\begin{equation}
|
||
\label{eq:49}
|
||
\begin{aligned}
|
||
ω_{n}^{X} &= ω_{s} + n Ω_{X} & k_{n}^{X} = \frac{2π}{L_{B}}
|
||
n_{0}^{X} + \frac{2π}{L_{B}} n = k_{0} + \frac{2π}{L_{B}} n,
|
||
\end{aligned}
|
||
\end{equation}
|
||
where \(X=S,B\).
|
||
|
||
\begin{figure}[H]
|
||
\centering
|
||
\includegraphics[width=.9\textwidth]{walker_setup}
|
||
\caption{\label{fig:schematic}Schematic of the experimental setup
|
||
and the different frames that the input-output theory deals
|
||
with. Two fibre loops \(B\) (Big) and \(S\) (small) are being
|
||
coupled with strength \(δ\). The golden parts correspond to the
|
||
bare modes (\(ω_{m}^{B},ω_{s}\)) being coupled to form the
|
||
eigenmodes with frequencies \(ω_{m}^{0}\) of the composite \(B+S\)
|
||
system. We then transition into a rotating frame (Blue) with
|
||
respect to \(H_{0}\) and the \(ω^{0}_{m}\) and consider a
|
||
modulation through the EOM to give us an effectively
|
||
time-independent target Hamiltonian \(H^{T}_{mn}\). Diagonalizing
|
||
this Hamiltonian gives us the energies \(ω_{λ}\) that we can
|
||
detect when measuring the transmission (Green). Damping terms have
|
||
been left out of the picture for simplicity.}
|
||
\end{figure}
|
||
These loops are coupled to each other with amplitude \(δ\eu^{\iu ϕ}\) which can
|
||
possibly contain a phase due to the choice of the coordinate
|
||
origin. Let us now assume that
|
||
\begin{equation}
|
||
\label{eq:50}
|
||
\frac{Ω_{S}}{Ω_{B}} = 2N+1
|
||
\end{equation}
|
||
with \(N\in \NN\). We denote by \(a\) the annihilation operator for the
|
||
mode with \(ω=ω_{s}\) in the small loop and suppress all other modes
|
||
in the small loop, as we will not populate them. The annihilation
|
||
operators \(f_{n}\) destroy modes with frequencies \(ω^{B}_{n}\) in
|
||
the big loop, where we limit \(n\) to the range \([-N, N]\) for the
|
||
same reasons as above.
|
||
This leads to the Hamiltonian
|
||
\begin{equation}
|
||
\label{eq:51}
|
||
H_{A}= ω_{s}a^†a + ∑_{n=-N}^{N} ω_{n}^{B}f_{n}^†f_{n} + δ \pqty{\eu^{\iu
|
||
ϕ}f_{0}^†a + \eu^{-\iu ϕ}a^†f_{0}} + ∑_{mn}J_{mn}(t) f_{m}^†f_{n},
|
||
\end{equation}
|
||
where \(J_{mn}(t)\) is the coupling mediated by the EOM in the big
|
||
loop. The phase \(ϕ=k_{0}L_{S}/2\) of the coupling stems from the
|
||
location of the fibre coupler along the loop but can potentially also
|
||
contain other contributions.
|
||
|
||
To be bring the Hamiltonian into the form \cref{eq:1}, we define
|
||
\begin{equation}
|
||
\label{eq:52}
|
||
\begin{aligned}
|
||
c_{\pm} &= \frac{1}{\sqrt{2}}\pqty{f_{0}\eu^{-\iu ϕ} \pm a} &
|
||
c_{n\neq
|
||
0}&=f_{n}\\
|
||
ω_{\pm}^{0}&=ω_{s}\pm δ & ω_{n\neq 0}&= ω_{s} + Ω_{B} n\\
|
||
V_{\pm,\pm}&=\frac{J_{00}}{2} & V_{\pm, n\neq0} &=
|
||
\frac{J_{0n}}{\sqrt{2}}\eu^{\iu
|
||
ϕ}\\
|
||
V_{n\neq \pm, m\neq \pm} &= J_{nm}
|
||
\end{aligned}
|
||
\end{equation}
|
||
and obtain
|
||
\begin{equation}
|
||
\label{eq:54}
|
||
H_{A} = ∑_{n} ω_{n}^{0}c_{n}^†c_{n} + ∑_{nm} V_{nm}(t) c_{n}^†c_{m},
|
||
\end{equation}
|
||
where the index \(n\) can take on the values in
|
||
\(\pqty{[-N,N]\setminus \{0\}} \cap \{+, -\}\). The spectra of the
|
||
coupled and uncoupled systems are visualized in \cref{fig:spectra}.
|
||
|
||
|
||
Upon transitioning into the rotating frame and applying the RWA we
|
||
arrive at a target Hamiltonian
|
||
\begin{equation}
|
||
\label{eq:111}
|
||
H^{T}_{nm}=V^{0}_{n,m} + \pqty{ε_{m} -\iu \pqty{η^{0}_{m} + η_{m}}
|
||
δ_{nm}},
|
||
\end{equation}
|
||
where we can control the \(V^{0}_{n,m}\) through the drive amplitudes
|
||
and the \(ε_{m}\) through the drive
|
||
detunings.
|
||
|
||
The choice of \(δ\) will be discussed in
|
||
\cref{sec:choice-hybridization} and the origin of the \(η^{0}_{m}\),
|
||
as well as the effects of asymmetric loss in the two loops are
|
||
investigated in \cref{sec:effects-asymm-damp}.
|
||
\Cref{sec:rotat-wave-inter} is concerned with relating the
|
||
\(V^{0}_{nm}\) with the drive amplitudes, frequencies and phases,
|
||
whereas \cref{sec:steadyst-transm,sec:steady-state-transm} will deal
|
||
with the transmission through transmission lines attached to either
|
||
the small or the big loop.
|
||
|
||
\tikzmath{
|
||
\Nmodes = 5;
|
||
integer \NmodesBetween;
|
||
\NmodesBetween = \Nmodes - 1;
|
||
\delta = .2;
|
||
}
|
||
\begin{figure}[p]
|
||
\centering
|
||
\begin{tikzpicture}[y=-1.5cm]
|
||
\draw[->] (-1.5, -1) -- node[left] {\(ω\)} ++(0,2);
|
||
|
||
\foreach \y in {-\Nmodes,...,\Nmodes}
|
||
\draw (0, \y) node[left] {\(ω^{B}_{\y}\)} -- ++(1, 0);
|
||
|
||
\foreach \y/\name in {-\Nmodes/-1,0,\Nmodes/1}
|
||
\draw (1.1, \y) -- ++(1, 0) node[right] {\(ω^{S}_{\name}\)};
|
||
\end{tikzpicture}
|
||
\begin{tikzpicture}[y=-1.5cm]
|
||
\foreach \y in {-\NmodesBetween,...,-1}
|
||
\draw (0, \y) node[left] {\(ω^{0}_{\y}\)} -- ++(1, 0);
|
||
|
||
\foreach \y in {1,...,\NmodesBetween}
|
||
\draw (0, \y) node[left] {\(ω^{0}_{\y}\)} -- ++(.5, 0) node(mc\y){}
|
||
-- ++(.5, 0) node(mr\y){};
|
||
|
||
\foreach \y in {-\Nmodes,\Nmodes} {
|
||
\foreach \sub/\name in {-\delta/-, \delta/+}
|
||
\draw (0, \sub+\y) -- ++(.5, 0) -- ++(.5, 0);
|
||
}
|
||
|
||
\foreach \y in {0} {
|
||
\foreach \sub/\name in {-\delta/-, \delta/+}
|
||
\draw (0, \sub+\y) node[left] {\(ω^{0}_{\name}\)} -- ++(.5, 0) node(pmc\name){} -- ++(.5, 0) node[right] (pmr\name){};
|
||
}
|
||
|
||
\foreach \y in {-\Nmodes,0,\Nmodes}
|
||
\draw[dashed,color=gray] (0, \y) -- ++(1, 0);
|
||
|
||
\draw[<->] (pmc+) -- node[right] {\(2δ\)} (pmc-);
|
||
\draw[<->] (pmc+) -- node[right] {\(Ω_{B} - δ\)} (mc1);
|
||
\draw[<->] (mc1) -- node[right] {\(Ω_{B}\)} (mc2);
|
||
\end{tikzpicture}
|
||
\caption{\label{fig:spectra}The spectra of the uncoupled loops (left) \(B,S\) and the resulting
|
||
spectrum after coupling the loops (right) for \(N=2\) and \(δ=Ω_{B}/5\).}
|
||
\end{figure}
|
||
|
||
\subsection{The Choice of the Hybridization Amplitude}
|
||
\label{sec:choice-hybridization}
|
||
|
||
It remains to be discussed which choice of \(δ\) is suitable. By
|
||
modulating \(V(t)\) and applying the RWA we can couple certain levels
|
||
in the spectrum of the system. To be able to couple one level to many
|
||
and implement the non-Markovian quantum walk, we have to select out
|
||
unique level-spacings. At the same time, we want to maximize the
|
||
frequency of the residual rotating terms.
|
||
|
||
A suitable mode for this one-to-many coupling is the \(c_{+}\) mode
|
||
with frequency \(ω_{+}^{0}=ω_{s}+δ\). We identify the \(A\) site of
|
||
the non-Markovian Quantum Walk in \(k\)-space with \(c_{+}\) and the
|
||
\(j\)th bath level with \(c_{j}\) (\(j\in [1, N]\)).
|
||
|
||
Let us denote the frequency
|
||
difference of the \(m\)th and \(n\)th mode by
|
||
\(Δ_{nm}=ω^{0}_{n}-ω^{0}_{m}\).
|
||
For \(n>m\) we have
|
||
\begin{align}
|
||
\label{eq:56}
|
||
Δ_{+,-}&=2δ\\
|
||
\label{eq:57}Δ_{n>0,-} &= n Ω_{B} + δ\\
|
||
\label{eq:58}Δ_{n>0,+} &= n Ω_{B} - δ\\
|
||
\label{eq:59}Δ_{n>0,m>0} &= (n-m) Ω_{B}.
|
||
\end{align}
|
||
To couple exactly one other mode with \(n>0\) to the \(+\) mode, the
|
||
RWA requires that \(\abs{Δ_{n,+}}\neq \abs{Δ_{kl}}\) for \(k\neq n\), \(l\neq +\).
|
||
This requirement yields the following restrictions on the value of
|
||
\(δ\)
|
||
\begin{align}
|
||
\label{eq:61}
|
||
δ&\neq \frac{n}{2}Ω_{B} & \text{\cref{eq:58,eq:57}}\\
|
||
δ&\neq \frac{n}{3}Ω_{B} & \text{\cref{eq:58,eq:56}}\\
|
||
δ&\neq {n}Ω_{B} & \text{\cref{eq:58,eq:59}\footnote{Duh!}}.
|
||
\end{align}
|
||
To maximize the residual rotating terms, the minimum of the
|
||
\cref{eq:56,eq:57,eq:58,eq:59} has to be maximized for \(δ \in [0, Ω_{B}]\)
|
||
\begin{equation}
|
||
\label{eq:62}
|
||
Δ_{\max}\equiv \max_{δ}Δ_{\min}(δ) = \max_{δ}\min\Bqty{2δ, \abs{Ω_{B}-δ}, \abs{Ω_{B}-3δ},
|
||
\abs{2Ω_{B}-3δ}, \abs{Ω_{B}-2δ}}.
|
||
\end{equation}
|
||
We find that \(Δ_{\max}=2Ω_{B}/5\) for
|
||
\begin{equation}
|
||
\label{eq:63}
|
||
δ_{\mathrm{opt}}=Ω_{B}/5,
|
||
\end{equation}
|
||
as can be ascertained from \cref{fig:delta_choice}.
|
||
\begin{figure}[H]
|
||
\centering
|
||
\begin{tikzpicture}
|
||
\begin{axis}[
|
||
scale only axis=true,
|
||
width=.8\columnwidth,
|
||
height=.2\columnwidth,
|
||
xmin = 0, xmax = 1,
|
||
ymin = 0, ymax = .5,
|
||
axis lines* = left,
|
||
xtick = {0}, ytick = \empty,
|
||
clip = false,
|
||
xtick={},ytick={},
|
||
minor tick num=5,
|
||
grid=both,
|
||
grid style={line width=.1pt, draw=gray!10},
|
||
major grid style={line width=.2pt,draw=gray!50},
|
||
axis lines=middle,
|
||
ylabel = {\(Δ_{\min}/Ω_{B}\)},
|
||
xlabel = {\(δ/Ω_{B}\)},
|
||
x label style={at={(axis description cs:0.5,-0.2)},anchor=north},
|
||
y label style={at={(axis description cs:-0.06,.5)},rotate=90,anchor=south},
|
||
]
|
||
\addplot[domain = 0:1, restrict y to domain = 0:1, samples =
|
||
1000]{min(2*x, 1-x, abs(1-3*x), abs(2-3*x),
|
||
abs(1-2*x))};
|
||
\addplot[color = black, mark = *, only marks, mark size = 3pt]
|
||
coordinates {(.2, .4)};
|
||
\addplot[color = black, dashed, thick] coordinates {(.2, 0) (.2,
|
||
.4) (0, .4)};
|
||
|
||
\addplot[color = gray, mark = *, only marks, mark size = 3pt]
|
||
coordinates {(.25, .25)};
|
||
\addplot[color = gray, dashed, thick] coordinates {(.25, 0) (.25,
|
||
.25) (0, .25)};
|
||
\end{axis}
|
||
\end{tikzpicture}
|
||
\caption{\label{fig:delta_choice} The minimal rotating frequencies
|
||
\cref{eq:62} for the range of possible \(δ\). The black marker
|
||
highlights \(δ=Ω_{B}/5\) and the grey marker marks \(δ=Ω_{B}/4\).}
|
||
\end{figure}
|
||
|
||
|
||
|
||
|
||
\subsection{Effects of Asymmetric Damping}
|
||
\label{sec:effects-asymm-damp}
|
||
|
||
In the above, we have not accounted for the apriori damping in the
|
||
fibre loops. As they are of vastly differing lengths, it is sensible
|
||
to expect different damping rates for each.
|
||
|
||
To account for this, we modify \cref{eq:51} to
|
||
\begin{equation}
|
||
\label{eq:89}
|
||
\begin{aligned}
|
||
H_{A}= ω_{s}a_{S,0}^†a_{S,0} &+ ∑_{n=-N}^{N} ω_{n}^{B}a_{B,n}^†a_{B,n} + δ \pqty{\eu^{\iu
|
||
ϕ}a_{B,0}^†a_{S,0} + \eu^{-\iu ϕ}a_{S,0}^†a_{B,0}} + ∑_{mn}J_{mn}(t) a_{B,m}^†a_{B,n}
|
||
\\&- \iu η_{S}a_{S,0}^†a_{S,0} - \iu \pqty{η_{S}+2Δδ} ∑_{n}a_{B,n}^†a_{B,n}
|
||
\end{aligned}
|
||
\end{equation}
|
||
with the damping rates \(η_{S}\) and the damping asymmetry
|
||
\(\abs{Δ}< 1\).
|
||
|
||
|
||
The diagonalizing transformation is then
|
||
{\renewcommand\arraystretch{1.5}
|
||
\begin{equation}
|
||
\label{eq:90}
|
||
\begin{aligned}
|
||
T_{i,0;\mp} &\triangleq \frac{1}{\sqrt{2}}
|
||
\begin{pmatrix}
|
||
-\eu^{-\iu ψ} & \eu^{\iu ψ} \\
|
||
\eu^{\iu ϕ} & \eu^{\iu ϕ}
|
||
\end{pmatrix}
|
||
& T^{-1}_{\mp;i,0} &\triangleq
|
||
\frac{1}{\sqrt{2}\cos(ψ)}
|
||
\begin{pmatrix}
|
||
-1 & \eu^{-\iu (ϕ-ψ)}\\
|
||
1 & \eu^{-\iu (ϕ+ψ)}
|
||
\end{pmatrix}
|
||
\end{aligned}
|
||
\end{equation}}
|
||
with \(i=B,S\) and
|
||
\begin{equation}
|
||
\label{eq:93}
|
||
\eu^{\iu ψ} \equiv \iu Δ + \sqrt{1-Δ^{2}} \implies ψ = \sin^{-1}(Δ).
|
||
\end{equation}
|
||
For all other matrix elements \(T\) equals the
|
||
identity
|
||
\begin{equation}
|
||
\label{eq:95}
|
||
T_{i,α;m} = T^{-1}_{m;i,α} = δ_{i,B} δ_{αm}
|
||
\end{equation}
|
||
for \(m\neq\pm\).
|
||
|
||
|
||
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} &= ω_{s} \pm δ \cos(ψ)\\
|
||
η^{0}_{m\neq\pm} &= η_{B} = η_S+2δΔ & η^{0}_{\pm} &= η_{S} + δΔ =
|
||
\frac{η_{S}+ η_{B}}{2}.
|
||
\end{aligned}
|
||
\end{equation}
|
||
Note that the eigenenergies of the \(\pm\) modes are slightly shifted.
|
||
|
||
Next, we compute the transformation of the interaction
|
||
\(T^{-1}VT\) (see \cref{eq:74}) and find
|
||
\begin{equation}
|
||
\label{eq:96}
|
||
\begin{aligned}
|
||
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\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 \(Δ\to 0\).
|
||
|
||
In all of the above one can set \(\cos(ψ)=1\) to only account for
|
||
\(Δ\) in leading order.
|
||
|
||
|
||
\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 \({h}_{m}\) (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} + ε_{m}-ε_{n}}}t} \eu^{\iu \varphi_{j}}
|
||
- \eu^{-i \pqty{\hat{ω_{j}} +
|
||
\hat{δ}_{j}+\pqty{ω^{0}_{n}-ω^{0}_{m} + ε_{m}-ε_{n}}}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 Loop}
|
||
\label{sec:steadyst-transm}
|
||
|
||
We can now proceed to calculate the steady state transmission for the
|
||
transmission line attached to the small loop (see
|
||
\cref{fig:schematic}).
|
||
|
||
As our input field has only right-moving components and our detector
|
||
is situated to the right of the fibre-coupler, we can set \(δ,σ=+\) in
|
||
\cref{eq:87} and will suppress those indices in the following.
|
||
|
||
Using \cref{sec:effects-asymm-damp}, we identify
|
||
\begin{equation}
|
||
\label{eq:97}
|
||
\begin{aligned}
|
||
T_{i_{0},β;m} &\to T_{S,0;\pm} &\implies U_{β,m} &\to U_{0,m} =
|
||
T_{S,0;m} =
|
||
\frac{1}{\sqrt{2}}
|
||
\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}\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_{T}}{2c\cos(ψ)} \pqty{δ_{m,+}
|
||
\eu^{\iu ψ} + δ_{m,-}\eu^{-\iu ψ}},
|
||
\end{equation}
|
||
where \(n_{T}\) is the refractive index of the transmission line
|
||
(replace \(n_{B}\to n_{T}\) in \cref{sec:micr-deriv}).
|
||
|
||
To obtain the \(ω_{γ}\), \(λ_{γ}\) and \(O_{m,γ}\) we have to
|
||
diagonalize the target Hamiltonian
|
||
\begin{equation}
|
||
\label{eq:99}
|
||
H^{T}_{nm}=V^{0}_{n,m} + \pqty{ε_{m} -\iu \pqty{η^{0}_{m} + η_{m}} δ_{nm}}.
|
||
\end{equation}
|
||
Both apriori loss \(η_{m}^{0}\) and the loss induced by the coupling
|
||
to the transmission line \(η_{m}\), as well as the detunings \(ε_{m}\)
|
||
enter the final interaction hamiltonian.
|
||
|
||
As
|
||
\begin{equation}
|
||
\label{eq:100}
|
||
\Im η_{\pm}= \pm \frac{\abs{κ} πn_{T}}{2c\cos(ψ)}Δ,
|
||
\end{equation}
|
||
we obtain a correction to the on-site energies \(ε_{\pm}=\pm ε_{A}\),
|
||
whereas
|
||
\begin{equation}
|
||
\label{eq:101}
|
||
\Re η_{\pm} = \frac{\abs{κ} πn_{T}}{2c}
|
||
\end{equation}
|
||
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_{T}}{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. Also, the magnitude of
|
||
\cref{eq:100} is likely negligible. If it is required, we can
|
||
determine it by choosing \(N=1\) and measuring the eigenenergies with
|
||
the result obtained below.
|
||
|
||
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}^0_{\pm} \equiv ω_{σ}^{0}-ε_{σ} = ω_{S} \pm δ\cos(ψ) - ε_{A}
|
||
+ \frac{\abs{κ}
|
||
πn_{T}}{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}^0_{σ}t -
|
||
\tilde{ω}^0_{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}^0_{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}^0_{σ} +
|
||
ω_{γ}} + λ_{γ}}t} \eu^{\bqty{\iu\pqty{\tilde{ω}^0_{\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. Persistent oscillations in the output intensity therefore
|
||
would likely signal a breakdown of the RWA.
|
||
|
||
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}^0_{σ} + ω_{γ}-ω} +
|
||
λ_{γ}}\frac{\eu^{\iu σψ}}{\cos(ψ)}\equiv \eu^{-\iu ω t}F_{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{F_{S}(ω)} + \abs{F_{S}(ω)}^{2}} \approx
|
||
I_{0}\pqty{1-2\Re{F_{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{F_{S}(ω)} = χ_{0} ∑_{σγ}\frac{λ_{γ}\abs{O_{σ,γ}}^{2}}{\pqty{\tilde{ω}^0_{σ} + ω_{γ}-ω}^{2} +
|
||
λ_{γ}^{2}},
|
||
\end{equation}
|
||
whereas \(Δ\neq 0\) will very slightly shift the peak locations 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. The
|
||
locations of the peaks (dips in the transmission) are the eigenenrgies
|
||
\(ω_{γ}\) of the target Hamiltonian \(H^{T}\) relative to the
|
||
eigenenrgies of the unmodulated system and the drive detunings
|
||
\(\tilde{ω}^0_{σ}=ω^{0}_{σ}-ε_{σ}\). Wheras the \(ω^{0}_{σ}\) define
|
||
the ``zero'' of energy, the shifts by \(ε_{σ}\) can be interpreted as
|
||
an effect similar to the AC stark shift that arises due to the drive
|
||
detuning. Another effect of the drive, namely persistent oscillations
|
||
of the output intensity, is not observed here, as we don't couple the
|
||
\(σ=\pm\) states with the drive. Finally, the peak heights and widths are
|
||
controlled by the loss rates \(λ_{γ}\).
|
||
|
||
\subsection{Steady-state Transmission on the Big Loop}
|
||
\label{sec:steady-state-transm}
|
||
|
||
To probe the structure of the states in bath, we have to probe the big
|
||
loop.
|
||
|
||
Analogous to \cref{sec:steadyst-transm} we can also obtain the
|
||
transmission for a transmission line (inclusive laser) attached to the
|
||
big loop. Note, that the phase \(ϕ\) is now \(ϕ=k_{0}L_{B}/2\),
|
||
whereas all the other model parameters retain their meaning. The main
|
||
difference to the calculations for the small loop is the number of
|
||
modes that interact with the transmission line, leading to
|
||
non-vanishing stationary (low-frequency) oscillations in the output
|
||
intensity.
|
||
|
||
Just as in \cref{sec:steadyst-transm}, we begin by identifying the
|
||
relation between the bare modes in the big loop and the eigenmodes of
|
||
the unmodulated system.
|
||
\begin{equation}
|
||
\label{eq:55}
|
||
\begin{aligned}
|
||
T_{i_{0},β;m} &\to T_{B,β;\pm} &\implies U_{β,m} &= T_{S,0;m} = δ_{β0}
|
||
\frac{\eu^{\iu ϕ}}{\sqrt{2}}
|
||
∑_{σ=\pm} δ_{m,σ}
|
||
+ ∑_{j\neq 0}δ_{βj}δ_{mj}\\
|
||
&& U^{-1}_{m,β} &=
|
||
T^{-1}_{m;S,0}=δ_{β,0}\frac{\eu^{-\iu ϕ}}{\sqrt{2}\cos(ψ)}
|
||
∑_{σ=\pm} δ_{m,σ}\eu^{-\iu σ ψ} +∑_{j\neq 0}δ_{βj}δ_{mj}
|
||
\end{aligned}
|
||
\end{equation}
|
||
where \(j\in [-N,N] \setminus 0\) and \(β\in [-N,N]\).
|
||
|
||
The decay rate introduced by the coupling to the transmission line
|
||
works out to be
|
||
\begin{equation}
|
||
\label{eq:71}
|
||
\begin{aligned}
|
||
η_{\pm} = \frac{\abs{κ} πn_{T}}{2c\cos(ψ)} \eu^{\mp \iu ψ}
|
||
\end{aligned}
|
||
\end{equation}
|
||
where the sign in the exponent is \emph{inverted} compared to
|
||
\cref{eq:98}. Therefore the effective on-site energy for the \(\pm\)
|
||
states will be shifted in the inverse direction. Note however, that it
|
||
is not guaranteed that \(κ\), \(η_{S}\), and \(Δ\) will be the same as
|
||
in \cref{sec:steadyst-transm}.
|
||
|
||
Using the fact that either \(O_{σ,γ}=0\) or \(O^{-1}_{\bar{σ},γ} =0\)
|
||
for any value of \(σ\) we find for the transmission
|
||
\begin{equation}
|
||
\label{eq:79}
|
||
T_{B}(ω,t) = θ(s) χ_{0}' ∑_{σ=\pm}\bqty{T^{B}_{σ,σ}(ω) + ∑_{n\neq\pm}\pqty{T^{B}_{σ,n}(ω,t) + T^{B}_{n,σ}(ω,t)} + ∑_{n,m\neq\pm} T^{B}_{σ,m,n}(ω,t)},
|
||
\end{equation}
|
||
with
|
||
\begin{equation}
|
||
\label{eq:91}
|
||
\begin{aligned}
|
||
χ_{0}' &= \abs{κ'}\frac{π n_{T}}{c} & \tilde{ω}_{m} = ω^{0}_{m} - ε_{m}
|
||
\end{aligned}
|
||
\end{equation}
|
||
and
|
||
\begin{align}
|
||
\label{eq:85}
|
||
T^{B}_{σ,σ}(ω) &= ∑_{γ}\frac{O_{σ,{γ}}O^{-1}_{γ,σ}\eu^{\iu σψ}}{2\cos(ψ)}\frac{1}{\iu\pqty{\tilde{ω}_{σ} + ω_{γ}-ω} +
|
||
λ_{γ}}\\
|
||
\label{eq:92}
|
||
T^{B}_{σ,n}(ω,t) &=δ_{σ,\sgn(n)}\frac{O_{σ,{γ}}O^{-1}_{γ,n}\eu^{\iu ϕ}}{\sqrt{2}}
|
||
\frac{\eu^{-\iu(\tilde{ω}_{σ}-\tilde{ω}_{n})t}}{\iu\pqty{\tilde{ω}_{n}+ ω_{γ}-ω} +λ_{γ}}\\
|
||
\label{eq:102}
|
||
T^{B}_{n,σ}(ω,t) &= δ_{σ,\sgn(n)}\frac{O_{n,{γ}}O^{-1}_{γ,σ}\eu^{-\iu (ϕ+ψ)}}{\sqrt{2}}
|
||
\frac{\eu^{-\iu(\tilde{ω}_{n}-\tilde{ω}_{σ})t}}{\iu\pqty{\tilde{ω}_{σ}+ ω_{γ}-ω} + λ_{γ}}\\
|
||
\label{eq:106}
|
||
T^{B}_{σ,m,n}(ω,t) &= δ_{σ,\sgn(n)} δ_{σ,\sgn(m)}O_{nγ}O^{-1}_{γm} \frac{\eu^{\iu(\tilde{ω}_{m}-\tilde{ω}_{n})t}}{\iu\pqty{\tilde{ω}_{m}+ ω_{γ}-ω} + λ_{γ}}.
|
||
\end{align}
|
||
|
||
The stationary transmission peaks around \(\tilde{ω}_{\pm}\) and has
|
||
subpeaks shifted by \(ω_{γ}\) just as in \cref{sec:steadyst-transm},
|
||
where the peak height is roughly proportional to the overlap of the
|
||
\(γ\) and \(\pm\) states. In the regime we're interested in, there
|
||
will only be one state with substantial overlap with the \(\pm\)
|
||
states (a.k.a. the \(A\) site). In the same frequency region,
|
||
\cref{eq:102} will also have peaks. Those peaks however will be
|
||
suppressed, as their height is proportional to the overlap of the
|
||
\(\pm\) states and the \(n\neq \pm\), i.e. the \(A\)-site, and bath
|
||
sates with the eigenstate \(γ\). The frequency of the steady state
|
||
oscillations of \cref{eq:102} allows to tune the relative energies of
|
||
the \(A\) sites and the bath site. The same signal may be retrieved
|
||
more cleanly from \cref{eq:92}, where the peaks are situated around
|
||
\(\tilde{ω}_{n}\). The transmission component \cref{eq:106} will only
|
||
be significant if \(m=n\), as the \(γ\) states that don't overlap with
|
||
the \(\pm\) states are almost identical to the \(n\neq\pm\),
|
||
i.e. bath, states. In this case the transmission does not exhibit
|
||
oscillations making the signal from \cref{eq:92} even clearer.
|
||
Comparing \cref{eq:92} and \cref{eq:102}, we can extract the damping
|
||
asymmetry \(Δ = \sin(ψ)\).
|
||
|
||
Time-averaging \cref{eq:79} leaves us with the stationary transmission
|
||
\begin{equation}
|
||
\label{eq:110}
|
||
T_{B}(ω) = θ(s) χ_{0}' ∑_{σ=\pm}\bqty{T^{B}_{σ,σ}(ω) + ∑_{n}T^{B}_{σ,n,n}(ω)}.
|
||
\end{equation}
|
||
|
||
\newpage
|
||
\printbibliography{}
|
||
\end{document}
|
||
|
||
|
||
%%% Local Variables:
|
||
%%% mode: latex
|
||
%%% TeX-master: t
|
||
%%% TeX-output-dir: "output"
|
||
%%% TeX-engine: luatex
|
||
%%% jinx-languages: "en_CA"
|
||
%%% End:
|