mirror of
https://github.com/vale981/notes_io_loop
synced 2025-03-04 17:11:40 -05:00
rework derivation of i/o to remove \tilde{c}_m + explicit sqrt{w^B_k}
This commit is contained in:
parent
04b9fde5dd
commit
6a6e345b03
1 changed files with 179 additions and 137 deletions
|
|
@ -84,40 +84,53 @@ 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{equation}
|
||||
\begin{multline}
|
||||
\label{eq:12}
|
||||
\tilde{c}(t)_{m} = c_{m}(t)\eu^{\iu ω^{0}_{m}t} \implies H_{A} \to \tilde{H}_{A}=
|
||||
∑_{mn}V_{mn}(t) \eu^{-\iu t (ω^{0}_{n}-ω^{0}_{m})}
|
||||
\tilde{c}_{m}^†\tilde{c}_{n} \approx ∑_{mn}V^{0}_{mn} \eu^{\iu (ε_{m}-ε_{n})t}\tilde{c}_{m}^†\tilde{c}_{n},
|
||||
\end{equation}
|
||||
where \(\abs{ε_{m}-ε_{n}}\ll\abs{ω^{0}_{m} - ω^{0}_{n}}\).
|
||||
Upon changing into another rotating frame we can remove this residual
|
||||
time dependence
|
||||
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
|
||||
\(V(t)\).}
|
||||
|
||||
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}
|
||||
h_{n}(t) = \tilde{c}_{n}\eu^{-\iu ε_{n}t} \implies \tilde{H}'_{A} =
|
||||
∑_{mn}\bqty{V^{0}_{mn} + δ_{mn}{ε_{m}-\iu η_{m}}} h_{m}^†h_{n},
|
||||
\iu \dot{h}_{m} = ∑_{n}\bqty{V^{0}_{mn} + δ_{mn}{ε_{m}-\iu η_{m}}}h_{n}.
|
||||
\end{equation}
|
||||
where we've added an ad-hock decay rate due to the coupling to the
|
||||
transmission line that will be introduced more rigorously later on.
|
||||
|
||||
We can subsequently find a unitary transformation that diagonalizes
|
||||
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'} = ω_{γ} δ_{γ,γ'}.
|
||||
∑_{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.}.
|
||||
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}(0) h_{n} = ∑_{n}O^{-1}_{γn}(t) \eu^{-\iu
|
||||
ε_{n}t}\tilde{c}_{n} \implies \iu \dot{d}_{γ} = ω_{γ}d_{γ}
|
||||
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.
|
||||
|
||||
|
|
@ -126,27 +139,33 @@ 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 eigenmodes in the rotating frame of
|
||||
\(H_{0}\) are called \(\tilde{c}_{m}\). The frame in which the
|
||||
equations of motion for the modes reflects the target Hamiltonian
|
||||
\(\tilde{H}_{A}'\) is given by rotating away the slow-oscillating
|
||||
terms 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
|
||||
Hamiltionaion \(d_{γ}\) including damping.
|
||||
\(α_{j,α}\) by \cref{eq:43}. The frame in which the equations of
|
||||
motion for the modes reflects the target Hamiltonian
|
||||
\(\tilde{H}_{A}'\) 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.
|
||||
|
||||
Let us list the relation between the \(a\), \(c\), \(h\) and \(d\) operators
|
||||
for later reference
|
||||
\begin{align}
|
||||
\label{eq:15}
|
||||
c_{n} &= \eu^{-\iu
|
||||
ω^{0}_{n} t}\tilde{{c}}_{n} = \eu^{-\iu
|
||||
(ω^{0}_{n}-ε_{n}) t} h_{n}= \eu^{-\iu
|
||||
(ω^{0}_{n}-ε_{n}) t} ∑_{γ} O_{nγ}d_{γ} \\
|
||||
a_{β} &= ∑_{m} U_{β,m} c_{m} = ∑_{m} U_{β,m} \tilde{c}_{m}\eu^{-\iu
|
||||
ω^{0}_{m} t} = ∑_{mγ} U_{β,m}O_{mγ} \eu^{-\iu
|
||||
(ω^{0}_{m}-ε_{m}) t} d_{γ}
|
||||
\end{align}
|
||||
\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}
|
||||
|
|
@ -271,57 +290,61 @@ 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
|
||||
\(\tilde{c}_{m}\) and \(\tilde{f}_{ω} = f_{ω}\eu^{\iu \abs{ω}t}\) to
|
||||
\(h_{m}\) 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
|
||||
(ω^{0}_{m}-\abs{ω}) t}
|
||||
U_{β,m} \tilde{f}_k^†\tilde{c}_{m} \dd{ω} + \hc
|
||||
(\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
|
||||
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 \(ω_{β}
|
||||
\approx ω_{0}^{m}\) for some small range of \(m\).}
|
||||
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 \(g \ll ω_{m}^{0}\) each \(\tilde{c}_{m}\) in \cref{eq:10} only
|
||||
For \(g \ll \tilde{ω}_{m}^{0}\) each \(h_{m}\) in \cref{eq:10} only
|
||||
interacts with non-overlapping sub-intervals
|
||||
\([ω^{0}_{m}-λ_{m}, ω^{0}_{m}+λ_{m}]\) of the transmission frequency axis
|
||||
(rotating wave approximation) with \(g\ll λ_{m} \ll ω_{m}^{0}\). This
|
||||
\([\tilde{ω}^{0}_{m}-λ_{m}, \tilde{ω}^{0}_{m}+λ_{m}]\) 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
|
||||
\(ω^{0}_{m}\) are to the \(ω_{k_{β}}\). We obtain
|
||||
\(\tilde{ω}^{0}_{m}\) 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}∫_{ω^{0}_{m}-λ_{m}}^{ω^{0}_{m}+λ_{m}}
|
||||
\sqrt{L_{A}}} ∑_{β,m}∫_{\tilde{ω}^{0}_{m}-λ_{m}}^{\tilde{ω}^{0}_{m}+λ_{m}}
|
||||
\eu^{-\iu
|
||||
(ω^{0}_{m}-\abs{ω}) t}
|
||||
(\tilde{ω}^{0}_{m}-\abs{ω}) t}
|
||||
U_{β,m} \pqty{G'_{β}(ω) \tilde{f}_{ω}^† + G'_{β}(-ω)
|
||||
\tilde{f}_{-ω}^†}\tilde{c}_{m} \dd{ω} + \hc
|
||||
\tilde{f}_{-ω}^†}h_{m} \dd{ω} + \hc
|
||||
\end{equation}
|
||||
For any finite \(Δx\) and
|
||||
\(ω_{0}^{m},ω_{k_{β}}\gg \frac{2πc}{Δx n_{A}}\) we can assume
|
||||
\(\tilde{ω}_{0}^{m},ω_{k_{β}}\gg \frac{2πc}{Δx n_{A}}\) we can assume
|
||||
\begin{equation}
|
||||
\label{eq:44}
|
||||
G'_{β}\pqty{-\sgn(β) ω}\approx 0
|
||||
\end{equation}
|
||||
in \cref{eq:16}.
|
||||
|
||||
As each \(\tilde{c}_{m}\) is now interacting with non-overlapping
|
||||
As each \({h}_{m}\) is now interacting with non-overlapping
|
||||
transmission-line field modes, we can introduce a separate field for
|
||||
each \(\tilde{c}_{m}\) that commutes with all other fields and extend
|
||||
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},
|
||||
|
|
@ -330,8 +353,8 @@ Care has to be taken to maintain consistency with \cref{eq:44},
|
|||
\tilde{H}_{I}= \frac{gΔx}{
|
||||
\sqrt{L_{A}}} ∑_{β,m}∫_{0}^{∞}
|
||||
\eu^{-\iu
|
||||
(ω^{0}_{m}-\abs{ω}) t}
|
||||
U_{β,m} G'_{β}(\sgn({β})ω) \tilde{f}^{m,†}_{\sgn({β})ω}{c}_{m} \dd{ω} + \hc
|
||||
(\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}
|
||||
|
|
@ -339,39 +362,47 @@ which becomes\footnote{A lot of discussion for a simple result :).}
|
|||
H_{I}= ∑_{m}∫_{-∞}^{∞}
|
||||
\tilde{G}_{m}(k) {b}^{m,†}_{k}{c}_{m} \dd{k}
|
||||
\end{equation}
|
||||
upon transitioning out of the rotating frame with \(\tilde{G}_{m}(k) =
|
||||
upon transitioning out of the rotating frame with
|
||||
\begin{equation}
|
||||
\label{eq:15}
|
||||
\tilde{G}_{m}(k) =
|
||||
\frac{gΔx}{
|
||||
\sqrt{L_{A}}} ∑_{β\gtrless 0}U_{β,m} G_{β}(k)δ_{\sgn(β),\sgn(k)}\). The equation of motion
|
||||
\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) c_{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)} c_{m}(s)\dd{s}.
|
||||
ω_{k}^{B}(t-s)} \eu^{-\iu ω_{k}^{B}s}h_{m}(s)\dd{s}.
|
||||
\end{gather}
|
||||
The equation of motion for \(\tilde{c}_{m}\) is
|
||||
The equation of motion for \(h_{m}\) is
|
||||
\begin{equation}
|
||||
\label{eq:21}
|
||||
\iu\dot{\tilde{c}}_{m} = ∑_{n}V^{0}_{mn} \tilde{c}_n +
|
||||
\underbrace{\eu^{\iu ω_{m}^{0}t}∫_{-∞}^{∞}\tilde{G}^\ast_{m}(k)
|
||||
\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 ω_{m}^{0}t} ∫_{-∞}^{∞}\tilde{G}^\ast_{m}(k)
|
||||
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} - ω^{0}_{m})t}\dd{k} -\iu ∫_{0}^{t}∫_{-∞}^{∞}\abs{\tilde{G}_{m}(k)}^{2}
|
||||
\tilde{c}_{m}(s)\eu^{-\iu ω^{B}_{k}(t-s)} \eu^{\iu
|
||||
ω^{0}_{m}(t-s)}\dd{k}\dd{s}\\
|
||||
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}
|
||||
Inspired by the RWA, we now assume
|
||||
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}
|
||||
|
|
@ -379,24 +410,21 @@ Inspired by the RWA, we now assume
|
|||
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} U_{βm}δ_{\sgn(β),\sgn(k)} \equiv g_{m, \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 \([ω^{0}_{m}-λ_{m}, ω^{0}_{m}+λ_{m}]\) (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}.
|
||||
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}.
|
||||
|
||||
Additionally we resurrect\footnote{Within
|
||||
the RWA this is all equivalent, but I prefer having the input field
|
||||
proportional to the electric field!} the \(ω_{k}^{B}\) dependence of
|
||||
\(G_{m}(k)\) in \(I\) to obtain
|
||||
Using this in \(I\), we obtain
|
||||
\begin{equation}
|
||||
\label{eq:24}
|
||||
\begin{aligned}
|
||||
II &= \frac{\eu^{\iu ω_{m}^{0}t}}{\sqrt{ω_{m}^{0}}} \bqty{g_{m,+}^\ast ∫_{0}^{∞}\sqrt{ω^{B}_{k}}b^{m}_{k}(0)\eu^{-\iu
|
||||
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 \frac{\eu^{\iu ω_{m}^{0}t}}{\sqrt{ω_{m}^{0}}}\pqty{
|
||||
&\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}
|
||||
|
|
@ -405,36 +433,45 @@ 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 ω^{0}_{m}(t-s)}\tilde{c}_{m}(s)
|
||||
\bqty{ \abs{g_{m,+}}^{2} ∫_{0}^{∞}\eu^{-\iu ω^{B}_{k}(t-s)} \dd{k} + \abs{g_{m,-}}^{2} ∫_{-∞}^{0}\eu^{-\iu ω^{B}_{k}(t-s)} \dd{k}}\dd{s}.
|
||||
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}^{∞}\eu^{-\iu ω^{B}_{k}(t-s)} \dd{k} = \frac{n_{B}}{c}
|
||||
∫_{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}
|
||||
III= -2\iu η_{m}∫_{0}^{t}\eu^{\iu ω^{0}_m(t-s)}\tilde{c}_{m}(s)
|
||||
δ(t-s)\dd{s} = -\iu η_{m} \tilde{c}_{m}(t),
|
||||
\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{n_{B}}{c}\bqty{\abs{g_{m,-}}^{2}+\abs{g_{m,+}}^{2}}.
|
||||
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}
|
||||
\dot{\tilde{c}}_{m} = -\iu\bqty{∑_{n}V^{0}_{mn} \tilde{c}_n +
|
||||
\frac{\eu^{\iu ω_{m}^{0}t}}{\sqrt{ω_{m}^{0}}}
|
||||
∑_{σ=\pm}g_{m,σ}^\ast b_{\inputf,σ}^{m}(t)} - η_{m}\tilde{c}_{m}.
|
||||
\iu\dot{h}_{m} = {∑_{n}\bqty{V^{0}_{mn} + \pqty{ε_{m}-\iu η_{m}}δ_{nm}}{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.
|
||||
|
|
@ -445,13 +482,12 @@ Integrating \cref{eq:19} over all \(k\) yields
|
|||
\begin{equation}
|
||||
\label{eq:29}
|
||||
\begin{aligned}
|
||||
\frac{b_{\outputf}^{m}(x,t)}{\sqrt{ω^{0}_{m}}} &\equiv
|
||||
\frac{1}{\sqrt{ω_{m}^{0}}}∫ \sqrt{ω^{B}_{k}} b_{k}^{m}(t) \eu^{\iu k
|
||||
{b_{\outputf}^{m}(x,t)} &\equiv
|
||||
∫ \sqrt{ω^{B}_{k}} b_{k}^{m}(t) \eu^{\iu k
|
||||
t}\dd{k}\\
|
||||
&=
|
||||
\frac{1}{\sqrt{ω_{m}^{0}}} b_{\inputf}^{m}(x, t) -\iu
|
||||
g_{m,\sgn(x)}\frac{π n_{B}}{c}
|
||||
\tilde{c}_{m}(τ(x,t))\eu^{-i ω^{0}_{m}τ(x,t)}Θ(τ(x,t)),
|
||||
&= 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
|
||||
|
|
@ -469,11 +505,12 @@ the time argument is properly retarded. We defined
|
|||
used that
|
||||
\begin{equation}
|
||||
\label{eq:42}
|
||||
∫_{0}^{∞}\eu^{-\iu ω^{B}_{k}(t-s)}\eu^{\pm\iu k x} \dd{k} =
|
||||
\frac{n_{B}}{c}
|
||||
∫_{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}}}.
|
||||
\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}
|
||||
|
|
@ -483,7 +520,7 @@ The case of \(x=0\) is recovered by defining
|
|||
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 \(k\)-depnedence of
|
||||
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.
|
||||
|
|
@ -496,14 +533,15 @@ 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 \tilde{n}_{γ})
|
||||
δ_{γγ'}(ω_{γ}-\iu λ_{γ})
|
||||
\end{equation}
|
||||
to obtain \(O_{mγ}(t)\) and find
|
||||
using the transformation \(O_{nγ}\) (see \cref{sec:rotating-frames})
|
||||
and find
|
||||
\begin{equation}
|
||||
\label{eq:32}
|
||||
\dot{d}_{γ} = ∑_{m}\pqty{O^{-1}(t)}_{γm}\dot{\tilde{c}}_{m} =
|
||||
-\iu\bqty{\pqty{ω_{γ} - \iu \tilde{η}_{γ}}d_{γ} +
|
||||
∑_{σ=\pm}∑_{m}\pqty{O^{-1}(t)}_{γm}\frac{g_{m,σ}^\ast }{\sqrt{ω_{m}^{0}}} \eu^{\iu ω_{m}^{0}t}
|
||||
\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}
|
||||
|
||||
|
|
@ -511,11 +549,11 @@ 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 \(g_{β}^{0}\approx \sqrt{κ}\) and
|
||||
\(\sqrt{ω^{0}_{m}}\approx\sqrt{ω_{0}}\), where \(ω_{0}\) is a typical
|
||||
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:32} to
|
||||
\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}
|
||||
|
|
@ -524,9 +562,9 @@ and
|
|||
\begin{gather}
|
||||
\label{eq:34}
|
||||
\dot{d}_{γ} =
|
||||
-\iu\bqty{\pqty{ω_{γ}-\iu \tilde{η}_{γ}}d_{γ} + \sqrt{κ^\ast} ∑_{σ=\pm}
|
||||
U^{σ}_{γ}(t) \frac{b_{\inputf}(t)}{\sqrt{ω_{0}}}}\\
|
||||
U^{σ}_{γ}(t) = ∑_{m,β} δ_{\sgn({β}),σ}U^\ast_{βm}\pqty{O^{-1}(t)}_{γm} \eu^{\iu ω_{m}^{0}t}= ∑_{m,β} δ_{\sgn({β}),σ}U^\ast_{βm}\pqty{O^{-1}}_{γm}\eu^{\iu (ω_{m}^{0}-ε_{m})t}.
|
||||
-\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
|
||||
|
|
@ -535,14 +573,14 @@ 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} + \tilde{η}_{γ}}t} -
|
||||
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)
|
||||
\frac{b_{\inputf,σ}(t)}{\sqrt{ω_{0}}}\dd{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} + \tilde{η}_{γ}}t}.
|
||||
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
|
||||
|
|
@ -559,8 +597,8 @@ whole copy of the input field.
|
|||
This leads us to
|
||||
\begin{equation}
|
||||
\label{eq:38}
|
||||
\frac{b_{\outputf}(x,t)}{\sqrt{ω_{0}}} \equiv
|
||||
\frac{1}{\sqrt{ω_{0}}} b_{\inputf}(x, t) -i θ(τ(x,t)) \frac{\sqrt{κ}πn_{B}}{c}
|
||||
{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}
|
||||
|
||||
|
|
@ -585,7 +623,8 @@ position \(x>0\) we have
|
|||
\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 \(b_{\inputf}(s) = b_{\inputf,+}(s) + b_{\inputf,-}(s) = b_{\inputf,+}(s)\).
|
||||
with \(b_{\inputf}(s) = b_{\inputf,+}(s) + b_{\inputf,-}(s) =
|
||||
b_{\inputf,+}(s)\).
|
||||
|
||||
\subsection{Langevin-Equations for Lossy Oscillators}
|
||||
\label{sec:lang-equat-lossy}
|
||||
|
|
@ -644,7 +683,7 @@ terms and express everything in terms of the \(c_{m}\) using
|
|||
\(T\). Subsequently, we change into a rotating frame
|
||||
\begin{equation}
|
||||
\label{eq:66}
|
||||
\tilde{c}_{m} = c_{m}\eu^{\iu ω^{0}_{m}t},
|
||||
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
|
||||
|
|
@ -660,34 +699,25 @@ replacements along the way
|
|||
\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 g^{-1}_{m,σ}=∑_{β}\pqty{g^{0}_{β}}^\ast U^{-1}_{mβ}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{π n_{B}}{c} ∑_{σ} g_{mσ}g^{-1}_{mσ},
|
||||
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}
|
||||
\dot{\tilde{c}}_{m}= -\iu\bqty{∑_{n}V^{0}_{mn}\eu^{\iu
|
||||
\pqty{ε_{m}-ε_{n}}t} \tilde{c}_{n} + \frac{\eu^{\iu ω^{0}_{m}t}}{\sqrt{ω^{0}_{m}}}∑_{σ=\pm}g_{mσ}^{-1}b_{\inputf,σ}^{m}} - \pqty{η_{m} +
|
||||
η_{m}^{0}}\tilde{c}_{m}.
|
||||
\end{equation}
|
||||
To remove the residual explicit time dependence in \cref{eq:73} we
|
||||
define
|
||||
\begin{equation}
|
||||
\label{eq:75}
|
||||
h_{m}=\tilde{c}_{m}\eu^{-\iu ε_{m}t}
|
||||
\end{equation}
|
||||
and find
|
||||
\begin{equation}
|
||||
\label{eq:76}
|
||||
\dot{h}_{m}= -\iu\bqty{∑_{n}\Bqty{V^{0}_{mn}+ \bqty{ε_{m}-\iu
|
||||
\pqty{η_{m}^{0}+η_{m}}}δ_{nm}}h_{m}} + \frac{\eu^{\iu \pqty{ω^{0}_{m}-ε_{m}}t}}{\sqrt{ω^{0}_{m}}}∑_{σ=\pm}g_{mσ}^{-1}b_{\inputf,σ}^{m}.
|
||||
\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}
|
||||
|
|
@ -698,20 +728,19 @@ Diagonalizing
|
|||
and defining
|
||||
\begin{equation}
|
||||
\label{eq:78}
|
||||
d_{γ} = ∑_{n}O^{-1}_{γn}h_{n} = ∑_{n}O^{-1}_{γn}\eu^{-\iu
|
||||
ε_{n}t}\tilde{c}_{n}\implies h_{n}=∑_{γ}\eu^{\iu ε_{n}t}O_{nγ}d_{γ}
|
||||
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}\frac{b_{\inputf,σ}}{\sqrt{ω_{0}}}}d_{γ}
|
||||
\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 \(g_{β}^{0}=\sqrt{κ}\) and defined
|
||||
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\pqty{ω^{0}_{m}-ε_{m}}t}O^{-1}_{γm}U^{-1}_{mβ}δ_{\sgn(β),σ}.
|
||||
∑_{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}
|
||||
|
|
@ -722,7 +751,7 @@ This also simplifies \cref{eq:64} to
|
|||
Further defining
|
||||
\begin{align}
|
||||
\label{eq:83}
|
||||
\bar{U}^{σ}_{γ}&=∑_{mβ}\eu^{-\iu\pqty{ω^{0}_{m}-ε_{m}}t}O_{mγ}U_{βm}δ_{\sgn(β),σ}\qq{and}&
|
||||
\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
|
||||
|
|
@ -740,7 +769,20 @@ input fields
|
|||
|
||||
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 \(\tilde{c}_{m}\).
|
||||
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}
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue