diff --git a/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/graphics/walker_setup.pdf b/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/graphics/walker_setup.pdf index 1705bc4..2b06ea7 100644 Binary files a/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/graphics/walker_setup.pdf and b/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/graphics/walker_setup.pdf differ diff --git a/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/graphics/walker_setup.pdf_tex b/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/graphics/walker_setup.pdf_tex index 41ede6b..c9adcc7 100644 --- a/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/graphics/walker_setup.pdf_tex +++ b/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/graphics/walker_setup.pdf_tex @@ -50,7 +50,7 @@ \global\let\svgwidth\undefined% \global\let\svgscale\undefined% \makeatother% - \begin{picture}(1,0.84005445)% + \begin{picture}(1,0.84005373)% \lineheight{1}% \setlength\tabcolsep{0pt}% \put(0,0){\includegraphics[width=\unitlength,page=1]{walker_setup.pdf}}% diff --git a/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/graphics/walker_setup.png b/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/graphics/walker_setup.png index bb6e7a9..d764a9c 100644 Binary files a/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/graphics/walker_setup.png and b/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/graphics/walker_setup.png differ diff --git a/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/graphics/walker_setup.svg b/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/graphics/walker_setup.svg index a6fff3d..c9ff3c7 100644 --- a/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/graphics/walker_setup.svg +++ b/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/graphics/walker_setup.svg @@ -2,16 +2,16 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + @@ -3278,697 +3385,6 @@ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex b/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex index d83e3a8..dc4768e 100644 --- a/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex +++ b/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex @@ -68,7 +68,7 @@ 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 time is measured in units of inverse energy. + = 1\) such that energy is measured in units of frequency. \subsection{Transformations and Rotating Frames} \label{sec:rotating-frames} @@ -145,14 +145,16 @@ 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 -\(\tilde{H}_{A}'\) are reached through a transformation into a +\(H^{T}\) 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. +\(α_{β} = α_{i_{0},β}\) 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 @@ -841,7 +843,8 @@ where \(X=S,B\). 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).} + 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 @@ -1065,7 +1068,7 @@ To account for this, we modify \cref{eq:51} to \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 a_{S,0}^†a_{S,0}η_{S} a^†a - \iu \pqty{η_{S}+2Δδ} ∑_{n}a_{B,n}^†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 @@ -1113,6 +1116,8 @@ With this, we have for \(m=\pm,-N,-N+1,\ldots,-1,1,2,\ldots,N\) \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} @@ -1128,6 +1133,9 @@ Next, we compute the transformation of the interaction 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} @@ -1149,14 +1157,14 @@ Let us now assume that the voltage modulation takes the form of \pqty{\hat{ω}_{j}+\hat{δ}_{j}}t}\eu^{-\iu \varphi_{j}}}. \end{equation} -Transforming into the rotating frame of the \(\tilde{c}_{m}\) (see +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}}}t} \eu^{\iu \varphi_{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}}}t} \eu^{-\iu \varphi_{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 @@ -1294,7 +1302,7 @@ To calculate the susceptibility (see \cref{eq:87}), we evaluate with \begin{equation} \label{eq:117} - \tilde{ω}_{\pm} \equiv ω_{σ}^{0}-ε_{σ} = ω_{S} \pm δ\cos(ψ) - ε_{A} + \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} @@ -1304,8 +1312,8 @@ Finally we arrive at \label{eq:118} \begin{aligned} χ(t,s) &= χ_{0}Θ(s) - ∑_{γ,σ,σ'}\eu^{{-\iu\bqty{\tilde{ω}_{σ}t - - \tilde{ω}_{σ'}s + ω_{γ}(t-s)} - + ∑_{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). @@ -1320,11 +1328,11 @@ The stationary and non-stationary susceptibilities work out to be \begin{align} \label{eq:121} χ_{1}(t) &= χ_{0} - ∑_{γ,σ}\eu^{{-\bqty{\iu\pqty{\tilde{ω}_{σ} + ω_{γ}} + + ∑_{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}_{σ} + - ω_{γ}} + λ_{γ}}t} \eu^{\bqty{\iu\pqty{\tilde{ω}_{\bar{σ}} + + 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, @@ -1344,8 +1352,8 @@ coherent input beam with frequency \(ω\) in the limit of \(t\gg λ_{γ}\) \begin{equation} \label{eq:123} - ∫_{0}^{t}\eu^{-\iu ω s} χ(t-s)\dd{s} = \eu^{\iu ω t} χ_{0} ∑_{σγ}\frac{O_{σ,{γ}}O^{-1}_{γ,σ}}{\iu\pqty{\tilde{ω}_{σ} + ω_{γ}-ω} + - λ_{γ}}\frac{\eu^{\iu σψ}}{\cos(ψ)}\equiv \eu^{-\iu ω t}T_{S}(ω). + ∫_{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 @@ -1354,21 +1362,30 @@ vector, e.g. the intensity, averaged over the oscillation period of the input light becomes \begin{equation} \label{eq:128} - \bar{I} = I_{0}\pqty{1-2\Re{T_{S}(ω)} + \abs{T_{S}(ω)}^{2}} \approx - I_{0}\pqty{1-2\Re{T_{S}(ω)}} , + \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{T_{S}(ω)} = χ_{0} ∑_{σγ}\frac{λ_{γ}\abs{O_{σ,γ}}^{2}}{\pqty{\tilde{ω}_{σ} + ω_{γ}-ω}^{2} + + \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 peaks and influence -the peak heights. We also see, that we only have a good signal on the -states that have some overlap with the small loop. +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}