text:
\section{Characterics of optical speckle}\label{characteristics}

Optical speckle can be understood as the self-interfering wave field of a laser after acquiring random phase by reflection off rough surfaces or transmission through disordered media, called a diffuser~\cite{goodman2007speckle}. We will focus on the transmission case and assume that the spatial scale of the disorder $\sigma$ is small in comparison to the laser beam size and that the diffuser transmits light uniformly. The transmitted field can be intuitively thought of as of many waves scattered from microscopic elements comprising the diffuser.  So randomness arises.  As a disordered field, optical speckle is characterized by its intensity distribution, spatial intensity correlation function and power spectral density (PSD).

As shown in Fig.~\ref{fig:speckle_model} shows, ray optics in the paraxial limit provides a simple and useful approach to estimating the on-axis beam properties of a speckle beam a distance $z$ beyond a diffuser.  As a collimated laser beam of wavelength $\lambda$ travels through a diffuser of diameter $D_d$, it acquires a local divergence angle $\theta_d\simeq \lambda / (2 \sigma)$.

Figure~\ref{fig:speckle_model}a depicts the most simple case consisting of an isolated diffuser, for which there are two qualitatively different regimes: A near-field regime with $z < D_d / (2 \theta_d)$, where the typical length scale of optical speckle is $\sigma$, and a far-field regime where the NA of the diffuser increases the speckle scale to $(\lambda / 2)\times(D_d / 2 z)$.  This simple approach is insufficient because we will be interested in micrometer scale speckle, which is far smaller than the 10 to 100 micrometer scale of $\sigma$ for commercial  diffusers.

In Fig.~\ref{fig:speckle_model}b we add a lens with diameter $D_L$ and focal length $f$  just after the diffuser.  In the focal plane of the lens, the speckle scale is set by the lens NA, giving a speckle length scale $\lambda f / D_L$, independent of $\sigma$. In contrast, the beam width at the focal plane $w(f) \simeq 2 f \theta_d$ is set by the speckle scale $\sigma$ and not the lens diameter.

In the following section we will derive the origin of these design guidelines from  the paraxial wave equation.

\subsection{Gaussian beam equations with speckle}

\begin{figure}[htbp]
    \begin{center}
        \includegraphics{Fig1_Example.pdf}
    \end{center}

    \caption{Optical speckle schematic. (a) A collimated beam is transmitted through a rough medium and its intensity is measured in plane $z$. (b) The diverged beam after the rough medium is imaged by a lens at plane $z=z_L$ and $f$ is the focal point of the lens. (c) Field-field correlation length for a Gaussian speckle beam initially with $\sigma=100\ \mu{\rm m}$ and $w=25\ {\rm mm}$ as a function of propagation distance. The red curves plot $c_E(z)$ computed with (solid) and without (dashed) a lens with focal length $f=100\ {\rm mm}$ at $z_L=25\ {\rm mm}$.
    }
    \label{fig:speckle_model}
\end{figure}

We focus on monochromatic optical electric fields $E({\bf x}, t)$ with angular frequency $\omega$ traveling predominantly along ${\bf e}_z$.  Such waves can be decomposed as $E({\bf x}, t) = E_\perp({\bf r}; z) \exp[i(k_0 z -\omega t)]$, where $E_\perp({\bf r}; z)$ describes the transverse structure of the electric field with the high spatial frequencies associated with the nominal propagation along ${\bf e}_z$ factored out.  For spatial scales in excess of the optical wavelength the transverse field obeys the paraxial wave equation
\begin{equation}\label{para equation}
    -2i k_0 \partial_z E_\perp({\bf r}; z) = \left[-\nabla_\perp^2 + k_0^2 \chi({\bf r}; z)\right] E_\perp({\bf r}; z)
\end{equation}
traveling in a material with relative susceptibility $\chi({\bf r}; z)$.  We will suppress the $\perp$ subscript in the remainder of our discussion.

Upon traversing through a thin but disordered material with susceptibility $\chi({\bf r}) $ and thickness $\delta z$, an initially Gaussian wave field $E^-({\bf r}, 0) = E_0\exp{-{\bf r}^2/w^2}$ acquires a position dependent complex phase $\phi({\bf r}) = \chi({\bf r}) k_0 \delta z/2$.  The resultant field
\begin{equation}
    E^+({\bf r}, 0) = E^-({\bf r}, 0)\exp[- i \phi({\bf r})]
\end{equation}
carries the imprint of the disordered medium. The field a distance $z$ beyond the speckle plate follows from
\begin{align}\label{para field}
    E({\bf r}; z) &= \frac{-i k_0}{2\pi z}\int d^2{\bf r'} E^+({\bf r'}; 0) e^{-i k_0|{\bf r}-{\bf r'}|^2/2z},
\end{align}
the formal solution to the paraxial wave equation Eq.~(\ref{para equation}).  We model typical diffusion plates, for which: (1) the correlation function of the susceptibility $\langle \chi({\bf r_1})\chi({\bf r_2}) \rangle$ depends only on relative distance $|{\bf r_1}-{\bf r_2}|$, where $\langle ... \rangle$ denotes the ensemble average over disorder realizations. (2) the variation of the imprinted phase $\phi({\bf r})$ is much larger than $2\pi$ with
\begin{align}\label{eq:zeromean}
\langle \exp\left[- i \phi({\bf r}_1)\right]\rangle &= 0,
\end{align}
i.e., $\phi({\bf r})$ is uniformly distributed over the interval $[-\pi,\pi]$.

We turn to the field-field correlation function
\begin{equation}\label{C_E}
\begin{aligned}
    C_E({\bf r}_1,{\bf r}_2;z) &=\langle E({\bf r}_1; z)E^\ast({\bf r}_2; z)\rangle\! -\! \langle E({\bf r}_1; z)\rangle\langle E^\ast({\bf r}_2; z)\rangle
\end{aligned}
\end{equation}
to characterize the statistical properties of the disordered electric field.  Equation~\eqref{eq:zeromean} implies that the second term is zero.  At $z=0$, the uniform phase distribution implies $\langle E^+({\bf r}; 0) \rangle = 0$, giving
\begin{align*}
\frac{C_E({\bf r}_1,{\bf r}_2;0)}{E_0^2 } &= \exp\left(-\frac{{\bf r}_1^2 + {\bf r}_2^2}{w^2}\right) \langle \exp\left\{- i \left[\phi({\bf r}_1)-\phi({\bf r}_2)\right]\right\}\rangle.
\end{align*}
Under the assumptions of the typical diffusion plates, we model the phase-phase correlation function
\begin{align}\label{gaussian C_E}
\langle \exp\left\{- i \left[\phi({\bf r}_1)-\phi({\bf r}_2)\right]\right\}\rangle &= \exp(-\frac{|{\bf r}_1-{\bf r}_2|^2}{\sigma^2}),
\end{align}
with a Gaussian decay of width $\sigma$ that is amenable to the following analytic treatments.  The relation
\begin{align}\label{eq:sum_CE_zero}
\langle \exp\left\{- i \left[\phi({\bf r}_1)+\phi({\bf r}_2)\right]\right\}\rangle &= 0,
\end{align}
that follows from Eq.~\eqref{eq:zeromean}, in conjunction with the assumption that the correlation function depends only on relative distance, will be useful as well.

\support{
Lets have a look at the computation of this quantity.  As a first step we will re-express the coordinates in terms of relative and center of mass like coordinates, specifically defining
\begin{align*}
\bar {\bf r} &= \frac{{\bf r}_1 + {\bf r}_2}{\sqrt{2}} && {\rm and} & {\boldsymbol \delta} {\bf r} &= \frac{{\bf r}_1 - {\bf r}_2}{\sqrt{2}}.
\end{align*}
I added the $\sqrt{2}$ terms to make the Jacobin of the transformation equal to 1.  In this basis we have the replacements
\begin{align*}
{\bf r}_1 &= \frac{\bar {\bf r} +  {\boldsymbol \delta} {\bf r}}{\sqrt{2}} && {\rm and} & {\bf r}_2 &= \frac{\bar {\bf r} - {\boldsymbol \delta} {\bf r}}{\sqrt{2}}.
\end{align*}
Now preparing for the integrals we need the two terms
\begin{align*}
|{\bf r}_1 - {\bf r}_2|^2 &= 2|{\boldsymbol \delta} {\bf r}|^2 && {\rm and} & |{\bf r}_1|^2 + |{\bf r}_2|^2 &= \frac{1}{2}\left[\left(|\bar {\bf r}|^2 + 2\bar {\bf r}\cdot{\boldsymbol \delta} {\bf r} + |{\boldsymbol \delta} {\bf r}|^2 \right) + \left(|\bar {\bf r}|^2 - 2\bar {\bf r}\cdot{\boldsymbol \delta} {\bf r} + |{\boldsymbol \delta} {\bf r}|^2 \right)\right]. \\
&&&&&= |\bar {\bf r}|^2 + | {\boldsymbol \delta} {\bf r}|^2
\end{align*}
to insert into the argument of the Gaussians.  All together the integral in the denominator then becomes
\begin{align*}
\iint |C_E({\bf r}_1,{\bf r}_2;0)|d^2{\bf r}_1d^2{\bf r}_2 &= \iint d^2\bar {\bf r} d^2{\boldsymbol \delta} {\bf r} \exp(-\frac{|\bar {\bf r}|^2}{w^2}) \exp\left[-\left(\frac{2}{\sigma^2} + \frac{1}{w^2}\right)|{\boldsymbol \delta} {\bf r}|^2\right] = \left[\pi D^2\right]\left[\pi \frac{\sigma^2 w^2}{2w^2 + \sigma^2}\right].
\end{align*}
I added the square brackets at the end to make clear that the $\bar r$ integral doesn't even matter since it is common to the numerator and denominator defining $c_E(0)$.  The integral in the numerator is
\begin{align*}
\iint |C_E({\bf r}_1,{\bf r}_2;0)| |{\bf r}_1 - {\bf r}_2|^2 d^2{\bf r}_1d^2{\bf r}_2 &= \left[\pi w^2\right]\iint d^2{\boldsymbol \delta} {\bf r} \left[2 |{\boldsymbol \delta} {\bf r}|^2\right] \exp\left[-\left(\frac{2}{\sigma^2} + \frac{1}{w^2}\right)|{\boldsymbol \delta} {\bf r}|^2\right] = \left[\pi w^2\right]\left[2\pi\left(\frac{\sigma^2 w^2}{2w^2 + \sigma^2}\right)^2\right].
\end{align*}
Finally giving
\begin{align*}
c_E(0)^2 &= \frac{2 w^2 \sigma^2 }{2w^2 + \sigma^2} = \sigma^2\left(1 + \frac{2 w^2}{\sigma^2}\right)^{-1} \approx \sigma^2\left(1 - \frac{2 w^2}{\sigma^2}\right) \xrightarrow[w\rightarrow\infty]{} \sigma^2
\end{align*}
So we see that for for large beam waists, we recover the intrinsic correlation length.  Great!  When it comes to know the correlation function at all positions, this set of variables will continue to be handy!

How shall we proceed, and what to expect?  Well the next task is to compute the correlation function $C_E({\bf r}_1,{\bf r}_2;z)$ at all $z$, from physical arguments, we expect a result of the same mathematical form as in Eq.~\eqref{gaussian C_E}, with a Gaussian width with a new width $D(z)$, and a correlation length with a new scale $\sigma(z)$.  I.e., we expect a solution of the form (up to a complex phase)
\begin{align*}
\frac{C_E({\bf r_1},{\bf r_2};z)}{E_0^2} &\propto \exp(-\frac{{\bf r_1}^2 + {\bf r_2}^2}{w(z)^2}) \exp(-\frac{|{\bf r_1}-{\bf r_2}|^2}{\sigma(z)^2}).
\end{align*}
Now that we know the shape of the answer, lets find it!

There are two important terms in the correlation function, and we need to know both for arbitrary $z$ (don't worry the first is trivial!)
\begin{align*}
\langle E({\bf r}; z)\rangle &=  \frac{-i k_0}{2\pi z}\int d^2{\bf r'} \langle E^+({\bf r'}; 0)\rangle e^{-i k_0|{\bf r}-{\bf r'}|^2/2z} = 0
\end{align*}
the next term is less nice, but not {\it too} horrible
\begin{align*}
\langle E({\bf r}_1; z)E({\bf r}_2; z)^*\rangle &= \frac{k_0^2}{4\pi^2 z^2} \int d^2{\bf r}^\prime_1 \int d^2{\bf r}^\prime_2 \langle E({\bf r}_1^\prime; 0)E({\bf r}_2^\prime; 0)^*\rangle \exp\left[{-\frac{i k_0}{2z}\left(|{\bf r}_1-{\bf r}^\prime_1|^2 - |{\bf r}_2-{\bf r}^\prime_2|^2\right)} \right]
\end{align*}
As a sanity check, $C_E({\bf r_1},{\bf r_2};z)$ reduces to $C_E({\bf r_1},{\bf r_2};0)$ when $z=0$.

checked with the modified definition of $z_R$, $w(z)$, $R(z)$ can be expressed in the same way as usual Gaussian beam.

In case one
\begin{equation}
    \begin{cases}

    &1/R(z) = 1/z - k_0^2w^2/8Az^3\\
    &1/\sigma^2(z)=k_0^2w^2/8z^2 - k_0^4w^4/64Az^4 - k_0^2/16Az^2\\
    &1/w^2(z) = k_0^2/8Az^2\\
    &A = 1/2w^2 + 1/\sigma^2 + k_0^2w^2/8z^2
    \end{cases}
\end{equation}
These quasi-Gaussian coefficients are expressed in terms of the effective Rayleigh length(s?) $z_{\rm R}$, $z_{{\rm R}, \sigma}$.  Depending on the equations, these may be related to the usual Rayleigh range by a $M^2$ parameter.
}

We first consider the case illustrated by Fig.~\ref{fig:speckle_model}(a) where a Gaussian beam goes through a large disordered medium. The field-field correlation function at all positions following the disordered medium  can be exactly computed and takes the form
\begin{align}
\frac{C_E({\bf r}_1,{\bf r}_2;z)}{E_0^2} =& \left[\frac{w}{w(z)}\right]^2 \exp(-ik_0\frac{{\bf r}_1^2 - {\bf r}_2^2}{2 R(z)})\label{eq:C_E}\\
&\times \exp(-\frac{{\bf r}_1^2 + {\bf r}_2^2}{w(z)^2})\exp(-\frac{|{\bf r}_1-{\bf r}_2|^2}{\sigma(z)^2}) \nonumber
\end{align}
reminiscent of that of Gaussian beams.

This correlation function is characterized in terms of three $z$-dependent functions: the beam waist $w(z)$, the radius of curvature $R(z)$, and the correlation length $\sigma(z)$.  Each of these is simply related to a reduced Rayleigh range $z_{\rm R}^* = z_{\rm R}/M$, with conventional Rayleigh range $z_{\rm R} = k_0 w^2/2$ and beam quality factor $M^2 = 1+2w^2/\sigma^2$.  The resulting coefficients
\begin{align}
    \left[\frac{w(z)}{w}\right]^2 &= \left[\frac{\sigma(z)}{\sigma}\right]^2 = 1+\left(\frac{z-z_0}{z_{\rm R}^*}\right)^2\label{eq:rayleigh}
\end{align}
and
\begin{align}
    \frac{R(z)}{z-z_0} = 1 +\left(\frac{z_{\rm R}^*}{z-z_0}\right)^2
\end{align}
take the same form as a usual Gaussian beams focused at $z_0$.  Lastly, as in Fig.~\ref{fig:speckle_model}(b), an ideal lens with focal length $f$ at position $z_L$ gives new Gaussian beam parameters defined by
\begin{align}\label{lens making}
\frac{w^\prime}{w} &= \frac{\sigma^\prime}{\sigma} = f \left[\left(z_0'-z_L-f\right)^2+z_{\rm R}^{*2}\right]^{-1/2}
\end{align}
and
\begin{align*}
\left(z_0^\prime-z_L\right)^{-1} &=  f^{-1} - \left[\left(z_L-z_0\right) + \frac{z_{\rm R}^{*2}}{z_L-z_0-f} \right]^{-1}
\end{align*}
\support{
results from the paper, for large $z_{\rm R}^*$, $w'$ is complex. $z_0'$ agrees.
\begin{align}
\frac{w^\prime}{w} = \frac{\sigma^\prime}{\sigma} &= \left[1 + \left(\frac{z_L-z_0}{f} \right)^2 - \left(\frac{z_R^*}{f}\right)^2\right]^{-1/2}, & {\rm and} && \left(z_0^\prime-z_L\right)^{-1}  &=  f^{-1} - \left[\left(z_L-z_0\right) + \frac{(z^*_R)^2}{z_L-z_0-f} \right]^{-1},
\end{align}
}
where the first expression defines the magnification and the second is analogous to the usual lens makers equation~\cite{Self1983}.  While this leaves $M^2$ unchanged, the Rayleigh range is altered owing to the change in $w$.
All together these relations fully define field-field correlation function $C_E$ throughout an ideal imaging system.

\support{
Lets have a go at simplifying $\sigma^2(z)$.
\begin{align*}
 \sigma^2(z)&=2w^2\frac{z^2+(z_R^*)^2}{z_R^2-(Z_R^*)^2} = 2w^2\frac{1+z^2/(z_R^*)^2}{z_R^2/(z_R^*)^2-1} = 2w^2\frac{1+z^2/(z_R^*)^2}{M^2-1}= 2w^2\frac{1+z^2/(z_R^*)^2}{2w^2/\sigma^2}\\
 &= \sigma \left(1+z^2/(z_R^*)^2\right).
\end{align*}
Remarkable: Even $\sigma$ takes same form as for the waist.  So I will simplify the equation to take this structure instead.
}

In most quantum-gas experiments, optical potentials are created using laser light in the far detuned limit, thereby experiencing a potential proportional to the optical intensity
\begin{equation}\label{intensity}
    I({\bf r}; z) = \frac{c \epsilon_0}{2} \left|E({\bf r}; z)\right|^2
\end{equation}
not the electric field directly.  The ensemble-averaged intensity
\begin{align}
\langle I({\bf r}; z) \rangle &= \frac{c \epsilon_0}{2} C_E({\bf r},{\bf r}; z), \label{eq:intensity}
\end{align}
simply related to the field-field correlation function in Eq.~(\ref{eq:C_E}), contains no information about the optical speckle excepting for the changed $M^2$.

As discussed in the next section, the power spectral density (PSD) of the intensity
\begin{align}
    \rho({\bf k}; z) &= \langle \tilde{I}({\bf k};z)\tilde{I}^*({\bf k};z)\rangle\nonumber \\
    &=  \frac{\pi^2w^2(z)}{4M^2}\exp{-\frac{{\bf k}^2w^2(z)}{4M^2}},\label{psd}
\end{align}
computed using Eq.~\eqref{eq:C_E}, describes the momentum-change imparted by the speckle potential to a moving atomic wavepacket.

\subsection{Correlation length}
The field-field correlation length
\begin{align}
    c_E(z)^2 &= \frac{\iint |C_E({\bf r}_1,{\bf r}_2; z)||{\bf r}_1-{\bf r}_2|^2d^2{\bf r}_1d^2{\bf r}_2}{\iint |C_E({\bf r}_1,{\bf r}_2 z)|d^2{\bf r}_1d^2{\bf r}_2}\\
    &= \frac{2 w(z)^2 \sigma(z)^2}{2w(z)^2 + \sigma(z)^2} \approx \sigma(z)^2
\end{align}
obtained from Eq.~(\ref{eq:C_E}), sets the scale over which the electric field retains its spatial coherence.  The field-field correlation length is minimized at $z=z_0$, and is always larger than $\sigma$.  Generally speckle beams operate in the regime $w \gg \sigma$, where there are many speckle grains within a large beam, giving the final approximate relation.

As was already noted in our ray-optics discussion, this has important implications for experiment design.  For cold atom experiments such as ours, the large momentum-change imparted by short-length scale speckle is essential, where a correlation length at or below the micron scale is desirable.  Since the correlation length available for typical commercial diffusers ranges form $10\ {\rm \mu m}$ to $100\ {\rm \mu m}$, an additional focusing stage is required .

A focusing lens can easily take the $10\ {\rm \mu m}$ to $100\ {\rm \mu m}$ correlation length available for typical commercial diffusers and create a beam with sub-micrometer correlation length at its focus.  Figure~\ref{fig:speckle_model}c compares the correlation length of a beam with (red solid) and without (red-dashed) a focusing lens for the specific case of an initial laser beam of wavelength $\lambda = 532\ {\rm nm}$ with input Gaussian beam parameters: focal point $z_0=0$, beam waist $w = 25\ {\rm mm}$ and correlation length $\sigma = 100\ \mu{\rm m}$.
This beam is focused by a lens of focal length $f=100\ {\rm mm}$, the correlation length at the focus is $c_E=0.96\ \mu{\rm m}$.  The remaining derived beam parameters are $M^2 \approx 1.25\times10^5$, $z_R \approx 3.7\ {\rm km}$, and $z_R^* \approx 10.4\ {\rm m}$.

\subsection{Impact of apertures}

In the case of focusing optical speckle as shown in Fig.~\ref{fig:speckle_model}(b), a lens of focal length $f$ and diameter $D_L \ll w$ is placed at $z=z_L \leq k_0\sigma^2$. The field in the plane $z=z_L$ before the lens, $E^-({\bf r};z_L)$ is essentially unchanged from field $E^+({\bf r};0)$. The field $E^-({\bf r};z_L)$ passes through the lens aperture, where it acquires a position dependent phase and is truncated outside the lens. The emerging field $E^+({\bf r};z_L)$ propagates to the focal plane $z=f+z_L$ where it is
\begin{equation}\label{field in z}
    E_f({\bf r}) = \frac{-i k_0}{2\pi f}e^{-i k_0{\bf r}^2/2f}\int \displaylimits_{|{\bf r'}|<\frac{D_L}{2}} d^2{\bf r'} E^+({\bf r'}; 0) e^{i k_0{\bf r}\cdot{\bf r'}/f}.
\end{equation}
When $\sigma \ll D_L \ll w$, the field-field correlation function at the focal plane is
\begin{align}
C_{E,f}({\bf r}_1,{\bf r}_2) \approx & C_0\exp[-\frac{ik_0({\bf r}_1^2-{\bf r}_2^2)}{2f}]\\
&\times\exp[\frac{-k_0^2\sigma^2({\bf r}_1 + {\bf r}_2)^2}{16f^2}]\frac{J_1(k_c \Delta r/2)}{k_c\Delta r /2}.\nonumber
\end{align}
Here $C_0 =  k_0^2E_0^2D_L^2\sigma^2 / 8f^2$ is the peak correlation amplitude; $\Delta r = |{\bf r}_1-{\bf r}_2|$ is the relative position coordinate; and $J_1$ is a Bessel function of the first kind.  The ratio
\begin{align}
k_c &= k_0\frac{D_L}{f}\label{kc}
\end{align}
is a cutoff above which the PSD of the intensity
\begin{align}
    \rho_f(k) = C_0^2 \frac{2}{\pi k_c^2}\left[ \cos^{-1}\left(\frac{k}{k_c}\right)
     -\frac{k}{k_c}\sqrt{1-\frac{k^2}{k_c^2}}\right]\label{psd_image}
\end{align}
is strictly zero.  Equation~\eqref{psd_image} is valid near the optical axis where $|{\bf r}_1|,|{\bf r}_2| \ll w(z)$.

\begin{comment}
When $D \simeq D_L$, $c_I(f)$ and $k_c(f)$ can be expressed in terms of the NA of the lens which is defined as ${\rm NA} = nsin(\theta) \approx D_L/2f$, $n$ is the index of refraction of the medium in which the lens is working and $\theta$ is the maximal half-angle of the cone of light that can enter or exit the lens.
\begin{equation}
    \begin{aligned}
    &c_I(f) \approx \frac{1.22}{{\rm NA}}\lambda\\
    &k_c(f) \approx 2{\rm NA}/\lambda
    \end{aligned}
\end{equation}
For a fixed wavelength, correlation length $c_I(f)$ is inversely proportional to the NA of the lens while $k_c(f)$ is proportional to NA.  Intuitively, it can be understood as the finest structure of optical pattern in a image is dependent on the highest k vectors of a optical beam that can be focused by the lens, so the higher the NA is, the higher k vectors can be captured by the lens to form patterns with smaller $c_I(f)$. $k_c(f)$ corresponds to the largest possible $k$ exchange by two rays focused by the lens, so it is proportional to NA.
\end{comment}

\subsection{Field and intensity probability distribution}

In the previous sections, we focused on the average properties of speckle fields.
Here we extend this discussion to predict the probability distribution of the electric field strength $P(E)$ and intensity $P(I)$.
Our approach focuses first on $P(E)$, and consists of two steps: (1) we find the regime when the central limit theorem applies, thereby assuring a Gaussian probability distribution; and (2) we identify $\langle E \rangle$ and $\langle E^2 \rangle$ as the lowest moments of the distribution, fully defining the Gaussian distribution.

We now interpret the electric field
\begin{align*}
    E({\bf r}; z) &= \frac{-i k_0}{2\pi z}\int d^2{\bf r'} E^-({\bf r}')e^{-i \phi({\bf r'}) } e^{-k_0|{\bf r}-{\bf r'}|^2/2z},
\end{align*}
of Eq.~\eqref{para field} as a random variable constructed from a sum over incoherent complex phasors.  The cross correlation function (CCF) $\langle E({\bf r}_1; z) E({\bf r}_2; 0)\rangle$ specifies the range over which the initial random field contributes to the final field.  The closed form expression for this CCF is similar to the field-field correlation function in Eq.~\eqref{eq:C_E}; the length scale for the decay of correlations $\sigma_{\rm CCF}(z)$ again obeys  Eq.~\eqref{eq:rayleigh}, but with $M_{\rm CCF}^2=(1 + w^2/\sigma^2 )^2$.  When $w\gg\sigma$, i.e., the initial waist is much larger than the speckle size,  the resulting Rayleigh range reduces to $z_{\rm R, CCF} = k_0 \sigma^2/2$: as if each random source was an individual Gaussian beam with extent $\sigma$.  The criterion that a field $E({\bf r}; z)$ have contributions from many incoherence sources is therefore $\sigma_{\rm CCF}(z)/\sigma\gg1$, i.e., $z\gg z_{\rm R, CCF}$.

This identifies the central limit theorem's regime of applicability, and we now consider $E({\bf r}; z)$ as a complex valued Gaussian random variable.  The probability distribution for electric field is therefore a function of two independent degrees of freedom, here we select the quadrature variables $E$ and $E^*$, giving $P(E, E^*)$.   Most moments of this quantity are easy to identify using Eqs.~\eqref{para field}, \eqref{eq:zeromean}, \eqref{gaussian C_E} and \eqref{eq:sum_CE_zero}: $\langle E \rangle = \langle E^2 \rangle = 0$, and similarly for $E^*$.   Then Eqs.~\eqref{C_E} and the following discussion assure us that  $\langle E E^* \rangle = \langle |E|^2 \rangle$ takes on a non-zero value.  Together these fully define the Gaussian probability distribution for electric fields
\begin{align}
    P(E, E^*) &= \frac{1}{\pi \langle |E|^2 \rangle} \exp\left(-\frac{|E|^2}{\langle |E|^2 \rangle}\right)\label{eq:dist_fields},
\end{align}
and using Eq.~\eqref{eq:intensity}, the intensity distribution
\begin{align}
    P(I) &= \frac{1}{\langle I \rangle} \exp\left(-\frac{I}{\langle I \rangle}\right) \label{P of int}
\end{align}
follows directly.
The intensity of a speckle field obeys an exponential distribution and the mean speckle intensity $\langle I \rangle$ should be equal to its standard deviation $\sqrt{\langle I^2 \rangle}$.

\subsection{Simulated speckle and the comparison to experiment} \label{Numerical_speckle}
\begin{figure}[tbp]
    \begin{center}
    \includegraphics{optical_speckle.pdf}
    \end{center}

    \caption{Simulated and measured optical speckle. The columns in the figure correspond to: simulated speckle with uniform laser beam,  simulated speckle from a Gaussian laser beam and measured speckle. In each column, the first row shows the intensity of the optical speckle field. The second row shows the PSD of the intensity shown in the first row (symbols).  The red curve shows a fit of Eq.~(\ref{psd_image}) to the data, along with the resulting $k_c$. The third row histograms the  intensity from the first row.}
    \label{fig:Optical speckle}
\end{figure}

Having now fully set the stage for understanding and creating speckle laser beams, we turn to a laboratory confirmation of key prediction of these models relevant to cold atom experiment: the field-field correlation length $C_E$ and the distribution of intensities $P(I)$.

In our lab, we directed a collimated laser beam (waist $w\approx25\ {\rm mm}$) through a diffuser (divergence angle $\theta_d = 0.5^\circ$, and aperture $D=20\ {\rm mm}$) focused by immediately by a lens (focal length $f = 30\ {\rm mm}$) as depicted by Fig.~\ref{fig:speckle_model} and quantified the the optical speckle formed at the focal plane.
We then imaged the optical speckle onto a charge coupled device (CCD) camera using a Keplerian telescope with magnification $M=46$.
The CCD's $1024 \times 1280$ array of $4.8\ \mu {\rm m}$ pixels gave a $100\ \mu{\rm m} \times 130\ \mu{\rm m}$ magnified field of view with $0.1\ \mu {\rm m}$ pixels.

Our analytic results for $C_E$ are valid in the Gaussian beam limit ( $w\ll D$) or uniform illumination limit ($w\gg D$).  Because our experiment has $w\approx D$, we numerically simulated the optical speckle to compare with our measurements and both models.

For the numerical simulation, the desired optical speckle field $E_{i,j}$ is represented by a $1024 \times 1280$ array at the focal point of the lens.  We use the optical Fourier transform property of lenses to compute this efficiently, whereby the field a focal distance beyond the lens is related to the Fourier transform of the field a focal distance prior to the lens (which we will term the Fourier plane).  An important aspect of this method is that the $0.1\ \mu {\rm m}$ grid spacing in the focal plane transforms to a $1.5\ {\rm mm}$ grid spacing in the Fourier plane.

Our simulation progresses as follows.  (1) We first initialize $E_{i,j}(z=0)$ to the field of either a uniform field or a Gaussian beam.  (2) We then imprint random phases on each point~\footnote{The grid size is much larger than the correlation length of the diffuser, so the imprinted phase at each grid point is uncorrelated with all other points.}.  (3) We set the field outside our physical aperture to zero.  (4) Then we back-propagate the field to the Fourier plane and take the Fourier transform to obtain the field at the focal plane.

Figure~\ref{fig:Optical speckle} compares our measured speckle with numerics and our analytic model; the three columns depict: the case of a uniformly illuminated aperture, Gaussian illumination, and experiment.   The top row shows that intensity at the focal plane is qualitatively similar for all three cases. The middle row, the PSD (computed from the intensity in the top row, and plotted by blue symbols), highlights the differences.  In each case, we fit Eq.~(\ref{psd_image}) the PSD and extracted $k_c$ from the fits (red curves).  Because  Eq.~(\ref{psd_image}) was derived for a uniformly illuminated aperture it provides a good fit to the uniform illumination case, but deviates at large $k$ for Gaussian illumination and experiment.  In contrast, the numerics for Gaussian illumination and the experiment are indistinguishable.
The bottom row, we histogram the intensity distribution and verify that in all three cases we recover the expected exponential fall-off.

\section{Scattering of an SOBEC from a speckle potential} \label{Model}

\begin{figure}[htbp]
    \centering
    \includegraphics{Fig3_Dispersions.pdf}
    \caption{Fermi's Golden Rule. Momentum are expressed in units of the single-photon recoil momentum $\kr$ used to create SOC in (c). (a) Representative PSD for optical speckle with $k_c = 6 \kr$. (b) Free particle dispersion relation. The dashed arrow marks the boundary above which the FGR rate vanishes, while the solid arrow provide an example with non-zero rate. (c) SOC dispersion relations computed for $\delta = 0$ add $\Omega_{\rm R} = 1\Er$ colored according to the expectation value $\langle\sigma_z(q)\rangle$, with arrows marked as in (b).  Note the transition through the gap in the dispersion relation at $E\approx \Er$ where the FGR rate is nearly zero.}
    \label{fig:Dispersion_relations}
\end{figure}

We now focus on the motion of spin-orbit coupled bosons in a speckle-induced disorder potential.  In this section, we develop a Fermi's golden rule (FGR) approach for scattering from a disorder potential, both with and without SOC, schematically depicted in Fig.~\ref{fig:Dispersion_relations}.   The first order scattering processes captured by the FGR are possible when a matrix element (here from the disorder potential) can couple energetically degenerate initial and final states (here momentum or quasi-momentum states).  We will see that the strength of this coupling is proportional to the PSD of the speckle potential, an example of which is shown in Fig.~\ref{fig:Dispersion_relations}(a).  As depicted in  Fig.~\ref{fig:Dispersion_relations}(b), this implies an absence of scattering for momenta differences larger than the speckle-cutoff $k_c$.  Adding SOC, as in Fig.~\ref{fig:Dispersion_relations}(c), can suppress scattering for additional wavevectors.  Because a spin-independent speckle potential has no spin-changing matrix element, the energetically allowed transition at an energy $E/\Er \approx 1$ between states of opposite spin is strongly suppressed.  The following discussion quantifies these observations.

\subsection{Fermi's golden rule}

In this section we first develop our understanding of scattering from disorder potentials by deriving the FGR for spinless particles.  With that understanding in hand we turn to the case adding SOC.

\subsection{Spinless atoms}

For spinless free particles, the unperturbed Hamiltonian $H = \hbar^2 k^2 / 2m$ implies that we will study scattering between initial and final momentum states, labeled by  $\ket{k_0}$ and $\ket{k_f}$ respectively.  Figure~\ref{fig:Dispersion_relations}(b) depicts examples by open circles, with arrows connecting initial states to final states.

The time evolution of the initial state $\ket{\psi(0)} = \ket{k_0}$ subject to the speckle potential $V(x)$ may always be expressed as
\begin{align}
\ket{\psi(t)} &= \sum_{k}C_{k,k_0} (t) e^{-i\omega_k t}\ket{k}, \label{expand}
\end{align}
with $C_{k,k_0}(0) = \delta_{k,k_0}$ and $\hbar\omega_k = \hbar^2 k^2 / 2 m$.  The coefficients $C_{k,k_0} (t)$ are governed by the time-dependent Schr\"{o}dinger equation giving the exact expression
\begin{align}
    C_{k_f,k_0}(t) = &  C_{k_f,k_0}(0) + \label{allorder} \\
    & \frac{1}{i\hbar}\sum_k \bra{k_f}\hat V\ket{k} \int_0^t d\tau e^{i\omega_{k_f,k}\tau}C_{k,k_0}(\tau).\nonumber
\end{align}
with
\begin{align}
    \omega_{k,l} &= \omega_k-\omega_l, &{\rm and} && \hat V &= \sum_x V(x) \ket{x}\bra{x}.
\end{align}
An order-by-order perturbation theory is typically obtained by recursively inserting the integral expression for $C_{k_f,k_0}(t)$ back into the integrand; unfortunately, the general problem is intractable and we truncate the perturbation series at first order. This term is effectivly obtained by replacing $C_{k_f,k_0}(\tau)$ with $C_{k_f,k_0}(0) = \delta_{k_f,k_0}$, and find
\begin{equation}\label{firstC}
    C_{k_f,k_0} (t) = \delta_{k_f,k_0} + \frac{1}{i\hbar}\int_0^t d\tau\bra{k_f}\hat V\ket{k_0} e^{i\omega_{k_f,k_0}\tau}.
\end{equation}
Unfortunately we do not know $V(x)$ for any specific realization of the speckle potential.

In Sect.~\ref{characteristics} we characterized optical speckle in terms of second-order statistical metrics such as the PSD, here equal to $\rho(k_f-k_0) = \langle\bra{k_f}\hat V\ket{k_0}\bra{k_0}\hat V\ket{k_f}\rangle$, where the double-brackets indicate the ensemble average. The resulting ensemble averaged transition probability
\begin{equation}
 P_{f,0} (t) = \frac{\rho(k_f- k_0)}{\hbar^2}\left[\frac{2}{\omega_{f,0}} \sin\left(\frac{\omega_{f,0}t}{2}\right)\right]^2
\end{equation}
is a sharply peaked function centered at $\omega_{f,0} = 0$ with width $2\pi/t$, showing that a narrow range of energy matching states can be populated.  For long times, $\omega_{f,0}t \gg 1$ the quantity in square brackets converges to a scaled Dirac delta function $t\times\delta(\omega_{f,0})$.

Figure~\ref{fig:Dispersion_relations}(a) displays the normalized PSD for a speckle potential computed with $k_c = 6 \kr$, reminding us that $\rho(k) = 0$ for $k\geq k_c$.  Our FGR expression allows two types of scattering processes for the free particle dispersion shown in Fig.~\ref{fig:Dispersion_relations}(b).  In the first process, depicted by the black arrow, the atom's initial momentum is reversed, changed by $\Delta k = 2 k_0$; as indicated by the dashed line, this process is second-order forbidden for $k_0 \geq k_c/2$.  In the second process (not pictured), the atom's momentum is only infinitesimally changed: spreading the wave-packet, but leaving the average momentum unchanged.
This picture shows that back-scattering is essential for momentum-relaxation.

\subsection{Spin-orbit coupled atoms}

Our 1D SOC coupling~\cite{lin2011spin} is created by illuminating a two-level atom with a pair of counter-propagating lasers with wavelength $\lambdar$ tuned to drive stimulated Raman transitions between states $\{ \ket{q+\kr,\uparrow},\ket{q-\kr,\downarrow} \}$.  Here $\hbar\kr = 2\pi\hbar/\lambdar$ and $\Er = \hbar^2 \kr^2 / 2m$ are the single-photon Raman recoil momentum and energy respectively.
Subject to this Raman coupling, the atoms obey the 1D Hamiltonian
\begin{align}
\hat{H}(q) = \frac{\hbar^2 }{2m}\left(q\hat 1 + \kr\hat \sigma_z\right)^2 + \frac{\delta}{2}\hat \sigma_z - \frac{\hbar\Omega_{\rm R}}{2}\hat \sigma_x,\label{eq:soc}
\end{align}
where $\left\{\hat 1, \hat\sigma_x,\hat\sigma_y,\hat\sigma_z\right\}$ are the identity and Pauli operators, respectively.
Here $q$ is the quasi-momentum, $\Omega_{\rm R}$ is Raman coupling strength, and $\delta$ is the detuning from the two-photon Raman resonance condition.
The resulting dispersion relations, plotted in Fig.~\ref{fig:Dispersion_relations} for $\delta = 0$ and $\Omega_{\rm R} = 1\Er$, have energies $E^\pm(q)$  labeled by $q$ along with $\pm$ to indicate if they are in the upper or lower band.

These new energies and their associated amplitudes
\begin{align*}
\ket{q,\pm} &\propto a_\pm(q)\ket{q-\kr,\downarrow} + b_\pm(q)\ket{q+\kr,\uparrow}
\end{align*}
change the potential scattering processes, which we again compute using a FGR expression.  The coefficients
\begin{align*}
a_\pm(q) & = \mp\frac{\Omega_{\rm R}}{2}& {\rm and} && b_\pm(q) &= \pm\frac{\Delta(q)}{2} + \frac{\sqrt{\Delta^2(q) + \Omega_{\rm R}^2}}{2}.
\end{align*}
along with the quasi momentum dependent detuning
\begin{align}\label{delta}
    \Delta(q) &= \frac{2\hbar^2q\kr}{m} + \delta
\end{align}
fully define these superposition states.

Following the same FGR argument presented above for initial states $\ket{q_0,-}$ in the lower dispersion scattering from a spin-independent speckle potential, we find scattering probabilities
\begin{align}\label{transfer prob}
    P^\pm_{f, 0}(t) &= \frac{\rho(\Delta q)}{\hbar^2}\left|\frac{2\sin(\omega^\pm_{f, 0}t)}{\omega^\pm_{f, 0}}\!\matrixel{q_f,\pm}{e^{i\Delta q x}}{q_0,-}\right|^2
\end{align}
expressed in terms of the quasimomentum and energy differences $\hbar\Delta q = \hbar q_f-\hbar q_i$ and $\hbar \omega_{f,0}^\pm = E^\pm(q_f) - E^-(q_0)$.
For most initial states $\ket{q_0,-}$, such as the two higher energy states marked in Fig.~\ref{fig:Dispersion_relations}(c), the scattering is essentially unchanged from our spinless example, with scattering occurring between energy-matched states with the same initial and final spin.
In contrast, for initial states residing in the SOC energy gap there is no energy-matched state of the same spin available for back-scattering; as indicated by the dashed line, scattering is greatly suppressed.
We note that backscattering is not completely blocked, because the energy matching states $\ket{\pm q_0,-}$ are not spin-eigenstates and do have some spin-overlap.

\subsection{Computed scattering rates}

\begin{figure}[tbp]
    \centering
    \includegraphics{Fig4_Scattering_Rate.pdf}
    \caption{Fermi's golden rule scattering rate for $\Omega_{\rm R}/\Er = 0, 0.5, 2.5$ and $4.0$.
    Left column: SOC dispersion relations computed for each $\Omega_{\rm R}$, colored as in Fig.~\ref{fig:Dispersion_relations}.
    Right column: normalized scattering rate as a function of initial energy for the initial state $\ket{q_0, -}$ with $q_0 \geq q_{\rm min}$, i.e., in the bottom dispersion and to the right of the higher momentum local energy minimum.
    }
    \label{fig:scatteringRate}
\end{figure}

We now use these FGR expressions to compute the scattering rates for both forward scattering and back scattering processes.  Because we are interested in transport properties, we define forward scattering processes as those that leave the sign of the group velocity unchanged and back scattering processes and those that do reverse the direction of motion.
We therefore consider initial states $\ket{q_0,-}$ in the lower band with positive group velocity.  Because the lower energy SOC dispersion plotted in Fig.~\ref{fig:scatteringRate} can have a pair of minima located at $\pm q_{\rm min}$, we always select $q_0 > q_{\rm min}$ to assure positive group velocity.  We numerically evaluated the FGR for $^{87}{\rm Rb}$ atoms illuminated with $\lambdar=790\ {\rm nm}$ Raman lasers, giving $\Er = h \times 3.7\ {\rm kHz}$, and for speckle with $k_c = 6 \kr$.  The $t=13.4\ {\rm ms}$ interaction time was selected to be experimentally relevant.

The right panels of Fig.~\ref{fig:scatteringRate} show the normalized scattering rate computed for four different values of $\Omega_{\rm R}$, with the back-scattering rate plotted in black and forward scattering plotted in gray.  These rates combine the contributions from the $\pm$ bands in Eq.~(\ref{transfer prob}).

Panel (a), computed for $\Omega_{\rm R}=0$ (equivalent to the case with no SOC), shows two key effects.  Firstly, the diverging forward and back scattering rates at low energy follow from the diverging density of states (DoS) in 1D.  Secondly, the rate of back scattering (red) falls to zero when $\delta q > k_c$, while forward scattering (black) simply falls with the DoS.  Panels (b) and (c) show cases with a well resolved SOC energy gap.  As expected, back scattering is nearly completely suppressed for initial energies in the energy gap, while forward scattering is hardly changed.  In addition, a pair of singular features boarder the energy gap, resulting from the diverging DoS the local extrema of the dispersions.  Panel (d) shows the same phenomena, but just as the two minima at $\pm q_{\rm min}$ have merged into a single minimum at $q_{\rm min}=0$.

We therefore conclude, for non-interacting particles back scattering and momentum relaxation is nearly completely suppressed for atoms starting in the SOC energy gap.

\section{Numerical simulation of GPE}\label{GPE_simulation}

Our single particle FGR results only describe the short-time scattering from a disorder potential, they cannot describe the full approach to equilibrium.
To bridge the gap between the FGR and the physical system, we need to account for both higher order scattering processes and interparticle interactions.
In our proposed SOBEC realization, all aspects of SOC Hamiltonian and the speckle potential are tunable, making SOBECs an ideal system for exploring enhanced transport in 1D quantum wires.
\newline
\subsection{Gross-Pitaevskii equations}
Here we numerically study the deceleration of an SOBEC initially moving in a speckle potential using the time-dependent Gross-Pitaevskii equation (GPE).
The time-dependent GPE
\begin{align}\label{gpe}
i\hbar\partial_t\Psi({\bf r},t) &= \left[-\frac{\hbar^2}{2m}\nabla^2 + V({\bf r}) + g_{\rm 3D}|\Psi({\bf r},t)|^2\right]\Psi({\bf r},t)
\end{align}
is a non-perturbative dynamical description ~\cite{erdHos2010derivation} of a large number of interacting identical bosons occupying the same spatial mode $\Psi({\bf r}, t)$, normalized to the total atom number, $N=\int d^3{\bf r} |\Psi({\bf r}, t)|^2$.
The interaction strength $g_{\rm 3D} = 4\pi\hbar^2a_s/m$ can be expressed in terms of the $s$-wave scattering length $a_s$.
This GPE provides a good description of low-temperature spin-polarized BECs, with negligible thermal excitations~\cite{Dalfovo1999}.

Since our focus is on 1D transport, we must first obtain a 1D description of our 3D system~\cite{bao2003numerical}.
Here we, we assume that the potential $V({\bf r}) = V(x) + V_\perp(y,z)$ can be separated into a weak longitudinal potential $V_\parallel(x)$ along with a strongly confining transverse potential $V_\perp(y,z)$.
When the single-particle energy spacing from $V_\perp(y,z)$ greatly exceeds all other energy scales, the 3D wavefunction can be factorized into
\begin{equation}
    \Psi({\bf r},t) = \psi(x,t)\phi(y, z),
\end{equation}
containing a longitudinal term of interest giving the 1D density $n(x) = |\psi(x,t)|^2$, and a transverse term $\phi(y, z)$, normalized to unity, assumed to be the ground state of the transverse potential.
Inserting this ansatz into the GPE and integrating out the transverse degrees of freedom, gives the 1D GPE
\begin{equation}
    i\hbar\partial_t\psi = \left[-\frac{\hbar^2}{2m}\partial_x^2 + V(x) + g|\psi|^2\right]\psi,
\end{equation}
suitable for studying single-component 1D bosons in a speckle potential with 1D interaction strength
\begin{equation}
    g = g_{\rm 3D}\int dy dz |\phi(y,z)|^4.
\end{equation}
For compactness of notation, here and below, we shall omit the functional dependance of $\psi$ on $x$ and $t$.

\begin{widetext}
The two component 1D spinor GPE describing SOBECs extends Eq.~\eqref{eq:soc} to include interactions, and consists of a pair of coupled non-linear differential equations
\begin{align}
i\hbar\partial_t\psi_\uparrow &= \left[\frac{\hbar^2}{2m}\left(-i\partial_x + \kr\right)^2 + \frac{\delta}{2} + V(x) + g_{\uparrow\uparrow} |\psi_\uparrow|^2 + g_{\uparrow\downarrow}|\psi_\downarrow|^2\right]\psi_\uparrow + \frac{\Omega_{\rm R}}{2}\psi_\downarrow \\
i\hbar\partial_t\psi_\downarrow &= \left[\frac{\hbar^2}{2m}\left(-i\partial_x - \kr\right)^2 - \frac{\delta}{2} + V(x) + g_{\downarrow\downarrow} |\psi_\downarrow|^2 + g_{\uparrow\downarrow}|\psi_\uparrow|^2\right]\psi_\downarrow + \frac{\Omega_{\rm R}}{2}\psi_\uparrow
\end{align}
including the interaction strengths $g_{\uparrow\uparrow}$, $g_{\uparrow\downarrow}$, and $g_{\downarrow\downarrow}$.
Here we focus on the specific case of $^{87}{\rm Rb}$ atoms~\cite{kawaguchi2012spinor} in the $f=1$ ground state manifold and have selected $\ket{\uparrow} = \ket{m_F=0}$ and $\ket{\downarrow} = \ket{m_F = -1}$.
The interactions can be parameterized in terms of an $s$-wave pseudo-potential $(g_{0, {\rm 3D}} + g_{2, {\rm 3D}}\Vec{F}_\alpha\cdot \Vec{F}_\beta) \delta({\bf r}_i - {\bf r}_j)$ now dependent on spin.
In $^{87}{\rm Rb}$'s $f = 1$ manifold $g_{0, {\rm 3D}} = 100.86 \times 4\pi\hbar^2 a_{\rm B}/m$ is vastly larger than $g_{2, {\rm 3D}}\approx-4.7\times10^{-3} \times g_{0, {\rm 3D}}$, where $a_{\rm B}$ is the Bohr radius~\cite{vanKempen2002,Widera2006}.
The interaction coefficients reduce to effective 1D interaction strengths just as in the single component case, and are related to the generic coefficients~\cite{Ho1998,Ohmi1998} via $g_{\uparrow\uparrow} = g_0$ and $g_{\downarrow\downarrow}  = g_{\uparrow\downarrow} = g_0 + g_2$.
Table \ref{table:1} summarizes the parameters used in our simulations.
\end{widetext}

Our simulation results are divided into two sections: Sec.~\ref{single} hones our understanding by considering a single-component BEC evolving in a speckle potential, and then in Sec.~\ref{soc} we contrast to the case with SOC.
In both sections, we simulate initially trapped BECs accelerated to an initial momentum $k_0$ or quasi-momentum $q_0$ and we study their deceleration.
All the results are averaged over 20 speckle realizations, as in  Fig.~\ref{fig:Optical speckle}(b).
The average speckle potential $h\times 200\ {\rm Hz} \approx 0.05\Er$ was selected to be weak enough to cause no trapping effect yet strong enough to produce significant deceleration within $15\ {\rm ms}$.

\begin{table}
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{l c c}
\hline\hline
Description & Symbol & Value \\
\hline
$^{87}{\rm Rb}$ atomic mass & $m$ & $1.42\times10^{-25}~{\rm kg}$\\
Raman laser wavelength & $\lambdar$ & $790~{\rm nm}$ \\
Recoil energy & $\Er$ & $h\times 3.678\ {\rm kHz}$\\
Dipole trap frequency & $\omega/2\pi$ & $10~{\rm Hz}$  \\
Angle of Raman beams & $\theta_{\rm R}$& $180^{\circ}$ \\
Speckle potential cut off &  $k_c$ & $6\kr$\\
Average speckle potential & $\overline{V(x)}$ & $0.05\Er$\\
Grid spacing  & $\delta x$ & $66\ {\rm nm}$\\
Grids points (single-component)& $N_x$ & $2^{14} + 1$\\
Grids points (SOC)& $N_x$ & $2^{13} + 1$\\
Atom number & $N$ & $2\times 10^5$ \\
Chemical potential & $\mu$ & $h\times 300\ {\rm Hz}$ \\
\hline\hline
\end{tabular}
\normalsize
\caption{Simulation parameters}
\label{table:1}
\end{table}

\subsection{Single component systems}\label{single}

\begin{figure}[tbp]
    \centering
    \includegraphics{Fig5_Single_Localization.pdf}
    \caption{Single-component GPE simulation with $k_c / \kr = 6$.
    (a) Density distributions.  The filled red curve depicts the initial density distribution, while the black and red curves show the final-state density distributions for initial momenta $\ket{k_0=0.2\kr}$ and $\ket{k_0=3.1\kr}$,above and below $k_c/2$, respectively.
    (b) Mean momentum. $\langle k(t) \rangle$ averaged over 20 speckle realizations is plotted for a range of initial momentum in the range of $0$ to $3.3\kr$, the $t=0$ point of each curve marks the initial $k_0$.
    (c) Deceleration.  The colored symbols plot $k_f = \langle k(t=16\ {\rm ms}) \rangle$ as a function of $k_0$ along with their standard deviations, and the black line marks $k_f = k_0$ corresponding to ballistic motion.
    }
    \label{fig:single}
\end{figure}

The simulations are performed in three steps to as accurately as possible model a realistic experimental sequence.
First, we initialize a ground state BEC in a harmonic trap using imaginary time evolution~\cite{chiofalo2000ground}, giving the density distribution plotted in black in Fig.~\ref{fig:single}(a)], and follow with real-time evolution.
Second, because the BEC's narrow momentum distribution is centered at $k=0$, we briefly apply a linear potential $\alpha x$ with time-evolution approximately described by the phase factor $\exp(i k_0 x)$, a momentum translation operator that transforms $\ket{k=0}$ to $\ket{k_0}$.
Third, having prepared our $\ket{k_0}$initial state, we replace the harmonic potential with a speckle potential (with $k_c/\kr = 6$) and follow the time evolution for $16\ {\rm ms}$.
Figure~\ref{fig:single}(b) captures the main result of this section: when $k_0 > k_c / 2$ the time-evolution is almost unchanged by the speckle potential, while slowly moving initial states are both decelerated and exhibit considerable interference.

Figure~\ref{fig:single}(b) plots the ensemble-averaged momentum $\langle k(t)\rangle$ as a function of time for a range of initial states with $k_0$ from near-zero to $k_0 / \kr = 3.3$, and Fig.~\ref{fig:single}(c) plots the final momentum $k_f$ as a function of $k_0$.
At $t=0$, the average momentum is $\langle k(t)\rangle = k_0$; for $k_0\gtrsim k_c/2$ the BEC evolves ballistically, leaving $\langle k(t)\rangle$ unchanged, while $\langle k(t)\rangle = k_0$ falls rapidly for smaller $k_0$.
Both of these observations are consistent with our FGR analysis which showed a complete absence of momentum changing back-scattering for $k_0 \geq k_c/2$, and with rapidly increasing back-scattering as $k$ falls to zero.

\subsection{SOBECs}\label{soc}

\begin{figure}[t]
    \centering
    \includegraphics{Fig6_SOC_Localization.pdf}
    \caption{Motion in the presence of speckle and SOC.
    The left column was computed without interactions and the right column added interactions.
    (a) and (b) Density distributions colored by their magnetization according to the color bar in Fig.~\ref{fig:Dispersion_relations}.
    The shaded curve depicts the initial density distribution, while the remaining red and and black curves were computed for  $q_0=2.0k_{\rm R}$ (in the SOC gap) and $q_0=1.2k_{\rm R}$ (below the SOC gap), respectively.
    (c) and (d) Ensemble averaged final group velocity plotted as a function of initial group velocity for coupling strengths  from $0.5E_{\rm R}$ to $7.5E_{\rm R}$, spaced by $1.0E_{\rm R}$.
    The results in (c) and (d) were averaged over 20 random speckle realizations.
       }
    \label{fig:SOC}
\end{figure}

As in the single-component case, simulations with SOC begin with three steps aligned with experiment, however, the process of preparing the initial quasimomentum state $\ket{q_0,-}$ is considerably more elaborate than preparing a momentum state $\ket{k_0}$ in a single component system.
(1) As before, we initialize a ground state BEC in a harmonic trap using imaginary time evolution, spin polarized in state $\ket{k_0 = 0, \downarrow}$.
(2) We then use a combination of adiabatic and unitary evolution (described below) to transform this state into $\ket{q_0,-}$ for $\delta=0$ and $\Omega_{\rm R}$ ranging from $0.5\Er$ to $8\Er$.
(3) Lastly, we again remove the harmonic potential and again follow the time evolution with a speckle potential ($k_c/\kr = 6$) for $16\ {\rm ms}$.

Our procedure (2) begins with the observation that in a frame moving with velocity $\hbar \delta k / m$, the detuning $\delta$ present SOC Hamiltonian in Eq.~\eqref{eq:soc} is Dopper-shifted~\cite{Cheuk2012, Valdes-Curiel2017} to $\delta + 2\hbar^2 \delta k \kr/m$.
Our first task is to adiabatically transform the initial state $\ket{k_0 = 0\downarrow}$ into $\ket{q_{\rm min}, -}$, a ground state SOBEC with quasi-momentum centered at $q=q_{\rm min}$, the global minima of the SOC dispersion, but with $\delta = 2\hbar^2 (q_0 - q_{\rm min}) \kr/m$.
We achieve this by ramping up the Raman coupling strength from zero to $\Omega_{\rm R}$ on a time scale slow compared to $\hbar/\Delta(q_0,0)$.
In the slow ramp up process, the harmonic trap provides the restoring force required to keep the state at a local minima of the dispersion~\cite{lin2011spin}, i.e., with zero group velocity.
Lastly, we diabatically set $\delta=0$ and apply momentum kick $\exp[i(q_0-q_m)x]$, giving the desired state $\ket{q_0, -}$.

Figures~\ref{fig:SOC}(a) and (b) show representative density distributions $n(x) = |\psi_\uparrow(x,t)|^2 +|\psi_\downarrow(x,t)|^2$ before and after a $16\ {\rm ms}$ time evolution with $\Omega_{\rm R} = 2\Er$, both (a) with no interactions and (b) with interactions.
In both cases the pink shaded curve depicts the initial density distribution, while the density distributions for initial quasimomemta of $q_0 = 1.2 \kr$ and $2.0\kr$ are shown by the black and red curves respectively.
In both cases the momentum exchange for back-scattering is below $k_c$, however, as with the FGR results, these direct simulations show that initial states prepared with energy within the SOC energy gap experience negligible change in velocity, independent of the presence of interactions.

While the free particle group velocity is simply related to wave-vector by $v = \hbar k / m$, atoms evolving according to the SOC dispersions, as in Fig.~\ref{fig:scatteringRate}, have group velocity given by the more complex relation
\begin{align}
\frac{v_\pm}{\vr} &= \frac{q}{\kr} \left\{1 \pm \left[\left(\frac{q}{\kr}\right)^2 + \left(\frac{\Omega_{\rm R}}{4 \Er},\right)^2\right]^{-1/2}\right\},
\end{align}
for atoms in state $\ket{q, \pm}$, expressed in units of the recoil velocity $\vr = \hbar \kr/m$.
Because we are interested in transport phenomena, it is this group velocity not the quasimomentum, that is the quantity of primary interest.

Figures~\ref{fig:SOC}(c) and (d) plot the final group velocity $v_f$ as a function of initial group velocity $v_0$ after a $16\ {\rm ms}$ period of free evolution, both (c) with no interactions and (d) with interactions, and with $\Omega_{\rm R}$ from $0.5\Er$ to $7.5\Er$.
As compared to the simulations without SOC in Fig.~\ref{fig:SOC}(c), these curves show a near-complete suppression of relaxation for velocities near $v_0 \approx \vr$, in the SOC energy gap, and with an increasing window of suppression with increasing $\Omega_{\rm R}$.

Lastly, we see that interaction effects do play a role, leading to more rapid deceleration.
The origin of this effect can be understood by comparing the red curves in Figs.~\ref{fig:SOC}(a) and (b): adding interactions leads a mean-field driven expansion of the BEC, increasing the range of velocities present.
As a result, when the SOC energy gap is small (small $\Omega_{\rm R}$), a significant fraction of the BEC's velocity distribution falls outside the SOC energy gap, thereby sampling points in the dispersion where first-order backscattering is allowed.
At larger $\Omega_{\rm R}$, motion is near-ballistic near the center of the SOC gap, but the transition from ballistic to decelerated is smoothed as compared to the case with no interactions.

\section{Conclusion} \label{Conclusion}

Our analytical and numerical studies of the transport of SOBECs in disorder potentials clearly show dramatically enhanced transport for initial states in the SOC energy gap.
The enhanced transport described here results from the same physics giving rise to a spin transistor in Ref.~\cite{Mossman2019}, which also relied on a combination if kinematic and matrix-element effects to yield non-reciprocal appearing transport behavior.
In the appendix, we describe an explicit experimental proposal using laser speckle derived from $532\ {\rm nm}$ green laser and an off-the-shelf optical diffuser.
In this proposal, SOC is generated from  a pair of $790\ {\rm nm}$ laser beams intersecting at the atoms, and initial states would be prepared as described above.
The protection from back scattering is independent of quantum statistics: non-interacting fermions would experience a conductivity increased by the factor predicted by the FGR when the Fermi energy resides in the SOC gap.
As with the interacting SOBEC we analyzed, we expect that fermionic systems with moderate interactions would show gains in conductivity, however, the details of this latter case would necessitate future study.

Reference~\cite{Hugel2014} showed that in lattices, the type of 1D SOC in Eq.~\eqref{eq:soc} has the same dispersion as the edge modes of 2D ${\rm Z}_2$ topological insulators~\cite{Kane2005}.
Together with our finding, this indicates that 1D nanowires with SOC either of the Rashba~\cite{Bychkov1984} or linear-Dresselhaus~\cite{Dresselhaus1955} type should provide the same protection to backscattering from spin-independent disorder as would be observed at the edge of a topological insulators.

\section{acknowledgments} \label{acknowledgements}

This work was partially supported by the AFOSRs Quantum Matter MURI, NIST, and the NSF through the PFC at the JQI.
We benefited from laboratory assistance from E.~M.~Altuntas, C.~J.~Billington, and F.~Salces-Carcoba.
We are grateful for the meticulous reading of our manuscript by the afore noted individuals along with W.~McGehee and G.~H.~Reid.
\newline

redpajama_set_name: RedPajamaArXiv


New technology has given a wide option to the UK homeowner to renovate their home. Sometimes, they even go in for new products that give an exciting and valuable appearance to their homes. Replacement Windows Kent talks about windows and window designs that everyone knows of.
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Ever since our founders added a fiery orange pigment to their enamel glaze, color has been at the heart of our work. Inspirational and impactful, our colors have become as central to our character as our dependable quality. Today, Le Creuset's range of signature shades continues to evolve, building on the best of the past and shaping the trends of tomorrow.


NASA's Mars InSight Lander, which cost over 800 million dollars and has travelled an impressive 483 million miles on its journey from Earth, has finally landed on Mars. The Interior Exploration using Seismic Investigations, Geodesy, and Heat Transport (InSight) was launched on May 5 and successfully landed on Mars on November 26. The InSight was sent to Mars in order to study the deep interior of the still relatively unknown planet.
The mission was initially scheduled for March 2016 but was delayed for more than two years due to vacuum failures in the Seismic Experiment for Interior Structure (SEIS), a round instrument that measures the movement of the planet with a ground-motion detection sensor that detects the vibrations, or seismic waves, of Mars. Even with the reuse of materials from the 2008 Mars Phoenix lander, the mission cost increased from 675 million dollars to 830 million dollars due to the faulty SEIS instrument.
There are three main instruments on the InSight Lander. One goal of this Mars mission is to place the SEIS instrument on the surface of Mars in order to measure the planet's seismic activity and to 3D model the interior of Mars. The SEIS instruments studies the seismic waves created from marsquakes (similar to earthquakes), meteorite impacts, and the weather. Hopefully, the scientists can use the SEIS measurements to study the material that formed the planets and what lies underneath the surface of Mars.
Additionally, NASA hopes to use HP3, a heat probe used to measure Mars' temperature, revealing both the magnitude and source of the heat coming out from the interior of the planet. Using this data, the NASA scientists can determine the material that formed Mars (as compared to that of the Earth and the Moon) and study Mars' early evolution.
Another goal of the NASA InSight Mission is to study the reflexes of Mars through the Rotation and Interior Structure Experiment (RISE), which tracks the movement of Mars' North Pole as the planet orbits the sun. With these measurements, the scientists can further the planet's core—specifically, the size and composition, or the state and the elements found in the core.


text:
Competitive Drive: More than just a game
by: Shaquira Martin
WICHITA, Kan. (KSNW) — Former Shockers and current Derby High School varsity coach, Jodie Karsak, has had a lot of success at the school. In the last eight years the team has made five straight 6A appearances to the Final Four and they claimed the state title in 2018.
"I've only lost four games my whole high school career," said Derby High school senior Jaidyn Schomp.
"I'm a competitor," said Karsak. "You hate to lose."
Since accepting the job as Head coach in 2011, Karsak's mission has been to pour purpose into her athletes.
"Leadership creates the culture, which then creates the behavior," said Dan Harrison, who has coached under Karsak for over five years. "It was wonderful to see and become apart of something truly uncommon."
Karsak has been Derby's girl's basketball coach since 2011 and has made five straight appearances to the 6A state tournament.
"I think, at the end of the day, it's how do you define success? Society tells us it's about our record. We talk about being uncommon, and we want to be different," said Karsak, "If we do common things in an uncommon way, we're going to have really great success."
"It's exciting for kids to go, 'yeah, I'm a part of something really neat and fun and successful, but I get to be a part of it and I get to write my own chapter,'" said Harrison.
"They still have college to get to and through," said Karsak. "They're still finding themselves. But when they're old, what kind of adults are they? and then come to me and asked me 'do you think you've been successful' here."
Karsak helped Derby win their first state title since 1985.
"She's done a really good job of making sure that we stay hungry," said Harrison.
During Karsak's eighth and final season with the Panthers, the legacy she is leaving behind means more to her than the 121 wins in her stat book.
"My cup runneth over, right? Sometimes I'm humble and I say 'Lord, why me?' because this situation, I feel so undeserving because it has been such a blessing," said Karsak.
More Competitive Drive Stories
Competitive Drive: Tracey Mason Sr. uses boxing to inspire area youth
by Chris Arnold / Jan 15, 2021
WICHITA, Kan. (KSNW) - Condition, hear, discipline or CHD.
Those are words Tracey Mason Sr. lives by and a way of life that he is trying to instill in Wichita's youth through the sport of boxing.
One last hurrah: Dale Faber coaching his last season as Friends' men's basketball coach
by Taylor Rocha / Jan 8, 2021
WICHITA, Kan. (KSNW) – Beginning with the 2003-04 season, Dale Faber arrived on campus as the Friends University head men's basketball coach. Eighteen seasons and 294 wins later, Faber has decided that the 2020-21 campaign will be his last at the helm for the Falcons, as he will retire from coaching before transitioning into a fundraising role within the athletic department.
"We have been looking at a fundraising role for a while and am grateful for the support of Dr. Carey to start it," said Dr. Rob Ramseyer, Friends University Director of Athletics.
Competitive Drive: Julius "Juice" Bolden ready to be on the loose at the next level
by Chris Arnold / Jan 1, 2021
WICHITA, Kan. (KSNW) - Julius Bolden remembers the first touchdown he ever scored like it was yesterday.
He says it came in the first grade on an inside handoff. He would bounce it outside for a 60 yard run to the endzone.
Sports Tweets
Mostly Clear Precip: 23%

redpajama_set_name: RedPajamaCommonCrawl


Following in the footsteps of other cable TV channels, Fox News plans to launch a standalone subscription service that will be available to viewers without a cable package.
According to the New York Times, the streaming service — Fox Nation — will offer right-leaning commentary by way of original shows, as well as cameos from such Fox News staples as Sean Hannity. The network plans to develop several hours of new daily programming with a mostly fresh roster of anchors and commentators, and will not overlap with Fox News' regular cable broadcasts.
Fox News is expected to make a formal announcement later today about the streaming service, which is set to launch before the end of 2018. The cost of a subscription is reportedly still being discussed. In the meantime, tell us: Are you interested in Fox Nation?


text:
Lives in Key Periods
Tam Samson
Souter Johnnie
Maria Riddell
The Earl of Glencairn
Jean Armour
Frances Anna Dunlop
James MacKinlay
Gavin Hamilton
Willie Fisher
Robert Burns: People and Places
The McKie Collection
Collectors & Explorers
Soldiers, Sailors, Rebels & Outlaws
Writers & Poets
Science & Invention
Trade, Commerce & Industry
Politics & State
Sportsmen & Performers
Ellisland, c 1908
An etching by Francis S Walker of the walk along the banks of the River Nith close to Robert Burns farm.
Robert Burns took on the lease of Ellisland Farm from Patrick Miller of Dalswinton from Whitsunday 1788 for a rent of £50 per year. It was a small unimproved holding of 170 acres situated on the bank of the River Nith about 5 miles north of the town of Dumfries. Miller gave Burns £300 with which to build a farm house and enclose the fields.
As a respite from the demands of the farm, Burns would walk along the banks of the River Nith, often using this time to think over his writing. He wrote to George Thomson, his publisher, "I walk out, sit down now & then, look out for objects in Nature around me that are in unison or harmony with the cogitations of my fancy & workings of my bosom; humming every now & then the air with the verses I have framed".
It is believed that it was here that Burns composed "Tam O'Shanter" in a single day. Burns wrote the poem as a "witch story" to accompany the illustration of Alloway Kirk which was to be published by his friend, Captain Francis Grose in his "Antiquities of Scotland". It is both comic and epic, telling the story of Tam's encounter with a coven of witches on his ride home from a night's drinking at the inn.
width: 408 mm, length: 264 mm
Dumfries Museum & Camera Obscura
DUMFM:1988.24.1
DMBN280n
Dumfries & Galloway Council


text:
Public Holidays In Nigeria For 2019
Comments Off on Public Holidays In Nigeria For 2019
Discover the international public holiday dates you're looking for with us! Public Holidays Global seeks to provide accurate holiday information to help people across the world plan for their local festivities and make the most of their long weekends. Please choose your country in the lists below to view public holiday dates for 2018, 2019 and future years.
All text and image content copyrighted by HolidaysCalendar.com. All rights reserved.
• Armenia (Oct 10) Armenia October 11-12 Public Holidays Update • Bolivia (Oct 10) Bolivian Afternoon Public Holiday October 11 • Guatemala (Oct 10) Guatemalan Bank Holidays October 12 and 19 • Sénégal (Oct 10) Senegal 2018 Magal Public Holiday Date Announced • Spain (Oct 10) Spain 2019 Galicia Public Holidays Announced •.
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It eventually destroyed 3,800 shops, according to the Nigeria Emergency Management Administration. Boko Haram has carried out big operations before around the principal Christian holidays, and the.
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"It seems the whole economy is caught between strikes and slow international growth," he said. Gillis said although the BETI took public holidays into consideration, they alone could not be used as an.
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Slidemotion is an automated web service enabling users to create videos with music from users' photos, with no skills and in a few clicks. The service has many features and offers various possibilities to personalise the videos. Users can make unlimited number of free videos during the 30-days trial period. This free temporary version comes with a limit, so videos can be created with 15 visuals. After 30 day trial finishes users will still be able to play and share all of their videos but will not be able to create new ones. It is also possible to try the premium version by spending free credits, so users can decide if they like premium features and continue by purchasing a plan.
The project Videolectures.Net was created at Jozef Stefan Institute, Slovenia, Europe in early 2002 in synergy with the global trends and more formal efforts by the European Union in creating a Knowledge Economy and Information Society.
Nowadays VideoLectures.NET is a wide ranged free and open access educational video lectures repository. The lectures are given by distinguished scholars and scientists at the most important and prominent events like conferences, summer schools, workshops and science promotional events from many fields of Science. The portal is aimed at promoting science, exchanging ideas and fostering knowledge sharing by providing high quality didactic contents not only to the scientific community but also to the general public. All lectures, accompanying documents, information and links are systematically selected and classified through the editorial process taking into account also users' comments.
The training materials were initially limited to those being developed within the FP5, FP6, and FP7 European Framework Programs, where the web based portal Videolectures.Net is being used as an educational platform for several EU funded research projects such as PASCAL NoE, ECOLEAD NoE, SEKT IP and different organizations, among others Xerox Parc, British Telecom, Max Planck, Fraunhofer Institute, Australian National University and Carnegie Mellon. The range of countries involved and languages used varies from Europe, USA, Taiwan, Australia, Ukraine, Russia and Brazil.


text:
Q: Updating or adding multiple columns with pydatatable in style of R datable's .SDcols Given iris data I'd like to add new columns corresponding to all numeric columns found. I can do by explicitly listing each numeric column:
from datatable import fread, f, mean, update
iris_dt = fread("https://h2o-public-test-data.s3.amazonaws.com/smalldata/iris/iris.csv")
iris_dt[:, update(C0_dist_from_mean = dt.abs(f.C0 - mean(f.C0)),
                  C1_dist_from_mean = dt.abs(f.C1 - mean(f.C1)),
                  C2_dist_from_mean = dt.abs(f.C2 - mean(f.C2)),
                  C3_dist_from_mean = dt.abs(f.C3 - mean(f.C1)))]

But that way I hard-coded column names. More robust way is readily available with R datatable using .SDcols:
library(data.table)
iris = fread("https://h2o-public-test-data.s3.amazonaws.com/smalldata/iris/iris.csv")
cols = names(sapply(iris, class)[sapply(iris, class)=='numeric'])
iris[, paste0(cols,"_dist_from_mean") := lapply(.SD, function(x) {abs(x-mean(x))}),
     .SDcols=cols]

Is there a way to take similar approach with pydatatable today?
I do realize how to get all numeric columns in py-datatable, e.g. like this:
iris_dt[:, f[float]]

but it's the last part that uses .SDcols in R that evades me.

A: Create a dict comprehension of the new column names and the f expressions, then unpack it in the update method:
from datatable import f, update, abs, mean

aggs = {f"{col}_dist_from_mean" : abs(f[col] - mean(f[col]))
        for col in iris_dt[:, f[float]].names}

iris_dt[:, update(**aggs)]

UPDATE:
Using the Type properties in v1.1, this is an alternative approach :
aggs = {f"{col}_dist_from_mean" : dt.math.abs(f[col] - f[col].mean())
        for col, col_type
        in zip(iris_dt.names, iris_dt.types)
        if col_type.is_float}

You could also chunk the steps:
Create a Frame with the calculated values:
expression = f[float]-f[float].mean()
expression = dt.math.abs(expression)

compute = iris_dt[:, expression]

Rename the column labels for compute:
compute.names = [f"{name}_dist_from_mean" for name in compute.names]

Update iris_dt with compute (note that you could also use a cbind):
iris_dt[:, update(**compute)]

redpajama_set_name: RedPajamaStackExchange


I returned yesterday from the invitational conference held in Leipzig (28-30 September) and focused on the relationship of "history-of-religion" research and theology. The conference was scheduled in observance of the 100th anniversary of the death of Georg Heinrici, an eminent scholar in Leipzig who was himself very much involved in history-of-religion research, and also in the controversies about how this research related to traditional Christian theology.
Geurt Henk van Kooten (Groningen) proposed connections between the Gospel of John and Greek mythology and philosophy, urging that the author intended to make such connections as part of a concern to evangelize. Udo Schnelle (Halle-Wittenberg) focused on the NT idea of Jesus as "incarnate" deity, making comparisons and contrasts with other ancient ideas sometimes seen as analogies. Cilliers Breytenbach (Berlin) discussed the use (and misuse) of purported "analogies" and how dangerous it can be to construct "genealogies" from them. In short, similarities don't necessarily signal "borrowing" of one thing from the other.
Marco Frenschkowski (Leipzig) reviewed the stances of Heinrici, Harnack, Bousset on the relationship of history-of-religion and theological approaches of the early 20th century. John Fitzgerald (Notre Dame, USA) surveyed references to Heinrici in English-language scholarship, noting the unjustified neglect but also the particular instances where Heinrici has been acknowledged (e.g., Hans Dieter-Betz's notice of Heinrici's commentary on 2 Corinthians as what he judged the best case for the literary integrity of 2 Corinthians).
I was invited to explore possible theological implications arising from the sort of work I've been involved in over the past few decades on the origins of Jesus-devotion, which forms part of what is now sometimes referred to as the "new religionsgeschichtliche Schule." Noting that the term is misleading, and there isn't any one theological stance represented in the body of scholars sometimes so-designated, I then offered some preliminary suggestions of my own, which I'll briefly summarize here.
First, what implications arise from emerging consensus that the treatment of Jesus as somehow worthy of divine(like) honor originated in the earliest circles of the Jesus-movement? In my view, this doesn't decide the theological validity of traditional Christological claims, but it does remove one major reason that has sometimes been used as a basis for rejecting them. You can't relativize a "high" Christological stance by claiming that it reflects a secondary and late development. The early centrality of Jesus-devotion, in my view, makes it central also for theology today.
Moreover, the place of Jesus in earliest Christian proclamation and practice has theological implications for "God" as well as for Christology. The link of Jesus and God is the key distinguishing feature of discourse about "God" in the NT. Just as Jesus is consistently defined with reference to God (e.g., sent forth by God, raised/exalted by God, etc.), so "God" is rather consistently defined with reference to Jesus. I suggest that there are theological implications for virtually every traditional topic in theology.
I also discussed how the terminology used in early Christological discourse varies, focusing on the curious Christological usage of the Greek term pais ("child/servant"), which appears in only Acts and then a few other early Christian writings, and mainly in passages that reflect liturgical settings. Whatever the reason that the term had this limited usage, it illustrates how early Christians used various terms to express their faith. The basis implication is that theology has this continuing responsibility to find meaningful ways of communicating in different cultural settings and circumstances.
I also pointed to the importance of early Christian devotional/worship practices. My point was that the central place of the risen/exalted Jesus in earliest devotional practices has, I contend, profound implications for theology. There is a distinctive "dyadic" shape to earliest Christian devotional practice and to earliest Christian proclamation. Indeed, I propose that theologians might be advised to study earliest Christianity more closely, as perhaps a more fruitful resource for theologizing in the modern global and multi-faith context. For some 15 centuries, Christian theology has basically been an "in-house" discussion, Christian theologians arguing with one another. And the key starting points have been theological developments of the 4th century AD and later. But in the modern context in which again Christianity is simply one religious option among others, it is perhaps again important for theologians to work at articulating Christian faith in ways that can be engaged by people of other faiths. For that, I suggest that the texts of the first three centuries may now be more valuable than the later writings that have been more frequently the focus of theologians (e.g., Cappadocian Fathers, Augustine, etc.).


This document shows you how to write a simple, console-based Java SOAP client to access the Open Calais web service. The concepts presented in this tutorial are also useful when you write a SOAP client for any other web service.
According to http://opencalais.mashery.com/Overview "The Calais web service automatically attaches rich semantic metadata to the content you submit – in well under a second. Using natural language processing, machine learning and other methods, Calais categorizes and links your document with entities (people, places, organizations, etc.), facts (person 'x' works for company 'y'), and events (person 'z' was appointed chairman of company 'y' on date 'x'). The metadata results are stored centrally and returned to you as industry-standard RDF constructs accompanied by a Globally Unique Identifier (GUID). Using the Calais GUID, any downstream consumer is able to retrieve this metadata via a simple call to Calais.
This metadata gives you the ability to build maps (or graphs or networks) linking documents to people to companies to places to products to events to geographies to … whatever. You can use those maps to improve site navigation, provide contextual syndication, tag and organize your content, create structured folksonomies, filter and de-duplicate news feeds or analyze content to see if it contains what you care about. And, you can share those maps with anyone else in the content ecosystem. "
From a programmer's point of view, you construct a SOAP request and send it to the web service remote endpoint. The web service replies with the output in Resource Description Framework.
Create a Java application project. In this project, create a web service client that consumes the Open Calais enlighten service.
In the New Project wizard, create a Java application project. Name it contextualizer. Accept all default settings--it should create a Main class.
Right-click on the project node. The context menu opens.
Add the URL of the enlighten WSDL, http://api.opencalais.com/enlighten/?wsdl.
The Open Calais API provides both a SOAP 1.1 and a SOAP 1.2 function named "enlighten". We are going to use the SOAP 1.1 function in our client.
To understand the various parameters used to make the API call, please refer to the Open Calais API guide.
Use this RDF to obtain the returned information in a useful format.
Build an application called "contextualiser" (hence I chose this name for the Java application here) which returns contextual information about the query text.
The main objective of this tutorial was to show how you could create a JAX-WS client that interacts with the Open Calais Web Service. However, the concepts can also be used to write a client to interact with any other Web-Service instead.


The Management and Administration Division creates opportunities for its members to exchange ideas and build skills in leadership, planning, obtaining and allocating funds, management, and trend forecasting.
Goal #1: Provide high quality programs and training to educate and develop OLC members.
Objective #2: MAD will propose at least 3 sessions for the Annual Convention and Expo.


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Vor! (ungh) What Is It Good For?
Absolutely everything apparently. One of the things about being 'experienced' in the ways of Doctor Who is that you think of yourself accepting of the notion that you are incapable…
Dup-topia
Perhaps it's just the jaded nature of a long-term fan, but the Tour Honchos there's been only one truly-punch-the-air moment in the whole of nu-Who for us, and that was…
Come and See the Softer Side of Six
The Most Magnificent Caps in the History of History Itself
A Hill Not Worth Dying On
Headwaters of the River
The triumphant return of Modern Recap-itulation, something itself marks the beginning of the end, for a story itself that marks either the beginning or the end of a character in Doctor Who, depending on how you look at it.
WIth the production gaps between Series 9, 10, and 11 reaching inevitable, regrettable heights, it's easy to forget that The Husbands of River Song, Alex Kingston's last appearance as River, aired in 2015. It only seems more recent than that. But the Recap-itulation team go back to Kingston's beginning (2008!), and River's end for a new set of caps for Silence in the Library.
Silence and the back half are one of those mini-mid-season close quarters epics that the Tennant era dallied and excelled in. These were the last stories Steven Moffat penned before taking the reins and while it's essentially a BUS story, the character of River, someone from the Doctor's future who knows more about upcoming events than the Doctor, is such an intriguing and original concept that it's a wonder it hadn't been done before.
This is a different form of timey-wimey, and Moffat would be sure to mine it for all it's worth in the years to come.
Every Witch Way But Lose I Will Follow Hymn Better Belated than Never
Silence in the Library
New Who for You
Xmas Marks the Spot
Christmas (Again) Comes But Twice a Year
Matt Takes His Final Bow (Tie)
Beatle Ood
Classic Capitology begins the final push until Series 8 begins in …
Doin' the Polo Hop
Today we celebrate the 12th Anniversary of possibly the most …
Revolution Resolution
Yep. We're back on the clock again. One of the …


Legendary duelists drop sometimes when you have duels against them playing such as Kaiba, Weevil, Pegasus, Mako and even Paradox Brothers.
Unfortunately, this skill has never been used on any decks when it comes to PVP and farming. The reason is that a deck might not be viable when having one spell or one trap card. However, with the Yami Bakura event and Dark Necrofear being released; drawing Skull Lair and Chaos Zone might be worth taking a shot using this skill.
You can also add Yami Bakura to this list, too.


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The Billion-Dollar Misunderstanding: Ecommerce Merchants and Latin America
Influencer MarketingNimble Sales Intelligence
By STS News Desk On Dec 29, 2020
Retail ecommerce sales in Latin America will reach $86.6 billion in 2020, up almost 20% from 2019. Latin American ecommerce sales for 2023 are projected to reach $116.2 billion, representing 7.1% of total retail sales for the region.(1) For all its promise, however, many U.S. ecommerce merchants continue to be reluctant to enter the Latin American market. "There's an assumption," says A.J. Hernandez, president and CEO of international business-to-consumer shipping firm SkyPostal, Inc., "that Latin America is a prohibitively difficult place to do business. That's simply not true. It's complex, but the tools exist to make it manageable."
In fact, says Hernandez, the Latin American market represents a major opportunity for U.S. ecommerce. The region has a total population of more than 656 million(2); Brazil, where ecommerce sales jumped 57% in the first five months of 2020(3), is the world's seventh most populous country(4) and ranks among its top 10 economies in terms of GDP.(5)
Read More: SalesTechStar Interview With Helena Gamvros, Head Of Digital Sales South East Asia & Greater…
Much of the reputation for difficulty in this market, says Hernandez, stems from merchants' attempting to access it via the United States Postal Service, and encountering delivery times of three weeks and sometimes longer. He notes that USPS international service works very to places like western Europe or Japan, but badly for Latin America. "It's not USPS's fault," says Hernandez. "The post office is only as good as its partner in the destination country, and in Latin America they're dealing with bankrupt and dysfunctional postal authorities that provide, frankly, horrible service.(7)"
Encountering this situation, says Hernandez, has led many merchants to believe that their only alternative is to ship via the large, well-established—and very expensive—international carriers. In most cases, this is not a reasonable business proposition, as the shipping cost can easily outweigh the value of the parcel.
And yet, Hernandez notes, Latin America has flourishing economies and a major ecommerce boom. How do they do it? To fill the gap between unacceptable postal service and unaffordable international shipping rates, says Hernandez, merchants turn to private delivery networks that understand Latin America's complex taxation and custom issues and offer timely, trackable, reliable delivery service throughout the region, at rates far below those of the major international carriers.
Read More: 7 ECommerce Insights For 2020 Peak Season
Hernandez offers his own company, SkyPostal, Inc., the largest and best established of the private Latin American networks and part of Amazon's Seller Provider Network, as a case in point. "We handle all customs clearances, include payment of taxes and duties. Then we handle the last-mile delivery. Throughout, our clients and their customers can track delivery of the shipment online, in real time."
AmazoneCommerce merchantseCommerce salesLatin American marketNewspostal authoritiesSkyPostal
Digiscribe Receives SOC 2 Type 2 Report for Eighth Consecutive Year
Amware Fulfillment Acquires Moulton Logistics
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Top Sales Prospecting Challenges for B2B Sales Teams to Overcome
SalesTechStar Interview with Cory Cummings, CEO at Pack Digital


Vincent De Paul is the president and co-founder of Five Arts Productions. De Paul received his first Emmy in 2015 and second consecutive Emmy in 2016 for producing The Bay the series created by Gregori J. Martin.
As a setter, he participated in the 2004 Olympic games in Athens, Greece as well as the 2012 Olympic Games in London. Suxho and his family came to America in 1996 and lived in Natick, Massachusetts while searching for a college.
He eventually chose to play for the University of Southern California and became one of the most well known volleyball players to come out of USC.
At a book launch for his book "The Mask of Masculinity," Lewis Howes was asked the question he asks every guest: What are your three truths? I recorded his answer.
He has produced music for power players like Snoop Dogg, Jay-Z, 50 Cent and Big L.
He participated in national campaigns for Nike, Sprite, Lugz and FUBU.
He's sold out areas of more than 10,000.
She has won the hardest endurance mountain bike races around the globe including being crowned world champion in 2015. She is also a 4x USA National Champion. Sonya is passionate about helping and inspiring others to live healthier and more meaningful lives.
Inc Magazine named him one of the Top 100 Leadership Speakers, alongside such noteworthy giants as Richard Branson, Brené Brown, John Maxwell and Robin Sharma.
Bobby is a 5x TEDx speaker, and one of the top social media influencers in the world, with over half a million followers.
She advises her clients on a variety of issues that confront modern businesses, including cybersecurity and protecting their online content, services and intellectual property, implementing proper contracts to protect their work and relationships with clients, and asset and risk management.
Often times we fall into a place where we thing we have to do and do it all ourselves and it keeps.
Here's my experience this past week.
She's known nationwide for her "Trendsetting" and "Carefree" design skill set as well as for being the "Fabulously" famous Party Planner to the stars" as seen on the Style Networks, "Whose Wedding Is It Anyway," and various other networks.
To that end, everything that he and his business partner, Hall of Fame quarterback Warren Moon, take on has a charitable component, and this rule has only accelerated their business growth. Sports 1 Marketing is currently involved in projects surrounding many of the world's biggest sporting events, including the Super Bowl, the Pro Football Hall of Fame, The Masters, and countless others.
That same day I was supposed to interview him, needless to say, that did not happen.
I learned a very valuable lesson that day and I want to share that with you.
Over the years, Rebecca Minkoff has created a fashion empire. But like all women, she is many things; a wife, mother, designer, friend and an all around superwoman.
In her new podcast Superwoman, Rebecca talks to the ladies who are doing it all and shape our culture. From CEOs and chefs, to entrepreneurs and instructors, Rebecca sits down with the leading ladies of every industry to find out the stories behind their success.
These are the stories you don't see on instagram. These are the real women behind companies like Who What Wear and Allure, who forged their own paths to success and lifted each other up along the way.
You're about to hear a preview of Superwomen with Rebecca Minkoff, in which you'll sit down at the table with Rebecca and Michelle Lee, Editor In Chief of Allure.


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