text:
\section{Introduction}
Quantum vortices in Bose-Einstein condensates (BECs) are topologically stable excitations of the superfluid matter wave field that carry angular momentum~\cite{Fetter2001,*Fetter09a}. Unless the system is rotating sufficiently fast, vortices are thermodynamically unstable~\cite{Rokhsar97a}.  At finite temperatures the thermal cloud can provide a source of dissipation and noise, and hence allow vortex decay. However, there is currently little understanding of the decay mechanism and its timescale. The dynamics of vortices in such systems is therefore of interest as a robust test of non-equilibrium field theory. A number of different methods have been developed to describe the dynamics of vortices in finite-temperature systems. Single vortex decay has been simulated using the Zaremba-Nikuni-Griffin formalism \cite{Jackson09a}, the classical field method \cite{Schmidt2003}, and within a Gross-Pitaevskii theory with phenomenological damping~\cite{Madarassy08a}. The nature of the thermal instability has also been studied within a variational approach~\cite{Duine2004} to the stochastic Gross-Pitaevskii equation~\cite{Stoof1999,*Stoof2001}.

Classical vortices can support long-wavelength helical traveling waves known as Kelvin waves~\cite{Thompson1880}. Superfluid Kelvin waves, or Kelvons~\cite{Fetter04a}, have long be studied in Helium~\cite{Tsubota09a}, and more recently have become a topic of interest in BECs. The Kelvon excitation mechanism has been theoretically investigated \cite{Mizushima03a}, along with the microscopic dynamics of coherently excited Kelvin waves~\cite{Simula08a,*Simula08b}, with results consistent with the experimental evidence for Kelvons \cite{Bretin03a}.  Similar bending waves have also been shown to play an important role in the instability of vortex rings in trapped BECs \cite{Horng2006a}.

While the coherent excitation of Kelvin modes in a BEC is well understood, their role in the finite-temperature system is largely unexplored. Here we show that the thermal activation of Kelvin waves is a dominant factor in the decay of a vortex for a finite-temperature three-dimensional (3D) BEC. The underlying reason is that Kelvin waves cause the vortex to wobble and emit acoustic radiation, thus allowing the vortex to more effectively dissipate energy, and hence decay by moving out to the boundary of the BEC. As Kelvin waves are fundamentally 3D excitations of the vortex line, they may be suppressed by flattening the system  (see Fig.~\ref{fig1}).

Indeed, beyond a certain oblateness we find that the Kelvon mode contribution is essentially frozen out and the vortex decay rate reduces to a geometrically-invariant value, characterizing a regime of two-dimensional vortex (2DV) dynamics. We also find that for tighter traps and higher temperatures the matter wave itself crosses over to being quasi-2D and we find a dynamical signature of the phase-fluctuating condensate: an anomalous increase in the rate of vortex decay.

As a further application of these concepts, we model a recent experiment~\cite{Neely10a} studying vortex dipoles in the 2DV regime, confirming the 2DV behavior in finite temperature dynamics. The decay characteristics are found to be qualitatively very different from 3D dipole decay which exhibits vortex ring and loop formation, accelerating the damping process.

Our theoretical framework for studying this system is the stochastic projected Gross-Pitaevskii equation (SPGPE)~\cite{SGPEI,*SGPEII}, which is a grand-canonical c-field method~\cite{Blakie08a} (also see  \cite{Proukakis2006a,*Proukakis2006b,*Proukakis2008a,Nestor2011a} for related theories).  This formalism has been used to study vortex lattice formation from a rotating thermal cloud \cite{Bradley08a}, a quantitative model of spontaneous vortex formation in experiments~\cite{Bradley08a,Weiler08a}, and the decay of a single vortex \cite{Rooney10a} (also see \cite{Wright09a,*Wright08a,*Wright10a}).

\begin{figure}[!t]
\begin{center}
\includegraphics[width=\columnwidth]{fig1.eps}
\caption{Schematic of vortex (solid line) bending from the $z$-axis (dashed line) in a trapped BEC for (a) a spherical trap  (in terms of trap frequencies defined in Eq.~\ref{Vdef}, $\Lambda=\omega_z/\omega_r\sim1$), and (b) a highly oblate trap ($\Lambda\gg1$).
\label{fig1}}
\end{center}
\end{figure}

\begin{figure*}[!t]
\begin{center}
\includegraphics[width=\textwidth]{fig2.eps}
\caption{Density isosurfaces of the $\rC$-field after $t = 0.61~{\rm{s}}$ of SPGPE evolution, for a range of trap geometries. Red: the full density isosurface. Blue: isosurface restricted to the region of the vortex core. The system temperature is $T = 0.78T_c^0$.}
\label{figiso}
\end{center}
\end{figure*}
\section{System and equation of motion}
\subsection{Physical system}
We begin by describing the system of bosonic atoms in the cold-collision regime with the second-quantized many body Hamiltonian
\EQ{\label{fullHamil}
H&=&\int d^3{x}\;\Psi^\dag(\mathbf{x}) H_{\rm sp}\Psi(\mathbf{x})\nonumber\\
&&+\frac{g}{2}\int d^3{x}\;\Psi(\mathbf{x})^\dag\Psi(\mathbf{x})^\dag\Psi(\mathbf{x}) \Psi(\mathbf{x})
}
where the single-particle Hamiltonian is
\EQ{
H_{\rm sp}=-\frac{\hbar^2\nabla^2}{2m}+V(\mbf{x}),
}
the cylindrically symmetric harmonic trap is
\EQ{\label{Vdef}
V(\mbf{x}) = \frac{m}{2}\left[\omega_r^2(x^2+y^2)+\omega_z^2z^2\right],
}
and $g=4\pi\hbar^2a/m$, with $m$ the atomic mass and $a$ the s-wave scattering length. The field operators obey Bose commutation relations
\EQ{
[\Psi(\mathbf{x}),\Psi^\dag(\mathbf{y})]=\delta(\mathbf{x}-\mathbf{y}).
}
We define $\Lambda\equiv\omega_z/\omega_r$ as the degree of trap oblateness, with
$\omega_r = \Lambda^{-1/3} \bar{\omega},$
enforcing a constant geometric mean frequency $\bar{\omega}^3\equiv \omega_r^2\omega_z$ for arbitrary oblateness. This choice ensures the ideal gas transition temperature $T_c^0$ is invariant under scaling of $\Lambda$ at fixed total atom number $N$.

\subsection{Stochastic PGPE Formalism}
In this work we use the SPGPE formalism to describe the process of vortex decay in a finite temperature BEC.
The approach we follow in choosing parameters and preparing initial states for the SPGPE is extensively detailed in Ref.~\cite{Rooney10a}, hence we only briefly summarize it here.

The SPGPE is a c-field method where the system is decomposed in terms of the single-particle modes of the system satisfying $H_{\rm sp}\phi_n(\mathbf{x})=\epsilon_n\phi(\mathbf{x})$, where $n$ represents all quantum numbers required to specify the modes.
The system is divided into the coherent region ($\rC$) consisting of all states with  $\epsilon_n\leq \epsilon_{\rm{cut}}$, and the incoherent region ($\rI$) containing the remaining high energy modes \cite{Blakie08a}. This division is made
using a projection operator, $\mathcal{P}_{\rC}$ which defines the $\rC$-region field operator as
\EQ{
\psi(\mathbf{x})=\mathcal{P}_{\rC}\Psi(\mathbf{x})\equiv \sum_{\epsilon_n\leq \epsilon_{\rm{cut}}} \phi_n(\mathbf{x})\int d^3y\;\phi_n^*(\mathbf{y})\Psi(\mathbf{y}).
}
Equivalently, we have the expansion
\EQ{\label{fieldOp}
\psi(\mathbf{x})=\sum_{\epsilon_n\leq \epsilon_{\rm{cut}}} a_n\phi_n(\mathbf{x}),
}
where $[a_n,a_m^\dag]=\delta_{n,m}$.
The $\rI$-region acts as a thermal reservoir for the $\rC$-region, and is assumed to be in thermal equilibrium at a temperature $T$ and chemical potential $\mu$.  The  $\rC$-region dynamics are described by a stochastic differential equation for the c-field corresponding to \eref{fieldOp}:
\EQ{\label{cfield}
\psi_\rC(\mathbf{x})&\equiv& \sum_{\epsilon_n\leq \epsilon_{\rm{cut}}}\alpha_n\phi_n(\mathbf{x})
}
where $\alpha_n$ are ordinary c-numbers.
The equation of motion is known as the simple-growth SPGPE~\cite{Bradley08a}
\begin{equation}
d\psi_\rC=\mathcal{P}_{\rC}\Big\{ -\frac{i}{\hbar}L_\rC\psi_\rC dt
+\frac{\gamma}{k_BT}(\mu-L_\rC)\psi_\rC dt+dW\Big\},
\label{SGPEsimp}
\end{equation}
where $L_\rC$ is the Hamiltonian evolution operator for the $\rC$ region $L_\rC\psi_\rC \equiv \left(H_{\rm sp}+g|\psi_\rC|^2\right)\psi_\rC$, and the complex Gaussian noise, $dW$, satisfies $\langle dW^*(\mbf{x},t)dW(\mbf{x}^\prime,t)\rangle  =2\gamma\delta(\mbf{x}-\mbf{x}^\prime)dt,$
$\langle dW(\mbf{x},t)dW(\mbf{x}^\prime,t)\rangle =0$. For a full derivation of the SPGPE we refer the reader to \cite{SGPEI,*SGPEII}.

The first term of the SPGPE (\ref{SGPEsimp}) describes Gross-Pitaevskii evolution over the $\rC$ region, while the second and third terms account for scattering of two $\rI$-region atoms, resulting in growth of the $\rC$-region, and the corresponding time-reversed process. The growth rate, $\gamma$, can be calculated in near-equilibrium systems \cite{Bradley08a} and is independent of position over a large region of the trap with the value
\begin{eqnarray}\label{gamdef} \gamma&=&\frac{4m(ak_BT)^2}{\pi\hbar^3}\sum_{j=1}^\infty\; \frac{e^{\beta\mu(j+1)}}{e^{2\beta\epsilon_{\rm{cut}} j}}\Phi\left[\frac{e^{\beta\mu}}{e^{\beta\epsilon_{\rm{cut}}}},1,j\right]^2,
\end{eqnarray}
where $\beta=1/k_BT$ and $\Phi$ is the Lerch transcendent.

To obtain initial states for evolution according to (\ref{SGPEsimp}), we follow the procedure used in \cite{Rooney10a}, which we briefly summarize.  We generate (non-vortex) finite-temperature equilibrium states by evolving the SPGPE with an appropriate choice of $T$, $\mu$, and $\epsilon_{\rm{cut}}$\footnote{We require  $N$ to be fixed and the highest energy mode in  $\rC$   to be appreciably occupied ($\approx 2 $ atoms).}.
The values of $T$, $\mu$, and $\epsilon_{\rm{cut}}$ then determine $\gamma$ [Eq.~(\ref{gamdef})] and hence all SPGPE parameters are obtained in a physically consistent manner prior to simulation.
To create the vortex initial state we imprint a phase $\psi_\rC(\mathbf{x})\to \psi_\rC(\mathbf{x})e^{i\Theta(\mathbf{x})}$ onto the equilibrium state, where
$\Theta(\mathbf{x})$ is the azimuthal angle in the radial plane~\footnote{We do not find it necessary to imprint a vortex density profile because at the temperatures under consideration the extra energy of the core with phase-only imprinting is negligible compared with that from thermal fluctuations}. This creates a straight singly-charged vortex line along the $z$ axis of the trap. This procedure generates a non-equilibrium state because the reservoir parameters in (\ref{SGPEsimp}) are for a non-rotating reservoir.

\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.9\columnwidth]{fig3.eps}

\caption{Power spectra for a range of geometries as a function of $k_z / \bar{k}$, where $\bar{k} = \pi /\mathcal{W}$ corresponds to the largest wavelength resolvable for the chosen window $\mathcal{W}=R^z_{TF}/2$ .  The spectra are calculated after $t = 0.70$~s, and for $T=0.78T_c^0$.
\label{figspectra}}
\end{center}
\end{figure}
\section{Single Vortex Decay \label{results}}
In our simulations we use initial vortex states prepared for a system containing $5\times 10^5$ $^{87}$Rb atoms in a trap of (fixed) geometric mean trap frequency $\bar{\omega}=2\pi\times 19.7{\rm s^{-1}}$ [for which  $T_{\rm{c}}^0 = 69.7$nK], and a range of geometries from spherical to highly oblate.  For each parameter set we calculate vortex decay properties by evolving 10-30 trajectories of the SPGPE~\footnote{The number of trajectories required to obtain good statistics increases with the system temperature}.

Fig.~\ref{figiso} shows density isosurfaces of the vortex in a variety of condensate geometries after $t = 0.61 ~{\rm{s}}$ of evolution.  The phase imprinting technique leads to a straight vortex line along the $z$-axis, but under evolution the vortex rapidly thermalizes, typically within $0.1 ~{\rm{s}}$ (see discussion below).   We observe a significant amount of vortex bending in the spherical geometry and as the condensate becomes more two-dimensional,  the magnitude of bending lessens and the vortex behaves essentially as a straight line.  Although vortex bending on a short length scale is evident in spherical geometries, the vortex stays approximately aligned with the $z$-axis throughout its evolution.

\subsection{Kelvon power spectrum}
To obtain the vortex line coordinates during evolution we locate the radial position of the phase singularity $(x_v, y_v)$ in each plane (i.e.~$z$ value) and for all times $t$.  Fluctuations in the phase at the condensate boundary restrict the spatial range over which the vortex can be detected. This reduces the range of $z$-values for which we can obtain the vortex-line coordinates, and, as time evolves and the vortex precesses toward the radial condensate boundary, this range of $z$-values further reduces. For this reason we limit vortex detection to the range $z = \pm \mathcal{W}$, where $\mathcal{W}=\frac{1}{2} R^z_{{\rm{TF}}}$, with $R^z_{{\rm{TF}}}$ the Thomas-Fermi radius along the $z$ direction. We have found this to be the largest range which gives a well defined and unique curl signal over the vortex lifetime for our simulations.

We define  $u(z,t) = [x_v(z,t) - \bar{x}(t)] + i[y_v(z,t) - \bar{y}(t)]$ to be the (complex) vortex line displacement from its instantaneous mean, where the barred quantities are the vortex positions averaged over $z$.  To quantify the extent of vortex bending at a given time, we take the Fourier transform with respect to $z$ (denoted by $\mathcal{F}_z$) of $u(z,t)$, and calculate the power spectrum
 \begin{equation}
 P_u(k,t)=| \mathcal{F}_z [ u(z,t) ] | ^2.
 \end{equation}
 Note that this measure of the power spectrum is insensitive to varicose waves~\cite{Simula08b} and is thus entirely due to vortex bending.

\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.8\columnwidth]{fig4.eps}
\caption{(a) Total power as a function of trap oblateness $\Lambda$. (b) Effective vortex decay rate ($1/\bar{t}$) as a function of $\Lambda$. The disconnected points for $\Lambda=20$ break from the trend of vortex decay rate independent of $\Lambda$, as discussed in \sref{anomdecay}.}
\label{figratpow}
\end{center}
\end{figure}
Fig.~\ref{figspectra} shows the power spectra vs Kelvon wavenumber for a range of geometries at $t = 0.70$~s. We rescale the $k$~axis by $\bar{k} =   \pi / \mathcal{W}$, which is the wavenumber corresponding to the largest visible wavelength within our choice of window length.  In all geometries we see the power spectra take a thermalized form, with maximum occupation of the long-wavelength bending modes (low values of $k$).
 The thermalization of the vortex line from the straight initial conditions is most directly monitored using the Kelvon power spectrum. We find that typically within $\sim100$ ms the Kelvon spectrum reaches a stable state. This is much shorter than the vortex decay time (see Sec.~\ref{SecEffOfGeom}), so we can rule out the influence of any initial state artifacts on our predictions for the vortex decay process.

\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.95\columnwidth]{fig5.eps}
\caption{Vortex decay rate as a function of relative temperature, for a range of geometries.  Here the critical temperature is adjusted to account for interactions and finite size effects (\eeref{Tccorr}). The highest decay rate (for $\Lambda = 20$) is at $T/T_{c{\rm{(corr)}}} = 0.98$, giving $1/\bar{t} = 3.9\; {\rm s}^{-1}$.
\label{figvel}}
\end{center}
\end{figure}

\subsection{Effect of geometry: Kelvon power and vortex decay}\label{SecEffOfGeom}
 The effect of geometry on the importance of the Kelvon modes is clearly visible comparing  Figs.~\ref{figspectra} (a)-(c): The total power in the spectrum decreases significantly as the system becomes more oblate.
It is useful to calculate this total power in the bending modes, $P_{\rm{tot}}= \int P_u(k,t) {\rm{d}}k$, which by Parseval's theorem is related to the line-averaged deviation of the vortex from its mean position~\footnote{To obtain good statistics we integrate an instantaneous $P_u(k)$ curve (as in Fig.~\ref{figspectra}), and then time-average the result for $0.5{\rm s}$. This time is chosen to be much shorter than the vortex decay-time (which exceeds $5{\rm s}$ for all parameters), to give a short-time ensemble-average of the power data.}. Results for the total power are shown in Fig.~\ref{figratpow} (a). It decreases exponentially as the geometry becomes more two-dimensional, and we see that the total power is independent of oblateness for the most highly oblate condensates. We see that vortex bending is significantly reduced in the highly oblate systems, being almost entirely absent for $8<\Lambda$.

Fig.~\ref{figratpow}(b) shows the effect of condensate geometry on the vortex lifetime, for a range of system temperatures. The lifetime is quantified here in terms of the \emph{mean first exit time}~\cite{Rooney10a}, denoted by $\bar{t}$.
This is calculated as the ensemble average of the time it takes for the vortex to become indistinguishable from thermal fluctuations at the BEC edge (in the plane $z = 0$). The vortex decay rate, $1/\bar{t}$, decreases with increasing $\Lambda$, following the same trend as the total bending mode power [\fref{figratpow}(a)]. However, rather than decreasing to zero, the decay rate approaches a constant value of order $\sim 0.1{\rm s}^{-1}$, being almost independent of $\Lambda$ for  $8< \Lambda < 20$.
This constant is determined by purely 2D vortex dissipation processes which do not involve bending. Interestingly, for the highest temperatures considered, the decay rate starts to increase again at $\Lambda=20$. This increase is not associated with an increase of bending mode power [\fref{figratpow}(a)]. Aside from this anomaly, discussed below, we infer a critical oblateness of $\Lambda_c\approx 8$ for the onset of 2DV dynamics in our system.

\subsection{Effect of temperature: Anomalous decay in quasi-2D systems  } \label{anomdecay}

The results of the previous section show that the vortex enters a 2D dynamics regime at $\Lambda\sim 8$, associated with strong suppression of bending modes, and a diminished vortex decay rate. However, for the highest temperatures the decay rates exhibit a revival as $\Lambda$ increases. We interpret this as arising from the system entering a phase fluctuating quasi-2D regime.

First, to more accurately characterize the critical temperature we include the  downward shifts from $T_c^0$ due to interaction and finite-size effects \cite{Giorgini96a}. This gives a corrected critical temperature
\eqn{T_{c({\rm{corr}})} = T^0_c + \delta T^{\rm fs}_c + \delta T^{\rm{int}}_c  ,\label{Tccorr}}
where $\delta T^{\rm fs}_c= -0.73 \frac{\omega}{\bar{\omega}} N^{-1/3} T_c^0$ is the first-order correction due to finite-size effects, and $\delta T^{\rm{int}}_c = -1.33 \frac{a}{\bar{a}} N^{1/6} T_c^0$ is the first order correction due to interactions, with $\omega = (2\omega_r + \omega_z)/3$ and $\bar{a} = (\hbar /m \bar{\omega})^{1/2}$.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.85\textwidth]{fig6.eps}
\caption{Density isosurfaces of the $\rC$-field showing the SPGPE evolution of a vortex dipole with initial vortex locations $(x,y_\pm)=(0,\pm s/2)$, for separation $s=3{\mu \rm m}$. Red: the $\rC$-field isosurface. Blue: isosurface restricted to the region of the vortex cores.}
\label{sphSeq}
\end{center}
\end{figure*}
In  Fig.~\ref{figvel} we show the decay rates versus ${T_{c({\rm{corr}})}} $   verifying that these systems at the highest temperatures remain below the 3D critical temperature.  However, for oblateness $12\lesssim\Lambda$ we see anomalously rapid increase in the vortex decay rate for temperatures exceeding $\sim 0.95 T_{c({\rm{corr}})}$.  For these cases the system crosses over to being quasi-2D as the chemical potential becomes of the order of the energy to activate degrees of freedom in the tightly confined direction, i.e. $\mu\lesssim\hbar\omega_z+\epsilon_0$ (where $\epsilon_0$ is the single particle ground state energy). In particular, the highest temperature results for $\Lambda=8,12,15,20$ have $\mu/\hbar\omega_z=2.2,1.69,1.42,1.19$  (c.f.~$\hbar\omega_z+\epsilon_0\approx3\hbar\omega_z/2$). In the quasi-2D regime the dynamics of vortices are not well-understood, but strong phase fluctuations will likely facilitate the rate of vortex damping as we observe in our results \footnote{The system still has a phase space density sufficiently high that spontaneous vortices are not expected to be significant (e.g.~see \cite{Hadzibabic2006a,*Simula2006a,*Bisset09d}).}.  A systematic investigation  of vortex decay in a system more deeply in the 2D regime is therefore needed, but is beyond the scope of our work here.

\section{Dipole decay}
Having systematically studied the role of geometry and temperature in single vortex decay, we now consider two particular cases of the decay of a vortex dipole at finite temperature.

\subsection{Spherical geometry: decay via ring and loop formation}\label{vring}
We first consider dipole decay in a spherical trap. We use SPGPE simulation parameters $(T, \mu, \epsilon_{\rm{cut}}) = (52~{\rm{nK}}, 37\hbar\bar{\omega},71\hbar\bar{\omega})$, which give a total atom number $N=2\times10^6$ and dimensionless damping rate (\ref{gamdef}) of $\hbar \gamma/ k_B T = 3\times10^{-4}$. These parameters give a system with the same atom number and temperature as the oblate configuration considered in the experiment of Ref.~\cite{Neely10a} and in the next section, so the system serves as a benchmark for the effect of geometry on vortex dipole dynamics in this system. We create a finite-temperature vortex dipole state by imposing a vortex dipole phase pattern on a SPGPE equilibrium state via $\psi_\rC(\mathbf{x})\to \psi_\rC(\mathbf{x})e^{i\Theta(\mathbf{x})}$, where $\Theta(\mathbf{x})=\arctan{\left(\frac{y-y_1}{x}\right)} - \arctan{\left(\frac{y-y_2}{x}\right)}$.  This creates an axially aligned vortex dipole with positive and negative vortices located at $(x,y_\pm) = (0,\pm s/2)$, with $s=3{\mu \rm m}$. Figure \ref{sphSeq} shows the decay sequence for the vortex dipole. The initially axially aligned vortices rapidly curve as the vortices precess, such that the vortex cores are normal to the surface. This long-wavelength bending leads to formation of a vortex ring, which then subsequently dissociates into two vortices at $t=0.5{\rm s}$. As the vortices closely approach again at $t=0.62{\rm s}$, both loops and rings form, reducing the total vortex length, and leading to total decay in less than 1s. We thus observe rapid vortex dipole decay via intermediate vortex ring and loop formation.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=\columnwidth]{fig7.eps}
\caption{Average number of vortices as a function of time during vortex dipole decay for numerical simulations and experimental data. The dashed lines show SPGPE results for a range of initial vortex dipole separations. Circles show experimental data of Ref.~\cite{Neely10a}. The blue diamond at $t = 0$~s shows the average number of vortices observed in the experimental initial state due to spontaneous vortex formation~\cite{Weiler08a}.}
\label{figlifetime}
\end{center}
\end{figure}
\begin{figure*}[!t]
\begin{center}
\includegraphics[width=0.85\textwidth]{fig8.eps}
\caption{Column densities (top) and phase slices for the $\rC$-field showing the SPGPE evolution of a vortex dipole with initial vortex locations $(x,y_\pm) = (0,\pm s/2)$, for separation $s=3~\mu{\rm{m}}$. The series spans two orbits of the dipole, where a single orbital period is $\sim 1.2{\rm s}$, culminating in mutual vortex annihilation towards the completion of the second orbit. The peak density is renormalized to unity at each time (upper colorbar), and the phase is represented on the interval $[-\pi,\pi)$ (lower colorbar).}
\label{figsequence}
\end{center}
\end{figure*}
\subsection{2DV regime: comparison of experiment and theory}
In this section we use (\ref{SGPEsimp}) to model the dynamics of a charge-one vortex dipole in a highly oblate finite-temperature Bose-Einstein condensate, as in the experiment by Neely {\emph{et al.}}~\cite{Neely10a}.  The system consists of $\sim 2\times 10^6$ atoms at temperature $\sim 52$nK, held in a harmonic trap with frequencies $(\omega_{r},\omega_z) = 2\pi\times(8,90)$Hz. The trap aspect ratio $\Lambda=11.25$ exceeds $\Lambda_c\approx 8$ identified in Sec.~\ref{SecEffOfGeom} for closely related parameters. We can therefore expect that the experimental regime is also approximately that of 2DV. Experimentally, a single vortex dipole is created by pushing the BEC around an past an obstacle above a critical velocity. Ideally, we would like to compare the distance $s$ between the two vortices upon the creation of the dipole in both experiment and simulations, and to relate $s$ to the vortex dipole lifetime in each case. However, the experimental spatial resolution was $\sim 6\mu m$, which is also the approximate vortex separation found immediately after dipole creation in Gross-Pitaevskii simulations of the expriment~\cite{Neely10a}. Since the vortices are not individually resolvable immediately after creation, the experimental data can only place an upper bound on $s$. We are thus unable to carry out quantitative comparison at this point. In this work we instead map out the dependence of dipole lifetime on $s$ by computing the decay dynamics for a range of initial vortex separations.

As in the spherical geometry of Sec.~\ref{vring} we use SPGPE simulation parameters $(T, \mu, \epsilon_{\rm{cut}}) = (52~{\rm{nK}}, 37\hbar\bar{\omega},71\hbar\bar{\omega})$, which again gives a total atom number $N=2\times10^6$ and dimensionless damping rate (\ref{gamdef}) of $\hbar \gamma/ k_B T = 3\times10^{-4}$.
As in the spherical case, we phase imprint a vortex dipole into the BEC. We choose the initial separation between vortices as $ s = 1, 2,2.5,3,$ and $3.5~\mu{\rm{m}}$. Note that the condensate radius is $\sim 50\mu{\rm{m}}$, so these separations are small relative to the system size, but larger than the healing length $\xi\approx0.2 \mu\rm m$.
Qualitatively, our simulations differ from the experiment in several ways: in our simulations we have two vortices initially, a high degree of cylindrical and mirror symmetry, and a well defined total atom number. In the experiment there is occasionally a vortex present from the BEC formation process~\cite{Weiler08a}, an irreducible (albeit very small) cylindrical asymmetry of the harmonic potential, a mirror asymmetry of the stirring potential, and uncertainty in the total atom number. We do not attempt to model all of this complexity here, confining our study to the basic decay process of internal annihilation in the symmetric system, with fixed total atom number~\footnote{Our simulation initial conditions correspond to a grand canonical ensemble for the $\rC$-field region, with well defined average total atom number $N=2\times 10^6=N_{\rI}+N_{\rC}$. We made use of a Hartree-Fock scheme for SPGPE parameter estimation developed in Ref.~\cite{Rooney10a}.}.

Figure \ref{figlifetime} shows the mean vortex number over time from the experimental data~\cite{Neely10a}, compared with the results of our simulations. In our simulations we always find that the vortices mutually annihilate each other in the center of the BEC, rather than damping at the boundary. The vortices move rapidly when closely approaching each other, nearing the speed of sound $c=0.2 \rm cm/s$. Consequently, most of the orbital time is spent at the outer edge of the BEC, and the time interval in which annihilation may occur is a small fraction of the orbital period ($\sim 10\%$). Thus the vortex number has an abrupt time dependence for a given initial separation, with characteristic timescale given by the orbital period of the vortex dipole, found numerically and experimentally to be $\sim 1.2\rm s$ and independent of $s$ over a wide range~\cite{Neely10a}. In general, the $s$ value selects a given orbit of annihilation, within which there is very small variation of lifetime.

In detail, we find that for $s=(1,2,2.5,3,3.5)\mu\rm m$ the mean lifetime is $\tau=(0.01,0.86,2.04,2.07,3.30)\rm s$, corresponding to $N_0=(0,1,2,2,3)$ dipole orbits before annihilation. For example, with $s=1\mu{\rm m}$, the dipole self annihilates immediately after creation. The separations $s=2.5\mu\rm m$ and $s=3\mu\rm m$ lead to almost identical lifetimes of about $2\rm s$, corresponding to the significant drop in vortex number seen experimentally for $2{\rm s}\lesssim t \lesssim 2.5{\rm s}$ which we interpret as an experimental indication of the dipole lifetime.

In Figure \ref{figsequence} we plot a time series of simulation data for $s=3\mu\rm m$ showing internal annihilation at the end of the second orbit. Interestingly, we observe a tilt in the axis of dipole precession caused by thermal fluctuations, which is also observed in experimental absorption images.

\section{Conclusions and outlook}
Here we summarize our results on single vortex and vortex dipole decay.

\emph{Sub-critical Regime.---}For spherical systems at $T = 0.78T_c^0$, a lone vortex is freely deformed from its initial straight-line configuration by the activation of vortex bending modes.  This effect diminishes as the condensate is flattened, and bending modes become energetically inaccessible. The total bending mode power decreases exponentially as the condensate geometry becomes increasingly two-dimensional.  For geometries with oblateness $\Lambda \geq \Lambda_c (\approx 8)$, the total power asymptotes to zero, indicating bending is strongly inhibited.  The effective vortex decay rate also decreases as $\Lambda$ increases, approaching a constant two-dimensional value for $\Lambda_c\leq\Lambda$. $\Lambda_c$ can therefore be identified as the critical oblateness required for the system to enter the 2D vortex regime.

\emph{Critical Regime.---}The foregoing comments hold for temperatures outside an observed critical regime. When critical fluctuations associated with the phase transition become significant, we find that the vortex decay rate is highly sensitive to temperature and geometry. In particular, we see an anomalously large increase in the decay rate for geometries with $\Lambda_c< \Lambda$, and $0.95 T_{c \rm (corr)}\lesssim T$ that we attribute to phase-fluctuations associated with the system entering the quasi-2D regime.

\emph{Vortex Dipole Decay.---}
We have modeled the experiment of Neely {\em et al}~\cite{Neely10a} which is in the 2DV regime ($\Lambda=11.25$), and found that the approximate time for decay of a vortex dipole in the experiment ($\sim 2.25$s) is consistent with numerical simulations with an initial vortex separation of $s\sim 2.5-3\mu$m. In the simulations with these initial conditions mutual vortex annihilation occurs after two complete orbits, at $t\sim 2$s. In this work we have used $s$ as the fitting parameter; improved experimental resolution would be required for a quantitative comparison without any fitting parameters. In similar computations of spherical ($\Lambda=1$) BECs, the vortices decay much more quickly to rings and other bent geometries that would not be readily visible after only $\sim 100{\rm ms}$ using the experimental methods of Ref.~\cite{Neely10a}. We thus infer from the experiments of Ref. \cite{Neely10a} and the calculations presented here that the experimental system consisting of a BEC with oblateness $\Lambda\approx 11.25$ is within the 2D vortex dynamics regime, consistent with our calculations suggesting a critical oblateness of $\Lambda_c\approx 8$.

{\bf Acknowledgements:}
We thank Simon Gardiner for discussions regarding this work.

\bibliographystyle{prsty}

redpajama_set_name: RedPajamaArXiv


We all need a break from the stresses of city life, and escape on our own for a little peace of quiet. And there's no better place for escape than the beautiful Gippsland ranges in the quiet surroundings of Mirador Springs Retreat located just two hours from Melbourne, it's a short drive to the heart of serenity.
The property is located on 15 acres of botanical gardens, with bright flowers and lush greenery. Meandering paths spiral across the acreage, making it the perfect place to get lost and let your mind wander.
Elegance and luxury is the emphasis of the retreat, with three spectacular suites to choose from. The Cottage accommodation combines the comfort and charm of a rustic farmhouse with indulgent luxuries, making it the perfect home away from home. Snuggle up by the French fireplace, relax in the private garden, or unwind with an on-demand movie in the expansive living room.
Romanticists will love the glamping experience, full of sweet charm. The secluded bell tent is fitted with dangling plants, atmospheric fairy lights, and lounge chairs. The tent features a private bath house, complete with a deep oval tub and natural bath salts, overlooking the surrounding tropical gardens.
For those looking for a truly indulgent experience, The Residence is the place to feel like royalty. This historic manor home has been superbly renovated, ensuring the finest amenities and touches of luxuries in every corner. Ideal for a large family holiday or wedding party, the picture-perfect property boasts six bedrooms with three ensuites, a sauna, open plan living and dining area along with an outdoor entertaining area with valley views.
Spend your days reading by the babbling natural spring, wandering through the gardens, or visiting the character-filled miniature donkeys.


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Welcome to the most patriotic village of India! Presenting India's Military Village - Madhavaram in AP. Where every man strives to join the armed forces with 700+ serving the forces and 1500 ex servicemen. In total population of only 7000.
Its military history dates back to the 17th century. Its soldiers have been a part of every war India has fought.
There is a war memorial here to commemorate the martyrs. Madhavaram's military tradition is the pride of the nation.


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Gazooba LLC
Toyota Corolla Axio Japan : Here are some talking points
When it is a question of which sedan compact car to buy in Japan, Toyota Corolla Axio is the definite answer. Its station wagon version is called Toyota Corolla Fielder and both these units are manufactured by Central Motors which is Toyota subsidiary and is located in the Miyagi prefecture on the east coast of Honshu Island.
Corolla Axio is a 4 door sedan and comes in both terrains of 2wd and 4wd with 1300 and 1500 cc engines in the petrol versions and 1500 cc engines in the Hybrid version. Corolla Axio sales crossed 1.2 million units in 2013 which confirms its popularity and loyalty of its customers worldwide.
Why is Toyota Corolla Axio so popular in the world
Toyota Corolla Axio has become the most adored vehicle all around the world, from Africa to South America, Arab nations to Southeast Asia, New Zealand to countries in the Pacific all prefer this smaller sized yet commodious sedan. Are there any particular reasons for its popularity why? Yes, there are plenty. Its fuel tank has a capacity of 50 liters. With a tank full, it can run for 900 km. This makes it the most eminent car particularly among cab and uber drivers who prefer simple car design and fuel efficient vehicles. Its fuel efficiency makes it the choice of car for city dwellers and those who drive on long routes.
In Japan, most of the Corolla Axio cars are owned by Rent-a-Car companies and Vehicle Leasing Companies who rent out the car for a fixed period of time to companies and individuals, specially to foreign expats living in Japan for a short period of 2 to 3 years. These are very well maintained by rental or leasing companies and therefore largely come with spotless interior and decent exterior. Most of these are never even smoked in.
Axio has a plain yet alluring peculiar look. Modish and trendy design and excellent road handling, and its handleability even on the rough roads, this car is designed to last for a long time. Because it has strong demand among car lovers all over the world, the depreciation rate of its price is very low. Axio holds its value for long and has the best resale price in the market among used car buyers.
Front seats are sufficiently roomy and even the rare seats have comfy leg space. Plenty of luggage space is available in the trunk.
Some of the features like power steering, electric windows, electric side mirrors, driver Airbag, front passenger Airbag, FM radio, air conditioning are available as basics on Axio with plenty of other extra options to choose from.
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redpajama_set_name: RedPajamaCommonCrawl


I have recently been given the opportunity by FlashSticks for my family to learn a second language. With currently eight different languages to choose from, we chose British Sign Language and for 2 reasons. Firstly, when Max was still learning to speak he would say please and thank you in sign, which he'd learnt at nursery, and I thought this was wonderful. Secondly, ever since I learnt the alphabet in Guides (I know, going back a few years now....) I've always thought of BSL as an important language to learn. To be able to just say 'Hi, how you doing', to someone who can not communicate in the more conventional way, rather than just smile or nod would mean so much for me to be able to do.
The way in which FlashSticks works is you have a packet of post-it notes sent with a description and picture on for you to learn, and because these lessons are on post-its they can be stuck all around the house in key areas for learning.
We have only just started our journey with FlashSticks and have our first few worlds stuck on the mantelpiece. All three of us now know the sign for a hospital, happy, eat and mother and therefore, today we have moved on to the next 4 post-its. As we get familiar with the new words I'll move the first ones we learnt to a less dominate space in the house and add 4 new post-its each week.
The program also comes with an App that you can download via the Apple Store or Google play which will help with pronouncing foreign language words perfectly.
I will write up another post in a couple of month's time to show how far we have come with this form of learning, which will give you an idea of how easy it is to learn with FlashSticks.... In the meantime, you can find out more via their website, Twitter and Facebook sites.
I received BSL learning from FlashSticks to review.


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« Sense on Cents Media Appearances
"I Just Can't Trust Them…" »
If It Sounds Too Good To Be True…
Posted by Larry Doyle on January 18, 2011 9:04 PM |
The market feels firm. Your broker calls with an opportunity that sounds very appealing. You are sitting on cash and feel you may have missed the run in the market. Your cash is generating little to no return. You are contemplating taking a flyer on this "deal."
Your broker is really pushing you and indicating a lot of people are getting involved in these deals. The highlighted returns look too good to be true. But wait, haven't you seen this movie before? Remember, 'if it sounds too good to be true…' it typically is.
Another play on this same theme was highlighted at Bloomberg a few weeks back. This story deserves serious attention. Pay particular attention to the commissions and fees earned by Wall Street and the risks absorbed by you, the investor.
Bloomberg wrote, Wall Street Turns Stock Gains Into Investor Losses with Structured Notes:
Banks sold more than $6 billion of bonds linked to the performance of stocks last year, promising returns of as much as 64 percent at a time when interest rates were at historic lows.
Instead of reaping such extraordinary gains, reverse convertibles, as the products are known, lost 1 percent on average, according to data compiled by Bloomberg on 1,481 of the securities sold in the U.S. last year that matured by Nov. 30. The Standard & Poor's 500 Index returned 8 percent during that period and corporate bonds gained 11.1 percent, including reinvested interest, Bank of America Merrill Lynch index data show.
"Here's an asset class that's basically failed on all counts," said Glenn Tongue, a money manager who oversees about $200 million with Whitney Tilson at T2 Partners LLC in New York. "I doubt these are being pitched as an opportunity to lose 1 percent."
Barclays Plc, based in London, and UBS AG in Zurich led more than a dozen banks selling reverse convertibles, which are short-term bonds generally marketed to individuals that convert into stock if a company's share price plummets. The securities are being created as part of a boom in structured notes, or bonds packaged with derivatives whose values are derived from assets including stocks, bonds, currencies and commodities, or from events such as changes in interest rates.
Structured note sales rose 46 percent last year to a record $49.4 billion in the U.S., Bloomberg data show. The securities fed demand from individual investors frustrated with record low rates on everything from certificates of deposit to money market funds with the Federal Reserve holding its target interest rate for overnight loans between banks in a range of zero to 0.25 percent since 2008. Banks issued $33.9 billion in 2009, according to StructuredRetailProducts.com, a database used by the industry.
Royal Bank of Scotland Group Plc sold $1.15 million in three-month notes tied to Rochester, New York-based Eastman Kodak Co. on June 10 that paid 24 percent annualized interest, a filing with the U.S. Securities and Exchange Commission shows. That's 24 times the average rate on one-year certificates of deposit, according to data from Bankrate Inc. in North Palm Beach, Florida.
Buyers couldn't lose money unless shares of the camera maker fell to below $3.54 from $5.06. Kodak dropped to $3.50 on Aug. 31 in New York trading. RBS converted the bonds into stock and investors lost about 18 percent even with the high interest rate.
Investors lost an average of 2 percent on the 180 reverse convertibles issued by RBS last year that matured by Nov. 30, Bloomberg data show. The securities had a total face value of $69.3 million. Pholida Phengsomphone, an RBS spokeswoman, declined to comment.
"This isn't something that a retail investor calls up and asks for," said Marilyn Cohen, who oversees $250 million as chief executive officer of Envision Capital Management in Los Angeles. "Is it ever explained to them that you might end up with the stock and there's a large probability that will happen?"
Buyers profited on 75 percent of the notes analyzed by Bloomberg. The average return was negative because investors lost more on the unprofitable ones than they gained on those that made money. Investors also took on credit risk because the notes are unsecured debt.
"There's no free lunch," Tongue said. "If you're going to get super-standard returns, that means you're taking super- standard risks."
Reverse convertibles aren't traded on exchanges and their performance isn't reported publicly. Investors lost $27 million on the $2.19 billion of securities compiled by Bloomberg. Banks also sold $4.56 billion of reverse convertibles whose returns weren't included because they didn't mature by Nov. 30.
Kenneth Lench, head of the SEC's structured products unit that investigated Goldman Sachs Group Inc.'s subprime-mortgage investments, said in a September interview that the agency was examining whether brokers overcharged investors for the notes. He declined to comment for this article. SEC Enforcement Director Robert Khuzami said at a Senate Judiciary Committee hearing the same month it has looked at reverse convertibles.
"The reason these are popular is they're kind of an easy sale," said Charlie O'Flaherty, who used to oversee U.S. structured products and derivatives at Bank of Ireland. The products are marketed as a way to earn higher yields during times when stock prices are stable and interest rates are low, he said. Buyers aren't told how they have performed in the past, he said.
Banks charged fees that average 1.6 percent on a three- month reverse convertible, or about 6 percent a year, the data show. The average annual fee on stock mutual funds is 1 percent, according to the Investment Company Institute, a trade group in Washington.
The three-month reverse convertibles sold by Edinburgh- based RBS and linked to Kodak, a photography company founded more than 100 years ago, paid brokers a 2.75 percent commission.
The fees, which sometimes exceeded the securities' maximum possible yield, cut into returns and probably account for the losses, said O'Flaherty, who left the industry last year to manage his own investments.
Professional money managers generally don't buy reverse convertibles because they can pay less in fees by trading derivatives directly, according to Tongue and O'Flaherty. Small investors would need a computer model and access to options pricing data to see if they're being compensated fairly for the risks they're taking on, O'Flaherty said.
"If you were a big-boy investor, you'd just do it yourself," O'Flaherty said. "You'd structure it yourself and take all the middlemen out."
Leroy and Carol Conklin, a retired couple in Sun City Center, Florida, said they were pitched an investment in reverse convertibles by James Tuberosa, a broker at H&R Block Financial Advisors whom they met at the retirement community's annual FunFest fair.
They bought reverse convertibles from Tuberosa in 2007 and 2008, including $20,000 of notes linked to Mooresville, North Carolina-base home improvement retailer Lowe's Cos. that paid 9 percent and that the broker called corporate bonds, the Conklins said in an interview.
"I said, 'Boy, this is a good investment,'" said Leroy Conklin, 80, who earned a Silver Star in the 1950-1953 Korean War and worked as a high school principal.
The Lowe's notes converted into stock at a $4,000 loss, said Jason Doss, the Conklins' lawyer at Doss Firm LLC in Marietta, Georgia. The couple lost more than $130,000 on reverse convertibles overall, Doss said.
"We feel so stupid," Carol Conklin said, adding that she still has "no idea" what derivatives are.
The Conklins filed an arbitration claim that's pending against Ameriprise Financial Inc., which bought H&R Block's financial advisory business in 2008. Tuberosa, who works for Ameriprise in Sun City Center, declined to comment, as did Benjamin Pratt, an Ameriprise spokesman. A hearing is scheduled for May.
State regulators are seeing an "influx of concerns and complaints" from individual investors about structured products, including reverse convertibles, said Joseph Borg, director of the Alabama Securities Commission. Brokers who sell the investments sometimes don't understand them, Borg said in a telephone interview.
Barclays sold $757.4 million of reverse convertibles in 588 offerings in the U.S. last year that matured by Nov. 30, the most of any bank, Bloomberg data show. Investors lost about 2 percent on average. Kristin Friel, a Barclays spokeswoman, declined to comment.
Royal Bank of Canada's 505 reverse convertibles with a total face value of $372.9 million lost 1 percent on average, as did Wells Fargo & Co.'s 27 with a face value of $156.8 million, Bloomberg data show. Morgan Stanley issued $294.9 million in seven deals that lost an average of 6 percent.
Citigroup Inc. sold $274.1 million of reverse convertibles in 15 deals that made an average of 6 percent, the data show. Investors made an average of 1 percent on the 114 JPMorgan Chase & Co. reverse convertibles with a face value of $85.2 million that Bloomberg analyzed. UBS sold 44 with a total face value of $177.7 million that also made about 1 percent on average.
Spokesmen for the banks declined to comment.
JPMorgan, based in New York, charged about 3 percent on average in fees, the data show. In some cases, the fixed fees gave banks and brokers more profit than investors could earn on the securities.
The bank sold a three-month note on April 27 linked to West Chester, Ohio-based steelmaker AK Steel Holding Corp. that paid 3.5 percent in interest over the term of the security, or 14 percent on an annualized basis, according to an SEC filing. JPMorgan charged 5.2 percent in fees.
The bond couldn't be converted into stock unless shares of the company fell more than 20 percent from their initial level of $17.43. AK Steel plummeted and triggered the conversion when it dropped to $13.40 on May 20. Investors lost 13 percent. Justin Perras, a JPMorgan spokesman, declined to comment.
"It sounds too good to be true, but the markets are stacked against you," said Envision's Cohen. Reverse convertibles are "really a toxic product," she said.
On May 11, JPMorgan sold $800,000 of reverse convertibles linked to TiVo Inc. that paid 64 percent a year in interest, according to a prospectus the bank filed with the SEC. Three days later, the Alviso, California-based digital-video recording pioneer dropped 42 percent after an adverse court ruling. Investors ended up losing 42 percent, including interest. The bank charged 1.75 percent in commission on the two-month notes.
While banks must pay competitive commissions to brokers, the fees on some short-term reverse convertibles are "very high" compared with other investments, said Richard Sandulli, a former head of structured products at Morgan Stanley and Wells Fargo who left the industry last year and is now a consultant.
Most structured products are "valuable tools" that spread the fee over a longer period of time, he said. Paying too much commission cuts into investors' returns and encourages brokers to act against their clients' best interests, he said.
"Just because something can be sold doesn't mean it should be," Sandulli said.
Are you ready to vomit? Wall Street earns fees of 3, 4, 5, 6%. You, the investor, are truly nothing more than the pawn. Where are the regulators?
Remember, if it sounds too good to be true . . .
Navigate accordingly.
Please subscribe to all my work via e-mail, an RSS feed, on Twitter or Facebook.
I have no affiliation or business interest with any entity referenced in this commentary. The opinions expressed are my own and not those of Greenwich Investment Management. As President of Greenwich Investment Management, an SEC regulated privately held registered investment adviser, I am merely a proponent of real transparency within our markets so that investor confidence and investor protection can be achieved.
This entry was posted on Tuesday, January 18th, 2011 at 9:04 PM	and is filed under General, Structured Notes. You can follow any responses to this entry through the RSS 2.0 feed. You can skip to the end and leave a response. Pinging is currently not allowed.


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Home Business Companies Construction industry development authority a priority for us –Contractors
Construction industry development authority a priority for us –Contractors
The leadership of the Contractors Association have said that an industry development authority is crucial if the sector has to be transformed and contribute meaningfully to development of the country.
The construction sector is currently reeling under several challenges – key among them delayed payments for works done, preference for foreign contractors over indigenous ones, and low capacity of indigenous construction firms. These challenges have existed over several years and are becoming the norm.
But the new President of the Association of Building and Civil Engineering Contractors of Ghana (ABCECG), Prosper Yao Ledi, said these negative practices must give way to a new phase wherein the indigenous contractor can also rub shoulders with contractors anywhere on the globe.
He explained that even when contracts are given to foreign companies ahead of Ghanaian firms, in most of the cases the foreign contractor sub-contracts the job to indigenous contractors. This, Mr. Ledi said, clearly shows that the Ghanaian contractor is capable and should therefore be given the opportunity to prove himself.
The President of ABCECG stated that transformation of the construction industry can only be achieved through a holistic approach. He however explained that the holistic approach can be attained through the establishment of an industry development authority that will regulate and ensure the sector proper transformation.
See Also: Russia rubbishes London over Sergey Skripal Case
Contractors have mooted the idea of an industry regulator, which they have christened the Construction Industry Development Authority (CIDA). The CIDA will regulate the activities of contractors in the industry to ensure high standard of performance. It will also prescribe and enforce professional requirements according to the category of contractors. The Authority will also monitor and evaluate the performance and efficiency of registered contractors.
ABCECG is receiving funding support from the Business Sector Advocacy Challenge (BUSAC) Fund to make establishment of the Authority a reality. The Authority will not only seek to develop the industry but also ensure that all stakeholders, including government, act appropriately.
It is anticipated that the Authority will ensure delayed payments are reduced to the barest minimum, and when they occur interest will be paid to the contractor. In other jurisdictions, such an authority also ensures that no project begins unless funding has been duly secured – and contractors in Ghana are hoping that such a policy will become reality in Ghana as well.
Mr. Ledi explained that efforts to establish the Authority are far advanced, as the draft bill is being worked on. He is grateful to (BUSAC) for their support and said he is going to work with other stakeholders to complete work on the draft bill. He therefore called on the sector-minister to throw his weight behind the Authority's establishment.
See Also: Africans fleeing war top list of worlds' most neglected, says charity
The ABCECG President also took the opportunity to urge government to pay the arrears owed contractors. He said contractors have been unfairly treated and some urgency should be attached to efforts to pay up the arrears, explaining that the delay is becoming unbearable.
He noted also that that current complaints of economic hardship in the country cannot be delinked from the contractors' arrears, stating: "Let government pay the contractors and see if the complaints will not come down".
SOURCEthebftonline
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In 2014, two business partners with a history of successful projects in the fields of marketing, international trade and services established Sovereign Group, as a means to further their progress and collaboration.
Our aim was to create a multi-purpose platform which promotes and enhances the supply of services and products with reputation in mind, as our investment is based on long-term strategies and success.
We have many projects which will determine the creation of new services, at the right time and if they prove to be doable. Our pledge is to maintain a healthy, organic growth, based on strategic, long-term views.