We propose in this White Paper a concept for a space experiment using cold atoms to search for ultra-light dark matter, and to detect gravitational waves in the frequency range between the most sensitive ranges of LISA and the terrestrial LIGO/Virgo/KAGRA/INDIGO experiments. This interdisciplinary experiment, called Atomic Experiment for Dark Matter and Gravity Exploration (AEDGE), will also complement other planned searches for dark matter, and exploit synergies with other gravitational wave detectors. We give examples of the extended range of sensitivity to ultra-light dark matter offered by AEDGE, and how its gravitational-wave measurements could explore the assembly of super-massive black holes, first-order phase transitions in the early universe and cosmic strings. AEDGE will be based upon technologies now being developed for terrestrial experiments using cold atoms, and will benefit from the space experience obtained with, e.g., LISA and cold atom experiments in microgravity.KCL-PH-TH/2019-65, CERN-TH-2019-126

We present C programming language versions of earlier published Fortran programs (Muruganandam and Adhikari (2009) [1]) for calculating both stationary and non-stationary solutions of the time-dependent Gross-Pitaevskii (GP) equation. The GP equation describes the properties of dilute Bose-Einstein condensates at ultra-cold temperatures. C versions of programs use the same algorithms as the Fortran ones, involving real-and imaginary-time propagation based on a split-step Crank-Nicolson method. In a one-spacevariable form of the GP equation, we consider the one-dimensional, two-dimensional, circularly-symmetric, and the three-dimensional spherically-symmetric harmonic-oscillator traps. In the two-space-variable form, we consider the GP equation in two-dimensional anisotropic and three-dimensional axially-symmetric traps. The fully-anisotropic three-dimensional GP equation is also considered. In addition to these twelve programs, for six algorithms that involve two and three space variables, we have also developed threaded (OpenMP parallelized) programs, which allow numerical simulations to use all available CPU cores on a computer. All 18 programs are optimized and accompanied by makefiles for several popular C compilers. We present typical results for scalability of threaded codes and demonstrate almost linear speedup obtained with the new programs, allowing a decrease in execution times by an order of magnitude on modern multi-core computers. [14][15][16]. This new version represents translation of all programs to the C programming language, which will make it accessible to the wider parts of the corresponding communities. It is well known that numerical simulations of the GP equation in highly experimentally relevant geometries with two or three space variables are computationally very demanding, which presents an obstacle in detailed numerical studies of such systems. For this reason, we have developed threaded (OpenMP parallelized) versions of programs imagtime2d, imagtime3d, imagtimeaxial, realtime2d, realtime3d, realtimeaxial, which are named imagtime2d-th, imagtime3d-th, imagtimeaxial-th, realtime2d-th, realtime3d-th, realtimeaxial-th, respectively. Figure 1 shows the scalability results obtained for OpenMP versions of programs realtime2d and realtime3d. As we can see, the speedup is almost linear, and on a computer with the total of 8 CPU cores we observe in Fig. 1(a) a maximal speedup of around 7, or roughly 90% of the ideal speedup, while on a computer with 12 CPU cores we find in Fig. 1(b) that the maximal speedup is around 9.6, or 80% of the ideal speedup. Such a speedup represents significant improvement in the performance.

Many of the static and dynamic properties of an atomic Bose-Einstein condensate (BEC) are usually studied by solving the mean-field Gross-Pitaevskii (GP) equation, which is a nonlinear partial differential equation for short-range atomic interaction. More recently, BEC of atoms with long-range dipolar atomic interaction are used in theoretical and experimental studies. For dipolar atomic interaction, the GP equation is a partial integro-differential equation, requiring complex algorithm for its numerical solution. Here we present numerical algorithms for both stationary and non-stationary solutions of the full threedimensional (3D) GP equation for a dipolar BEC, including the contact interaction. We also consider the simplified one-(1D) and two-dimensional (2D) GP equations satisfied by cigar-and disk-shaped dipolar BECs. We employ the split-step Crank-Nicolson method with real-and imaginary-time propagations, respectively, for the numerical solution of the GP equation for dynamic and static properties of a dipolar BEC. The atoms are considered to be polarized along the z axis and we consider ten different cases, e.g., stationary and non-stationary solutions of the GP equation for a dipolar BEC in 1D (along x and z axes), 2D (in x-y and x-z planes), and 3D, and we provide working codes in Fortran 90/95 and C for these ten cases (twenty programs in all). We present numerical results for energy, chemical potential, rootmean-square sizes and density of the dipolar BECs and, where available, compare them with results of other authors and of variational and Thomas-Fermi approximations. Program summary Program title: (i) imag1dZ, (ii) imag1dX, (iii) imag2dXY, (iv) imag2dXZ, (v) imag3d, (vi) real1dZ, (vii) real1dX, (viii) real2dXY, (ix) real2dXZ, (x) real3d