We present an open source computational framework geared towards the efficient numerical investigation of open quantum systems written in the Julia programming language. Built exclusively in Julia and based on standard quantum optics notation, the toolbox offers speed comparable to low-level statically typed languages, without compromising on the accessibility and code readability found in dynamic languages. After introducing the framework, we highlight its features and showcase implementations of generic quantum models. Finally, we compare its usability and performance to two well-established and widely used numerical quantum libraries. Nature of problem: Dynamics of open quantum systemsSolution method: Numerically solving the Schrödinger or master equation or a Monte Carlo wave-function approach. Additional comments including Restrictions and Unusual features:The framework may be used for problems that fulfill the necessary conditions such that they can be described by a Schrödinger or master equation. Furthermore, the aim is to efficiently and easily simulate systems of moderate size rather than pushing the limits of what is possible numerically.

Abstract. We numerically study the collective coherent and dissipative dynamics in spin lattices with long range interactions in one, two and three dimensions. For generic geometric configurations with a small spin number, which are fully solvable numerically, we show that a dynamical mean-field approach based upon a spatial factorization of the density operator often gives a surprisingly accurate representation of the collective dynamics. Including all pair correlations at any distance in the spirit of a second order cumulant expansion improves the numerical accuracy by at least one order of magnitude. We then apply this truncated expansion method to simulate large numbers of spins from about ten in the case of the full quantum model, a few thousand, if all pair correlations are included , up to several ten-thousands in the mean-field approximation. We find collective modifications of the spin dynamics in surprisingly large system sizes. In 3D, the mutual interaction strength does not converge to a desired accuracy within the maximum system sizes we can currently implement. Extensive numerical tests help in identifying interaction strengths and geometric configurations where our approximations perform well and allow us to state fairly simple error estimates. By simulating systems of increasing size we show that in one and two dimensions we can include as many spins as needed to capture the properties of infinite size systems with high accuracy. As a practical application our approach is well suited to provide error estimates for atomic clock setups or super radiant lasers using magic wavelength optical lattices.

Atoms trapped in magic wavelength optical lattices provide a Doppler-and collision-free dense ensemble of quantum emitters ideal for fast high precision spectroscopy and thus they are the basis of the best optical clock setups to date. Despite the minute optical dipole moments the inherent long range dipole-dipole interactions in such lattices at high densities generate measurable line shifts, increased dephasing and modified decay rates. We show that these effects can be resonantly enhanced or suppressed depending on lattice constant, geometry and excitation procedure. While these interactions generally limit the accuracy and precision of Ramsey spectroscopy, under optimal conditions collective effects can be exploited to yield zero effective shifts and long dipole lifetimes for a measurement precision beyond a noninteracting ensemble. In particular, 2D lattices with a lattice constant below the optical wavelength feature an almost ideal performance.Since the turn of the century the technology of manipulating and controlling ultracold atoms and molecules with laser light has seen breathtaking advances [1][2][3]. Following the seminal first demonstration of a quantum phase transition in an optical lattice [4], nowadays the so-called Mott insulator phase for atoms in an optical lattice can be prepared almost routinely [5, 6] and experiments with photo-associated ultracold molecules have reached a comparable level of control [7][8][9][10]. Employing the particles' internal structure coherent interactions between the atoms at neighboring sites in such a lattice can be tailored to a large extend, e.g. via spin-dependent tunneling [11].In one of the typical experimental setups atoms possessing a long-lived clock transition are prepared in an optical lattice using a differential light shift free (magic) trapping wavelength [12, 13]. Its most prominent application is the implementation of the world's best optical clocks [14][15][16].When the atoms in such a lattice are excited on an optical or infrared transition they will interact stronger and on a much longer range via dipole-dipole energy exchange than via tunneling or collisions. At sufficient densities the dipole interaction strength surpasses the excited state lifetime and resonantly enhanced collective excitations analogous to excitons in solid state physics appear [17, 18]. For polar molecules in optical lattices, such long wavelength collective excitations even dominate the dynamics [19] and can form the basis for studying generic phenomena of solid state physics by means of tailored toy models [1]. In the case of clock transitions, the extremely tiny dipole moment keeps these interactions small in absolute magnitude. Nevertheless, for densities close to unit filling the exciton's effective transition frequencies and their spontaneous decay is governed by dipole-dipole interaction [20] and deviates from the bare atom case.In a standard Ramsey interrogation sequence used as the generic basis for a clock setup, the first pulse creates a product state of all s...

A dilutely filled N -site optical lattice near zero temperature within a high-Q multimode cavity can be mapped to a spin ensemble with tailorable interactions at all length scales. The effective full site to site interaction matrix can be dynamically controlled by the application of up to N (N +1)/2 laser beams of suitable geometry, frequency and power, which allows for the implementation of quantum annealing dynamics relying on the all-to-all effective spin coupling controllable in real time. Via an adiabatic sweep starting from a superfluid initial state one can find the lowest energy stationary state of this system. As the cavity modes are lossy, errors can be amended and the ground state can still be reached even from a finite temperature state via ground state cavity cooling. The physical properties of the final atomic state can be directly and almost non-destructively read off from the cavity output fields. As example we simulate a quantum Hopfield associative memory scheme.

We study light induced spatial crystallization of ultracold quantum particles confined along the axis of a high-Q linear cavity via a transverse multicolor pump using numerical simulations.Whenever a pump frequency is tuned close to resonance with a longitudinal cavity mode, the dynamics favors bistable spatial particle ordering into a Bragg grating at a wavelength distance.Simultaneous pumping at several resonant frequencies fosters competition between the different spatial lattice orders, exhibiting complex nonlinear field dynamics involving several metastable atom-field states. For few particles even superpositions of different spatial orders entangled with different light mode amplitudes appear. By a proper choice of trap geometry and pump frequencies a broad variety of many particle Hamiltonians with a nontrivial long range coupling can be emulated in such a setup. When applying quantum Monte Carlo wave function simulations to study time evolution we find simultaneous super radiant scattering into several light modes and the buildup of strong non-classical atom field correlations.