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.

An array of N closely spaced dipole coupled quantum emitters exhibits super-and subradiance with characteristic tailorable spatial radiation patterns. Optimizing their geometry and distance with respect to the spatial profile of a near resonant optical cavity mode allows to increase the ratio between light scattering into the cavity mode and free space by several orders of magnitude. This leads to a distinct nonlinear particle number scaling of the relative strength of coherent light-matter interactions versus decay. In particular, for subradiant states the collective cooperativity increases much faster than the typical linear ∝ N scaling of independent emitters. This extraordinary collective enhancement is manifested both in the intensity and phase profile of the sharp collective emitter antiresonances detectable at the cavity output port via transmission spectroscopy.PACS numbers: 42.50. Ar, 42.50.Lc, The confinement of atoms and photons in small volumes with very low loss has been a renowned success [1][2][3] as it allows for tests of light-matter interactions where the quantum nature of both comes into play. In a cavity quantum electrodynamics (CQED) setup the photon-emitter interaction strength g ∝ µE for a particle with dipole moment µ is strongly enhanced by decreasing the field mode volume and thus increasing the local field per photon E. In a standard Fabry-Pérot cavity geometry this is achieved by closely surrounding the emitter with two high-reflectivity mirrors. The atom-photon interaction time is then enhanced by a factor roughly proportional to the cavity finesse characterizing the number of round trips a photon can make before escaping to the environment at a rate κ. At the single quantum emitter level, this has facilitated experimental progress towards strong coupling allowing the study of single photon nonlinear effects, such as the photon blockade regime [4], of vacuum Rabi splittings and other tests of fundamental quantum optics effects [5,6].As a characteristic quantity of coupling strength, the single emitter cooperativity C = g 2 /(κγ) (where γ is the rate of spontaneous decay into modes outside the cavity) is a well established figure of merit for strong light-matter interactions when C 1. Since for a single emitter the dipole moment matrix element µ of the coupled transition appears both in g ∝ µ and γ ∝ µ 2 , the cooperativity C is merely a geometric factor independent of µ [7]. This means that cavity design (increasing the finesse and decreasing the transverse mode area) is the central aspect for reaching high single emitter cooperativity. In an alternative approach, in the case of a lossy cavity where κ is large, one often tries to reach a large effective cooperativity by coupling N emitters simultaneously to a cavity mode such that the cooperativity scales like C eff ∝ CN . An implicit important assumption is that the emitter-cavity coupling increases proportionally to N while the free space emission stays constant (independent emitter approach). In reality, especially for small parti...

Inherent binary or collective interactions in ensembles of quantum emitters induce a spread in the energy and lifetime of their eigenstates. While this typically causes fast decay and dephasing, in many cases certain special entangled collective states with minimal decay can be found, which possess ideal properties for spectroscopy, precision measurements or information storage. We show that for a specific choice of laser frequency, power and geometry or a suitable configuration of control fields one can efficiently prepare these states. We demonstrate this by studying preparation schemes for strongly subradiant entangled states of a chain of dipole-dipole coupled emitters. The prepared state fidelity and its entanglement depth is further improved via spatial excitation phase engineering or tailored magnetic fields.

A ring of sub-wavelength spaced dipole-coupled quantum emitters possesses only few radiant but many extraordinarily subradiant collective modes. These exhibit a 3D-confined spatial radiation field pattern forming a nano-scale high-Q optical resonator. We show that tailoring the geometry, orientation and distance between two such rings allows for increasing the ratio of coherent ring-toring coupling versus free-space emission by several orders of magnitude. In particular we find that subradiant excitations, when delocalized over several ring sites, are effectively transported between the rings with a high fidelity.

A full quantum mechanical treatment of open quantum systems via a Master equation is often limited by the size of the underlying Hilbert space. As an alternative, the dynamics can also be formulated in terms of systems of coupled differential equations for operators in the Heisenberg picture. This typically leads to an infinite hierarchy of equations for products of operators. A well-established approach to truncate this infinite set at the level of expectation values is to neglect quantum correlations of high order. This is systematically realized with a so-called cumulant expansion, which decomposes expectation values of operator products into products of a given lower order, leading to a closed set of equations. Here we present an open-source framework that fully automizes this approach: first, the equations of motion of operators up to a desired order are derived symbolically using predefined canonical commutation relations. Next, the resulting equations for the expectation values are expanded employing the cumulant expansion approach, where moments up to a chosen order specified by the user are included. Finally, a numerical solution can be directly obtained from the symbolic equations. After reviewing the theory we present the framework and showcase its usefulness in a few example problems.