Author: Thomas Niederprüm

**Compressibility and the Equation of State of an Optical Quantum Gas in a Box**

Erik Busley, Leon Espert Miranda, Andreas Redmann, Christian Kurtscheid, Kirankumar Karkihalli Umesh, Frank Vewinger, Martin Weitz, Julian Schmitt

A collective description of macroscopic systems in terms of thermodynamic quantities allows to understand large-scale many-body systems without the need to keep track of each individual particle and its microscopic interaction. In particular in quantum systems, such quantities use to reveal fascinating collective behaviour and exotic phases. In their work E. Busley and coworkers report on the realization of a photon gas inside of a two-dimensional potential box where they manage to Bose-Einstein condense the gas and extract the compressability of the gas and its equation of state. In their study, they demonstrate the phase transition to the 2D BEC in the finite-sized potential box by observing the real space and k-space distribution of the photons. From these measurments they successfully deduce the scaling of the central gas density, the occupation of the lowest trap mode, and the internal energy with the mean photon number in the gas of which all nicely show the BEC phase transition. To get access to the compressability, a tilt is added to the box potential and the shift in the center of mass of the photon gas is measured. This, in turn, allows to determine the compressability and the equation of state of the gas in terms of the density response to the chemical potential. The study has direct consequences for thermodynamic machines with light as working medium and pave the way to study sound and universal phenomena in two dimensional photon gases.

**Variational truncated Wigner approximation for weakly interacting Bose fields: Dynamics of coupled condensates**

Christopher D. Mink, Axel Pelster, Jens Benary, Herwig Ott, Michael Fleischhauer

Efficient theoretical predictions of weakly interacting Bose gases still comprise a substantial challenge today. Due to the important role of quanutm fluctuation in these systems, mean-field methods fail to predict the correct behaviour. In contrast, the truncated Wigner approximation (TWA) includes quantum fluctions to lowest order but comes with a large computational effort. To overcome these constraints, C. Mink and coworkers developed a method called variational truncated Wigner approximation (VTWA) that drastically reduces the required simulation time. Here, a variational ansatz for the bosonic fields is introduced for which the functional Focker-Plank equation can be transformed onto a multivariate one in the variational parameters. To demonstrate the potential of the newly developed method, it is applied to describe an paradigmatic experimental system consisting of a 1D chain of coupled 3D Bose-Einstein condensates where a central site is emtied and the refilling dynamics is observed. Due the variational approach, the system dynamics can be deduced by simply numerically propagating a set of four variational parameters of the system. This makes large scale numerical simulations for the studied system feasable on commodity hardware. The obtained predictions for the refilling dynamics as well as the number fluctions of the density in the emptied lattice site are in very good agreement with the experimental data. In contrast to a mean-field approach (GPE) or an approximation of the system as 1D Bose-Hubbard chain (BHM), the new method ephasizes the importance of quantum fluctuations and the dimensionality for the studied system of coupled condensates. By applying the Lindblad master equation on top of their variational approach, the authors show how to extend their method to apply for situations with gain and loss mechanisms. Even further, the oulined variational approach can straight-forwardly be extended to include, for instance, dissipation or thermal effects in the future.

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**Geometric control of next-nearest-neighbor coupling in evanescently coupled dielectric waveguides**

Julian Schulz, Christina Jörg, Georg von Freymann

Discrete lattice models are one of the working horses in physics. Ranging from solid state systems over ultracold gases to photonics, complex systems can be idealized to points on a lattice. Cooparative effects in such many-body systems then arise from assuming a nearest neighbor coupling in these lattices. While sch an approximation makes the system theoratically tractable, it might also miss relevant physical effects. A particular example is the coupling to next-nearest neighbors (NNN) which is studied by J. Schulz and corworkers in coupled dielectric waveguides. To that end they create an array of waveguides in a zig-zag shape which via the angle in the zig-zag allows them to change the distance (and thus the coupling) to the next-nearest neighbor while keeping the nearest-neighbor coupling constant. Taking into account the non-orthogonality of the eigenmodes in the waveguide, a numeric solution of the coupled mode equations reveals that the relative sign of coupling to the next-nearest neighbor depends on the zig-zag angle. While the NNN coupling is negative for a linear chain of waveguides, it is positve large zig-zag angles. In between a sample-specific zig-zag angle exists at which the NNN coupling vanishes. The authors point out that this behavior can be understood from the shape of the corrected coupled eigenmodes of the system that show oscillations on the length scale of the array spacing, remaniscent of the Wannier functions. Using a direct laser writing technique, samples are prepared with the zig-zag angles of 0°, 40°, and 50° representing the cases of negative, vanishing and positive NNN coupling, respectively. By measuring the diffraction of the light coupled into the wave guides the effect of the sign of the NNN coupling can be clearly demonstrated in comparison to the model calculations. The ability to tune the NNN coupling and even to change its sign makes the results of the study relevant for other discrete systems like e.g. cold gases in optical lattices in which the tight-binding model is often used.