The color encodes the tilt strength of О” = h Г— 1.1 kHz (blue), О” = h Г— 1.4 kHz (red), О” = h Г— 1.7 kHz (black). (b)В Dynamics of the density-wave contrast c for different evolution times after the quench into the tilted system showing the fast buildup and slow decay of the patterns. The dashed lines mark k a Лњ / ( 2 ПЂ ) = В± 1 / 5, which is used in the computation of the density-wave contrast (see main text). The dots show the exact parameters for which data were taken. (a)В The normalized power spectral density of the column occupations for a tilt with energy offset О” = h Г— 1.4 kHz initially develops a peak at k a Лњ / ( 2 ПЂ ) = В± 1 / 2 (period 2), which splits and moves to k a Лњ / ( 2 ПЂ ) в€ј В± 1 / 3 (period 3) at longer time. Reuse & PermissionsДЇormation dynamics and decay of the density wave. The simulation does not model the density modulation within the lattice sites and the density distribution is therefore shown as tiled hexagons. (e) c-field simulation for the same parameters as in (d). The tunnel coupling is J = h Г— 13 Hz throughout the paper. The two example images are for identical parameters of a hold time of 60В ms and tilt with energy offset О” = h Г— 1.4 kHz, but one shows a domain wall in the center of the cloud and the other does not, directly reflecting the spontaneous nature of the pattern. (d)В Experimental images of the spatial density show the spontaneous formation of charge density waves with wave vector parallel to the tilt direction. Tunneling perpendicular to the tilt J вЉҐ and the effective on-site interaction U eff remain approximately unchanged. (c)В In the large tilt limit the system can be described by an effective Hamiltonian with no tilt and no single-particle tunneling in the tilt direction, but pair tunneling P and nearest-neighbor interaction V with a strength set by s U with s = ( J / О” ) 2. (b)В The system can be described by a Hubbard model with tunnel coupling J, on-site interaction U, and energy offset О” along the tilt direction. (a)В The system consists of a triangular lattice of tubes with a tilt applied perpendicularly to a lattice vector. The observed density-wave patterns already give a good taste of what exciting phenomena could be discovered in such tailored systems.Д«ose-Einstein condensates in strongly tilted optical lattices. This study presents a way to induce such processes, which are important for studying further tailored models in the future. For the theoretical description of the tilted lattice, we derive an effective analytical model that replaces the original single-particle tunneling process by a correlated-pair tunneling process of two atoms moving in opposite directions along the tilt. By employing a novel microscopy technique based on matter-wave optics, which magnifies the density distribution of the atomic cloud before taking an optical image via resonant light, we overcome the depth-of-focus limitation and image 3D systems with single-site resolution. We perform our experiments in an artificial quantum system formed by ultracold atoms trapped in a standing wave of laser light. This pattern spontaneously breaks the symmetry of the lattice and forms domains, with the larger density on either the odd or even sites. Surprisingly, we observe the formation of a pronounced density wave-a stripe pattern perpendicular to the tilt with an enhanced density on every second lattice site. Here, we study what happens when a strong tilt is added to Bose-Einstein condensates in a triangular lattice. Quantum particles in lattices form benchmark systems with intriguing phenomena.

More broadly, we establish the far out-of-equilibrium regime of lattice models subjected to a strong dc field, as an exemplary and paradigmatic scenario for transient pattern formation. Our study suggests that the short-time dynamics arises from resonant pair tunneling processes within an effective description of the tilted Hubbard model. We observe the emergence of pronounced density-wave patterns-which spontaneously break the underlying lattice symmetry-using a novel single-shot imaging technique with two-dimensional single-site resolution in three-dimensional systems, which also resolves the domain structure. Here, we experimentally and theoretically study the out-of-equilibrium dynamics of a bosonic lattice model subjected to a strong dc field, realized as ultracold atoms in a strongly tilted triangular optical lattice. The understanding of these phenomena expands the scope of established thermodynamics into the out-of-equilibrium domain. Driving a many-body system out of equilibrium induces phenomena such as the emergence and decay of transient states, which can manifest itself as pattern and domain formation.

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