Coupling two order parameters in a quantum gas.

Controlling matter to simultaneously support coupled properties is of fundamental and technological importance1 (for example, in multiferroics2-5 or high-temperature superconductors6-9). However, determining the microscopic mechanisms responsible for the simultaneous presence of different orders is difficult, making it hard to predict material phenomenology10,11 or modify properties12-16. Here, using a quantum gas to engineer an adjustable interaction at the microscopic level, we demonstrate scenarios of competition, coexistence and mutual enhancement of two orders. For the enhancement scenario, the presence of one order lowers the critical point of the other. Our system is realized by a Bose-Einstein condensate that can undergo self-organization phase transitions in two optical resonators17, resulting in two distinct crystalline density orders. We characterize the coupling between these orders by measuring the composite order parameter and the elementary excitations and explain our results with a mean-field free-energy model derived from a microscopic Hamiltonian. Our system is ideally suited to explore quantum tricritical points18 and can be extended to study the interplay of spin and density orders19 as a function of temperature20.

Monitoring and manipulating Higgs and Goldstone modes in a supersolid quantum gas.

Higgs and Goldstone modes are collective excitations of the amplitude and phase of an order parameter that is related to the breaking of a continuous symmetry. We directly studied these modes in a supersolid quantum gas created by coupling a Bose-Einstein condensate to two optical cavities, whose field amplitudes form the real and imaginary parts of a U(1)-symmetric order parameter. Monitoring the cavity fields in real time allowed us to observe the dynamics of the associated Higgs and Goldstone modes and revealed their amplitude and phase nature. We used a spectroscopic method to measure their frequencies, and we gave a tunable mass to the Goldstone mode by exploring the crossover between continuous and discrete symmetry. Our experiments link spectroscopic measurements to the theoretical concept of Higgs and Goldstone modes.

Supersolid formation in a quantum gas breaking a continuous translational symmetry.

The concept of a supersolid state combines the crystallization of a many-body system with dissipationless flow of the atoms from which it is built. This quantum phase requires the breaking of two continuous symmetries: the phase invariance of a superfluid and the continuous translational invariance to form the crystal. Despite having been proposed for helium almost 50 years ago, experimental verification of supersolidity remains elusive. A variant with only discrete translational symmetry breaking on a preimposed lattice structure-the 'lattice supersolid'-has been realized, based on self-organization of a Bose-Einstein condensate. However, lattice supersolids do not feature the continuous ground-state degeneracy that characterizes the supersolid state as originally proposed. Here we report the realization of a supersolid with continuous translational symmetry breaking along one direction in a quantum gas. The continuous symmetry that is broken emerges from two discrete spatial symmetries by symmetrically coupling a Bose-Einstein condensate to the modes of two optical cavities. We establish the phase coherence of the supersolid and find a high ground-state degeneracy by measuring the crystal position over many realizations through the light fields that leak from the cavities. These light fields are also used to monitor the position fluctuations in real time. Our concept provides a route to creating and studying glassy many-body systems with controllably lifted ground-state degeneracies, such as supersolids in the presence of disorder.

Ultra-precise holographic beam shaping for microscopic quantum control.

High-resolution addressing of individual ultracold atoms, trapped ions or solid state emitters allows for exquisite control in quantum optics experiments. This becomes possible through large aperture magnifying optics that project microscopic light patterns with diffraction limited performance. We use programmable amplitude holograms generated on a digital micromirror device to create arbitrary microscopic beam shapes with full phase and amplitude control. The system self-corrects for aberrations of up to several λ and reduces them to λ/50, leading to light patterns with a precision on the 10-4 level. We demonstrate aberration-compensated beam shaping in an optical lattice experiment and perform single-site addressing in a quantum gas microscope for 87Rb.

Strongly correlated quantum walks in optical lattices.

Full control over the dynamics of interacting, indistinguishable quantum particles is an important prerequisite for the experimental study of strongly correlated quantum matter and the implementation of high-fidelity quantum information processing. We demonstrate such control over the quantum walk-the quantum mechanical analog of the classical random walk-in the regime where dynamics are dominated by interparticle interactions. Using interacting bosonic atoms in an optical lattice, we directly observed fundamental effects such as the emergence of correlations in two-particle quantum walks, as well as strongly correlated Bloch oscillations in tilted optical lattices. Our approach can be scaled to larger systems, greatly extending the class of problems accessible via quantum walks.