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.

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. Here we demonstrate such control over the quantum walk -the quantum mechanical analogue of the classical random walk -in the strong interaction regime. Using interacting bosonic atoms in an optical lattice, we directly observe 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.Quantum walks are the quantum-mechanical analogues of the classical random walk process, describing the propagation of quantum particles on periodic potentials [1,2]. Unlike classical objects, particles performing a quantum walk can be in a superposition state and take all possible paths through their environment simultaneously, leading to faster propagation and enhanced sensitivity to initial conditions. These properties have generated considerable interest in using quantum walks for the study of position-space quantum dynamics and for quantum information processing [3]. Two distinct models of quantum walk with similar physical behavior were devised: The discrete time quantum walk [1], in which the particle propagates in discrete steps determined by a dynamic internal degree of freedom, and the continuous time quantum walk [2], in which the dynamics is described by a time-independent lattice Hamiltonian.Experimentally, quantum walks have been implemented for photons [4], trapped ions [5,6], and neutral atoms [7][8][9], among other platforms [4]. Until recently, most experiments were aimed at observing the quantum walks of a single quantum particle, which are described by classical wave equations.An enhancement of quantum effects emerges when more than one indistinguishable particle participates in the quantum walk simultaneously. In such cases, quantum correlations can develop as a consequence of Hanbury Brown-Twiss (HBT) interference and quantum statistics, as was investigated theoretically [10,11] and experimentally [12][13][14][15][16][17]. In the absence of interactions or auxiliary feed-forward measurements of the KnillLaflamme-Milburn type [18] this problem is believed to lack full quantum complexity, although it can still become intractable by classical computing [11].The inclusion of interaction between indistinguishable quantum walkers [19,20] may grant access to a much wider class of computationally hard problems, such as many-body localization and the dynamics of interacting quantum disordered systems [21]. Similarly, in the presence of interactions the quantum walk can yield universal Starting from a localized initial state (I), individual atoms perform independent quantum walks in an optical lattice (II). Right: The single-particle density distribution ex...

Access to collective excitations lies at the heart of our understanding of quantum many-body systems. We study the Higgs and Goldstone modes in a supersolid quantum gas that is created by coupling a Bose-Einstein condensate symmetrically to two optical cavities. The cavity fields form a U(1)-symmetric order parameter that can be modulated and monitored along both quadratures in real time. This enables us to measure the excitation energies across the superfluid-supersolid phase transition, establish their amplitude and phase nature, as well as characterize their dynamics from an impulse response. Furthermore, we can give a tunable mass to the Goldstone mode at the crossover between continuous and discrete symmetry by changing the coupling of the quantum gas with either cavity.Collective excitations are crucial for describing the dynamics of quantum many-body systems. They provide unified explanations of phenomena studied in different disciplines of physics, such as in condensed matter [1] or particle physics [2], or in cosmology [3]. The symmetry of the underlying effective Hamiltonian determines the character of the excitations, which changes in a fundamental way when a continuous symmetry is broken at a phase transition. Excitations can now appear both at finite and zero energy.In the paradigmatic case of models with U(1)-symmetry breaking, the system can be described by a complex scalar order parameter in an effective potential as illustrated in Fig. 1(A-B) [4]. In the normal phase, the potential is bowl-shaped with a single minimum at vanishing order parameter, and correspondingly two orthogonal amplitude excitations. Within the ordered phase, the potential shape changes to a 'sombrero' with an infinite number of minima on a circle. Here, fluctuations of the order parameter reveal two different excitations: a Higgs (or amplitude) mode, which stems from amplitude fluctuations of the order parameter and shows a finite excitation energy, and a Goldstone (or phase) mode, which stems from phase fluctuations of the order parameter and has zero excitation energy. The former should yield correlated fluctuations in the two squared quadratures of the order parameter, whereas the latter should show anticorrelated behavior.Condensed matter systems typically do not provide access to both quadratures of the order parameter, and Higgs and Goldstone modes have to be excited and detected by incoherent processes. In addition, the idealized situation is often disguised by further interactions that reduce the number of distinct modes [1, 6]. For charged particles, the minimal coupling to a vector potential can even completely suppress the Goldstone mode through the Anderson-Higgs mechanism [2]. In chargedensity wave compounds, a persisting Higgs mode has been observed as a well-defined resonance [7][8][9]. In superfluid Helium [10] and Bose-Einstein condensates [11] * donner@phys.ethz.ch Illustration of the experiment. A Bose-Einstein condensate (blue stripes) cut into slices by a transverse pump lattice potential (red stripes)...

Abstract: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 87 Rb.

We investigate a Bose-Einstein condensate strongly coupled to an optical cavity via a repulsive optical lattice. We detect a stable self-ordered phase in this regime, and show that the atoms order through an antisymmetric coupling to the P-band of the lattice, limiting the extent of the phase and changing the geometry of the emergent density modulation. Furthermore, we find a non-equilibrium phase with repeated intense bursts of the intra-cavity photon number, indicating non-trivial driven-dissipative dynamics. arXiv:1905.10377v1 [cond-mat.quant-gas]