Synthetic dimensions alter one of the most fundamental properties in nature, the dimension of space. They allow, for example, a real three-dimensional system to act as effectively four-dimensional. Driven by such possibilities, synthetic dimensions have been engineered in ongoing experiments with ultracold matter. We show that rotational states of ultracold molecules can be used as synthetic dimensions extending to many – potentially hundreds of – synthetic lattice sites. Microwaves coupling rotational states drive fully controllable synthetic inter-site tunnelings, enabling, for example, topological band structures. Interactions leads to even richer behavior: when molecules are frozen in a real space lattice with uniform synthetic tunnelings, dipole interactions cause the molecules to aggregate to a narrow strip in the synthetic direction beyond a critical interaction strength, resulting in a quantum string or a membrane, with an emergent condensate that lives on this string or membrane. All these phases can be detected using local measurements of rotational state populations.
We use network analysis to describe and characterize an archetypal quantum system -an Ising spin chain in a transverse magnetic field. We analyze weighted networks for this quantum system, with link weights given by various measures of spin-spin correlations such as the von Neumann and Rényi mutual information, concurrence, and negativity. We analytically calculate the spinspin correlations in the system at an arbitrary temperature by mapping the Ising spin chain to fermions, as well as numerically calculate the correlations in the ground state using matrix product state methods, and then analyze the resulting networks using a variety of network measures. We demonstrate that the network measures show some traits of complex networks already in this spin chain, arguably the simplest quantum many-body system. The network measures give insight into the phase diagram not easily captured by more typical quantities, such as the order parameter or correlation length. For example, the network structure varies with transverse field and temperature, and the structure in the quantum critical fan is different from the ordered and disordered phases. I. INTRODUCTIONNetwork analysis is a powerful technique to characterize the structure of connections between agents in a network [1,2]. Studies have shown that classical systems as diverse as the brain and the Internet have a complex network structure [3][4][5][6][7][8]. Quantum systems also show a wide variety of complexity emerging due to inter-particle interactions. Like classical systems, quantum systems have an interconnected web of correlations, and network analysis provides a powerful set of tools to study them. However, while complex networks are ubiquitous in classical systems with a sufficiently rich set of interacting components, it is an open question what the minimal interacting quantum many-body system is in which complex network structures can appear.In this paper, we address this question by studying the network of correlations that arises in the simplest of interacting quantum models, the one-dimensional transverse field Ising model (TIM). We introduce and calculate networks whose links are weighted by various measures of correlations and entanglement, and quantify their complexity. The emergence of network complexity illuminates the richness of the quantum system.Earlier works have studied complex networks in the context of quantum systems, but by enforcing complex network structure in the Hamiltonian, e.g, in interactions [9][10][11][12][13][14]. However, there is no need for this explicit enforcement, as one finds network structure already in quantum states even for simple models such as the nearest-neighbor TIM.
Motivated by experiments performed by Landig et al. [Nature (London) 532, 476 (2016)], we consider a two-dimensional Bose gas in an optical lattice, trapped inside a single mode superradiant Fabry-Perot cavity. The cavity mediates infinite-range checkerboard interactions between the atoms, which produces competition between Mott insulator, charge-density wave, superfluid, and supersolid phases. We calculate the phase diagram of this Bose gas in a homogeneous system and in the presence of a harmonic trap.
We consider ultracold polar molecules trapped in a unit-filled one-dimensional chain in real space created with an optical lattice or a tweezer array and illuminated by microwaves that resonantly drive transitions within a chain of rotational states. We describe the system by a two-dimensional lattice model, with the first dimension being a lattice in real space and the second dimension being a lattice in a synthetic direction composed of rotational states. We calculate this system's groundstate phase diagram. We show that as the dipole interaction strength is increased, the molecules undergo a phase transition from a two-dimensional gas to a phase in which the molecules bind together and form a string that resembles a one-dimensional object living in the two-dimensional (i.e., one real and one synthetic dimensional) space. We demonstrate this with two complementary techniques: numerical calculations using matrix product state techniques and an analytic solution in the limit of infinitely strong dipole interaction. Our calculations reveal that the string phase at infinite interaction is effectively described by emergent particles living on the string and that this leads to a rich spectrum with excitations missed in earlier mean-field treatments.arXiv:1812.02229v2 [cond-mat.quant-gas]
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