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arXiv:1411.3483 · Physical Review Letters

Observing Chiral Superfluid Order by Matter-Wave Interference

T. Kock, Matthias Ölschläger, A Ewerbeck, Wen-Min Huang, Ludwig Mathey, Andreas Hemmerich

Published Mar 19, 2015• 49 citations
arXivHTMLPDF

Abstract

The breaking of time-reversal symmetry via the spontaneous formation of chiral order is ubiquitous in nature. Here, we present an unambiguous demonstration of this phenomenon for atoms Bose-Einstein condensed in the second Bloch band of an optical lattice. As a key tool, we use a matter-wave interference technique, which lets us directly observe the phase properties of the superfluid order parameter and allows us to reconstruct the spatial geometry of certain low-energy excitations, associated with the formation of domains of different chirality. Our work marks a new era of optical lattices where orbital degrees of freedom play an essential role for the formation of exotic quantum matter, similarly as in electronic systems.

DOI
10.1103/physrevlett.114.115301

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The breaking of time-reversal symmetry via the spontaneous formation of chiral order is ubiquitous in nature. Here, we present an unambiguous demonstration of this phenomenon for atoms Bose-Einstein condensed in the second Bloch band of an optical lattice. As a key tool, we use a matter-wave interference technique, which lets us directly observe the phase properties of the superfluid order parameter and allows us to reconstruct the spatial geometry of certain low-energy excitations, associated with the formation of domains of different chirality. Our work marks a new era of optical lattices where orbital degrees of freedom play an essential role for the formation of exotic quantum matter, similarly as in electronic systems.
Abstract Topological superfluidity is an important concept in electronic materials as well as ultracold atomic gases1. However, although progress has been made by hybridizing superconductors with topological substrates, the search for a material—natural or artificial—that intrinsically exhibits topological superfluidity has been ongoing since the discovery of the superfluid 3He-A phase2. Here we report evidence for a globally chiral atomic superfluid, induced by interaction-driven time-reversal symmetry breaking in the second Bloch band of an optical lattice with hexagonal boron nitride geometry. This realizes a long-lived Bose–Einstein condensate of 87Rb atoms beyond present limits to orbitally featureless scenarios in the lowest Bloch band. Time-of-flight and band mapping measurements reveal that the local phases and orbital rotations of atoms are spontaneously ordered into a vortex array, showing evidence of the emergence of global angular momentum across the entire lattice. A phenomenological effective model is used to capture the dynamics of Bogoliubov quasi-particle excitations above the ground state, which are shown to exhibit a topological band structure. The observed bosonic phase is expected to exhibit phenomena that are conceptually distinct from, but related to, the quantum anomalous Hall effect3–7 in electronic condensed matter.
The study of superconductivity with unconventional order is complicated in condensed matter systems by their extensive complexity. Optical lattices with their exceptional precision and control allow one to emulate superfluidity avoiding many of the complications of condensed matter. A promising approach to realize unconventional superfluid order is to employ orbital degrees of freedom in higher Bloch bands. In recent work, indications were found that bosons condensed in the second band of an optical chequerboard lattice might exhibit p x ± i p y order. Here we present experiments, which provide strong evidence for the emergence of p x ± i p y order driven by the interaction in the local p-orbitals. We compare our observations with a multi-band Hubbard model and find excellent quantitative agreement.
Abstract This article provides a synopsis of our recent experimental work exploring Bose–Einstein condensation in metastable higher Bloch bands of optical lattices. Bipartite lattice geometries have allowed us to implement appropriate band structures, which meet three basic requirements: the existence of metastable excited states sufficiently protected from collisional band relaxation, a mechanism to excite the atoms initially prepared in the lowest band with moderate entropy increase, and the possibility of cross-dimensional tunneling dynamics, necessary to establish coherence along all lattice axes. A variety of bands can be selectively populated and a subsequent thermalization process leads to the formation of a condensate in the lowest energy state of the chosen band. As examples the 2nd, 4th and 7th bands in a bipartite square lattice are discussed. The geometry of the 2nd and 7th bands can be tuned such that two inequivalent energetically degenerate energy minima arise at the X ± -points at the edge of the 1st Brillouin zone. In this case even a small interaction energy is sufficient to lock the phase between the two condensation points such that a complex-valued chiral superfluid order parameter can emerge, which breaks time reversal symmetry. In the 4th band a condensate can be formed at the Γ-point in the center of the 1st Brillouin zone, which can be used to explore topologically protected band touching points. The new techniques to access orbital degrees of freedom in higher bands greatly extend the class of many-body scenarios that can be explored with bosons in optical lattices.
Ultracold atoms loaded into higher Bloch bands provide an elegant setting for realizing many-body quantum states that spontaneously break time-reversal symmetry through the formation of chiral orbital order. The applicability of this strategy remains nonetheless limited due to the finite lifetime of atoms in high-energy bands. Here we introduce an alternative framework, suitable for bosonic gases, which builds on assembling square plaquettes pierced by a $\ensuremath{\pi}$ flux (half a magnetic-flux quantum). This setting is shown to be formally equivalent to an interacting bosonic gas loaded into ppp orbitals, and we explore the consequences of the resulting chiral orbital order, both for weak and strong on-site interactions. We demonstrate the emergence of a chiral superfluid vortex lattice, exhibiting a long-lived gapped collective mode that is characterized by local chiral currents. This chiral superfluid phase is shown to undergo a phase transition to a chiral Mott insulator for sufficiently strong interactions. Our work establishes coupled $\ensuremath{\pi}$-flux plaquettes as a practical route for the emergence of orbital order and chiral phases of matter.
We investigate the unconventional Bose-Einstein condensations with the orbital degree of freedom in the three-dimensional cubic optical lattice, which gives rise to various exotic features absent in conventional scalar and spinor Bose-Einstein condensations. Orbital angular momentum moments are formed on lattice sites breaking time-reversal symmetry spontaneously. Furthermore, they exhibit orbital frustrations and develop a chiral ordering selected by the ``order-from-disorder'' mechanism.
We show that Bose–Einstein condensates in optical lattices with broken time-reversal symmetry can support chiral edge modes originating from nontrivial bulk excitation band topology. To be specific, we analyze a Bose–Hubbard extension of the Haldane model, which can be realized with recently developed techniques of periodically modulating honeycomb optical lattices. The topological properties of Bloch bands known for the noninteracting case are shown to be smoothly carried over to Bogoliubov excitation bands for the interacting case. We show that the parameter ranges that display topological bands enlarge with increasing the Hubbard interaction or the particle density. In the presence of sharp boundaries, chiral edge modes appear in the gap between topological excitation bands. We demonstrate that by coherently transferring a portion of a condensate into an edge mode, a density wave is formed along the edge owing to an interference with the background condensate. This offers a unique method of detecting an edge mode through a macroscopic quantum phenomenon.