Observation of Correlated Particle-Hole Pairs and String Order in Low-Dimensional Mott Insulators

Abstract

Quantum phases of matter are characterized by the underlying correlations of the many-body system. Although this is typically captured by a local order parameter, it has been shown that a broad class of many-body systems possesses a hidden non-local order. In the case of bosonic Mott insulators, the ground state properties are governed by quantum fluctuations in the form of correlated particle-hole pairs that lead to the emergence of a non-local string order in one dimension. Using high-resolution imaging of low-dimensional quantum gases in an optical lattice, we directly detect these pairs with single-site and single-particle sensitivity and observe string order in the one-dimensional case.The realization of strongly correlated quantum manybody systems using ultracold atoms has enabled the direct observation and control of fundamental quantum effects [1][2][3]. A prominent example is the transition from a superfluid (SF) to a Mott insulator (MI), occurring when interactions between bosonic particles on a lattice dominate over their kinetic energy [4][5][6][7][8]. At zero temperature, and in the limit where the ratio of kinetic energy over interaction energy vanishes, particle fluctuations are completely suppressed and the lattice sites are occupied by an integer number of particles. However, at a finite tunnel coupling, but still in the Mott insulating regime, quantum fluctuations create correlated particlehole pairs on top of this fixed-density background, which can be understood as virtual excitations. These particlehole pairs fundamentally determine the properties of the Mott insulator such as its residual phase coherence [9] and lie at the heart of superexchange-mediated spin interactions that form the basis of quantum magnetism in multi-component quantum gas mixtures [10][11][12].In a one-dimensional system, the appearance of correlated particle-hole pairs at the transition point from a superfluid to a Mott insulator is intimately connected to the emergence of a hidden string-order parameter O P [13,14]:Here δn j =n j −n denotes the deviation in occupation of the jth lattice site from the average background density, and k is an arbitrary position along the chain. In the simplest case of a Mott insulator with unity filling * Electronic address: manuel.endres@mpq.mpg.de (n = 1), relevant to our experiments, each factor in the product of operators in Eq. 1 yields −1 instead of +1, when a single-particle fluctuation from the unit background density is encountered. In the superfluid, particle and hole fluctuations occur independently and are uncorrelated, such that O P = 0. However, in the Mott insulating phase, density fluctuations always occur as correlated particle-hole pairs, resulting in O P = 0. For a homogeneous system, O P is expected to follow a scaling of Berezinskii-Kosterlitz-Thouless (BKT) type [15]. Non-local correlation functions, like the string-order parameter defined above, have been introduced in the context of low-dimensional quantum systems. They classify many-body quantum phases that are n...

Dynamical probing of a topological phase of bosons in one dimension

“…A second strategy consists in measuring non-local "string" order parameters, characteristic for example of topological phases in one dimension. A proof-of-principle experiment has already been successfully demonstrated using ultracold atoms in optical lattices 9 , following our predictions 10,11 .…”

“…Their experimental detection therefore requires the simultaneous measurement of all the sites in the lattice. Using new microscopic techniques for ultracold atoms 32,33 , the string order of the Mott insulator has been recently measured 9 .…”

“…For this task, we should compute the ma-trix element between the different excitations and the perturbation operators defined in Eq. (8) and (9). Unfortunately, these calculations are extremely complicated, due to the unknown interactions between the kinks.…”

Two-body correlations and natural-orbital tomography in ultracold bosonic systems of definite parity

“…The latter have theoretically been proven to give valuable physical insights in particular for low-dimensional systems [31][32][33][34][35][36] and are measurable both after ToF and in situ nowadays [37][38][39][40][41][42][43][44][45][46][47][48]. For this purpose, we derive the two-body density…”

Spin and charge modulations in a single-hole-doped Hubbard ladder: Verification with optical lattice experiments

“…On the other hand, heating effects can be reduced in small samples, as the low-energy excitations in bulk are gapped out. In addition, the extraordinary development of the atom microscope allows one to image specific atomic species with single site resolution [7][8][9][10][11] making quantum simulations with * jasontlho@gmail.com small systems a powerful way to explore strong correlation effects.…”