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arXiv:cond-mat/0703346 · Physical Review A

Spatial noise correlations of a chain of ultracold fermions: A numerical study

Andreas Lüscher, Andreas Martin Läuchli, R. M. Noack

Published Oct 12, 2007• 12 citations
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Abstract

We present a numerical study of noise correlations, i.e., density-density correlations in momentum space, in the extended fermionic Hubbard model in one dimension. In experiments with ultracold atoms, these noise correlations can be extracted from time-of-flight images of the expanding cloud. Using the density-matrix renormalization group method to investigate the Hubbard model at various fillings and interactions, we confirm that the noise correlations contain full information on the most important fluctuations present in the system. We point out the importance of the sum rules fulfilled by the noise correlations and show that they yield nonsingular structures beyond the predictions of bosonization approaches. Noise correlations can thus serve as a universal probe of order and can be used to characterize the many-body states of cold atoms in optical lattices.

DOI
10.1103/physreva.76.043614

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We present a numerical study of noise correlations, i.e., density-density correlations in momentum space, in the extended fermionic Hubbard model in one dimension. In experiments with ultracold atoms, these noise correlations can be extracted from time-of-flight images of the expanding cloud. Using the density-matrix renormalization group method to investigate the Hubbard model at various fillings and interactions, we confirm that the noise correlations contain full information on the most important fluctuations present in the system. We point out the importance of the sum rules fulfilled by the noise correlations and show that they yield nonsingular structures beyond the predictions of bosonization approaches. Noise correlations can thus serve as a universal probe of order and can be used to characterize the many-body states of cold atoms in optical lattices.
Time of flight images reflect the momentum distribution of the atoms in the trap, but the spatial noise in the image holds information on more subtle correlations. Using bosonization, we study such correlations in generic 1D systems of ultracold fermions. We show how pairing as well as spin and charge density wave correlations may be identified and extracted from time of flight images. These incipient orders manifest themselves as power-law singularities in the noise correlations, that depend on the Luttinger parameters, which suggests a general experimental technique to obtain them.
It is well established that the noise correlations measured by time-of-flight imaging in cold-atom experiments, which correspond to the density-density correlations in the momentum space of trapped atomic gases, can probe the spin structure factor deep in the Mott-insulating regime of SU(2) Hubbard models. We explicitly derive the mathematical relation between the noise correlations and the spin structure factor in the strong-interaction limit of SU(N)\mathrm{SU}(N)SU(N) Hubbard models at any integer filling $\ensuremath{\rho}$. By calculating the ground states of one-dimensional SU(N)\mathrm{SU}(N)SU(N) Fermi-Hubbard models for $2\ensuremath{\le}N\ensuremath{\le}6$ with use of the density-matrix renormalization-group method, we confirm the relation numerically in the regime of strong interactions $U\ensuremath{\gg}t$, where UUU and ttt denote the on-site interaction and the hopping energy. We show that the deviation between the actual noise correlations and those obtained from the spin structure factor scales as approximately (t/U)2{(t/U)}^{2}(t/U)2 for $\ensuremath{\rho}=1$ at intermediate and large lattice sizes on the basis of numeric and semianalytic arguments.
We study the ground state properties of ultracold fermions trapped in a one-dimensional double-well optical lattice by using the density-matrix renormalization group method. The system is described by an extended Hubbard model with alternating hopping integrals and an external harmonic confining potential. We clarify the characteristic features of the metal–insulator phase transition originating from the double-well structure of the lattices. Several different types of insulating regions coexist when the number of atoms at each site is an integer or a half integer. We found that each insulating phase except for a band insulator exhibits rather large local density fluctuations, reflecting the strong dimerization of atoms within the unit cells of double-well lattices. The phase characteristics are elucidated in detail by investigating the profiles of the local density of atoms, the local density (spin) fluctuations, the double-occupancy probability and the spatially extended sp in correlations.
This thesis reports on the microscopic investigation of antiferromagnetic order in Hubbard chains, realized with ultracold repulsively interacting fermions in an optical lattice. Extending a quantum gas microscope on spin resolution opened the possibility to Access the full charge and spin statistics of a strongly correlated fermionic many body system. Using this technique, we measured antiferromagnetic correlations over distances up to three lattice sites. Starting with a repulsive degenerate two component mixture of 6Li atoms trapped in a single plane of a vertical lattice, we loaded the Atoms into one-dimensional tubes of a transverse optical superlattice. By adiabatically ramping up the lattice potential along the tubes, we isentropically entered the strongly correlated regime of the one-dimensional Fermi-Hubbard Hamiltonian. We spatially separated the two spin states into opposite sites of local double wells orthogonal to the direction of the lattice before single atom and site sensitive imaging. In this way we extracted the spin information of every single lattice site and were able to evaluate spin-spin correlations. The two component degenerate gases were produced with sub-Poissonian atom number fluctuations by magnetically driven evaporative cooling in vicinity of a Feshbach resonance, used to set the scattering length. The dependence of antiferromagnetic correlations on the driving superexchange coupling was experimentally investigated by varying the interparticle scattering length before loading the atoms into the optical lattice. Taking advantage of the underlying trapping potential of the optical lattice, which shaped the filling and the entropy of the gases, the influence of density excitations and different entropies on spin correlations was observed. By post-selecting the Hubbard chains based on their average filling and local atom number fluctuations, spin correlations corresponding to 58% of the zero temperature predictions in the Heisenberg regime were measured. By comparing the obtained antiferromagnetic correlations to Quantum-Monte-Carlo predictions for our System the lowest measured entropy per particle could be stated as s = 0.51(4)kB, clearly below the critical value of s = ln(2)kB, which is required to form longer ranged correlations. Below this critical entropy, in contrast to local density fluctuations, these spin correlations strongly depend on the entropy of the system. In the future, such a spin thermometer can be used to benchmark novel cooling techniques, which are required to enter the temperature regime of d-wave superconducting states. The access to the full particle and spin statistics will enable us to read out multi-point correlation functions, which characterize such exotic states of matter.
A Fermi gas of noninteracting electrons, or ultracold fermionic atoms, has a quantum ground state defined by a region of occupancy in momentum space known as the Fermi sea. The Euler characteristic ${\ensuremath{\chi}}_{F}$ of the Fermi sea serves to topologically classify these gapless fermionic states. The topology of a DDD-dimensional Fermi sea is physically encoded in the D+1D+1D+1 point equal time density correlation function. In this work, we first present a simple proof of this fact by showing that the evaluation of the correlation function can be formulated in terms of a triangulation of the Fermi sea with a collection of points, links, and triangles and their higher dimensional analogs. We then make use of the topological D+1D+1D+1 point density correlation to reveal universal structures of the more general MMM point density correlation functions in a DDD-dimensional Fermi gas. Two experimental methods are proposed for observing these correlations in D=2D=2D=2. In cold atomic gases imaged by quantum gas microscopy, our analysis supports the feasibility of measuring the third-order density correlation, from which ${\ensuremath{\chi}}_{F}$ can be reliably extracted in systems with as few as around 100 atoms. For solid-state electron gases, we propose measuring correlations in the speckle pattern of intensity fluctuations in nonlinear x-ray scattering experiments.