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arXiv:0902.2586 · Physical Review Letters

Modulation Spectroscopy and Dynamics of Double Occupancies in a Fermionic Mott Insulator

Rajdeep Sensarma, David Pekker, Mikhail D. Lukin, Eugene Demler

Published Jul 17, 2009• 36 citations
arXivHTMLPDF

Abstract

Observing antiferromagnetic correlations in ultracold fermions on optical lattices is an important step towards quantum simulation of the repulsive Hubbard model. We show that optical lattice modulation spectroscopy can be used to detect antiferromagnetic order and probe the nature of quasiparticle excitations in a fermionic Mott insulator. At high temperatures, the rate of creation of double occupancies shows a broad peak at frequency of the on-site repulsion U, reflecting the incoherent nature of the hole excitations. At low temperatures, antiferromagnetic order leads to fine structure in the response consisting of a sharp absorption edge reflecting coherent propagation of holes and oscillations as a function of modulation frequency representing spin-wave shake-off processes.

DOI
10.1103/physrevlett.103.035303

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Observing antiferromagnetic correlations in ultracold fermions on optical lattices is an important step towards quantum simulation of the repulsive Hubbard model. We show that optical lattice modulation spectroscopy can be used to detect antiferromagnetic order and probe the nature of quasiparticle excitations in a fermionic Mott insulator. At high temperatures, the rate of creation of double occupancies shows a broad peak at frequency of the on-site repulsion U, reflecting the incoherent nature of the hole excitations. At low temperatures, antiferromagnetic order leads to fine structure in the response consisting of a sharp absorption edge reflecting coherent propagation of holes and oscillations as a function of modulation frequency representing spin-wave shake-off processes.
We propose an experimental setup of ultracold fermions in an optical lattice to determine the pairing gap in a superfluid state and the spin ordering in a Mott-insulating state. The idea is to apply a periodic modulation of the lattice potential and to use the thereby induced double occupancy to probe the system. We show by full time-dependent calculation using the adaptive time-dependent density-matrix renormalization-group method that the position of the peak in the spectrum of the induced double occupancy gives the pairing energy in a superfluid and the interaction energy in a Mott insulator, respectively. In the Mott insulator we relate the spectral weight of the peak to the spin ordering at finite temperature using perturbative calculations.
Exotic phenomena in strongly correlated electron systems emerge from the interplay between spin and motional degrees of freedom. For example, doping an antiferromagnet is expected to give rise to pseudogap states and high-temperature superconductors. Quantum simulation with ultracold fermions in optical lattices offers the potential to answer open questions about the doped Hubbard Hamiltonian, and has recently been advanced by quantum gas microscopy. In order to take advantage of these possibilities, a stable, high-power, 2D square lattice system was developed and coupled with an image plane based spatial light modulator. The ability to finely tune the atomic potential has made it possible to realize an antiferromagnet in a repulsively interacting Fermi gas on a 2D square lattice. At our lowest temperatures of T/t=0.25(2)T/t=0.25(2)T/t=0.25(2), antiferromagnetic long-range order (LRO) manifests through the divergence of the correlation length that reaches the size of the system, the development of a peak in the spin structure factor and a value of the staggered magnetization approaching the ground state value. Similarly, by carefully shaping the confinement, we have produced ultra-low entropy band insulators, which promise to be a perfect starting point for more advanced cooling schemes. In addition to the production of new states, Fermi gas microscopy is superbly well suited to studies of many-body dynamics. To this end I report on preliminary measurements of the propagation of holes in a Mott insualtor. These results demonstrate that Fermi gas microscopy can address open questions on the low-temperature Hubbard model.
In this thesis, I present the experimental realisation of the two-dimensional Hubbard model with ultracold fermionic Potassium-40 atoms in optical lattices. By tuning the dimensionality, kinetic and interaction in the optical lattices, we perform an analogue quantum simulation to explore the phase diagram of the Hubbard model. \n \nThe first key result is the experimental observation of particle-hole symmetry, namely a phase mapping between repulsive and attractive interactions. We compare a density-ordered, Mott-insulating phase with repulsive interaction to a spin-ordered, preformed pair phase and found excellent agreement with the theoretical prediction. The precise control and excellent detection capability of our quantum gas apparatus allow us to validate the particle-hole symmetry, and utilise it to explore quantum phases with a novel approach. \n \nNext, we investigate the spin-ordering on the repulsive side. By implementing a novel scheme based on coherent manipulation of spin correlations, we probe the anti-ferromagnetic ordering in the low-temperature phase diagram. The momentum-resolved spin correlations permit the reconstruction of spatial correlators without site-resolved imaging fidelity. \n \nFinally, we probe the attractive side of the phase diagram with a focus in pairing phenomenon, in which we draw a close analogy with BCS-BEC crossovers present in high-temperature superconducting cuprates and trapped ultracold Fermi gases. From the pair correlation function derived from thermodynamics observables, we observe the competition between the effective Pauli repulsion in fermionic systems and the on-site attraction we implemented.
Fermionic atoms in a periodic optical lattice provide a realization of the single-band Hubbard model. Using quantum Monte Carlo simulations along with the maximum-entropy method, we evaluate the effect of a time-dependent perturbative modulation of the optical lattice amplitude on atomic correlations, revealed in the fraction of doubly occupied sites. We find that the effect of modulation depends strongly on the filling---the response of the double occupation is significantly different in the half-filled Mott insulator from that in the doped Fermi liquid region.
The ability to cool fermions in optical lattices to ultra cold temperatures has led to an interdisciplinary area of research, that has attracted a lot of attention in recent years. An interesting development in this area is the possibility to realize models for strongly correlated fermions in the laboratory, such as the fermionic Hubbard Model. Determinantal Quantum Monte Carlo simulations have proven to be an important tool in the study of fermionic atoms. Nonetheless, it is important to compare the results and efficiency of different methods. Here comparisons with Numerical Linked Cluster Expansion and Dynamical Mean Field Theory data for double occupation and short range correlations, both relevant to current optical lattice experiments, will be presented and discussed. Another topic relevant in the context of optical lattice experiments is the study of metal insulator transitions. Indeed, the Mott insulating phase has been realized and observed in two-flavor mixtures of fermionic atoms loaded on optical lattices, being characterized both by the double occupation and the compressibility. An interesting point that has been addressed in the literature over the years is whether the same fermion-fermion interaction, responsible for the Mott insulating state, could drive an insulating system metallic. Here we show that, when fermions are loaded in optical lattices with spatially varying interactions a correlation induced Mott insulator to metal transition can take place. The spatial modulation of the interactions was recently demonstrated and opens the possibility for the experimental realization of such exotic phases.
We report on the site-resolved observation of characteristic states of the two-dimensional repulsive Fermi-Hubbard model, using ultracold ^{40}K atoms in an optical lattice. By varying the tunneling, interaction strength, and external confinement, we realize metallic, Mott-insulating, and band-insulating states. We directly measure the local moment, which quantifies the degree of on-site magnetization, as a function of temperature and chemical potential. Entropies per particle as low as 0.99(6)k_{B} indicate that nearest-neighbor antiferromagnetic correlations should be detectable using spin-sensitive imaging.