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arXiv:0808.2652 · Proceedings of the National Academy of Sciences

Squeezing out the entropy of fermions in optical lattices

Tin-Lun Ho, Qi Zhou

Published Apr 13, 2009• 76 citations
arXivHTMLPDF

Abstract

At present, there is considerable interest in using atomic fermions in optical lattices to emulate the mathematical models that have been used to study strongly correlated electronic systems. Some of these models, such as the 2-dimensional fermion Hubbard model, are notoriously difficult to solve, and their key properties remain controversial despite decades of studies. It is hoped that the emulation experiments will shed light on some of these long-standing problems. A successful emulation, however, requires reaching temperatures as low as 10(-12) K and beyond, with entropy per particle far lower than what can be achieved today. Achieving such low-entropy states is an essential step and a grand challenge of the whole emulation enterprise. In this article, we point out a method to literally squeeze the entropy out from a Fermi gas into a surrounding Bose-Einstein condensed gas, which acts as a heat reservoir. This method allows one to reduce the entropy per particle of a lattice Fermi gas to a few percent of the lowest value obtainable today.

DOI
10.1073/pnas.0809862105

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At present, there is considerable interest in using atomic fermions in optical lattices to emulate the mathematical models that have been used to study strongly correlated electronic systems. Some of these models, such as the 2-dimensional fermion Hubbard model, are notoriously difficult to solve, and their key properties remain controversial despite decades of studies. It is hoped that the emulation experiments will shed light on some of these long-standing problems. A successful emulation, however, requires reaching temperatures as low as 10(-12) K and beyond, with entropy per particle far lower than what can be achieved today. Achieving such low-entropy states is an essential step and a grand challenge of the whole emulation enterprise. In this article, we point out a method to literally squeeze the entropy out from a Fermi gas into a surrounding Bose-Einstein condensed gas, which acts as a heat reservoir. This method allows one to reduce the entropy per particle of a lattice Fermi gas to a few percent of the lowest value obtainable today.
We propose a method to prepare fermionic atoms in a three-dimensional optical lattice at unprecedentedly low temperatures and uniform filling factors. The process involves adiabatic loading of degenerate atoms into multiple energy bands of an optical lattice followed by a filtering stage whereby atoms from all but the lowest band are removed. Of critical importance is the use of a nonharmonic trapping potential to provide external confinement for the atoms. For realistic experimental parameters, this procedure will produce a Fermi gas in a lattice with a reduced temperature T/TF 0.003T/{T}_{F}~0.003T/TF​ 0.003 and an entropy per particle of s 0.02kBs~0.02 {k}_{B}s 0.02kB​.
An advanced cooling scheme, incorporating entropy engineering, is vital for isolated artificial quantum systems designed to emulate the low-temperature physics of strongly correlated electron systems. This study theoretically demonstrates a cooling method employing multicomponent Fermi gases with SU(N)-symmetric interactions, focusing on the case of ^{173}Yb atoms in a two-dimensional optical lattice. Adiabatically introducing a nonuniform state-selective laser gives rise to two distinct subsystems: a central low-entropy region, exclusively composed of two specific spin components, acts as a quantum simulator for strongly correlated electron systems, while the surrounding N-component mixture retains a significant portion of the entropy of the system. The total particle numbers for each component are good quantum numbers, creating a sharp boundary for the two-component region. The cooling efficiency is assessed through extensive finite-temperature Lanczos calculations. The results lay the foundation for quantum simulations of two-dimensional systems of Hubbard or Heisenberg type, offering crucial insights into intriguing low-temperature phenomena in condensed-matter physics.
Cold atomic gases have proven capable of emulating a number of fundamental condensed matter phenomena including Bose-Einstein condensation, the Mott transition, Fulde-Ferrell-Larkin-Ovchinnikov pairing, and the quantum Hall effect. Cooling to a low enough temperature to explore magnetism and exotic superconductivity in lattices of fermionic atoms remains a challenge. We propose a method to produce a low temperature gas by preparing it in a disordered potential and following a constant entropy trajectory to deliver the gas into a nondisordered state which exhibits these incompletely understood phases. We show, using quantum Monte Carlo simulations, that we can approach the Néel temperature of the three-dimensional Hubbard model for experimentally achievable parameters. Recent experimental estimates suggest the randomness required lies in a regime where atom transport and equilibration are still robust.
Strongly correlated systems have rich and intriguing behaviors that have been at the focal point of modern condensed-matter physics. One promising research direction is the study of the Fermi-Hubbard model, which is thought to describe the essential physics of the cuprates and is naturally realized with ultracold atoms in optical lattices. The development of single-site control and readout on this platform has allowed for new methods in creating and studying states in the Fermi-Hubbard model. In this thesis, we discuss the engineering of low-entropy correlated states using a novel technique. By preparing a gapped band-insulator in the presence of an entropy reservoir, we create states with an entropy per particle of 0.016(3) k_B. We then demonstrate that the low-entropy band insulator can be used as a resource for creating strongly correlated states; by dynamically manipulating the on-site potentials we are able to convert the uncorrelated band insulator into a correlated Mott insulator. We also investigate if heating is responsible for the increase in entropy when creating the correlated phase and determine that it does not play a dominant role. In a separate experiment, we leverage the single-site manipulation capabilities in a different way by creating a single localized hole dopant in an antiferromagnetic state. We then allow the hole to delocalize by suddenly removing its pinning potential and study the resulting density and spin correlation evolution. As the hole delocalizes, we find that it displaces the surrounding spin correlations in a trajectory that is initially controlled by quantum interference. At later times, the hole delocalizes at a rate that scales with the spin exchange energy, revealing that the presence of magnetism fundamentally alters charge transport. These developments demonstrate continued progress in optical lattice quantum simulators and how they can play a role in unraveling the mysteries of the Fermi-Hubbard model.
Ultracold fermionic atoms in optical lattices offer pristine realizations of Hubbard models1, which are fundamental to modern condensed-matter physics2,3. Despite notable advancements4–6, the accessible temperatures in these optical lattice material analogues are still too high to address many open problems7–10. Here we demonstrate a several-fold reduction in temperature6,11–13, bringing large-scale quantum simulations of the Hubbard model into an entirely new regime. This is accomplished by transforming a low-entropy product state into strongly correlated states of interest via dynamic control of the model parameters14,15, which is extremely challenging to simulate classically10. At half-filling, the long-range antiferromagnetic order is close to saturation, leading to a temperature of T/t=0.05−0.05+0.06T/t=0.0{5}_{-0.05}^{+0.06}T/t=0.05−0.05+0.06​ based on comparisons with numerically exact simulations. Doped away from half-filling, it is exceedingly challenging to realize systematically accurate and predictive numerical simulations9. Importantly, we are able to use quantum simulation to identify a new pathway for achieving similarly low temperatures with doping. This is confirmed by comparing short-range spin correlations to state-of-the-art, but approximate, constrained-path auxiliary-field quantum Monte Carlo simulations16–18. Compared with the cuprates2,19,20, the reported temperatures correspond to a reduction from far above to below room temperature, at which physics such as the pseudogap and stripe phases may be expected3,19,21–24. Our work opens the door to quantum simulations that solve open questions in material science, develop synergies with numerical methods and theoretical studies, and lead to discoveries of new physics8,10. A several-fold reduction in temperature is accomplished using a neutral-atom Hubbard quantum simulator by transforming a low-entropy product state into strongly correlated states of interest via dynamic control of the model parameters.
A major motivation for cold atom experiments is the search for quantum ground states such as antiferromagnets and d-wave superfluids. The primary obstacle to this task is the difficulty of cooling to sufficiently low temperatures. We propose a way to achieve very low temperatures and entropies (∼0.03kB\sim 0.03k_B∼0.03kB​ per particle) by trapping fermions in a corral formed from another species of atoms. The Fermi system can then be used as a heat sink, or it can be adiabatically evolved into other desired states. In particular, we suggest methods for generating antiferromagnetism using this technique.