# Controlled hole doping of a Mott insulator of ultracold fermionic atoms 

Andr´ e Eckardt 1 ∗ and Maciej Lewenstein 1 , 2 

$^{1}$ICFO-Institut de Ci` encies Fot` oniques, Av. Canal Ol´ımpic s/n, E-08860 Castelldefels (Barcelona), Spain, and 

$^{2}$ICREA-Institucio Catalana de Recerca i Estudis Avan¸cats, Lluis Companys 23, E-08010 Barcelona, Spain 

(Dated: July 14, 2010) 

Considering a system of ultracold atoms in an optical lattice, we propose a simple and robust im- plementation of a quantum simulator for the homogeneous t - J model with a well-controlled fraction of holes x . The proposed experiment can provide valuable insight into the physics of cuprate super- conductors. A similar scheme applied to bosons, moreover, allows one to investigate experimentally the subtle role of inhomogeneity when a system passes from one quantum phase to another. 

Gases of ultracold atoms in optical lattice poten- tials provide extremely clean and controllable condi- tions for studying strongly correlated many-body physics [1]. Since for deep lattices these systems are described quantitatively by Hubbard-type Hamiltonians [2], they have great potential to serve as quantum simulators [3] for paradigmatic models of condensed matter physics. 

This prospect is fed by results for bosonic systems. In a seminal experiment, the quantum phase transi- tion from a superfluid of bosons to a strongly corre- lated Mott insulator—predicted for the bosonic Hubbard model [4]—has been observed [5]. Moreover, quantitative agreement between experiment and ab initio quantum Monte Carlo simulations clearly confirm the validity of the Bose-Hubbard description [6]. 

While the elementary Hubbard model for bosons is rather well understood, this is not the case for repul- sively interacting fermions: In a Mott insulator, with interaction localizing one particle of “spin” s = ↑ or ↓ at each site, Fermi statistics gives rise to an antiferro- magnetic superexchange coupling J between neighbor- ing spins. Intriguing physics is expected when such a quantum antiferromagnet is frustrated, either by a non- bipartite lattice geometry [7] or by doping it with holes (or fermions) displacing spins when moving around [10]. The latter scenario for a square lattice is conjectured to give rise to d$_{x}$ $^{2}$− y 2 -wave pair superfluidity and to explain basic properties of high-temperature cuprate supercon- ductors [8–10]. However, conclusive theoretical evidence of whether the plain Hubbard physics supports a super- conducting state is still lacking. As pointed out already in Ref. [11], here a cold atom realization of the fermionic Hubbard model (or its descendant, the t - J model) could provide critical insight. 

The recent observation of a Mott insulator of repul- sively interacting fermionic atoms in a deep optical lat- tice [12] is an important first step toward a clean cold atom realization of strongly correlated fermionic Hub- bard physics. However, further steps in that direction require solutions to two problems: (i) The temperatures 

PACS numbers: 03.75.Ss, 03.75.Lm, 67.85.-d, 05.30.Fk 

that can be achieved presently are still larger than (or at most comparable to) the energy scale of the superex- change spin-spin coupling [13]. Promising novel cooling procedures have been proposed to tackle this problem [14] (cf. also [15]). (ii) While it is relatively easy to cre- ate an incompressible Mott region with one atom per site in the center of a parabolic trap, it is very hard to dope such a trapped Mott insulator in a controlled way. When the particle number in the center of the trap is lowered, e.g. by slowly widening the trap, the central Mott-insulator phase will not be doped with holes ho- mogeneously. Rather, it will melt from the edge. Also simply switching off the trapping potential (i.e. compen- sating it with a blue-detuned laser) is difficult, since then particles may leak out and it will be hard to control the lattice filling. In this Rapid Communication we propose a robust solution to problem (ii). Our method allows one to both accurately control the fraction x of hole doping and effectively compensate the trapping potential. Both are crucial in order to learn about high-temperature su- perconductivity with ultracold atoms. Applied to bosons, our scheme, moreover, allows one to experimentally in- vestigate the non-trivial influence of spatial inhomogene- ity when a system passes from one quantum phase to another [16]. 

Our basic idea is to create holes in a fermionic Mott insulator by adding auxiliary bosonic particles to the sys- tem that interact repulsively both with the fermions and with each other. Each boson repels a fermion from one site. Experimentally, one has to create a mixed Mott phase with one particle, either fermion or boson, per site. This spatially rather well-defined “simulator region” re- sembles the system to be simulated. The degree of hole doping is well controlled by the number ratio between bosons and fermions. The price to be payed is that, un- like the tunneling of a real hole, the motion of the bosons happens via boson-fermion swaps, being slow second- order superexchange processes (cf. [17, 18] for the case of “spinless” Bose-Fermi mixtures). But this type of hole kinetics brings also a great advantage: Consider the trap being very similar for fermions and bosons (think of two Ytterbium isotopes in a far off-resonant dipole potential). 

Then such a boson-fermion swap on two neighboring sites will change the potential energy only marginally, even if 

