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arXiv:1002.3514

Simulation of the Laughlin state in an optical lattice

Arnau Riera

Published Feb 18, 2010• 0 citations
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Abstract

We analyze the proposal of achieving a Mott state of Laughlin wave functions in an optical lattice [M. Popp {\it et al.}, Phys. Rev. A 70, 053612 (2004)] and study the consequences of considering the anharmonic corrections to each single site potential expansion that were not taken into account until now. Our result is that, although the anharmonic correction reduces the maximum frequency at which the system can rotate before the atoms escape from each site (centrifugal limit), the Laughlin state can still be achieved for a small number of particles and a realistic value of the laser intensity.

DOI
10.48550/arxiv.1002.3514

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We analyze the proposal of achieving a Mott state of Laughlin wave functions in an optical lattice [M. Popp {\it et al.}, Phys. Rev. A 70, 053612 (2004)] and study the consequences of considering the anharmonic corrections to each single site potential expansion that were not taken into account until now. Our result is that, although the anharmonic correction reduces the maximum frequency at which the system can rotate before the atoms escape from each site (centrifugal limit), the Laughlin state can still be achieved for a small number of particles and a realistic value of the laser intensity.
An optical lattice is a periodic potential for atoms, created using a standing wave pattern of light. Due to the interaction between the light and the atoms, the atoms are attracted to either the nodes or the anti-nodes of the standing wave, depending on the exact wave lenght of the light. This means that if such a lattice is loaded with a sufficiently high number of ultracold atoms, a periodic array of atoms is obtained, we an interatomic distance of a few tenths of a micrometer. In order to obtain such a high number of cold atoms, one first has to create a so-called Bose-Einstein condensate. When an optical lattice is loaded from a Bose-Einstein condensate, it is possible to create a system in which every atom is in the lowest band of the lattice and there is on average one atom in each lattice site. Because the lattice potential is created with laser light, the depth of the lattice can easily be tuned by changing the intensity of the laser. When the intensity of the laser light is low, the atoms can tunnel from one site to the next. Due to this tunneling, the gas of atoms in the lattice will remain superfluid. However, if the intensity of the laser light is increased to above a certain critical value, a quantum phase transition occurs to a so-called Mott insulator. In this state, the atoms can no longer tunnel due to the fact that the on-site interaction between atoms becomes more important then the tunneling probability. In this PhD thesis, a description is given of the experimental setup that is being constructed in our group to create these systems in our lab. Also, a theoretical description is given of these systems and several important quantities our derived, such as the gap of the Mott-insulating state. Furthermore, an experiment is proposed that can be used to accurately measure this gap.
In this article we keep digging into the capabilities of optical lattices as simulators, showing how they can also be a powerful tool for simulating phenomena traditionally linked to quantum-optical systems (a study that we began in de Vega et al., (2008)). The basic idea consists of connecting an atomic state trapped in the optical lattice to a non trapped state through a Raman scheme. The coupling between these two types of atoms (trapped and free) turns out to be similar to that describing light-matter interaction within the rotating-wave approximation, the role of matter and photons being played by the trapped and free atoms, respectively.
THESIS SUMMARYTEXT:This thesis is made of two parts. In the first one, the issue of entanglement in many body systems is addressed. The concept of entanglement and some of the recent progress on the study of entropy of entanglement in many body quantum systems are reviewed. Emphasis is placed on the scaling properties of entropy for one-dimensional models at quantum phase transitions. Then, we focus on the area-law scaling of the entanglement entropy. An explicit computation in arbitrary dimensions of the entanglement entropy of the ground state of a discretized scalar free field theory that shows the expected area law result is also presented. For this system, it is shown that area law scaling is a manifestation of a deeper reordering of the vacuum produced by majorization relations.To finish this first part, the issue of how simple can a quantum system be such as to give a highly entangled ground state is addressed. In particular, we propose a Hamiltonian of a XX model with a ground state whose entropy scales linearly with the size of the block. It provides a simple example of a one dimensional system of spin-1/2 particles with nearest neighbour interactions that violates area-law for the entanglement entropy.The second part of this thesis deals with the problem of simulating quantum mechanics for highly entangled systems. Two different approaches to this issue are considered. One consists of using ultra-cold atoms systems as quantum simulators. With this aim, some experimental techniques related to cold atoms that allow to simulate strongly correlated many body quantum systems are reviewed an explicit example of simulation is presented. In particular, we analyze how to achieve a Mott state of Laughlin wave functions in an optical lattice and study the consequences of considering anharmonic corrections to each single site potential expansion that were not taken into account until now.Finally, a different approach to simulate strongly correlated systems is considered: to use small quantum computers to simulate them. An explicit quantum algorithm that creates the Laughlin state for an arbitrary number of particles n in the case of falling fraction equal to one is presented. We further prove the optimality of the circuit using permutation theory arguments and we compute exactly how entanglement develops along the action of each gate. We also discuss its experimental feasibility decomposing the qudits and the gates in terms of qubits and two qubit-gates as well as the generalization to arbitrary falling fraction.KEYWORDS: Entanglement, Many body quantum systems, Quantum Information Condensed Matter, Cold atoms, Spin chains, Quantum simulator, Quantum computation.
We study properties of a class of spin-singlet Mott states for arbitrary spin SSS bosons on a lattice, with particle number per cite n=S/l+1n=S/l+1n=S/l+1, where lll is a positive integer. We show that such a singlet Mott state can be mapped to a bosonic Laughlin wave function on a sphere with a finite number of particles at filling $\ensuremath{\nu}=1/2l$. Spin, particle, and hole excitations in the Mott state are discussed, among which the hole excitation can be mapped to the quasihole of the bosonic Laughlin wave function. We show that this singlet Mott state can be realized in a cold-atom system on an optical lattice and can be identified using Bragg spectroscopy and Stern-Gerlach techniques. This class of singlet Mott states may be generalized to map to bosonic Laughlin states with filling $\ensuremath{\nu}=q/2l$.