1. Arnau Riera, Grup QIC, Universitat de Barcelona Universität Potsdam 10 December 2009 Simulation of the Laughlin state in an optical lattice

2. 1. Introduction: quantum simulators. 2. Laughlin state in a harmonic trap. 3. Laughlin state in an optical lattice. 4. Conclusions. Outline

3. Introduction: quantum simulators

4. Simulation of Quantum Mechanics Classical computer Quantum computer It is only possible to simulate slightly entangled systems by means of MPS, PEPS, MERAs, etc. It is very powerfull but stil too far from our present techonology (8 perfectly controlled qubits).

5. Simulation of Quantum Mechanics Classical computer Quantum computer It is only possible to simulate slightly entangled systems by means of MPS, PEPS, MERAs, etc. It is very powerfull but stil too far from our present techonology (8 perfectly controlled qubits). What can we do in order to simulate strongly correlated and hightly entangled systems?

6. Simulation of Quantum Mechanics Small QC (QC with only tens of controlled qubits) Quantum Simulator Simulation of strongly correlated systems of few particles by means of exact circuits (FQHE, spin models, etc.) It consists of using experimentally controlled quantum systems to simulate other quantum systems (optical lattices, BEC's, etc.).

7. Exact circuits DISENTANGLER S STATE PREPARATION Verstraete, Cirac and Latorre ( arXiv:0804.188809, PRA 79, 032316 2009 ) Excited eigenstates, dynamics and finite temperature. Preparation of a state independently of the dynamics. Measure of correlations, understand the physics behind it.

8. Example of an exact quantum circuit The Laughlin state for m=1 (arXiv:0902.4797) :

9. Quantum simulator Atomic gases allow clean and controlled observation of many physical phenomena that have been studied in condensed systems. Atoms can be trapped, cooled, and manipulated with external electromagnetic fields, allowing many of the physical parameters that characterize their individual and collective behavior to be tuned.

10. Simulation of the Laughlin wave function in a harmonic potential

11. The Fractional Quantum Hall Effect In 1980, Tsui et al. measured the transversed resistivity of a 2D electron gas (at interfaces between semiconductors) under a strong transverse magnetic field at very low temperatures. They observed the formation of plateaus, where the Hall conductivity took values 1/3, 5/2, 2/3 of e 2 /h.

12. The Laughlin wave function The Laughlin wave function was postulated by Robert B. Laughlin in 1983 as the ground state of the FQHE. It is defined by where z i = x i + i y i. It takes into account: ● the single-electron Hilbert space is restricted to the subspace of zero energy (lowest Landau level). ● The statistics of the particles. ● The repulsive interaction between electrons. ● This state is is completely disordered and symmetry is not broken. ● States with a different m share the same symmetry and we cannot change one into another without finding a quantum phase transition. The Lauglin state has topological order:

13. Particles in a rotating harmonic potential The Hamiltonian of one particle confined in a harmonic potential rotating in the XY plane is Kinetic term Trap potential Rotation term

14. Particles in a rotating harmonic potential The Hamiltonian of one particle confined in a harmonic potential rotating in the XY plane is It can be diagonalized by means of

15. Lowest Landau Level

16. LLL In the fast rotating regime the effective Hamiltonian reads

17. Lowest Landau Level LLL The single particle states that form the LLL are We can write the wave function of these states where.

18. The interaction between atoms We describe the repulsive interaction between atoms by a contact potential (S-wave scattering): The whole Hamiltonian reads:

19. The interaction between atoms We describe the repulsive interaction between atoms by a contact potential (S-wave scattering): The whole Hamiltonian reads:

20. The interaction between atoms We describe the repulsive interaction between atoms by a contact potential (S-wave scattering): The whole Hamiltonian reads: Interaction energy is null iff when

21. The interaction between atoms We describe the repulsive interaction between atoms by a contact potential (S-wave scattering): The whole Hamiltonian reads: Interaction energy is null iff when This is precisely the Laughlin wave function for m=2!

22. Laughlin state in a harmonic trap The Laughlin wave function is the ground state of a set of bosons in a rotating harmonic trap in the fast rotation regime. Then... The problem is that if g is very weak, the gap is very small. Solution: to increase g! ● by means of Feshback resonances. ● using optical lattices [M. Popp et al, Phys. Rev. A 70, 053612 (2004).] Why is it so difficult to observe experimentally?

23. Simulation of the Laughlin wave function in an optical lattice

24. What is an optical lattice? Two counter propagating lasers form a standing wave. The atoms feel a periodic potential with the minima at the nodes of the standing wave.

25. The proposal by Popp et al. The idea by Popp, Paredes and Cirac (PRA 04) was to treat the lattice as a set of independent wells that can be approximated by a harmonic potential. The GS of the system is a porduct state of the Laughlin wave function in each site (Mott phase). Infinite barrier Finite barrier

26. The proposal by Popp et al. The idea by Popp, Paredes and Cirac (PRA 04) was to treat the lattice as a set of independent wells that can be approximated by a harmonic potential. The GS of the system is a porduct state of the Laughlin wave function in each site (Mott phase). Should we consider the quartic term of the potential expansion?

27. Our system We consider a system of bosonic atoms loaded in a 2D triangular optical lattice. The intensity of the laser is much larger than the recoil energy. Thus, tunneling of atoms between different sites is forbidden. where

28. The single particle spectra We can compute the corrections to the energy of the LLL states at first order in perturbation theory.

29. The centrifugal limit The centrifugal limit is the maximum rotation frequency at which the system can rotate before the particles escape from the trap.

30. The centrifugal limit We have thought about this question from different point of views ● Semiclassical interpretation. ● Slope of the single particle spectrum criterion. When one considers the anharmonic correction, the centrifugal limit depends on which is the maximum angular momentum of a particle that we want to keep trapped. Which is the angular momenutm limit m L in our case? It is the maximum single particle angular momentum of the Laughlin state, that is, N(N-1)/2.

31. Numerical results In which conditions the Laughlin state would be the GS of the system? Phase diagram. Fidelity between the Laughlin state and the GS.

32. Numerical results A) An abrupt transition.

33. Numerical results B) A well characterized boundary. We understand well the two limit cases: : GS is the Laughlin state. : GS is a product state with L=0. Then, the transition is expected when.

34. Numerical results C) The dependence on the number of particles. As larger is the number of particles, a higher laser intensity is needed to achive the Laughlin state.

35. Rotation The Hamiltonian of the system is rotational invariant [H,L]=0. Therefore, the angular momentum is conserved. In order to increase the angular momentum of the system we have to introduce a perturbation that breaks the rotational symmetry.

36. Measurement The Hamiltonian of the system is rotational invariant [H,L]=0. Therefore, the angular momentum is conserved. In order to increase the angular momentum of the system we have to introduce a perturbation that breaks the rotational symmetry. We can extract a lot of information from the time of flight (TOF) experiments: ● Density profile. ● Angular momentum. Furthermore, considering gases of 2 spicies, we can also determine correlation functions.

37. Conclusions 1) We have seen that the Laughlin state is the GS of a system of bosons rotating in a parabolic trap. 2) We have studied the method for achieving the Laughlin state in an optical lattice presented in [ Popp et al PRA (2006) ] and we have realized that it is essential to take into account the anharmonic corrections. 3) The consequence of the anharmonic correction is a smaller centrifugal limit. 4) Although the introduction of this more restrictive centrifugal limit makes it more difficult to drive the system into the Laughlin state, since it requires higher values of the laser intensity, for systems with a small number of particles the Laughlin state is perfectly achievable experimentally.