ANTIFERROMAGNETIC INTERACTIONS AT ROTATING SPIN-1 BOSONS

1. ANTIFERROMAGNETIC INTERACTIONS AT ROTATING SPIN-1 BOSONS S.ORAL E. AYDOGDU A.KILIC Anadolu University, Eskişehir, Turkey secilo@anadolu.edu.tr , ertanaydogdu@yahoo.com , abkilic@anadolu.edu.tr ANADOLU UNIVERSITY Eskisehir-TURKEY SCIENCE IS THE MOST GENUINE GUIDE IN LIFE Ultra cold atoms offer ultimate control over external and internal degrees of freedom facilitating a very pure experimental implementation of quantum mechanical models. They are therefore an ideal testing ground for the study of quantum magnetism phenomena. What is Bose-Einstein Condensation? Bose-Einstein Condensation derives from the fact that, in limit of an infinite three dimensional volume, the total number of states at vanishing energy becomes exceedingly small. Thus, there is not room for all particles when the temperature is decreased, and the system can only accumulate all the superfluous particles in its very ground state; they condense into the lowest energy state. In the thermodinamic limit, when both the particle number and the volume grow to infinity, the system enters into a different state, thus undergoing a phase transition. Subsequently the density drops and the energy scale of the interaction becomes small compared to the energy scale associated with the angular velocity, gn<<ħ. The dominant energy of the cloud in the rotating frame then becomes equivalent to the energy of an electron in a effective magnetic field B=−2mb/|e| in the symmetric gauge. The spectrum of the latter is organized in terms of degenerate harmonic oscillator states or Landau levels. Ho [41] predicted that atoms in a rapidly rotating Bose gas should condense in the lowest Landau level (LLL). The approximate many-body ground state solution of the Gross-Pitaevskii equation in the LLL regime describing a vortex lattice, is a coherent (superfluid) state given by The ground state of a rotating gas with N spin-1 bosons in the LLL and a spinindependent (c2=0) interaction is formed by a sequence of type I or II states lying on a certain path in (p, q)-space as L increases. To find the ground state in a rotating frame of reference, we need to find the ground state of H. Notion of rotation is such that the SO(3)orb-angular momentum decreases as the system rotates faster and faster. With three orbitals (Nv=2) we have . At low rotation rates, the typical boson occupation numbers of theoccupiedsingle-particle states are large compared with 1. In this situation, a meanfield (or classical) approach to the problem is generally expected to be quantitativelyaccurate. In such anapproach, the boson operators are replaced by expectation values,which are complex c-numbers: ,and the second-quantized Hamiltonianis then minimized with respect to both the magnitude and phase of thesenumbers to find the ground states. States with definite values of the good quantum numberssuch as N, S, L can be obtained afterwards by applying a projection to the mean fieldquantum state.In the case of very low rotation, where L  N( ) . For very small interaction ratios ||<<1, the total densities in the orbitals remainthe same as for =0, but there is non-trivial structure in the spin dependence,leading to spin transitions at critical values of , as we will describe shortly. Anti-ferromagnetic interactions The mean field ground states, given in the form of a three-component condensate wave function, for small, positive g=+e, and for . The condensate wave function is a vector in the a=,0, basis. Occupation numbers of the LLL mean field ground state with small antiferromagnetic interaction g=+e. For ,the condensate can be represented by with .Applying SO(3)spin rotations, one finds alternative representations such as For, the integrated value of for this state is non-zero and there is a spontaneous magnetization. The state at a can be viewed as a configuration of two -disclinations off the center of the trap, while the state in the regime (or possibly even as far as ) can be understood as a single -disclination in the polar state. The angular momentum for which the m = 2 orbital is first occupied in the mean field ground state, , is robust against small anti-ferromagnetic interactions. Upon increasing g further, the semi-plateau becomes flatter and the width decreases, until for g larger than some critical value gc≈1.19, jumps from to an plateau at a critical frequency c given by 0 − c≈0.15c0N. This is a transition from the non-rotating state to the polar vortex, analogous to what occurs in the scalar boson case. Observation of BEC in Rubidium by the JILA Group. Response of a Condensate to Rotation The quantum mechanical nature of a Bose-Einstein condensate becomes even more pronounced when an external torque is applied. Storing angular momentum in a condensate leads to very interesting states formed by quantized vortices. The appearance of quantized vortices in rotating ultra cold dilute gases has first been observed by Matthews et al. [2]. Bose-Einstein condensate are created by applying a laser beam with a fixed direction,and a center which is rotating in space. This “stirring” rotation field interacts with the atoms and induces a torque on the trapped cloud. The size of the vortex core is set by the healing length over which the condensate density heals back from zero (at the center of the vortex) to the bulk value of the condensate. In atomic condensates, this spatial scale is too small to directly resolve it optically. The stabilization of (a lattice of) singly quantized vortices in a rotating harmonically trapped ultra cold gas with repulsive atom-atom interactions over the uniform irrotational ground state can best be understood by comparing the energy of a single vortex state Evortex with that of the uniform condensate Euniform in the rotating frame where E’ = E − L· . The energy E is given and  is the external rotation frequency or angular velocity. In the rotating frame, E’vortex can be smaller than E’uniform for a rotation exceeding c. This is in marked contrast with vapors of atoms with attractive interactions, where the angular momentum is carried by center-of-mass motion of the cloud, and the system remains a compact blob, without any vortices, all the way up to the maximum rotation frequency. As the healing length becomes comparable with the mean distance between the vortices, the vortex cores start to shrink with increasing  [4, 5].The radius of the vortex core scales with the inter-vortex separation and in the limit where  approaches the eigen frequency 0 of the trap, the ratio of the area per vortex and the core area becomes a constant [4, 5, 6]. At this point, the rotating atomic cloud enters the so-called mean field quantum Hall limit. In this limit, the transverse trapping potential is canceled by the centrifugal potential, and the condensate expands dramatically in the plane perpendicular to the axis of rotation, effectivelybecoming a two-dimensional system. • References • ReijndersJ. W., “Quantum Phases for rotating bosons”,Netherland,2005.4. U. R. Fischer and G. Baym, Phys. Rev. Lett. 90, 140402 (2003). • M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E.A.Cornell, 5. G. Baym and C. J. Pethick, Phys. Rev. A 69, 043619 (2004). • Phys. Rev. Lett. 83, 2498 (1999). 6. V. Schweikhard, I. Coddington, P. Engels, V. P. Mogendorff and E. A. Cornell, • 3. Pethick,C.J.,Smith,H., “Bose Einstein Condensation in Dilute Gases”, Cambridge, 2002. Phys. Rev. Lett. 92, 040404 (2004).