Emergent Phases in Optical Lattice

Optical lattice formed by counter propagating laser beams provides us a new platform to study condensed matter physics. Cold atoms moving in an optical lattice could be described by BoseHubbard Model. By controlling laser intensity, we could change interaction and tunneling strength at will. Thus, different quantum phases could occur in optical lattice. In this review, I will give an overview of different phases (i.e. Superfluid, Mott Insulating) in optical lattice. I will focus on the phase diagram and the characteristics of different phases. Introduction: Due to ac stack effect, atoms moving in optical fields will experience a potential, and the strength of potential is dependent on the strength of the local optical field. Thus, a threedimensional optical lattice potential could be formed by aligning three optical standing waves orthogonal to each other. In a typical experiment, each standing wave laser field is created by focusing a laser beam to a waist of about 100 μμμμ at the position of the condensate. A second lens and a mirror are then used to reflect the laser beam back onto itself, creating an interference pattern of standing optical wave. When the field strength of the three standing waves is the same, the three-dimensional optical lattice potential has the simple form [1]: VV0(xx) = VV0(ssssss(kkkk)+ssssss(kkkk) + ssssss2(kkkk)) with wave vector kk = 2ππ/λλ, λλ is the wave length of the laser light, corresponding to a lattice period aa = λλ/2. VV0 is the maximum depth of the optical lattice potential, which is proportional to the laser light intensity and the polarity of atoms. This depth is conveniently measured in units of recoil energy EERR = ħ2kk2/2μμ. In the low temperature limit, this optical lattice potential could be approximated by a harmonic potential with trapping frequency ννRR ≈ (ħkk2 2ππμμ ⁄ )�VV0/EERR. When the pair interaction is not taken into account, the motion of atoms in periodic lattice potential is well described within the band theory. This consists of vibrational motion within an individual well and tunneling between neighboring wells. At low enough temperatures, atoms will Bose condense and the condensate will be in a Superfluid state, where wave function exhibits long range coherence. In reality, there are repulsive pair interactions between atoms, and this may change the properties of the system dramatically depending on the strength of interaction. When the pair interaction strength is small compared to the tunneling strength, the condensate will remain in the superfluid state. A delocalized wave function will minimize the energy of the system and atoms could hop around freely. However, in the case of strong repulsive interactions and commensurate filling (each site has the same number of atoms), atoms will not be able to hop around freely as before. There will be a large energy cost for an atom to hop from one site to another due to the strong repulsive interaction between atoms (see Figure 1). Thus, the condensate will be in a mott-insulating state characterized by the existence of a gap for particlehole excitations and by the zero compressibility [2]. Thus, the competition between tunneling and repulsive interaction will result in different emergent quantum phases. Optical lattice is a great system to study such phenomenon, because the relative strength of tunneling and repulsive interaction could be tuned by changing the brightness of laser light. In fact, a superfluid to mottinsulting quantum phase transition has been realized experimentally in 2002 [3]. Figure 1: If the atom in site 1 hops to site 2, there will be an energy cost of U due to the repulsive interaction between two atoms in site 2 [3]. In this experiment, the superfluidity near the mott-insulating transition was inferred indirectly from coherence measurements. Since the observed excitation spectrum and atomic inference pattern did not change abruptly, the precise location of phase transition could not be determined. Later experiment extended the early work by studying stability of superfluid current as a function of momentum and lattice depth [5, 6]. The superfluid regime could be better characterized by observing a critical current for superfluid flow through the onset of dissipation. In this review, I will first introduce the Bose-Hubbard Model and the phase diagram of atoms in optical lattice. Then, the experiments on quantum phase transition from superfluid to mott-insulating will be introduced and compared. Bose-Hubbard Model for atoms moving in optical lattice: Cold atoms moving in periodic optical lattice potential could be described by Bose-Hubbard Model. The starting point is the Hamiltonian operator [1] HH = ∫dd3 xxψψ†(xx)�− ħ 2 2μμ ∇2 + V0(xx) + VT(xx)�ψψ(xx) + 1 2 4ππaassħ μμ ∫dd 3 xxψψ†(xx)ψψ†(xx)ψψ(xx)ψψ(xx) (1) with ψψ(xx) a boson field operator for atoms in a given internal atomic state, V0(xx) is the optical lattice potential, and VT(xx) is the slowly varying external harmonic tramping potential, e.g., a magnetic trap. In the low energy regime, the pair interaction between cold atoms could be described with a single s-wave scattering length aass. This effective interaction is of contact type and isotropic with the form of 4ππaassħ μμ ⁄ . The single particle wave function in a periodic potential is given by Bloch wave function, and a proper recombination of Bloch wave function would yield a set of well localized Wannier functions. In the cold atomic system, the energies involved in system dynamics are small compared to the excitation energies to the second band. So, we could assume there are no excitations to the second band. Thus we could expand our field operators in the set of Wannier functions formed only by first band Bloch wave functions. Then, field operator could be written as ψψ(xx) = ∑ bbss ss ww(xx − xxss). Substitute this into equation (1), we have the following Bose-Hubbard Hamiltonian HH = −JJ∑ bbss bbjj + ∑ εεssssss ss + 1 2 UU∑ ssss ss (ssss − 1), (2) Where the operators ssss = bbss bbss count the number of bosonic atoms at site i; the annihilation and creation operators obey canonical commutation relations �bbss , bbjj � = δδssjj . The parameter UU = 4ππaassħ ∫dd3 xx|ww(xx)|4/m corresponds to the strength of the onsite repulsion of two atoms at site i. J = ∫dd3 xxww∗(xx − xxss) �− ħ2 2μμ ∇2 + V0(xx)�ww�xx − xxjj � is the hopping matrix between two sites i and j. < ss, jj > denotes the nearest neighbors. εεss = ∫dd3 xxVVTT(xx)|ww(xx − xxss)| ≈ VVTT(xxss) is the energy offset of each lattice site. For a given optical potential, J and U could be evaluated numerically [1]. For the optical lattice potential given above, the Wannier function could be written as ww(xx) = ww(kk)ww(kk)ww(zz) which can be determined from band structure calculations. Figure 2 shows J and U as a function of the parameter VV0 in units of recoil energy EERR = ħ2kk2/2μμ. A larger lattice depth will lead to smaller tunneling J because of the higher energy barrier between neighboring sites. On the other hand, the atomic wave function will be more localized, and this leads to a stronger onsite repulsive pair