Light scattering detection of quantum phases of ultracold atoms in optical lattices Jinwu Ye 1 , $^{2}$, J.M. Zhang $^{3}$, W.M. Liu $^{3}$, Keye Zhang $^{4}$, Yan Li4 and Weiping Zhang 4 1 Department of Physics, Capital Normal University, Beijing, 100048 China 2 Department of Physics, The Pennsylvania State University, University Park, PA, 16802, USA $^{3}$Institute of Physics, Chinese Academy of Sciences, Beijing, 100080, China 4 Department of Physics, East China Normal university, Shanghai, 200062, China (Dated: November 17, 2018) Ultracold atoms loaded on optical lattices can provide unprecedented experimental systems for the quantum simulations and manipulations of many quantum phases. However, so far, how to detect these quantum phases effectively remains an outstanding challenge. Here, we show that the optical Bragg scattering of cold atoms loaded on optical lattices can be used to detect many quantum phases which include not only the conventional superfluid and Mott insulating phases, but also other important phases such as various kinds of density waves (CDW), valence bond solids (VBS), CDW supersolids and VBS supersolids. arXiv:1001.3230v3 [physics.atom-ph] 11 Mar 2011 Various kinds of strongly correlated quantum phases of matter may have wide applications in quantum infor- mation processing, storage and communications [1]. It was widely believed and also partially established that due to the tremendous tunability of all the parameters in this system, ultracold atoms loaded on optical lat- tices (OL) can provide an unprecedented experimental systems for the quantum simulations and manipulations of these quantum phases and quantum phase transitions between these phases. For example, Mott and superfluid phases [2] may have been successfully simulated and ma- nipulated by ultra-cold atoms loaded in a cubic optical lattice [3]. However, there are still at least two outstand- ing problems remaining. The first is how to realize many important quantum phases [1]. The second is that as- suming the favorable conditions to realize these quantum phases are indeed achieved in experiments, how to detect them without ambiguity. In this paper, we will focus on the second question. So far the experimental way to de- tect these quantum phases is mainly through the time of flight (TOF) measurement [1, 3] which simply opens the trap and turn off the optical lattice and let the trapped atoms expand and interfere, then take the image. The atom Bragg spectroscopy is based on stimulated mat- ter waves scattering by two incident laser pulses [4, 5] through the TOF measurements. The momentum [4] transfer Bragg spectroscopy was used to detect the Bo- goliubov mode inside an BEC condensate. The energy transfer [5] Bragg spectroscopy was used to detect the Mott gap in a Mott state in an optical lattice. Opti- cal Bragg scattering (Fig.1) has been used previously to study periodic lattice structures of cold atoms loaded on optical lattices [6]. It was also proposed as an effective method for the thermometry of fermions in an optical lattice [7] and to detect putative anti-ferromagnetic (AF) ground state of fermions in OL [8]. There are very recent optical Bragg scattering experimental data from a Mott state, a BEC and AF state [9]. The atom Bragg spec- troscopy and Optical Bragg scattering are two different, but complementary experimental methods. In this paper, we will develop a systematic theory of using the optical Bragg scattering ( Fig.1) to detect the nature of quantum phases of interacting bosons loaded in optical lattices. We show that the optical Bragg scatter- ing not only couples to the density order parameter, but also the valence bond order parameter due to the hop- ping of the bosons on the lattice. At integer fillings, when ⃗ q matches a reciprocal lattice vector ⃗ K of the underlying OL, there is an increase in the optical scattering cross section as the system evolves from the Mott to the SF state due to the increase of hopping in the SF state. At 1 / 2 filling, in the CDW state, when ⃗ q matches the CDW ordering wavevector ⃗ Q$_{n}$ and ⃗ K , there is a diffraction peak proportional to the CDW order parameter squared and the density squared respectively (Fig.3a), the ratio of the two peaks is a good measure of the CDW order param- eter. In the VBS state, when ⃗ q matches the VBS or- dering wavevector ⃗ Q$_{K}$ , there is a much smaller, but de- tectable diffraction peak proportional to the VBS order parameter squared, when it matches ⃗ K , there is also a diffraction peak proportional to the uniform density in the VBS state (Fig.3b). All the diffraction peaks scale as the square of the numbers of atoms inside the trap. All these characteristics can determine uniquely CDW and VBS state at 1 / 2 filling and the corresponding CDW su- persolid and VBS supersolid slightly away from the 1 / 2 filling. In the following, we just take 2d optical lattices as examples. The 1d and 3d cases can be similarly dis- cussed. The Extended Boson Hubbard Model (EBHM) with various kinds of interactions, on all kinds of lattices and at different filling factors is described by the following Hamiltonian [2, 10–17]: H$_{BH}$ = − t ∑ 〈 ij 〉 ( b † $_{i}$b$_{j}$ + h.c. ) − µ ∑ i n$_{i}$ + U 2 ∑ i n$_{i}$ ( n$_{i}$ − 1) + V$_{1}$ ∑ <ij> n$_{i}$n$_{j}$ + V$_{2}$ ∑ 〈 ik 〉 n$_{i}$n$_{k}$ + · · · (1) where n$_{i}$ = b † $_{i}$b$_{i}$ is the boson density, t is the nearest