Quantum simulation of ultracold atoms in optical lattice based on dynamical mean-field theory

Tan Hui; Cao Rui; Li Yong-Qiang

doi:10.7498/aps.72.20230701

Abstract

With the development of atomic cooling technology and optical lattice technology, the quantum system composed of optical lattice and ultracold atomic gas has become a powerful tool for quantum simulation. The purity and highly controllable nature of the optical lattice give it a strong regulatory capability. Therefore, more complex and interesting physical phenomena can be simulated, which deepens the understanding of quantum many-body physics. In recent years, we have studied different Bose systems with strong correlations in optical lattice based on the bosonic dynamical mean-field theory, including multi-component system, high- orbit bosonic system, and long-range interaction system. In this review, we introduce the research progress of the above mentioned. Through the calculation by using bosonic dynamical mean-field theory which has been generalized to multi-component and real space versions, a variety of physical phenomena of optical crystal lattice Bose system in weak interaction intervals to strong interaction intervals can be simulated. The phase diagram of spin-1 ultracold bosons in a cubic optical lattice at zero temperature and finite temperature are drawn. A spin-singlet condensate phase is found, and it is observed that the superfluid can be heated into a Mott insulator with even (odd) filling through the first (second) phase transition. In the presence of a magnetic field, the ground state degeneracy is broken, and there are very rich quantum phases in the system, such as nematic phase, ferromagnetic phase, spin-singlet insulating phase, polar superfluid, and broken-axisymmetry superfluid. In addition, multistep condensations are also observed. Further, we calculate the zero-temperature phase diagram of the mixed system of spin-1 alkali metal atoms and spin-0 alkali earth metal atoms, and find that the system exhibits a non-zero magnetic ordering, which shows a second-order Mott insulation-superfluid phase transition when the filling number is $n=1$, and a first-order Mott insulation-superfluid phase transition when the filling number is $n=2$. The two-step Mott-insulating-superfluid phase transition due to mass imbalance is also observed. In the study of long-range interactions, we first use Rydberg atoms to find two distinctive types of supersolids, and then realize the superradiant phase coupled to different orbits by controlling the reflection of the pump laser in the system coupled to the high-finesse cavity. Finally, we study the high-orbit Bose system. We propose a new mechanism of spin angular-momentum coupling with spinor atomic Bosons based on many-body correlation and spontaneous symmetry breaking in a two-dimensional optical lattice, and then study the orbital frustration in a hexagonal lattice. We find that the interaction between orbital frustration and the strong interaction results in exotic Mott and superfluid phases with spin-orbital intertwined orders.

Keywords:

Authors and contacts

1.
College of Science, National University of Defense Technology, Changsha 410073, China
2.
Hunan Key Laboratory of Extreme Matter and Applications, National University of Defense Technology, Changsha 410073, China

Corresponding author: Li Yong-Qiang, li_yq@nudt.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12074431, 12374252) and the Science Foundation for Distinguished Young Scholars of Hunan Province, China (Grant No. 2021JJ10044).

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Figure 1. Schematic picture of BDMFT. In BDMFT, the many-body lattice problem is reduced to a single lattice problem coupling with normal Bosonic reservoir and reservoir of Bose-Einstein condensate (BEC)^[96]

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图 2 Anderson杂质模型下动力学平均场方法的自洽循环示意图. 给Anderson参数初值, 利用杂质求解器求解Anderson杂质模型, 得到物理量和自能, 通过自能得到格点格林函数, 利用Dyson方程得到杂质函数, 从而得到新的Anderson参量, 构成自洽过程

Figure 2. Schematic picture of BDMFT loop in Anderson impurity model. For an initial value of Anderson parameters, physical quantities and self-energy are obtained by solving the Anderson impurity model. After obtain lattice Green function through self-energy, impurity functions are attained. Finally, the loop is complete by fetched new Anderson parameters from impurity functions

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图 3 三维光晶格中自旋-1超冷玻色子在不同反铁磁相互作用下的零温相图^[119], $ U_2/U_0 $分别为0.01, 0.04 (²³Na), 0.3, 和2.0. 数据来源于BDMFT(黑线), Gutwiller(红线)以及文献[117](蓝线)中的计算. 体系中存在4种不同的相, 即超流相(SF)、向列绝缘相(NI)、自旋单态绝缘相(SSI)和自旋单态凝聚相(SSC)

Figure 3. Zero-temperature phase diagram for spin-1 ultracold bosons in a 3D cubic lattice^[119] for different antiferromagnetic interactions $ U_2/U_0 $ = 0.01, 0.04 (²³Na), 0.3, and 2.0, respectively, obtained via BDMFT (black circle), Gutzwiller (red cross) and in Ref. [117] (blue dashed). There are four different phases in these diagrams: superfluid (SF), nematic insulator (NI), spin-singlet insulator (SSI) and spin-singlet condensate (SSC)

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图 4 三维光晶格中由BDMFT计算得到的自旋-1 ⁸⁷Rb($ U'_1/U_1=-0.0046 $)和自旋-0 ⁸⁴Sr异核玻色子混合体系在不同种间相互作用$ U_{12}/U_1=0.2, 0.5, 0.935 $和2下的基态相图^[133]. 格点上粒子填充数为1时, 系统处在铁磁绝缘相. 格点上粒子填充数为2时, 体系为无序绝缘相. 此外随着隧穿振幅的增大会出现两种不同的超流相. 其中具有铁磁相互作用的三组分自旋-1 ⁸⁷Rb原子也处在铁磁相. 注意, 当种间相互作用特别大($ U_{12}/U_1=2 $)时, 系统中只有自旋1的玻色子. 作为比较, 红色线条是Gutzwiller平均场理论的计算结果. 其他参数为$ t \equiv t_{1_\sigma}\approx 0.97 t_{2_0} $, $ U_2/U_1=1.26 $

Figure 4. Phase diagrams of heteronuclear mixtures of ultracold spin-1 ⁸⁷Rb (spin-dependent interaction $ U_1'/U_1=-0.0046 $) and spin-0 ⁸⁴Sr bosons in a three dimensional (3D) cubic lattice for different interspecies interactions $ U_{12}/U_1 = 0.2, 0.5, 0.935 $ and 2, obtained by BDMFT^[133]. The system favors ferromagnetic insulating phase (FM) at filling $ n = 1 $, unorder insulating phase (UI) at $ n = 2 $, and two types of superfluid ($ {\rm{MI_{Sr} + SF_{Rb}}} $, and 2SF), where the three-components of spin-1 ⁸⁷Rb demonstrate ferromagnetic order as a result of ferromagnetic interactions. Note here that the system favors phase separation for $ U_{12}/U_1 = 2 $, and here we only show the phase diagram of spin-1 bosons. For comparisons, the red cross is obtained by Gutzwiller mean-field theory. The other parameters $ t=t_{1\sigma}\approx0.97 t_{20} $, and $ U_2/U_1 = 1.26 $.

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图 5 (a)考虑两个电子基态$ |b\rangle $(蓝色), $ |d\rangle $(红色)和一个里德伯态$ |r\rangle $. 一束非共振激光(拉比频率为Ω, 失谐量为Δ)将态$ |d\rangle $与$ |r\rangle $耦合. (b)里德伯态$ |d\rangle $间的软核型相互作用势$ V_{i j} $(红线). 软核半径$R_{\rm{c}}$可以大于晶格间距a, 图中展示的是$R_{\rm{c}}=2 a$的情形. (c)被修饰原子处于有序密度波(DW)时的裸态处于SS. (d)裸态的Roton不稳定性. 声子的Bogoliubov色散关系(沿$ k_x $轴)被种间相互作用显著地改变. 当种间相互作用$U_{{\rm{bd}}}$增加时, 会出现类Roton不稳定性, 表明基态相由均匀的超流体转变为超固体. 图中$U_{{\rm{b d}}} / U= $$ 0$(点线), $U_{{\rm{b d}}} / U=0.45$(虚线), $U_{{\rm{b d}}} / U=1$(实线), 其他参数为$ k_y=0, V / U=0.4 $, 和$ t / U=0.04 $ ^[151]

Figure 5. (a) Two electronic ground states $ |b\rangle $ (blue) and $ |d\rangle $ (red) and a Rydberg state $ |r\rangle $ are considered. An off-resonant laser (with Rabi frequency Ω and detuning Δ) weakly couples the state $ |d\rangle $ to $ |r\rangle $. (b) The soft-core shape interaction potential $ V_{i j} $ (red) between atoms in the Rydberg dressed state $ |d\rangle $. The soft-core radius $R_{\rm{c}}$ can be larger than the lattice spacing a. Here, $R_{\rm{c}}=2 a$ is shown. (c) SS of the bare state when dressed atoms are in an ordered density wave (DW). (d) Roton instability of the bare species. The Bogoliubov dispersion relation (along the $ k_x $ axis) of phonons is significantly modified by the interspecies interaction. A rotonlike instability emerges when the interspecies interaction $U_{{\rm{b d}}}$ is increased, indicating that the groundstate phase changes from a homogeneous superfluid to supersolid. We show $U_{{\rm{b d}}} / U=0$ (dotted line), $U_{{\rm{b d}}} / U= $$ 0.45$ (dashed line), and $U_{{\rm{b d}}} / U=1$ (solid line). Other parameters are $ k_y=0, V / U=0.4 $, and $ t / U=0.04 $^[151]

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图 6 原子在光腔中耦合高轨道态示意图^[167]　(a)原子被陷俘在光腔中, 由一束不平衡因子$ \eta= E_-/E_+ $的横向泵浦光驱动; (b)四方晶格的布里渊区示意图, 原子从动量态$ {\boldsymbol{k}}= $$ (0, 0) $被散射到$ (\pi, \pi) $, 右侧上下两幅图分别为p-轨道和d-轨道能带原子的动量分布图; (c), (d)腔模和泵浦光之间的主要散射过程, 其引起了高轨道激发. 通过控制参数$ \eta=1 $(c), 原子可以选择性地被散射到偶宇称的d-轨道态, 当$ \eta<1 $ (d), 原子被散射到奇宇称的p-轨道态. 此处$ J^{ij}_{{\rm{sd}}} $, $ J^{ij}_{{\rm{p}}_x{\rm{p}}_y} $,和$ J^{ij}_{{\rm{p}}_y{\rm{d}}} $分别表示s-轨道和$ {\rm{d}}_{xy} $-轨道, $ {\rm{p}}_x $-轨道和$ {\rm{p}}_y $-轨道, s-轨道和$ {\rm{p}}_x $-轨道, $ {\rm{p}}_y $-轨道和$ {\rm{d}}_{xy} $-轨道在格点i和格点j间由散射引起的轨道反转跃迁

Figure 6. Populating higher-orbital states with ultracold atoms in an optical cavity^[167]: (a) Atoms are prepared in an opticalcavity, pumped by a blue-detuned laser in the transverse direction with an imbalance parameter $ \eta= E_-/E_+ $. (b) Brillouin zone of the square lattice, where atoms are scattered from the quasimomentum state $ {\boldsymbol{k}}=(0, 0) $to the excite state $ (\pi, \pi) $, with quasimomentum distributions for the p- and d-orbital bands shown in right upper and lower panels, respectively; (c), (d) dominating scattering processes of atoms induced by cavity, leading to higher-orbital excitations. By controlling $ \eta $, atoms can be selectively scattered into the even-parity $ {\rm{d}}_{xy} $-orbital state with a single node in both x and y directions for $ \eta=1 $ (c), or into the odd-parity p-orbital state with a single node only in one direction for $ \eta<1 $ (d). Here, $ J^{ij}_{{\rm{sd}}} $, $ J^{ij}_{{\rm{p}}_x{\rm{p}}_y} $, $ J^{ij}_{{\rm{sp}}_x} $, and $ J^{ij}_{{\rm{p}}_y{\rm{d}}} $ denote cavity induced orbital-flip hoppings between sites i and j for the s- and $ {\rm{d}}_{xy} $-orbitals, $ {\rm{p}}_x $- and $ {\rm{p}}_y $-orbitals, s- and $ {\rm{p}}_x $-orbitals, and $ {\rm{p}}_y $- and $ {\rm{d}}_{xy} $-orbitals, respectively.

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图 7 (a) SAI相的示意图, 在SAI相中, 由于自发的自旋-轨道耦合, 粒子的自旋自由度和轨道自由度相互锁定; (b)在不同相互作用强度下, SAI相的稳定性; (c)在不同的温度下, SAI相的稳定性; (d)粒子数填充$ \langle n \rangle=2 $时, 玻色动力学平均场方法得到的两组分p轨道玻色系统基态相图, 相互作用强度设置为$U_{{/ /}}=U_{\bot}$; (e)表为不同相之间序参量的表征, 下左图和右图分别为$t_{{/ /}}/t_{\bot}=1$和10时, 不同填充数情况下基态相图, 下左图的插图为相变时序参量的变化. 相互作用强度设置为$U_{{/ /}}=U_{\bot}$ ^[194]

Figure 7. (a) Pictorial illustration of SAI order. In presence of spontaneous spin angular-momentum coupling, the phase of spatial wave-function is entangled with the internal degrees of freedom of an atom in each optical lattice site. (b) Stability of SAI order against interaction quantum fluctuations. (c) Stability of SAI order against thermal fluctuations. (d) Phase diagram of the spinful p-orbital system with an even integer filling. The phase diagram is obtained via BDMFT. The atomic filling is fixed at $ \langle n \rangle=2 $, we set $U_{{/ /}}=U_{\bot}$. (e) Table is the characterization of different quantum phase. Left and right picture are phase diagrams of spinful p-orbital bosons at generic fillings for $t_{{/ /}}/t_{\bot}=1$ and $ 10 $. The inset in left shows the evolution of the order parameters. We use interaction strengths $U_{{/ /}}=U_{\bot}$ ^[194]

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图 8 (a)左图为二维六角晶格的几何结构, 晶格的格矢为$ {{{\boldsymbol{e}}}}_m $, 右图为晶格的第一布里渊区. (b)强关联区间轨道极化矢量$ {\bf {\cal{P}}} $在实空间的分布图. 左图为无自旋玻色子的Ising型结构, 中间和右图分别为自旋向上、自旋向下玻色子的平面内轨道涡旋结构. (c)粒子填充数$ \langle n \rangle=2 $时, 实空间玻色动力学平均场得到的两组分六角晶格p轨道玻色系统多体基态相图. 左图和右图的相互作用分别为$ U_{\uparrow}=U_{\downarrow}=U_{\uparrow \downarrow} $和$ U_{\uparrow}=U_{\downarrow}=2 U_{\uparrow \downarrow} $. 右图的插图为右图灰色垂直线路径下, 序参量的相应变化. (d)不同自旋组分在动量空间下密度的分布^[202]

Figure 8. (a) Geometry of two-dimensional hexagonal lattice with lattice vector $ {{{\boldsymbol{e}}}_m} $ (left), and the first Brillouin zone (right). (b) Cartoons of real-space orbital polarization $ {\bf {\cal{P}}} $ for strongly interacting many-body phases in p-orbital bands of the two-dimensional (2D) hexagonal lattice, where left picture is spinless bosons demonstrate out-of-plane Ising-type orbital order, middle and right are spinful case in-plane orbital textures. (c) Hopping-dependent phase diagrams of spinful bosonic gases in p-orbital bans of a 2D hexagonal lattice for fixed filling $ \langle n \rangle=2 $, obtained via real-space bosonic dynamical mean-field theory. The left and right are set $ U_{\uparrow}=U_{\downarrow}=U_{\uparrow \downarrow} $ and $ U_{\uparrow}=U_{\downarrow}=2 U_{\uparrow \downarrow} $. Inset picture is the evolution of order parameter along the gray vertical line. (d) Momentum-space distributions of density $n_{\sigma, {\boldsymbol k}}$^[202]

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