⇦Chinese Physics Letters, 2020, Vol. 37, No. 4, Article code 040301 Superfluid-Mott-Insulator Transition in an Optical Lattice with Adjustable Ensemble-Averaged Filling Factors * Shifeng Yang (杨仕锋)1, Tianwei Zhou (周天伟)1, Chen Li (李辰)2, Kaixiang Yang (杨凯翔)1, Yueyang Zhai (翟跃阳)3**, Xuguang Yue (乐旭广)4, Xuzong Chen (陈徐宗)1** Affiliations 1School of Electronics Engineering and Computer Science, Peking University, Beijing 100871 2Vienna Center for Quantum Science and Technology, Atominstitut, TU-Wien, Stadionallee 2, 1020 Vienna, Austria 3Innovative Research Institute of Frontier Science and Technology, Beihang University, Beijing 100191 4Wuhan National Laboratory for Optoelectronics (WNLO), Wuhan 430074 Received 7 December 2019, online 24 March 2020 *Supported by the National Natural Science Foundation of China (Grant Nos. 61703025, 91736208, 11504328, and 11920101004), the National Program on Key Basic Research Project of China (Grant Nos. 2016YFA0301501 and 2017YFA0304204).
**Corresponding author. Email: yueyangzhai@163.com; xuzongchen@pku.edu.cn Citation Text: Yang S F, Zhou T W, Li C, Yang K X and Di Y Y et al 2020 Chin. Phys. Lett. 37 040301    Abstract We study the quantum phase transition from a superfluid to a Mott insulator of ultracold atoms in a three-dimensional optical lattice with adjustable filling factors. Based on the density-adjustable Bose–Einstein condensate we prepared, the excitation spectrum in the superfluid and the Mott insulator regime is measured with different ensemble-averaged filling factors. We show that for the superfluid phase, the center of the excitation spectrum is positively correlated with the ensemble-averaged filling factor, indicating a higher sound speed of the system. For the Mott insulator phase, the discrete feature of the excitation spectrum becomes less pronounced as the ensemble-averaged filling factor increases, implying that it is harder for the system to enter the Mott insulator regime with higher filling factors. The ability to manipulate the filling factor affords further potential in performing quantum simulation with cold atoms trapped in optical lattices. DOI:10.1088/0256-307X/37/4/040301 PACS:03.75.Hh, 37.10.Jk, 03.65.Nk © 2020 Chinese Physics Society Article Text Quantum gases trapped in periodic lattice potentials have opened innovative manipulation possibilities in an almost pure environment[1] and they have become a fascinating platform for performing quantum simulations.[2,3] With high controllability that would be unreachable in other conventional systems,[4,5] ultracold atoms in optical lattices can realize a variety of ideal Hamiltonians. The prediction[6] and the observation[7] of the quantum phase transition from a superfluid to a Mott insulator in degenerate Bose gases has offered an insight into the physics of the Bose–Hubbard model.[8] In the superfluid phase, the atoms are phase coherent with a long-range order, but in the Mott insulator phase, each atom is localized on individual lattice site and all phase coherence is lost. By dynamically adjusting the lattice depth, the ratio between the interaction strength and the tunneling rate can be continuously controlled to achieve this superfluid-Mott-insulator transition.[9] Considering the process of loading a Bose–Einstein condensate (BEC) into a three-dimensional (3D) optical lattice potential, the atom number density of the condensate is typical to obtain an ensemble-averaged filling factor around 1 or 2 atoms per lattice site.[10] In this Letter, based on a density-adjustable atomic BEC we produced, the superfluid-Mott-insulator transition with different filling factors is experimentally investigated. Through the excitation spectrum probed in both superfluid regime and Mott insulator regime, both phases are compared under low atom occupancy and high atom occupancy up to 7 atoms per lattice site. In our experiment, an atomic BEC of $1.5\times10^5$ weakly repulsive $^{87}$Rb atoms is produced in the $\vert F=1,\,m_{F}=-1\rangle$ state with no discernible thermal fraction. Here $F$ and $m_{F}$ denote the total angular momentum and the magnetic quantum number of the atomic hyperfine state, respectively. The BEC is prepared in a crossed-beam optical dipole trap with a nearly isotropic trapping frequency.[11] A cubic optical lattice comprises three mutually orthogonal retroreflected laser beams each with a beam waist of $\sim $145 µm.[12] During evaporation cooling, the trapping frequency of the optical dipole trap is enhanced by the gravitational tilting, especially when the effective trap depth is very low, which improves the efficiency of evaporation cooling. In our system, a nearly pure BEC is achieved with an average trapping frequency of $2\pi\times 80\,\mathrm{Hz}$, this is larger than the frequency of most magnetic traps, and the collision rate of the atoms is still very high. After the BEC is achieved, we gradually switch on the levitation magnetic field and decrease the dipole trap laser intensity at the same time to keep the effective trap depth constant. This prevents the atoms from being heated or spilling out of the trap before they are rethermalized. Through this adiabatic decompression process, the atom number density of the condensate can be manipulated with the condensate fraction remaining constant. The peak atom number density can be obtained from $n_{\rm c}=N\omega_x\omega_y\omega_z(m\lambda_{\rm dB}/h)$, where $N$ is the atom number, $m$ is the atom mass, $\lambda_{\rm dB}=h(2\pi mk_{_{\rm B}}T)^{-1/2} $ is the thermal de Broglie wavelength and $h$ is the Planck constant. Here $\omega_x$, $\omega_y$ and $\omega_z$ are the trapping frequencies in the three directions, the confinement comes from both the optical dipole trap and the magnetic field gradient in the transverse direction. In the following experiments, the control of the average filling number is realized by adjusting the trapping frequencies ($\omega_x$, $\omega_y$, $\omega_z$). Compared with adjusting the overall atom number, the average filling number can be controlled on a wider scale in this way. Ultracold bosonic atoms with repulsive interactions trapped in the lowest band of a 3D optical lattice can be well described by the Bose–Hubbard model. Considering only the nearest neighboring sites tunneling and the on-site interaction, the Bose–Hubbard Hamiltonian reads $$ \hat{H}=-t\sum_{\langle ij\rangle}\hat{b}^†_{i}\hat{b}_{j}+\dfrac{U}{2}\sum_{i}\hat{n}_i(\hat{n}_i-1)+\sum_{i}(\epsilon_i-\mu)\hat{n}_i\,.~~ \tag {1} $$ Here $\langle ij\rangle$ means that the summation is only carried out over the nearest neighboring sites, $\hat{b}^†_{i}$ and $\hat{b}_{i}$ denote the bosonic creation and annihilation operators on the $i$th lattice site, $\hat{n}_i=\hat{b}^†_{i}\hat{b}_i$ is the atomic number operator measuring the occupation number on the $i$th lattice site, $\mu$ is the chemical potential; and $\epsilon_i$ is the energy offset of the $i$th lattice site arising from the external confinement, which is zero for a homogenous system. The hopping of the atoms between the nearest neighboring sites is characterized by the tunneling rate $t$, and the interaction energy per atom pair for two atoms on a single site is given by the on-site interaction $U$. The physics of the Bose–Hubbard model is governed by the ratio between $U$ and $t$, containing the quantum phase transition from a superfluid state to a Mott insulator state. If the ratio $U/t$ is below a critical value, the system is in the superfluid regime. Above the critical value, the system enters the Mott insulator phase. Now we consider two limiting cases of the Bose–Hubbard model. In the limit of $U\to0$, where the tunneling term dominates the Hamiltonian, the ground state is a BEC in the lowest band with quasi-momentum $q=0$. In this case, each atom is delocalized over the entire lattice and the many-body ground state for a homogeneous system can be written as a product of identical single-particle Bloch waves with zero quasi-momentum $$ \vert{\it\Psi}\rangle_{U\to0}\propto\frac{1}{\sqrt{N!}}\left(\frac{1}{\sqrt{M}}\sum^{M}_{i=1}\hat{b}^†_{i}\right)^N\vert 0\rangle ,~~ \tag {2} $$ where $N$ is the total atom number and $M$ is the number of lattice sites. This state can be described by a macroscopic wave function with long-range phase coherence, which will give rise to the narrow momentum peaks observed in the atomic interference pattern after switching off the lattice potential. The excitation spectrum is gapless because arbitrarily small excitations causing phase differences between neighboring sites can be excited. In the other limit of $t\to0$, the interaction term in the Bose Hubbard Hamiltonian is dominant. Each lattice site becomes independent and each atom is localized on one single site. The many-body ground state for a homogeneous system is then a product of local Fock states with a fixed atom number on each lattice site $$ \vert{\it\Psi}\rangle_{t\to0}\propto\Big(\frac{1}{\sqrt{n!}}\Big)^M\prod^{M}_{i=1} \big(\hat{b}^†_{i}\big)^n\vert 0\rangle ,~~ \tag {3} $$ where $n$ is the filling factor. Different from the $U\to0$ case, this state cannot be described by a macroscopic wave function, thus no phase coherence can be observed in the atomic interference pattern. In addition, a gap will appear in the excitation spectrum corresponding to the minimum energy difference between neighboring sites.
cpl-37-4-040301-fig1.png
Fig. 1. Absorption images of atomic interference patterns for different lattice depths with an ensemble-averaged filling factor $\bar{n}=2$, captured after a time-of-flight (TOF) period of $18\,\mathrm{ms}$. In the superfluid regime, narrow interference peaks are clearly visible. For lattice depths ($26 E_\mathrm{r}$ and larger) deep in the Mott insulator regime, no interference pattern is visible at all.
To adiabatically load the BEC into the 3D optical lattice, the intensities of the lattice laser beams are exponentially ramped up to their final values in $250\,\mathrm{ms}$ with a time constant of $62.5\,\mathrm{ms}$. The optical dipole trap is ramped down to zero at the same time. After a holding time of $10\,\mathrm{ms}$ in the optical lattice, the lattice potentials are shut off and the absorption images are captured after a time-of-flight (TOF) period of $18\,\mathrm{ms}$. Here $E_\mathrm{r}=h^2/2m\lambda^2$ is the recoil energy and $\lambda=1064\,\mathrm{nm}$. The depth of the optical lattice is calibrated with a calibration uncertainty of less than $0.6\%$ via a method we used previously.[13] Figure 1 shows the absorption images of atomic interference patterns for different lattice depths with an ensemble-averaged filling factor $\bar{n}=2$. The momentum distribution in the TOF images reads $$\begin{alignat}{1} \!\!\!\!\!\!n({\boldsymbol k})&=\langle\hat{b}^†_{{\boldsymbol k}}\hat{b}_{{\boldsymbol k}}\rangle=\vert w({\boldsymbol k})\vert^2\sum^{M}_{i,j}\langle\hat{b}^†_{i}\hat{b}_{j}\rangle{e}^{{i}{\boldsymbol k}\cdot({\boldsymbol r}_i-{\boldsymbol r}_j)}\\ \!\!\!\!\!\!&=\vert w({\boldsymbol k})\vert^2\Big[\sum^{M}_{i}\langle\hat{n}_{i}\rangle+\sum^{M}_{i\neq j}\langle\hat{b}^†_{i}\hat{b}_{j}\rangle{e}^{-{i}{\boldsymbol k}\cdot({\boldsymbol r}_i-{\boldsymbol r}_j)}\Big],~~ \tag {4} \end{alignat} $$ where $w({\boldsymbol k})$ is the Wannier wave function for the lowest band of the momentum space, $\sum^{M}_{i}\langle\hat{n}_{i}\rangle=N$ is the total atom number, and $\vert w({\boldsymbol k})\vert^2\sum^{M}_{i}\langle\hat{n}_{i}\rangle$ is referred to as the Wannier background.
cpl-37-4-040301-fig2.png
Fig. 2. The representative momentum distribution during the superfluid-Mott-insulator transition with an ensemble-averaged filling factor $\bar{n}=2$.
In the superfluid regime, the state has a long-range order, therefore the second term dominates Eq. (4) as $n({\boldsymbol k})\approx\vert w({\boldsymbol k})\vert^2NM\sum_{{\boldsymbol G}}\delta_{{\boldsymbol k},{\boldsymbol G}}$, where ${\boldsymbol G}$ is a vector of the reciprocal lattice. The momentum distribution presents discrete interference peaks in the TOF image as can be seen from Fig. 2(a). As the lattice depth increases (see Fig. 2(b)), $w({\boldsymbol k})$ becomes wider and the high order interference peaks becomes much clearer. When the system enters the Mott insulator regime, the long-range order of the state has totally vanished, i.e., $\langle\hat{b}^†_{i}\hat{b}_{j}\rangle=0$. The Wannier background in Eq. (4) is dominant as $n({\boldsymbol k})\approx N\vert w({\boldsymbol k})\vert^2$ and the momentum distribution is the distribution of $\vert w({\boldsymbol k})\vert^2$. For the deep lattice, the potential on each lattice site is approximately an isotropic harmonic oscillator potential. The ground state wave function is the Gaussian wave packet, as well as $\vert w({\boldsymbol k})\vert^2$. There is no discrete interference peaks in the TOF image, instead of a global Gaussian distribution, as shown in Fig. 2(c). The fundamental properties of the quantum phases in the superfluid-Mott-insulator transition can be revealed by the excitation spectrum.[14] Under the same loading sequence, the excitation is carried out by applying amplitude modulation of the vertical lattice potential after the $10\,\mathrm{ms}$ holding time. The modulated lattice potential takes the form $V_z(z,t)=V_z[1+0.2\sin(\omega_{\rm mod}t)]\sin^2(k_0 z)$ with a modulation duration of $20\,\mathrm{ms}$, where $k_0$ is the lattice laser wave number. This modulation introduces two sidebands with frequencies of $\pm\omega_{\rm mod}$ relative to the lattice laser frequency and defines the excitation energy. The measurement method of the excitation spectrum we use is first performed in the previous experiments,[15,16] we evaluate the width of the central momentum peak in the TOF images as a measure of the introduced energy. As mentioned earlier, the atom number density of the condensate can be manipulated by applying different degrees of vertical magnetic field gradients after the BEC is achieved. Different atom number densities can lead to different ensemble-averaged filling factors when the BEC is loaded into the 3D optical lattice. In our experiments, three different configurations with the ensemble-averaged filling factor $\bar{n}=2.0, 4.5$ and $7.0$ are experimentally investigated by probing the excitation spectrum in both superfluid regime and Mott insulator regime.
cpl-37-4-040301-fig3.png
Fig. 3. The measured excitation spectra for different values of ensemble-averaged filling factor $\bar{n}$ in two kinds of situations: $(a)$ $U/t=2$, $(b)$ $U/t=23$.
Based on a mean-field theory, the superfluid-Mott-insulator phase transition is expected to occur at $U/t=2\bar{n}+1+\sqrt{(2\bar{n}+1)^2-1}\approx 4\bar{n}$ for an ensemble-averaged filling of $\bar{n}\geq2$ per lattice site.[17] The changes in the excitation spectrum from continuous to discrete when the system undergoes the superfluid-Mott-insulator transition can be seen in Fig. 3. Figure 3(a) displays the situation of $U/t=2$ with a series of spectra for different values of the ensemble-averaged filling factor $\bar{n}$. The gapless excitation spectra for all the three filling factors prove that the states are superfluid states with a weak interaction, which are in accordance with the prediction $U/t\approx 4\bar{n}$ for the quantum critical point. The excitation here is the Bogoliubov excitation that the system responses to infinitesimal introduced energy. Due to the Bragg condition, the atoms scattering two photons receive a momentum transfer of $0\hbar{\boldsymbol k}$ or $\pm2\hbar{\boldsymbol k}$.[15,16] There is a positive correlation between the center of the excitation spectrum and the ensemble-averaged filling factor, which indicates a higher sound speed of the system. Unlike the superfluid situation, for the situation shown in Fig. 3(b) with $U/t=23$, the discrete feature of the excitation spectrum becomes less pronounced as the ensemble-averaged filling factor increases. Here the first resonant peak for the ensemble-averaged filling factor $\bar{n}=2.0$ and $\bar{n}=4.5$ appears close to the calculated value of $U$, which means that it requires a critical introduced energy for the system to response. A second resonant peak occurs at 1.8(6) times the energy of the first resonance. This is consistent with the system entering a gapped Mott insulator phase at large interactions. For the ensemble-averaged filling factor $\bar{n}=2.0$ and $\bar{n}=4.5$, $U/t=23$ is larger than the prediction quantum critical point $U/t\approx 4\bar{n}$, the energy gap $U$ opens up as the quantum critical point is crossed. For $\bar{n}=7.0$, the system has not yet fully entered the Mott insulator regime, therefore the excitation spectrum does not exhibit a clear gap. The larger the filling factor is, the harder it is for the system to enter the Mott insulator phase. In conclusion, we have measured the momentum distribution and the excitation spectrum in the superfluid-Mott-insulator transition by loading an atomic BEC into a 3D optical lattice potential. The BEC we prepared in our experiments is density-adjustable, the excitation spectrum in the superfluid and the Mott insulator regime is observed with different filling factors. For the superfluid phase, the excitation spectrum is gapless and the excitation is the Bogoliubov excitation. The center of the excitation spectrum is related to the ensemble-averaged filling factor. For the Mott insulator phase, the resonant peaks of the excitation spectrum become less pronounced as the ensemble-averaged filling factor increases, which means that it is harder for the system to enter the Mott insulator regime. The ability to manipulate the ensemble-averaged filling factor affords further potential in performing quantum simulation with cold atoms trapped in optical lattices. References Optical latticesQuantum simulations with ultracold quantum gasesTopological quantum matter with ultracold gases in optical latticesUltracold quantum gases in optical latticesMany-body physics with ultracold gasesCold Bosonic Atoms in Optical LatticesQuantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atomsBoson localization and the superfluid-insulator transitionRevealing the superfluid–Mott-insulator transition in an optical latticeMott insulators in an optical lattice with high filling factorsDeep cooling of optically trapped atoms implemented by magnetic levitation without transverse confinementObservation of atom-number fluctuations in optical lattices via quantum collapse and revival dynamicsHigh precision calibration of optical lattice depth based on multiple pulses Kapitza-Dirac diffractionTransition from a Strongly Interacting 1D Superfluid to a Mott InsulatorSuperfluid to Mott insulator transition in one, two, and three dimensionsQuantum phases in an optical lattice
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