Nonlinear Bloch-Zener oscillations for Bose-Einstein condensates in a Lieb optical lattice

Two-dimensional condensed matter such as graphene and similar materials Liu2011 ; Mal2012 ; Zhou2014 with Dirac cones at Fermi energy, in which low-energy excitations at high symmetry points can be described by the Dirac equation, are of great theoretical and experimental interests due to their unique electronic properties. Recent advances in exploring spin-orbit coupling and artificial gauge fields for neutral atoms DWZhang2018AIP ; Dali2011 ; Glodman2014 ; Galtski2013 ; SLZhu2006 ; SLZhu2011 make possible the creation and manipulation of Dirac cones in optical lattices Tarr2012 ; Monta2009 ; SLZhu2007 ; DWZhang2012 . Different lattice geometries of one or two dimensions in ultracold atomic gases Sol2011 ; Sol2012 ; Tarr2012 have been realized experimentally, such as honeycomb lattice with Dirac points at the corner of Brillouin zone (BZ) Wun2008 , the Lieb lattice with a flat band Taie2015 , and the kagome lattice Jo2012 ; SZhang2019 , etc. Furthermore, some well-known models in condensed matter physics such as Harper-Hofstadter model and Haldane model Miv2013 ; Aidelsburger2013 ; Aidelsburger2015 ; Jotzu2014 ; DWZhang2017 ; LBShao2008 have been theoretically investigated and experimentally realized in optical lattices with promising controllability. The band structure of the cold atom systems can be reconstructed by combining techniques of Bloch-Zenor oscillations with adiabatic mapping of cold atoms, and the existence of Dirac points can be revealed by the momentum-resolved inter-band transitions Mor2006 ; Sal2007 ; Kling2010 ; Lim2012 ; Lim2015 ; DWZhang2016 ; FMei2012 ; YQZhu2017 .

On the other hand, with high purity and tunability, the ultracold atomic system is one of the best quantum simulators  Lewe2007 ; DWZhang2018AIP ; Bloch2008 to study nonlinear dynamics for a BEC with weak interaction. For instance, the lattice structure can be changed by adjusting the laser intensity, while the interaction can be manipulated by tuning s-wave scattering length Bloch2008 . In this paper, a nonlinear Gross-Pitaevskii (GP) theory is derived by applying the mean-field approximation to a weakly coupling system. Besides the atomic systems, GP equation can also be used to study other nonlinear systems such as photonic systems with Kerr nonlinearity, exciton-polariton condensates, acoustic cavities, and circuit resonators Hennig1999 ; Lagoudakis2008 ; Ley2012 ; Ley2016 ; Hadad2016 ; Whittaker2018 ; Nejad2019 . Previous studies have shown that the mean-field Bloch bands for a BEC in an optical lattice possess unique loop structures Dia2002 ; Wu2002 ; Wang2006 ; Bona2017 . As predicted in Ref. Chen2011 , while Dirac points in a honeycomb lattice are broken by the nonlinearity and an intersecting tubular structure appears around the K point, the TB model, however, fails to describe the nonlinear dynamics in this system Chen2011 . In this article, we study BEC in an optical Lieb lattice. The Lieb lattice features a diabolic single dirac cone with a flat band, distinguishing from most systems in which the Dirac points come up in pairs (the so-called Fermion doubling) Shen2010 ; GSilva2014 ; JWei2019 ; Mukherjee2015 ; Poli2017 . The flat band is attributed to the bipatite nature of the lattice Mielke1991 ; ZLiu2014 , leading to exotic physics which is entirely determined by the interaction and topology due to constant dispersion caused by destructive interference Kopnin2011 ; JUlku2016 . Thus the nonlinear Lieb lattice may serve as a natural, and possibly indispensable extension of the honeycomb lattice considered in Ref. Chen2011 . Our results show that the nonlinear Dirac cone for the Lied lattice has a similar tubular structure to that of Ref. Chen2011 . Furthermore, the Dirac points in our model are protected by a plane point group symmetry. Any small interatomic interaction will break the symmetry, then the Dirac points will be lifted. And more importantly, the TB model works well in the vicinity of the M point for our system unexpectedly, which is sharply different with the results in the honeycomb lattice explored in Ref. Chen2011 .

The interplay of nonlinear superfluidity and relativistic Dirac dynamics leads to many novel effects, such as adiabatic failure, chaos Wu2000 ; Wang2006 and dynamic instability Bro2001 , which can be utilized to reveal information on the nonlinear energy bands. In this article we prove that the Bloch-Zener oscillations for the BEC in an optical lattice can be used as an efficient tool to study the structure of the nonlinear energy bands. The regime of fold bands can be determined by detecting the time evolution of the population of sublattices during Bloch oscillations, while the nonlinearity strength could be measured by the interband transition probability of specific tunneling process.

The paper is organized as follows. In Sec. II, we numerically calculate the energy spectra from both the GP Hamiltonian and the Bloch Hamiltonian of the TB model. And we discuss how the nonlinearity breaks the Dirac cone from the perspectives of symmetry. In Sec. III, we consider atoms performing Bloch-Zener oscillations. The transition probabilities for different tunneling process are given both in the adiabatic limit and sudden limit. Finally, a short conclusion is given in Sec. IV.

The mean-filed description of a condensate in superfluid phase leads to the Gross-Pitaevskii (GP) equation,

where $m$ is the mass of particle, $a_{s}$ is the scattering length and $V_{latt}(\mathbf{r})$ is the optical lattice potential of configurations of Lieb lattice (see Fig. 1). The nonlinear strength is denoted as $U\equiv 4\pi\hbar^{2}a_{s}/m$ in following discussions. The lattice $V_{latt}(\mathbf{r})$ could be experimentally realized by superimposing three pairs of laser beams with the following formation Taie2015 ; Shen2010 ; Weeks2010 ,

where $k_{L}=2\pi/\lambda$ is the wave number of the lattice and $\lambda$ the wavelength of the lasers. The potential amplitudes $V_{\mathrm{long}}^{(x,y)}$, $V_{\mathrm{short}}^{(x,y)}$ and $V_{\mathrm{diag}}^{(x,y)}$ could be tuned by adjusting the laser intensities, $\phi_{x,y}$ and $\varphi$ are the phase of laser beams. For simplicity we choose $(V_{long}^{(x)},V_{short}^{(x)},V_{diag}^{(x)})=(V_{long}^{(y)},V_{short}^{(y)},V_{diag}^{(y)})=(V_{long},V_{short},V_{diag})$ and $\phi^{(x,z)}=\varphi=0$.

The Landau instability of the BEC superfluid is not much affected by the lattice strength ZChen2010 . As the lattice strength increases, the system goes into the tight binding regime. Following the methods outlined in Ref. Smerzi2002 ; Trombettoni2001 ; Hu2015 , we map the GP equation (1) onto a discrete nonlinear Scr$\mathrm{\ddot{o}}$dinger equation to contruct an effective tight-binding Hamiltonian. We write the bosonic field as a sum over the three sub-lattices $\Psi=\sum_{i,\alpha}\Psi_{i,\alpha}u(\mathbf{r}-\mathbf{r}_{i,\alpha})$, where $u(\mathbf{r}-\mathbf{r}_{i,\alpha})$ is the Wannier function and $\mathbf{r}_{i,\alpha}$ ($\alpha=1,2,3$) are lattice vectors in the three sub-lattices of the $i$-th cell. Then the tight-binding Hamiltonian for our BEC system reads

where $\langle i,j\rangle$ runs over all nearest-neighboring sites, $\alpha/\beta$ includes the sublattices in one unit cell, $t_{\alpha,\beta}\simeq\int d\mathbf{r}[\frac{\hbar}{2m}(\nabla\Psi_{i\alpha}\cdot\nabla\Psi_{j\beta})+\Psi_{i\alpha}V_{latt}(\mathbf{r})\Psi_{j\beta}]$ is the hopping constant and $g$ is the nonlinear strength, for simplicity we take $t_{\alpha,\beta}=t$ and do not consider the next-nearest hopping in the main text. In experiments, one may upload $\mathrm{{}^{87}Rb}$ atoms which weigh $m=1.443\times 10^{-25}~{}\mathrm{kg}$ into the optical lattice. The lattice can be created by a laser of wavelength $\lambda=1064~{}\mathrm{nm}$. Thus the recoil energy is estimated by $E_{R}=\hbar^{2}k_{L}^{2}/2m\sim 2\pi\times 2~{}\mathrm{kHz}$. And the tunneling rate $t\sim 1~{}\mathrm{kHz}$ for a lattice depth $V=2E_{R}$ in the absence of nonlinearity Miyake2013 .

In momentum space, $\hat{H}_{k}=\sum_{\mathrm{k}}\Psi_{\mathrm{k}}^{\dagger}\mathcal{H}_{\mathrm{k}}\Psi_{\mathrm{k}}$ with $\Psi_{\mathbf{k}}=\left(\Psi_{1\mathbf{k}},\Psi_{2\mathbf{k}},\Psi_{3\mathbf{k}}\right)^{T}$, and

where $\mathbf{d}(\mathbf{k})=(d_{x},d_{y})$ denotes the Bloch vector with $d_{x}=2t\cos(k_{x}a/2)$ and $d_{y}=2t\cos(k_{y}a/2)$, and $\mathbf{S}=(S_{x},S_{y})$ are spin-one matrices with $S_{x}=\lambda_{1}$, $S_{y}=\lambda_{6}$ from the Gell-Mann matrices (for explicit form, see Appendix A), together with $S_{z}=\lambda_{5}$ which consist the $\mathrm{SU(2)}$ Lie algebra $[S_{m},S_{n}]=\mathrm{i}\epsilon_{mn\ell}S_{\ell}$. $\hat{N}=\operatorname{diag}(|\Psi_{1\mathbf{k}}|^{2},|\Psi_{2\mathbf{k}}|^{2},|\Psi_{3\mathbf{k}}|^{2})$ is the nonlinear term. The Bloch Hamiltonian $\mathcal{H}_{k}$ in absence of nonlinear term $g\hat{N}$ has energy dispersions,

as illustrated in Fig. 1 (c), which are in good agreement with Bloch bands of the continuous system Eq. (1), see Appendix B.

Note that the low energy effective Hamiltonian $\mathcal{H}_{0}(\mathbf{q})$ has a $\hat{C}_{4}^{z}$ rotation symmetry $\hat{C}_{4}^{z}\mathcal{H}_{0}(\mathbf{q})(\hat{C}_{4}^{z})^{-1}=\mathcal{H}_{0}(D_{\hat{C}_{4}^{z}}\mathbf{q})$, where $\hat{C}_{4}^{z}\equiv e^{-i\frac{\pi}{2}S_{z}}$ and $D_{\hat{C}_{4}^{z}}\mathbf{q}=(-q_{y},q_{x})$ and a sublattice symmetry $\hat{O}\mathcal{H}_{0}(\mathbf{q})\hat{O}^{-1}=-\mathcal{H}_{0}(\mathbf{q})$, where $\hat{O}\equiv\operatorname{diag}(1,-1,1)$.

In fact the Lieb lattice has a $4mm$ plane point group symmetry. The quasi-momentum $\mathbf{k}$ is invariant with operations in little group $4mm$ at $M$ and $2mm$ at $X$. And the invariant line $L_{\Sigma}$ and $L_{\Delta}$ are protected by mirror symmetry $m_{d}$ and $m_{x}$ respectively, thus indegenerate, while the Dirac point at M is protected by the 4mm symmetry featuring a nontrivial two dimensional irreducible representation. However, in presence of a nonzero nonlinear term $g\hat{N}$, even weak interaction will violate this symmetry, lifting the degeneracy at M and resulting in two more additional crossings, as shown in Fig. 2.

We numerically calculate the Bloch bands of the GP equation Eq. (1) with the Bloch wave solutions which are of the form $\psi_{\mathbf{k}}(\mathbf{r})=\sum_{m,n}c_{mn}e^{i\left(\mathbf{k}+\mathbf{G}_{mn}\right)\cdot\mathbf{r}}$, here $\mathbf{G}_{mn}=m\mathbf{b}_{1}+n\mathbf{b}_{2}$ is the reciprocal lattice vector. The results are shown in Fig. 2, which are in good agreement with that of the tight binding model $\mathcal{H}_{\mathrm{k}}$ in the vicinity of $M$. As predicted by previous works Chen2011 , the Bloch bands of a nonlinear two dimensional system possess tube structures and the Dirac point is extended into a closed self-crossing loop. The bands of our system have similar structures around the $M$ point. As illustrated in Fig. 2 (g), the energy bands consist of three ”tubes”, which intersect at point $M$. They have wedged cross-section with size increasing monotonically from $L_{\Sigma}$ to $\overline{L}_{\Delta}$ ($\overline{L}_{\Delta}$ denotes the invariant line cross $M$ and parallel to $k_{x}$-direction) as shown in Fig. 2 (g), (ii)-(iii), and overlap with each other to produce the structure of Fig. 2 (g), (i/iv). For $g=0$ the Dirac point at $M$ accidentally degenerates with the flat band, which produces a triple point. As shown before, the Dirac point at $M$ will be lifted to two crossing points $D_{1}$ and $D_{2}$ for any $g\neq 0$ as the nonlinearity breaks the symmetry. However, the loop structures at the zone of Brillouin zone, as illustrated in Fig. 2 (c) [or Fig. 2 (f)], only emerges as the nonlinear strength $U$ ($g$) exceeds a critical value $U_{c}$ ($g_{c}$). Specifically, the nonlinearity does not modify the flat band on the invariant line $L_{\Sigma}$, as shown in Fig. 2 (a) and (c). Furthermore, the flat bands of the tight binding model for $g=0$ and $g\neq 0$ on $L_{\Sigma}$ share the same eigen-states, only with a uniform shift of the eigen-value from $E=0$ to $E=g/2$ for any finite $g$.

Now we consider atoms performing Bloch oscillations with a constant applied force $\mathbf{F}$. As these atoms are accelerated in the vicinity of the triple Dirac point, the tunneling probability to other bands is finite. This is a problem first considered by Landau and Zener, usually referred to as Landau-Zener (LZ) transition LandauZener1932 ; XTan2014 ; Shevchenko2010 ; Oliver2005 ; Car1986 .

The effective Hamiltonian can be obtained from Eq. (4) by performing Peierls substitution $\mathbf{k}=\mathbf{F}t$,

where $\hat{H}=\mathcal{H}_{\mathrm{k}}-\bm{F}\cdot\hat{\bm{r}}$. The Peierls substitution can also be understood in a semiclassical way that the quasimomentums are subjected to a constant force $\dot{\mathbf{k}}=\mathbf{F}$ Lim2015 . An initial state $|\psi(t_{0})\rangle=|u_{n}(\mathbf{k}_{0})\rangle$ in the n-th band evolves as

where $\mathbf{k}(t)=\mathbf{k}_{0}+\mathbf{F}t$. The force $\mathbf{F}=\dot{\mathbf{k}}$ linearly increases momentum $\mathbf{k}$ of atoms and leads to a non-adiabatic transition to other bands near the crossing (or anti-crossing) point. From the finial state $|\psi(t_{f})\rangle$ the transition probability is defined as

where $m,n=1,2,3$ and $|u_{m}(\mathbf{k}(t))$ is m-th eigen-state of $\hat{H}_{F}[\mathbf{k}_{f}]$, with $\mathbf{k}_{f}=\mathbf{k}_{0}+\mathbf{F}t_{f}$ at the final time $t_{f}$.

In conclusion, we have numerically studied the energy band structure of a BEC in an optical Lieb lattice and have provided a useful TB model. It is shown that under nonlinearity, the Dirac points at M point are broken and extended into a closed curve, while the flat band still exists on the high-symmetry line. We have shown that the Bloch-Zener oscillations could be utilized as a useful tool to explore the nonlinear bands: (i) the size of metastable fold bands can be determined by detecting the time evolution of the population of sublattices during Bloch oscillations, (ii) the nonlinearity strength could be measured through the interband transition probability of specific tunneling process. Since the interatomic interaction can be easily manipulated, the observation of the phenomena studied here is expectable in the table-top ultracold atomic experiments.