Figure 1

A one-dimensional optical lattice created from counterpropagating laser beams (a) and with beams enclosing an angle $\theta$ (b). The parameters $V_0$ (lattice depth) and $d$ (lattice spacing) are defined in the text.Reuse & Permissions

Figure 2

Examples of two-dimensional potentials created by two alternative methods. In (a), the polarizations of the two waves are orthogonal, i.e., two noninterfering lattices are superimposed. In (b), on the other hand, the polarizations are parallel to each other, leading to four-beam interference. The potential shown in (a) can also be realized with parallel polarizations and a (large enough) frequency difference between the two standing waves.Reuse & Permissions

Figure 3

(Color) Realization of a spin-dependent potential (a) and resulting potential $V_{+}$ (b) for $\theta =0°$ (solid line) and $\theta =45°$ (dashed line). Adapted from 111.Reuse & Permissions

Figure 4

Band structure for different potential depths: (a) weak potential with $s=1$, (c) deep potential with $s=10$. In both cases, the closed formulas given in the text are depicted with the dotted line. In graphs (b),(d) we visualize the spatial dependence of the corresponding Bloch states. The periodic potentials are represented by the dashed lines. For each energy, the absolute square value of the corresponding Bloch states is depicted in the gray scale plot (high probability is represented by black). Additionally, the wave functions are shown for the energies at the gaps indicated with the arrows. One clearly sees that the wave functions at the first gap change sign from well to well, i.e., there is a phase slip of $\pi$. These modes are also known as “staggered modes.”Reuse & Permissions

Figure 5

Summary of the linear propagation of a wave packet in a weak periodic potential. (a) The band structure corresponding to $s=1$, which can be harmonically approximated at $q=0$ and $q=\pi ∕d$. This corresponds to the effective-mass approximation. (b) The group velocity (in units of $v_R=\hslash k∕m$) corresponding to the lowest band reveals that at the center and at the edge of the Brillouin zone the wave packet does not move. Additionally, the velocity in the lowest band is limited to a maximum velocity. (c) The spreading of a wave packet is a consequence of the group velocity dispersion described by the effective mass. The effective mass can be larger (and, indeed, even infinite at $q_{\infty }^{\pm }$) than the free mass, but also smaller and negative at $q=\pi ∕d$. The evolution of the wave packets for the momentum distributions indicated in black (gray) shading is shown in (d) [(e)]. (d) The real space evolution of the envelope of the wave packet prepared in the region of constant effective mass. The wave packet spreads without distortion. (e) If the broad quasimomentum distribution does not allow the quadratic approximation of the energy, higher-order terms in the Taylor expansion become relevant and lead to a distortion of the wave packet. (f) The evolution of a wave packet prepared at the infinite mass point $q_0=q_{\infty }^{+}$ for a propagation time 2.5 and 5 times longer than that of (d) and (e). This wave packet moves with the maximum velocity associated with the lowest band and reveals strongly suppressed spreading.Reuse & Permissions

Figure 6

Characteristic linear energies as a function of potential modulation depth. (a) Numerically calculated absolute value of the effective mass for the center $q=0$ (solid line) and the edge $q=\pi ∕d$ (dashed line) of the Brillouin zone where the mass is negative. The analytical results discussed in the text are represented by the dashed lines in (c) and (d). For potentials with deep modulation, the absolute values of the masses become equal and there is no quasimomentum dependence. (b) The width of the band decreases exponentially with increasing potential modulation depth. In the deep potential limit, this energy scale is associated with the tunneling rate between two adjacent wells.Reuse & Permissions

Figure 7

Wave-packet dynamics in a periodic potential in the presence of a constant force. (a),(b) An external force leads to a variation of the central quasimomentum $q=0$. Since the group velocity changes sign when the quasimomentum exceeds the Brillouin zone, the wave packet will show an oscillatory behavior in real space. This is known as Bloch oscillations and is one example of intraband dynamics. (c),(d) For strong external forces, nonadiabatic transitions to the first excited band can occur near the band edge. This is known as Landau-Zener tunneling and leads to a splitting of the wave packet. All graphs reveal clearly the Bloch state structure. Near $q=0$, the wave packet is only weakly modulated with the period of the periodic potential, while at the band edge it is fully modulated revealing the sinusoidal Bloch state at the Brillouin zone edge.Reuse & Permissions

Figure 8

Characteristic energies as a function of potential modulation depth. (a) $\overline{n}$ represents the number of atoms per site. (b) Definition of bandwidth and band gap. (c) The graph reveals that the linear energy scales—bandwidth and band gap—divide the parameter space into three distinct regimes (dark shading, energy is smaller than band gap and bandwidth; light shading, energy is between band gap and bandwidth; no shading, energy is higher than both characteristic linear energy scales). The on-site interaction energy for different atom numbers per site is also given (assuming a radial trapping frequency of ${\omega }_{\perp }=2\pi \times 200\mathrm{Hz}$). The solid (dotted) line indicates the on-site interaction energy associated with the Bloch state at the Brillouin zone center (edge). With 100 atoms per site, the regimes I and II exhibiting very different dynamics can be explored by simply changing the potential modulation depth.Reuse & Permissions

Figure 9

The temporal evolution of a condensate wave function of repelling atoms prepared in the negative and positive effective-mass regime [from 96]. The modulation of the condensate wave function reveals the sinusoidal spatial dependence of the Bloch state at the Brillouin zone edge. Clearly, the initial wave function in the negative mass regime is not stable and decays into bright solitons and background. It is important to note that although the atom-atom interaction is repulsive, nonspreading wave packets are formed. In the positive mass regime (i.e., in the second band at the Brillouin zone edge), the wave function is also spatially modulated, yet no instability is present.Reuse & Permissions

Figure 10

Classification of the nonlinear propagation in deep periodic potentials [from 174]. The propagation of a Gaussian wave packet initially centered around $q_0=p∕d$ in a given periodic potential depends critically on the total atom number $\Lambda \propto N_t$ and the quasimomentum $q_0$. For large nonlinearity, after some initial dynamics the wave packet stops expanding independently of the initial quasimomentum (self-trapping regime). For small nonlinearities (small atom number), the wave packet will expand indefinitely. This is also called the diffusive regime. The solitonic propagation or “breathers” (time periodic and spatially localized excitations) are only possible for quasimomenta with corresponding negative mass. Since this excitation relies on a delicate balance between nonlinearity and linear spreading, it only appears for very well-defined atom numbers.Reuse & Permissions

Figure 11

Selection of stationary solutions in the regime of intermediate nonlinear on-site interaction energy (106). Clearly, the atom-atom interaction leads to new stationary solutions whose energies lie in the energy gap of the linear system. Since these are nonlinear solutions, their energies $\mu$ depend on the atom number $N$. The main graph shows that there are different branches of solutions. In the lower three graphs, we show three solutions corresponding to the indicated energies. In contrast to the solutions expected from a tight-binding approximation treatment, they show structure within the potential minima.Reuse & Permissions

Figure 12

Evolution of perturbed trigonometric solutions $(k=0)$ with nontrivial phase (25). (a),(c) The temporal dynamics of a stable and unstable solution, respectively. The difference of the initial conditions is shown in (b),(d), where $r$ represents the absolute square value and $\theta$ the phase of the initial wave function. The main difference is the constant background, which is smaller in the case of the unstable mode. Taken from 25.Reuse & Permissions

Figure 13

Energy per particle as a function of the quasimomentum (108). The energy spectrum can be interpreted as a modified linear band structure. Significant modifications are observed for quasimomenta at which two bands of the linear system come close to each other. The structure becomes less pronounced when the potential modulation is increased (right graph).Reuse & Permissions

Figure 14

Stability diagram obtained by 180. The parameters describing the physical situation are the potential modulation $V_0$, the nonlinearity $nU$ for the homogeneous case, and the quasimomentum of the homogeneous condensate (flow of the condensate). The parameter $Q$ is the corresponding wave vector of the perturbation. It is important to note that $Q=\pi ∕d$ implies a modulation of the density with twice the period of the periodic potential. The regime in which the stationary states exhibit a Landau instability are indicated by the light shaded area and associated critical quasimomentum $q_e$. The dark shaded area represents the dynamically unstable regime with associated critical quasimomentum $q_d$.Reuse & Permissions

Figure 15

(Color) Interference pattern of a Bose-Einstein condensate released from a one-dimensional optical lattice of depth $V_0=10E_R$ after a time of flight of 20 ms. In (a), the lattice was at rest, whereas in (b) it had been accelerated to $v_R$, i.e., the quasimomentum of the condensate was at the edge of the Brillouin zone.Reuse & Permissions

Figure 16

Quantities used to characterize the interference pattern of a condensate released from an optical lattice. Shown here is an absorption image integrated perpendicular to the lattice direction. Similarly to Fig. 15, the condensate was accelerated to the edge of the Brillouin zone before being released. The $x$ axis of this figure has been rescaled (in units of the recoil momentum $p_{\mathrm{rec}}=m{v}_R$) to reflect the condensate momentum before the release from the lattice.Reuse & Permissions

Figure 17

Adiabaticity of the loading process. The faster the potential is ramped up, the larger the parameter $\zeta$ (ratio of the width of the interference peaks to the separation of the peaks) describing the dephasing of the condensate (from 132.Reuse & Permissions

Figure 18

Initial dephasing (a) and eventual rephasing (b) of a condensate nonadiabatically loaded into an optical lattice. In this experiment, the criterion describing the phase coherence of the condensate is the visibility of the interference pattern [from 128]. Note that (a) and (b) refer to different harmonic trapping frequencies, leading to different time scales for dephasing.Reuse & Permissions

Figure 19

Bloch oscillations (in momentum space) of a condensate in an optical lattice. If the instantaneous lattice velocity $u_{\mathrm{lat}}$ (indicated on the horizontal axis) is subtracted from the mean velocity of the condensate measured in the laboratory frame of reference (a), one clearly sees Bloch oscillations in the lattice frame (b). From 45.Reuse & Permissions

Figure 20

(Color) Coherent “droplets” tunneling out of a condensate held in a vertical 1D optical lattice. This effect can be interpreted in terms of the condensate undergoing Bloch oscillations under the influence of the gravitational force and part of the condensate leaving the lattice due to Landau-Zener tunneling at successive crossings of the Brillouin zone edge. Holding times in the lattice are (a) 0, (b) 3, (c) 5, (d) 7, and (e) 10 ms, respectively. In (f), an integrated profile of the absorption image (e) is shown together with a theoretical fit (solid line). Taken from 9.Reuse & Permissions

Figure 21

Variation of the effective potential $U_{\mathrm{eff}}$ (corresponding to $V_{\mathrm{eff}}$ in the notation of this review) with the nonlinear parameter $C$. The square symbols are experimental data points (obtained by measuring the tunneling probability and using the linear Landau-Zener formula to infer an effective lattice depth, i.e., the equivalent lattice depth in the linear problem giving the experimentally measured tunneling probability), and the solid and dashed lines are the theoretical prediction by 39 and a best fit with a rescaled nonlinearity parameter, respectively. From 129.Reuse & Permissions

Figure 22

Signatures of dynamical instability of a Bose condensate in an optical lattice. (a) The loss rates from a condensate held at a fixed quasimomentum $q$ of a lattice with $s=1.15$. (b) The theoretically calculated growth rates for the dynamically most unstable mode are plotted as a function of $q$. Taken from 61.Reuse & Permissions

Figure 23

(Color) Experimental demonstration of gap solitons (bright solitons for repulsive interaction in an optical lattice). (a) Absorption images revealing the in situ density distribution in a one-dimensional wave guide for different evolution times. Clearly, a nonspreading wave packet is formed after 25 ms. (b) The systematic measurement of the widths of the wave packets in the negative and positive mass regimes. While in the negative mass regime a soliton is formed whose width is constant, in the normal mass regime the initial atom distribution spreads out as expected. Adapted from 58.Reuse & Permissions

Figure 24

Variation of the sloshing frequency of a condensate in the presence of an optical lattice of depth $s$. The circles are experimental data points from 32, the triangles represent the theoretical prediction based on a Josephson model (discrete nonlinear Schrödinger equation), and the solid line is a calculation based on an effective-mass approach. Taken from 99.Reuse & Permissions

Figure 25

(Color) Interference pictures and integrated profiles for small (a,d), intermediate (b,e), and large (c,f) lattice depths in the experiment of 132. As the lattice depths increase, the interference patterns become more and more “smeared out.”Reuse & Permissions

Figure 26

The excitation spectrum of a superfluid (c) and a Mott insulator state (d)–(f) measured by applying an energy gradient between adjacent wells in the experiment of 72. From (c) to (f), the lattice depth is increased, and the discrete excitation spectrum of the Mott insulator becomes visible. The horizontal axes indicate the potential gradient expressed in kHz.Reuse & Permissions

Figure 27

(a) An array of “tubes” created by a two-dimensional optical lattice, as used in the experiment of 172 to realize the Mott-insulator transition in one dimension, and (b) the excitation spectrum of the 1D Mott insulator. The spacing between adjacent tubes in (a) is 413 nm. Taken from 124 and 172.Reuse & Permissions

Figure 28

(Color) Density (left) and phase profile (right) of a gap vortex in a two-dimensional optical lattice. The $x$ and $y$ axes are labeled in units of $d∕\pi$, where $d$ is the lattice spacing. Taken from 134.Reuse & Permissions